,

subject to the coupled boundary conditions

$$\begin{cases} \mathbf{x}^{(j)}(0) = 0, \; j = 0, \ldots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{x}(0) = 0, \; D\_{0+}^{a\_0} \mathbf{x}(1) = \sum\_{j=1}^{n} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{y}(\tau) \, d\mathfrak{H}\_j(\tau) + \mathfrak{c}\_{0+} \\\\ \mathbf{y}^{(j)}(0) = 0, \; j = 0, \ldots, q - 2; \; D\_{0+}^{\delta\_2} \mathbf{y}(0) = 0, \; D\_{0+}^{\delta\_0} \mathbf{y}(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{x}(\tau) \, d\mathfrak{H}\_j(\tau) + \mathfrak{c}\_{0+} \end{cases} \tag{9}$$

**Lemma 4.** *Under assumption* (*K*1)*, the unique solution* (*x*, *<sup>y</sup>*) <sup>∈</sup> (*C*[0, 1])<sup>2</sup> *of problem (8) and (9) is*

$$\mathbf{x}(t) = \frac{t^{\delta\_1 - 1}}{\Delta} \left( \mathfrak{c}\_0 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} + \mathfrak{d}\_0 \Delta\_1 \right), \ y(t) = \frac{t^{\delta\_2 - 1}}{\Delta} \left( \mathfrak{c}\_0 \Delta\_2 + \mathfrak{d}\_0 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \alpha\_0)} \right), \ t \in [0, 1], \tag{10}$$

*which satisfies the conditions x*(*t*) > 0 *and y*(*t*) > 0 *for all t* ∈ (0, 1]*.*

**Proof.** We note that *ϕ*<sup>1</sup> (*Dδ*<sup>1</sup> <sup>0</sup>+*x*(*t*)) = *<sup>φ</sup>*(*t*), *ϕ*<sup>2</sup> (*Dδ*<sup>2</sup> <sup>0</sup>+*y*(*t*)) = *ψ*(*t*). Therefore, the problem (8) and (9) is equivalent to the following three problems:

$$(I)\ \begin{cases} \ D\_{0+}^{\gamma\_1} \phi(t) = 0, \\ \phi(0) = 0, \end{cases} \quad (II)\ \begin{cases} \ D\_{0+}^{\gamma\_2} \psi(t) = 0, \\ \psi(0) = 0, \end{cases}$$

and

$$(III)\begin{cases} \begin{cases} D\_{0+}^{\delta\_1}\mathbf{x}(t) = \mathfrak{q}\_{\rho\_1}(\phi(t)), \ t \in (0,1), \\ D\_{0+}^{\delta\_2}\mathbf{y}(t) = \mathfrak{q}\_{\rho\_2}(\psi(t)), \ t \in (0,1), \\ \text{with} \\ \begin{cases} \mathbf{x}^{(j)}(0) = 0, \ j = 0, \dots, p-2, \ D\_{0+}^{\delta\_0}\mathbf{x}(1) = \sum\_{j=1}^{n} \int\_0^1 D\_{0+}^{\delta\_j}\mathbf{y}(\tau) \, d\mathfrak{H}\_j(\tau) + \mathfrak{q}\_0, \\ \mathbf{y}^{(j)}(0) = 0, \ j = 0, \dots, q-2, \ D\_{0+}^{\delta\_0}\mathbf{y}(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j}\mathbf{x}(\tau) \, d\mathfrak{H}\_j(\tau) + \mathfrak{q}\_0. \end{cases} & (III)\_2) \end{cases} (III)\_2$$

Problem (*I*) has the solution *φ*(*t*) = 0 for all *t* ∈ [0, 1], and problem (*I I*) has the solution *ψ*(*t*) = 0 for all *t* ∈ [0, 1]. Therefore, problem (*III*) can be written as

$$\begin{cases} \ D\_{0+}^{\delta\_1} \mathfrak{x}(t) = 0, \ t \in (0,1), \\\ D\_{0+}^{\delta\_2} \mathfrak{y}(t) = 0, \ t \in (0,1), \end{cases} \tag{11}$$

supplemented with the boundary conditions (*III*)2. The solutions of system (11) are

$$\begin{array}{l} \text{x}(t) = a\_1 t^{\delta\_1 - 1} + a\_2 t^{\delta\_1 - 2} + \dots + a\_p t^{\delta\_1 - p} \ \ t \in [0, 1] \text{} \\ y(t) = b\_1 t^{\delta\_2 - 1} + b\_2 t^{\delta\_2 - 2} + \dots + b\_q t^{\delta\_2 - q} \ \ t \in [0, 1] \end{array} \tag{12}$$

with *<sup>a</sup>*1, ... , *ap*, *<sup>b</sup>*1, ... , *bq* <sup>∈</sup> <sup>R</sup>. By using the boundary conditions *<sup>x</sup>*(*j*)(0) = 0, *<sup>j</sup>* <sup>=</sup> 0, ... , *<sup>p</sup>* <sup>−</sup> 2, *<sup>y</sup>*(*j*)(0) = 0, *<sup>j</sup>* <sup>=</sup> 0, ... , *<sup>q</sup>* <sup>−</sup> 2 (from (*III*)2), we obtain *<sup>a</sup>*<sup>2</sup> <sup>=</sup> ··· <sup>=</sup> *ap* <sup>=</sup> 0 and *<sup>b</sup>*<sup>2</sup> <sup>=</sup> ··· = *bq* = 0. Then, the functions in Equation (12) become *x*(*t*) = *a*1*t <sup>δ</sup>*1<sup>−</sup>1, *<sup>t</sup>* <sup>∈</sup> [0, 1], *y*(*t*) = *b*1*t <sup>δ</sup>*2<sup>−</sup>1, *<sup>t</sup>* <sup>∈</sup> [0, 1]. For these functions, we find

$$\begin{array}{l} D\_{0+}^{a\_{0}}\boldsymbol{x}(t) = a\_{1} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1}-\alpha\_{0})} t^{\delta\_{1}-a\_{0}-1}, \; D\_{0+}^{\beta\_{0}}\boldsymbol{y}(t) = b\_{1} \frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2}-\beta\_{0})} t^{\delta\_{2}-\beta\_{0}-1},\\ D\_{0+}^{a\_{j}}\boldsymbol{y}(t) = b\_{1} \frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2}-\alpha\_{j})} t^{\delta\_{2}-a\_{j}-1}, \; D\_{0+}^{\beta\_{j}}\boldsymbol{x}(t) = a\_{1} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1}-\beta\_{j})} t^{\delta\_{1}-\beta\_{j}-1}.\end{array}$$

Therefore, by now using the above fractional derivatives and the conditions *Dα*<sup>0</sup> <sup>0</sup>+*x*(1) = ∑*n j*=1 % 1 <sup>0</sup> *<sup>D</sup>α<sup>j</sup>* <sup>0</sup>+*y*(*τ*) *<sup>d</sup>*H*j*(*τ*) + <sup>c</sup><sup>0</sup> and *<sup>D</sup>β*<sup>0</sup> <sup>0</sup>+*y*(1) = <sup>∑</sup>*<sup>m</sup> j*=1 % 1 <sup>0</sup> *<sup>D</sup>β<sup>j</sup>* <sup>0</sup>+*x*(*τ*) *d*K*j*(*τ*) + d<sup>0</sup> (from (*III*)2), we deduce the following system for *a*<sup>1</sup> and *b*1:

$$\begin{cases} \begin{aligned} a\_1 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \alpha\_0)} &= \sum\_{j=1}^n \int\_0^1 b\_1 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \alpha\_j)} \tau^{\delta\_2 - a\_j - 1} \, d\mathfrak{H}\_j(\tau) + \mathfrak{e}\_0, \\\ b\_1 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} &= \sum\_{j=1}^m \int\_0^1 a\_1 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \beta\_j)} \tau^{\delta\_1 - \beta\_j - 1} \, d\mathfrak{H}\_j(\tau) + \mathfrak{e}\_{0,0} \end{aligned} \end{cases}$$

or equivalently

$$\begin{cases} \begin{array}{l} a\_1 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \alpha\_0)} = b\_1 \Delta\_1 + \mathfrak{c}\_{0\prime} \\ b\_1 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} = a\_1 \Delta\_2 + \mathfrak{d}\_0 . \end{array} \end{cases}$$

The determinant of the above system in the unknown *a*<sup>1</sup> and *b*<sup>1</sup> is

$$\begin{array}{c|c} \Gamma(\delta\_1) & -\Delta\_1 \\ \hline \Gamma(\delta\_1 - \alpha\_0) & \Gamma(\delta\_2) \\ -\Delta\_2 & \overline{\Gamma(\delta\_2 - \beta\_0)} \\ \hline \end{array} \bigg|\_{\Gamma(\delta\_2)} = \frac{\Gamma(\delta\_1)\Gamma(\delta\_2)}{\Gamma(\delta\_1 - \alpha\_0)\Gamma(\delta\_2 - \beta\_0)} - \Delta\_1 \Delta\_2 = \Delta\_1$$

Then, we obtain

$$a\_1 = \frac{1}{\Delta} \left( \mathfrak{c}\_0 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} + \mathfrak{d}\_0 \Delta\_1 \right), \ b\_1 = \frac{1}{\Delta} \left( \mathfrak{d}\_0 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \mathfrak{a}\_0)} + \mathfrak{c}\_0 \Delta\_2 \right).$$

Therefore, we deduce the solution (*x*(*t*), *y*(*t*)) of problem (8) and (9) presented in (10). By assumption (*K*1), we find that *x*(*t*) > 0 and *y*(*t*) > 0 for all *t* ∈ (0, 1].

We use the functions *x*(*t*) and *y*(*t*), *t* ∈ [0, 1] (given by (10)), and we make a change of unknown functions for our boundary value problem (1) and (2) such that the new boundary conditions have no positive parameters. For a solution (u, v) of problem (1) and (2), we define the functions *h*(*t*) and *k*(*t*), *t* ∈ [0, 1] by

$$\begin{aligned} h(t) &= \mathbf{u}(t) - \mathbf{x}(t) = \mathbf{u}(t) - \frac{t^{\delta\_1 - 1}}{\Delta} \left( \mathfrak{c}\_0 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} + \mathfrak{d}\_0 \Delta\_1 \right), \ t \in [0, 1], \\\ k(t) &= \mathbf{v}(t) - \mathbf{y}(t) = \mathbf{v}(t) - \frac{t^{\delta\_2 - 1}}{\Delta} \left( \mathfrak{c}\_0 \Delta\_2 + \mathfrak{d}\_0 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \alpha\_0)} \right), \ t \in [0, 1]. \end{aligned}$$

Then, problem (1) and (2) can be equivalently written as the system of fractional differential equations

$$\begin{cases} D\_{0+}^{\gamma\_1} (\varphi\_{\ell\_1}(D\_{0+}^{\delta\_1}h(t))) + \mathfrak{a}(t)\mathfrak{f}(k(t) + \mathfrak{y}(t)) = \mathbf{0}, & t \in (0,1), \\\ D\_{0+}^{\gamma\_2} (\varphi\_{\ell\_2}(D\_{0+}^{\delta\_2}k(t))) + \mathfrak{b}(t)\mathfrak{g}(h(t) + \mathfrak{x}(t)) = \mathbf{0}, & t \in (0,1), \end{cases} \tag{13}$$

with the boundary conditions without parameters

$$\begin{cases} \begin{aligned} \label{10.10} \begin{cases} \begin{array}{c} h^{(j)}(0) = 0, \; j = 0, \ldots, p-2; \; D\_{0+}^{\delta\_1} h(0) = 0, \; D\_{0+}^{\delta\_0} h(1) = \sum\_{j=1}^{n} \int\_{0}^{1} D\_{0+}^{\delta\_j} k(\tau) \, d\mathfrak{H}\_{\flat}(\tau), \\\ k^{(j)}(0) = 0, \; j = 0, \ldots, q-2; \; D\_{0+}^{\delta\_2} k(0) = 0, \; D\_{0+}^{\delta\_0} k(1) = \sum\_{j=1}^{m} \int\_{0}^{1} D\_{0+}^{\delta\_j} h(\tau) \, d\mathfrak{H}\_{\flat}(\tau). \end{cases} \end{cases} \end{cases} \end{cases} \tag{14}$$

Using the Green functions G*i*, *i* = 1, ... , 4 and Lemma 1, a pair of functions (*h*, *k*) is a solution of problem (13) and (14) if and only if (*h*, *k*) is a solution of the system of integral equations

$$\begin{split} h(t) &= \int\_{0}^{1} \mathfrak{G}\_{1}(t,\zeta) \, q\_{\rho\_{1}}(I\_{0+}^{\gamma\_{1}}(\mathfrak{a}(\zeta)\mathfrak{f}(k(\zeta)+y(\zeta)))) \, d\zeta \\ &+ \int\_{0}^{1} \mathfrak{G}\_{2}(t,\zeta) \, q\_{\rho\_{2}}(I\_{0+}^{\gamma\_{2}}(\mathfrak{b}(\zeta)\mathfrak{g}(h(\zeta)+x(\zeta)))) \, d\zeta, \; t \in [0,1], \\ k(t) &= \int\_{0}^{1} \mathfrak{G}\_{3}(t,\zeta) \, q\_{\rho\_{1}}(I\_{0+}^{\gamma\_{1}}(\mathfrak{a}(\zeta)\mathfrak{f}(k(\zeta)+y(\zeta)))) \, d\zeta \\ &+ \int\_{0}^{1} \mathfrak{G}\_{4}(t,\zeta) \, q\_{\rho\_{2}}(I\_{0+}^{\gamma\_{2}}(\mathfrak{b}(\zeta)\mathfrak{g}(h(\zeta)+x(\zeta)))) \, d\zeta, \; t \in [0,1]. \end{split} \tag{15}$$

We consider the Banach space X = *C*[0, 1] with the supremum norm *z* <sup>=</sup> sup*τ*∈[0,1] <sup>|</sup>*z*(*τ*)<sup>|</sup> for *z* ∈ X, and the Banach space Y = X ×X with the norm (*h*, *k*) <sup>Y</sup> = max{ *h* , *k* } for (*h*, *k*) ∈ Y. We define the set V = {(*h*, *k*) ∈ Y, 0 ≤ *h*(*t*) ≤ e0, 0 ≤ *k*(*t*) ≤ e0, ∀ *t* ∈ [0, 1]}. We also define the operator S : V→Y, S = (S1, S2),

$$\begin{split} \mathcal{S}\_{1}(\boldsymbol{h},\boldsymbol{k})(t) &= \int\_{0}^{1} \mathfrak{G}\_{1}(t,\boldsymbol{\zeta}) \boldsymbol{\varrho}\_{\rho\_{1}}(I^{\gamma\_{1}}\_{0+}(\mathfrak{a}(\boldsymbol{\zeta})\mathfrak{f}(\boldsymbol{k}(\boldsymbol{\zeta})+\boldsymbol{y}(\boldsymbol{\zeta})))) \, d\boldsymbol{\zeta} \\ &+ \int\_{0}^{1} \mathfrak{G}\_{2}(t,\boldsymbol{\zeta}) \boldsymbol{\varrho}\_{\rho\_{2}}(I^{\gamma\_{2}}\_{0+}(\mathfrak{b}(\boldsymbol{\zeta})\mathfrak{g}(\boldsymbol{h}(\boldsymbol{\zeta})+\mathfrak{x}(\boldsymbol{\zeta})))) \, d\boldsymbol{\zeta}, \; t \in [0,1], \\ \mathcal{S}\_{2}(\boldsymbol{h},\boldsymbol{k})(t) &= \int\_{0}^{1} \mathfrak{G}\_{3}(t,\boldsymbol{\zeta}) \boldsymbol{\varrho}\_{\rho\_{1}}(I^{\gamma\_{1}}\_{0+}(\mathfrak{a}(\boldsymbol{\zeta})\mathfrak{f}(\boldsymbol{k}(\boldsymbol{\zeta})+\mathfrak{y}(\boldsymbol{\zeta})))) \, d\boldsymbol{\zeta} \\ &+ \int\_{0}^{1} \mathfrak{G}\_{4}(t,\boldsymbol{\zeta}) \boldsymbol{\varrho}\_{\rho\_{2}}(I^{\gamma\_{2}}\_{0+}(\mathfrak{b}(\boldsymbol{\zeta})\mathfrak{g}(\boldsymbol{h}(\boldsymbol{\zeta})+\mathfrak{x}(\boldsymbol{\zeta})))) \, d\boldsymbol{\zeta}, \; t \in [0,1], \end{split}$$

for (*h*, *k*) ∈ V. We easily see that (*h*, *k*) is a solution of system (15) if and only if (*h*, *k*) is a fixed point of operator S. Therefore, our next task is the detection of the fixed points of operator S. The first result is the following existence theorem for problem (1) and (2):

**Theorem 1.** *We assume that assumptions* (*K*1) − (*K*3) *are satisfied. Therefore, there exist* c<sup>1</sup> > 0 *and* d<sup>1</sup> > 0 *such that for any* c<sup>0</sup> ∈ (0,c1] *and* d<sup>0</sup> ∈ (0, d1]*, the problem (1) and (2) has at least one positive solution.*

**Proof.** By assumption (*K*3) we deduce that there exist s<sup>0</sup> > 0 and t<sup>0</sup> > 0 such that f(*w*) ≤ <sup>e</sup> 1−1 0 *<sup>L</sup>* for all *w* ∈ [0, e<sup>0</sup> + s0], and g(*w*) ≤ <sup>e</sup> 2−1 0 *<sup>L</sup>* for all *w* ∈ [0, e<sup>0</sup> + t0]. We define now c<sup>1</sup> and d<sup>1</sup> as follows:

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_{1} = \min\left\{ \frac{\mathfrak{s}\_{0}\Delta}{2\Delta\_{2}}, \frac{\mathfrak{t}\_{0}\Delta\Gamma(\delta\_{2}-\beta\_{0})}{2\Gamma(\delta\_{2})} \right\}, \ \mathfrak{d}\_{1} = \min\left\{ \frac{\mathfrak{s}\_{0}\Delta\Gamma(\delta\_{1}-\mathfrak{a}\_{0})}{2\Gamma(\delta\_{1})}, \frac{\mathfrak{t}\_{0}\Delta}{2\Delta\_{1}} \right\}.$$

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_{1} = \min \left\{ \frac{\mathfrak{s}\_{0}\Delta}{2\Delta\_{2}}, \frac{\mathfrak{s}\_{0}\Delta\Gamma(\delta\_{2}-\beta\_{0})}{\Gamma(\delta\_{2})} \right\}, \ \mathfrak{d}\_{1} = \frac{\mathfrak{s}\_{0}\Delta\Gamma(\delta\_{1}-\mathfrak{a}\_{0})}{2\Gamma(\delta\_{1})}.$$

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_{1} = \frac{\mathfrak{e}\_{0}\Delta\Gamma(\delta\_{2}-\beta\_{0})}{2\Gamma(\delta\_{2})},\ \mathfrak{d}\_{1} = \min\left\{\frac{\mathfrak{s}\_{0}\Delta\Gamma(\delta\_{1}-\mathfrak{a}\_{0})}{\Gamma(\delta\_{1})},\frac{\mathfrak{s}\_{0}\Delta}{2\Delta\_{1}}\right\}.$$

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_1 = \frac{\mathfrak{t}\_0 \Delta \Gamma(\delta\_2 - \beta\_0)}{\Gamma(\delta\_2)}, \ \mathfrak{d}\_1 = \frac{\mathfrak{s}\_0 \Delta \Gamma(\delta\_1 - \mathfrak{a}\_0)}{\Gamma(\delta\_1)}.$$

Let c<sup>0</sup> ∈ (0,c1] and d<sup>0</sup> ∈ (0, d1]. Then, for (*h*, *k*) ∈ V and *ζ* ∈ [0, 1], we have

$$\begin{cases} k(\boldsymbol{\zeta}) + y(\boldsymbol{\zeta}) \leq \mathfrak{e}\_{0} + \frac{1}{\Delta} \Big( \mathfrak{e}\_{0}\Delta\_{2} + \mathfrak{d}\_{0} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1} - a\_{0})} \Big) \leq \mathfrak{e}\_{0} + \frac{1}{\Delta} \Big( \mathfrak{e}\_{1}\Delta\_{2} + \mathfrak{d}\_{1} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1} - a\_{0})} \Big) \leq \mathfrak{e}\_{0} + \mathfrak{e}\_{0},\\ h(\boldsymbol{\zeta}) + \mathfrak{x}(\boldsymbol{\zeta}) \leq \mathfrak{e}\_{0} + \frac{1}{\Delta} \Big( \mathfrak{e}\_{0} \frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2} - \tilde{\rho}\_{0})} + \mathfrak{d}\_{0}\Delta\_{1} \Big) \leq \mathfrak{e}\_{0} + \frac{1}{\Delta} \Big( \mathfrak{e}\_{1}\frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2} - \tilde{\rho}\_{0})} + \mathfrak{d}\_{1}\Delta\_{1} \Big) \leq \mathfrak{e}\_{0} + \mathfrak{e}\_{0}. \end{cases}$$

and so

$$f(k(\zeta) + y(\zeta)) \le \frac{\mathfrak{e}\_0^{\varrho\_1 - 1}}{L}, \quad \mathfrak{g}(h(\zeta) + x(\zeta)) \le \frac{\mathfrak{e}\_0^{\varrho\_2 - 1}}{L}.\tag{16}$$

By using Lemma 3, we deduce that S*i*(*h*, *k*)(*t*) ≥ 0, *i* = 1, 2 for all *t* ∈ [0, 1] and (*h*, *k*) ∈ V. By inequalities (16), for all (*h*, *k*) ∈ V, we obtain

$$\begin{split} &I\_{0+}^{\gamma\_{1}}(\mathfrak{a}(\boldsymbol{\zeta})\mathfrak{f}(k(\boldsymbol{\zeta})+\mathfrak{y}(\boldsymbol{\zeta}))) = \frac{1}{\Gamma(\gamma\_{1})} \int\_{0}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_{1}-1} \mathfrak{a}(\boldsymbol{\tau}) \mathfrak{f}(k(\boldsymbol{\tau})+\mathfrak{y}(\boldsymbol{\tau})) \, d\boldsymbol{\tau} \\ &\leq \frac{\mathfrak{e}\_{0}^{\varrho\_{1}-1}}{L\Gamma(\gamma\_{1})} \int\_{0}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_{1}-1} \mathfrak{a}(\boldsymbol{\tau}) \, d\boldsymbol{\tau} \leq \frac{\mathfrak{D}\_{1} \mathfrak{e}\_{0}^{\varrho\_{1}-1}}{L\Gamma(\gamma\_{1})} \int\_{0}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_{1}-1} \, d\boldsymbol{\tau} \\ &= \frac{\mathfrak{D}\_{1} \mathfrak{e}\_{0}^{\varrho\_{1}-1} \mathfrak{f}^{\gamma\_{1}}}{L\Gamma(\gamma\_{1}+1)} \, \, \forall \, \boldsymbol{\zeta} \in [0,1], \end{split}$$

and

$$\begin{split} &I\_{0+}^{\gamma\_{2}}(\mathfrak{b}(\boldsymbol{\zeta})\mathfrak{g}(h(\boldsymbol{\zeta})+\mathfrak{x}(\boldsymbol{\zeta}))) = \frac{1}{\Gamma(\gamma\_{2})} \int\_{0}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_{2}-1} \mathfrak{b}(\boldsymbol{\tau})\mathfrak{g}(h(\boldsymbol{\tau})+\mathfrak{x}(\boldsymbol{\tau})) \,d\boldsymbol{\tau} \\ &\leq \frac{\mathfrak{c}\_{0}^{\varrho\_{2}-1}}{L\Gamma(\gamma\_{2})} \int\_{0}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_{2}-1} \mathfrak{b}(\boldsymbol{\tau}) \,d\boldsymbol{\tau} \leq \frac{\Xi\_{2}\mathfrak{c}\_{0}^{\varrho\_{2}-1}}{L\Gamma(\gamma\_{2})} \int\_{0}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_{2}-1} \,d\boldsymbol{\tau} \\ &= \frac{\Xi\_{2}\mathfrak{c}\_{0}^{\varrho\_{2}-1}\tilde{\zeta}^{\gamma\_{2}}}{L\Gamma(\gamma\_{2}+1)}, \ \forall \boldsymbol{\zeta} \in [0,1]. \end{split}$$

Then, by Lemma 2 and the definition of *L* from (*K*3), we find

$$\begin{split} &\mathcal{S}\_{1}(k,k)(t) \leq \int\_{0}^{1} \mathfrak{J}\_{1}(\zeta) \, d\rho\_{\rho\_{1}} \left( \frac{\Xi\_{1}\mathfrak{e}\_{0}^{\rho\_{1}-1}\zeta^{\gamma\_{1}}}{L\Gamma(\gamma\_{1}+1)} \right) d\zeta + \int\_{0}^{1} \mathfrak{J}\_{2}(\zeta) \, d\rho\_{\rho\_{2}} \left( \frac{\Xi\_{2}\mathfrak{e}\_{0}^{\rho\_{2}-1}\zeta^{\gamma\_{2}}}{L\Gamma(\gamma\_{2}+1)} \right) d\zeta \\ &= \left( \frac{\Xi\_{1}\mathfrak{e}\_{0}^{\rho\_{1}-1}}{L\Gamma(\gamma\_{1}+1)} \right)^{\rho\_{1}-1} \int\_{0}^{1} \mathfrak{J}\_{1}(\zeta) \zeta^{\gamma\_{1}(\rho\_{1}-1)} \, d\zeta \\ &+ \left( \frac{\Xi\_{2}\mathfrak{e}\_{0}^{\rho\_{2}-1}}{L\Gamma(\gamma\_{2}+1)} \right)^{\rho\_{2}-1} \int\_{0}^{1} \mathfrak{J}\_{2}(\zeta) \zeta^{\gamma\_{2}(\rho\_{2}-1)} \, d\zeta \leq \frac{\mathfrak{e}\_{0}}{2} + \frac{\mathfrak{e}\_{0}}{2} = \mathfrak{e}\_{0}, \ \forall \, t \in [0,1], \end{split}$$

and

$$\begin{split} &\mathcal{S}\_{2}(k,k)(t) \leq \int\_{0}^{1} \mathfrak{J}\_{3}(\zeta) \, d\rho\_{\rho\_{1}} \left( \frac{\Xi\_{1}\mathfrak{c}\_{0}^{\rho\_{1}-1}\zeta^{\gamma\_{1}}}{L\Gamma(\gamma\_{1}+1)} \right) d\zeta + \int\_{0}^{1} \mathfrak{J}\_{4}(\zeta) \, \rho\_{\rho\_{2}} \left( \frac{\Xi\_{2}\mathfrak{c}\_{0}^{\rho\_{2}-1}\zeta^{\gamma\_{2}}}{L\Gamma(\gamma\_{2}+1)} \right) d\zeta \\ &= \left( \frac{\Xi\_{1}\mathfrak{c}\_{0}^{\rho\_{1}-1}}{L\Gamma(\gamma\_{1}+1)} \right)^{\rho\_{1}-1} \int\_{0}^{1} \mathfrak{J}\_{3}(\zeta) \zeta^{\gamma\_{1}(\rho\_{1}-1)} \, d\zeta \\ &+ \left( \frac{\Xi\_{2}\mathfrak{c}\_{0}^{\rho\_{2}-1}}{L\Gamma(\gamma\_{2}+1)} \right)^{\rho\_{2}-1} \int\_{0}^{1} \mathfrak{J}\_{4}(\zeta) \zeta^{\gamma\_{2}(\rho\_{2}-1)} \, d\zeta \leq \frac{\mathfrak{c}\_{0}}{2} + \frac{\mathfrak{c}\_{0}}{2} = \mathfrak{c}\_{0}, \ \forall \, t \in [0,1]. \end{split}$$

Therefore, we find that S(V) ⊂ V. By using a standard method, we conclude that S is a completely continuous operator. Therefore, by the Schauder fixed point theorem, we deduce that S has a fixed point (*h*, *k*) ∈ V, which is a non-negative solution for problem (15), or equivalently, for problem (13) and (14). Hence, (u, v), where u(*t*) = *h*(*t*) + *x*(*t*) and v(*t*) = *k*(*t*) + *y*(*t*) for all *t* ∈ [0, 1], is a positive solution of problem (1) and (2). This solution (u, v) satisfies the conditions *<sup>t</sup> δ*1−1 <sup>Δ</sup> (c<sup>0</sup> Γ(*δ*2) <sup>Γ</sup>(*δ*2−*β*0) <sup>+</sup> <sup>d</sup>0Δ1) <sup>≤</sup> <sup>u</sup>(*t*) <sup>≤</sup> *<sup>t</sup> δ*1−1 <sup>Δ</sup> (c<sup>0</sup> Γ(*δ*2) <sup>Γ</sup>(*δ*2−*β*0) <sup>+</sup> <sup>d</sup>0Δ1) + <sup>e</sup><sup>0</sup> and *t δ*2−1 <sup>Δ</sup> (c0Δ<sup>2</sup> + d<sup>0</sup> Γ(*δ*1) <sup>Γ</sup>(*δ*1−*α*0)) <sup>≤</sup> <sup>v</sup>(*t*) <sup>≤</sup> *<sup>t</sup> δ*2−1 <sup>Δ</sup> (c0Δ<sup>2</sup> + d<sup>0</sup> Γ(*δ*1) <sup>Γ</sup>(*δ*1−*α*0)) + <sup>e</sup><sup>0</sup> for all *<sup>t</sup>* <sup>∈</sup> [0, 1].

The second result is the following nonexistence theorem for the boundary value problem (1) and (2).

**Theorem 2.** *We assume that assumptions* (*K*1)*,* (*K*2)*, and* (*K*4) *are satisfied. Then, there exist* c<sup>2</sup> > 0 *and* d<sup>2</sup> > 0 *such that for any* c<sup>0</sup> ≥ c<sup>2</sup> *and* d<sup>0</sup> ≥ d2*, the problem (1) and (2) has no positive solution.*

**Proof.** By assumption (*K*2), there exist [*η*1, *η*2] ⊂ (0, 1), *η*<sup>1</sup> < *η*<sup>2</sup> such that *τ*1, *τ*<sup>2</sup> ∈ (*η*1, *η*2), and then

$$\begin{aligned} \Lambda\_1 &= \int\_{\eta\_1}^{\eta\_2} \mathfrak{J}\_1(\zeta) \left( \int\_{\eta\_1}^{\zeta} \mathfrak{a}(\tau) (\zeta - \tau)^{\gamma\_1 - 1} \, d\tau \right)^{\rho\_1 - 1} d\zeta > 0, \\ \Lambda\_4 &= \int\_{\eta\_1}^{\eta\_2} \mathfrak{J}\_4(\zeta) \left( \int\_{\eta\_1}^{\zeta} \mathfrak{b}(\tau) (\zeta - \tau)^{\gamma\_2 - 1} \, d\tau \right)^{\rho\_2 - 1} d\zeta > 0. \end{aligned}$$

We define the number

$$R\_0 = \max\left\{ \frac{2^{\varrho\_1 - 1} \Gamma(\gamma\_1)}{\eta\_1^{(\delta\_1 + \delta\_2 - 2)(\varrho\_1 - 1)} \Lambda\_1^{\varrho\_1 - 1}}, \frac{2^{\varrho\_2 - 1} \Gamma(\gamma\_2)}{\eta\_1^{(\delta\_1 + \delta\_2 - 2)(\varrho\_2 - 1)} \Lambda\_4^{\varrho\_2 - 1}} \right\}.$$

By using (*K*4), for *R*<sup>0</sup> defined above, we obtain that there exists *L*<sup>0</sup> > 0 such that <sup>f</sup>(*w*) <sup>≥</sup> *<sup>R</sup>*0*w*1−<sup>1</sup> and <sup>g</sup>(*w*) <sup>≥</sup> *<sup>R</sup>*0*w*2−<sup>1</sup> for all *<sup>w</sup>* <sup>≥</sup> *<sup>L</sup>*0. We define now <sup>c</sup><sup>2</sup> and <sup>d</sup><sup>2</sup> as follows: • If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_{2} = \max\left\{ \frac{L\_{0}\Delta\Gamma(\delta\_{2}-\beta\_{0})}{2\eta\_{1}^{\delta\_{1}-1}\Gamma(\delta\_{2})}, \frac{L\_{0}\Delta}{2\eta\_{1}^{\delta\_{2}-1}\Delta\_{2}} \right\}, \ \mathfrak{d}\_{2} = \max\left\{ \frac{L\_{0}\Delta}{2\eta\_{1}^{\delta\_{1}-1}\Delta\_{1}}, \frac{L\_{0}\Delta\Gamma(\delta\_{1}-\mathfrak{a}\_{0})}{2\eta\_{1}^{\delta\_{2}-1}\Gamma(\delta\_{1})} \right\}.$$

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_2 = \max \left\{ \frac{L\_0 \Delta \Gamma(\delta\_2 - \beta\_0)}{\eta\_1^{\delta\_1 - 1} \Gamma(\delta\_2)}, \frac{L\_0 \Delta}{2\eta\_1^{\delta\_2 - 1} \Delta\_2} \right\}, \ \mathfrak{d}\_2 = \frac{L\_0 \Delta \Gamma(\delta\_1 - \mathfrak{a}\_0)}{2\eta\_1^{\delta\_2 - 1} \Gamma(\delta\_1)}.$$

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_2 = \frac{L\_0 \Delta \Gamma(\delta\_2 - \beta\_0)}{2\eta\_1^{\delta\_1 - 1} \Gamma(\delta\_2)}, \ \mathfrak{d}\_2 = \max\left\{ \frac{L\_0 \Delta}{2\eta\_1^{\delta\_1 - 1} \Delta\_1}, \frac{L\_0 \Delta \Gamma(\delta\_1 - \mathfrak{a}\_0)}{\eta\_1^{\delta\_2 - 1} \Gamma(\delta\_1)} \right\}.$$

• If Δ<sup>1</sup> = 0 and Δ<sup>2</sup> = 0, then

$$\mathfrak{c}\_2 = \frac{L\_0 \Delta \Gamma(\delta\_2 - \beta\_0)}{\eta\_1^{\delta\_1 - 1} \Gamma(\delta\_2)}, \ \mathfrak{d}\_2 = \frac{L\_0 \Delta \Gamma(\delta\_1 - \mathfrak{a}\_0)}{\eta\_1^{\delta\_2 - 1} \Gamma(\delta\_1)}.$$

Let c<sup>0</sup> ≥ c<sup>2</sup> and d<sup>0</sup> ≥ d2. We assume that (u, v) is a positive solution of (1) and (2). Then, the pair (*h*, *k*), where *h*(*t*) = u(*t*) − *x*(*t*), *k*(*t*) = v(*t*) − *y*(*t*), *t* ∈ [0, 1], with *x* and *y* given by (10), is a solution of problem (13) and (14), or equivalently, of system (15). By using Lemma 3, we find that *h*(*t*) ≥ *t <sup>δ</sup>*1−<sup>1</sup> *h* , *k*(*t*) ≥ *t <sup>δ</sup>*2−<sup>1</sup> *k* for all *t* ∈ [0, 1]. Then, inf*s*∈[*η*1,*η*2] *<sup>h</sup>*(*s*) ≥ *<sup>η</sup> δ*1−1 1 *h* , inf*s*∈[*η*1,*η*2] *<sup>k</sup>*(*s*) ≥ *<sup>η</sup> δ*2−1 1 *k* . By the definition of the functions *x* and *y*, we obtain

$$\begin{split} \inf\_{s \in [\eta\_1 \eta\_2]} \mathfrak{x}(s) &= \frac{\eta\_1^{\delta\_1 - 1}}{\Delta} \left( \mathfrak{o}\_0 \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} + \mathfrak{o}\_0 \Delta\_1 \right) = \eta\_1^{\delta\_1 - 1} ||\mathfrak{x}||\_{\mathcal{H}}, \\ \inf\_{s \in [\eta\_1 \eta\_2]} \mathfrak{y}(s) &= \frac{\eta\_1^{\delta\_2 - 1}}{\Delta} \left( \mathfrak{o}\_0 \Delta\_2 + \mathfrak{o}\_0 \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \mathfrak{o}\_0)} \right) = \eta\_1^{\delta\_2 - 1} ||\mathfrak{y}||. \end{split}$$

Hence, we deduce

$$\begin{array}{lcl} & \inf\_{s \in [\eta\_1, \eta\_2]} (h(s) + \mathfrak{x}(s)) \ge \inf\_{s \in [\eta\_1, \eta\_2]} h(s) + \inf\_{s \in [\eta\_1, \eta\_2]} \mathfrak{x}(s) \ge \eta\_1^{\delta\_1 - 1} ||h|| + \eta\_1^{\delta\_1 - 1} ||\mathfrak{x}||\\ & = \eta\_1^{\delta\_1 - 1} (||h|| + ||\mathfrak{x}||) \ge \eta\_1^{\delta\_1 - 1} ||h + \mathfrak{x}||\\ & \inf\_{s \in [\eta\_1, \eta\_2]} (k(s) + \mathfrak{y}(s)) \ge \inf\_{s \in [\eta\_1, \eta\_2]} k(s) + \inf\_{s \in [\eta\_1, \eta\_2]} \mathfrak{y}(s) \ge \eta\_1^{\delta\_2 - 1} ||k|| + \eta\_1^{\delta\_2 - 1} ||\mathfrak{y}||\\ & = \eta\_1^{\delta\_2 - 1} (||k|| + ||\mathfrak{y}||) \ge \eta\_1^{\delta\_2 - 1} ||k + \mathfrak{y}||. \end{array}$$

In addition we have

$$\begin{split} &\inf\_{s\in[\eta\_{1},\eta\_{2}]} \left( h(s) + \mathfrak{x}(s) \right) \geq \eta\_{1}^{\delta\_{1}-1} ||\!|\!| \!| = \eta\_{1}^{\delta\_{1}-1} \frac{1}{\Delta} \left( \mathfrak{c}\_{0} \frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2}-\beta\_{0})} + \mathfrak{d}\_{0}\Delta\_{1} \right) \\ &\geq \eta\_{1}^{\delta\_{1}-1} \frac{1}{\Delta} \left( \mathfrak{c}\_{2} \frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2}-\beta\_{0})} + \mathfrak{d}\_{2}\Delta\_{1} \right) \geq L\_{0}. \end{split}$$
 
$$\begin{split} &\inf\_{s\in[\eta\_{1},\eta\_{2}]} \left( k(s) + y(s) \right) \geq \eta\_{1}^{\delta\_{2}-1} ||y|| = \eta\_{1}^{\delta\_{2}-1} \frac{1}{\Delta} \left( \mathfrak{c}\_{0}\Delta\_{2} + \mathfrak{d}\_{0} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1}-\alpha\_{0})} \right) \\ &\geq \eta\_{1}^{\delta\_{2}-1} \frac{1}{\Delta} \left( \mathfrak{c}\_{2}\Delta\_{2} + \mathfrak{d}\_{2} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1}-\alpha\_{0})} \right) \geq L\_{0}. \end{split}$$

By using Lemma 3 and the above inequalities we find

$$\begin{split} &\frac{1}{\Gamma} \sum\_{\begin{subarray}{c}\gamma\\\gamma\end{subarray}} \frac{1}{\Gamma(\gamma\_{1})} \int\_{\eta\_{1}}^{\zeta} (\widetilde{\zeta}-\tau)^{\gamma\_{1}-1} \mathfrak{a}(\tau) \mathfrak{f}(k(\tau)+y(\tau)) \,d\tau\\ &\geq \frac{R\_{0}}{\Gamma(\gamma\_{1})} \int\_{\eta\_{1}}^{\zeta} (\widetilde{\zeta}-\tau)^{\gamma\_{1}-1} \mathfrak{a}(\tau) (k(\tau)+y(\tau))^{\varrho\_{1}-1} \,d\tau\\ &\geq \frac{R\_{0}}{\Gamma(\gamma\_{1})} \int\_{\eta\_{1}}^{\zeta} (\widetilde{\zeta}-\tau)^{\gamma\_{1}-1} \mathfrak{a}(\tau) \left(\inf\_{\tau\in[\eta\_{1},\eta\_{2}]} (k(\tau)+y(\tau))\right)^{\varrho\_{1}-1} \,d\tau\\ &\geq \frac{R\_{0} L\_{0}^{\varrho\_{1}-1}}{\Gamma(\gamma\_{1})} \int\_{\eta\_{1}}^{\zeta} (\widetilde{\zeta}-\tau)^{\gamma\_{1}-1} \mathfrak{a}(\tau) \,d\tau, \,\,\forall \,\zeta \in [\eta\_{1},\eta\_{2}]. \end{split}$$

and then

$$\begin{split} &h(\eta\_{1}) \geq \int\_{0}^{1} \eta\_{1}^{\delta\_{1}-1} \mathfrak{J}\_{1}(\zeta) \, \upmu\_{\rho\_{1}}(I\_{0+}^{\gamma\_{1}}(\mathfrak{a}(\zeta)\mathfrak{f}(k(\zeta)+\mathfrak{y}(\zeta)))) \, d\zeta \\ &\geq \int\_{\eta\_{1}}^{\eta\_{2}} \eta\_{1}^{\delta\_{1}-1} \mathfrak{J}\_{1}(\zeta) \left(\frac{R\_{0}L\_{0}^{\theta\_{1}-1}}{\Gamma(\gamma\_{1})} \int\_{\eta\_{1}}^{\zeta} (\zeta-\tau)^{\gamma\_{1}-1} \mathfrak{a}(\tau) \, d\tau\right)^{\rho\_{1}-1} \, d\zeta \\ &= \frac{R\_{0}^{\rho\_{1}-1}L\_{0}\eta\_{1}^{\delta\_{1}-1}\Lambda\_{1}}{(\Gamma(\gamma\_{1}))^{\rho\_{1}-1}} > 0. \end{split}$$

We deduce that *h* ≥ *h*(*η*1) > 0. In a similar manner, we obtain

$$\begin{aligned} &I\_{0+}^{\gamma\_2}(\mathfrak{b}(\boldsymbol{\zeta})\mathfrak{g}(h(\boldsymbol{\zeta})+\mathfrak{x}(\boldsymbol{\zeta}))) \\ &\geq \frac{R\_0}{\Gamma(\gamma\_2)} \int\_{\eta\_1}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_2-1} \mathfrak{b}(\boldsymbol{\tau}) \left(\inf\_{\boldsymbol{\tau}\in[\eta\_1,\eta\_2]} (h(\boldsymbol{\tau})+\mathfrak{x}(\boldsymbol{\tau}))\right)^{\varrho\_2-1} d\boldsymbol{\tau} \\ &\geq \frac{R\_0 L\_0^{\varrho\_2-1}}{\Gamma(\gamma\_2)} \int\_{\eta\_1}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_2-1} \mathfrak{b}(\boldsymbol{\tau}) \, d\boldsymbol{\tau}, \;\,\forall \,\zeta \in [\eta\_1,\eta\_2]. \end{aligned}$$

and so

$$\begin{split} &k(\eta\_{1}) \geq \int\_{0}^{1} \eta\_{1}^{\delta\_{2}-1} \mathfrak{J}\_{4}(\zeta) \mathfrak{q}\_{\rho\_{2}} \left( I\_{0+}^{\gamma\_{2}} (\mathfrak{b}(\zeta)\mathfrak{g}(h(\zeta)+\mathfrak{x}(\zeta))) \right) d\zeta \\ & \geq \int\_{\eta\_{1}}^{\eta\_{2}} \eta\_{1}^{\delta\_{2}-1} \mathfrak{J}\_{4}(\zeta) \left( \frac{R\_{0}L\_{0}^{\theta\_{2}-1}}{\Gamma(\gamma\_{2})} \int\_{\eta\_{1}}^{\zeta} (\zeta-\tau)^{\gamma\_{2}-1} \mathfrak{b}(\tau) \, d\tau \right)^{\rho\_{2}-1} d\zeta \\ &= \frac{R\_{0}^{\rho\_{2}-1}L\_{0}\eta\_{1}^{\delta\_{2}-1}\Lambda\_{4}}{(\Gamma(\gamma\_{2}))^{\rho\_{2}-1}} > 0. \end{split}$$

We deduce that *k* ≥ *k*(*η*1) > 0. In addition, from the above inequalities we have

$$\begin{aligned} &I\_{0+}^{\gamma\_1}(\mathfrak{a}(\boldsymbol{\xi})\mathfrak{f}(k(\boldsymbol{\zeta})+\boldsymbol{y}(\boldsymbol{\zeta}))) \\ &\geq \frac{R\_0}{\Gamma(\gamma\_1)} \int\_{\eta\_1}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_1-1} \mathfrak{a}(\boldsymbol{\tau}) \left(\inf\_{\boldsymbol{\tau}\in[\eta\_1,\eta\_2]} (k(\boldsymbol{\tau})+\boldsymbol{y}(\boldsymbol{\tau}))\right)^{\varrho\_1-1} d\boldsymbol{\tau} \\ &\geq \frac{R\_0 \eta\_1^{(\delta\_2-1)(\varrho\_1-1)}}{\Gamma(\gamma\_1)} ||k+\boldsymbol{y}||^{\varrho\_1-1} \int\_{\eta\_1}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_1-1} \mathfrak{a}(\boldsymbol{\tau}) \, d\boldsymbol{\tau}, \;\,\forall \,\boldsymbol{\zeta} \in [\eta\_1,\eta\_2]. \end{aligned}$$

and so

$$\begin{split} &h(\eta\_{1}) \geq \int\_{\eta\_{1}}^{\eta\_{2}} \eta\_{1}^{\delta\_{1}-1} \mathfrak{I}\_{1}(\zeta) \left(\frac{\mathsf{R}\_{0}\eta\_{1}^{(\delta\_{2}-1)(\varrho\_{1}-1)}}{\Gamma(\gamma\_{1})}\right)^{\rho\_{1}-1} ||k+y|| \left(\int\_{\eta\_{1}}^{\zeta} (\zeta-\tau)^{\gamma\_{1}-1} \mathfrak{a}(\tau) \,d\tau\right)^{\rho\_{1}-1} d\zeta \\ &= \frac{\eta\_{1}^{\delta\_{1}+\delta\_{2}-2} \mathsf{R}\_{0}^{\rho\_{1}-1}}{(\Gamma(\gamma\_{1}))^{\rho\_{1}-1}} \Lambda\_{1} ||k+y|| \geq 2||k+y|| \geq 2||k||. \end{split}$$

Hence,

$$||k|| \le \frac{1}{2}h(\eta\_1) \le \frac{1}{2}||h||.\tag{17}$$

In a similar manner, we deduce

$$\begin{aligned} &I\_{0+}^{\gamma\_2}(\mathfrak{b}(\boldsymbol{\zeta})\mathfrak{g}(h(\boldsymbol{\zeta})+\mathfrak{x}(\boldsymbol{\zeta}))) \\ &\geq \frac{R\_0}{\Gamma(\gamma\_2)} \int\_{\eta\_1}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_2-1} \mathfrak{b}(\boldsymbol{\tau}) \left(\inf\_{\boldsymbol{\tau}\in[\eta\_1,\eta\_2]} (h(\boldsymbol{\tau})+\mathfrak{x}(\boldsymbol{\tau}))\right)^{\varrho\_2-1} d\boldsymbol{\tau} \\ &\geq \frac{R\_0 \eta\_1^{(\delta\_1-1)(\varrho\_2-1)}}{\Gamma(\gamma\_2)} ||h+\mathfrak{x}||^{\varrho\_2-1} \int\_{\eta\_1}^{\zeta} (\boldsymbol{\zeta}-\boldsymbol{\tau})^{\gamma\_2-1} \mathfrak{b}(\boldsymbol{\tau}) \, d\boldsymbol{\tau}, \;\,\,\forall \,\zeta \in [\eta\_1,\eta\_2]. \end{aligned}$$

and then

$$\begin{split} k(\eta\_{1}) &\geq \int\_{\eta\_{1}}^{\eta\_{2}} \eta\_{1}^{\delta\_{2}-1} \mathfrak{I}\_{4}(\zeta) \left( \frac{R\_{0}\eta\_{1}^{(\delta\_{1}-1)(\varrho\_{2}-1)}}{\Gamma(\gamma\_{2})} \right)^{\rho\_{2}-1} ||h+\mathfrak{x}|| \left( \int\_{\eta\_{1}}^{\zeta} (\zeta-\tau)^{\gamma\_{2}-1} \mathfrak{b}(\tau) \, d\tau \right)^{\rho\_{2}-1} d\zeta \\ &= \frac{\eta\_{1}^{\delta\_{1}+\delta\_{2}-2}R\_{0}^{\rho\_{2}-1}}{(\Gamma(\gamma\_{2}))^{\rho\_{2}-1}} \Lambda\_{4} ||h+\mathfrak{x}|| \geq 2||h+\mathfrak{x}|| \geq 2||h||. \end{split}$$

Therefore,

$$||h|| \le \frac{1}{2}k(\eta\_1) \le \frac{1}{2}||k||.\tag{18}$$

Hence, by (17) and (18), we conclude that *h* <sup>≤</sup> <sup>1</sup> 2 *k* <sup>≤</sup> <sup>1</sup> 4 *h* , which is a contradiction (we saw before that *h* > 0). Therefore, problem (1) and (2) has no positive solution.

#### **4. An Example**

We consider *γ*<sup>1</sup> = <sup>3</sup> <sup>4</sup> , *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>5</sup> , *<sup>δ</sup>*<sup>1</sup> <sup>=</sup> <sup>14</sup> <sup>3</sup> , (*<sup>p</sup>* <sup>=</sup> 5), *<sup>δ</sup>*<sup>2</sup> <sup>=</sup> <sup>11</sup> <sup>2</sup> , (*<sup>q</sup>* <sup>=</sup> 6), *<sup>n</sup>* <sup>=</sup> 2, *<sup>m</sup>* <sup>=</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>=</sup> <sup>17</sup> 8 , *β*<sup>0</sup> = <sup>19</sup> <sup>6</sup> , *<sup>α</sup>*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>2</sup> , *<sup>α</sup>*<sup>2</sup> <sup>=</sup> <sup>16</sup> <sup>7</sup> , *<sup>β</sup>*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>7</sup> , <sup>1</sup> <sup>=</sup> <sup>73</sup> <sup>12</sup> , <sup>2</sup> <sup>=</sup> <sup>59</sup> <sup>8</sup> , *<sup>ρ</sup>*<sup>1</sup> <sup>=</sup> <sup>73</sup> <sup>61</sup> , *<sup>ρ</sup>*<sup>2</sup> <sup>=</sup> <sup>59</sup> <sup>51</sup> , a(*t*) = 1, b(*t*) = 1 for all *<sup>t</sup>* <sup>∈</sup> [0, 1], <sup>H</sup>1(*t*) = <sup>91</sup> <sup>6</sup> *<sup>t</sup>* for all *<sup>t</sup>* <sup>∈</sup> [0, 1], <sup>H</sup>2(*t*) = <sup>1</sup> <sup>3</sup> , *t* ∈ 0 0, <sup>2</sup> 3 ; <sup>17</sup> <sup>15</sup> , *<sup>t</sup>* <sup>∈</sup> <sup>0</sup> <sup>2</sup> <sup>3</sup> , 11 , <sup>K</sup>1(*t*) = <sup>1</sup> <sup>2</sup> , *t* ∈ 0 0, <sup>8</sup> 11  ; <sup>33</sup> <sup>26</sup> , *<sup>t</sup>* <sup>∈</sup> <sup>0</sup> <sup>8</sup> <sup>11</sup> , 11 . We introduce the functions f, g; [0, ∞) → [0, ∞), <sup>f</sup>(*z*) = *<sup>ω</sup>*1*zσ*<sup>1</sup> , <sup>g</sup>(*z*) = *<sup>ω</sup>*2*zσ*<sup>2</sup> for all *<sup>z</sup>* <sup>∈</sup> [0, <sup>∞</sup>) with *<sup>ω</sup>*1, *<sup>ω</sup>*<sup>2</sup> <sup>&</sup>gt; 0, *<sup>σ</sup>*1, *<sup>σ</sup>*<sup>2</sup> <sup>&</sup>gt; 0, *<sup>σ</sup>*<sup>1</sup> <sup>&</sup>gt; <sup>61</sup> <sup>12</sup> , *<sup>σ</sup>*<sup>2</sup> <sup>&</sup>gt; <sup>51</sup> 8 . We have lim*z*→<sup>∞</sup> <sup>f</sup>(*z*) *<sup>z</sup>*1−<sup>1</sup> <sup>=</sup> <sup>∞</sup> and lim*z*→<sup>∞</sup> <sup>g</sup>(*z*) *<sup>z</sup>*2−<sup>1</sup> <sup>=</sup> <sup>∞</sup>.

We consider the system of Riemann–Liouville fractional differential equations

$$\begin{cases} D\_{0+}^{3/4} \left( \varphi\_{73/12} \left( D\_{0+}^{14/3} \mathbf{u}(t) \right) \right) + \omega\_1 (\mathbf{v}(t))^{\sigma\_1} = 0, & t \in (0,1), \\\ D\_{0+}^{2/5} \left( \varphi\_{59/8} \left( D\_{0+}^{11/2} \mathbf{v}(t) \right) \right) + \omega\_2 (\mathbf{u}(t))^{\sigma\_2} = 0, & t \in (0,1), \end{cases} \tag{19}$$

subject to the coupled boundary conditions

$$\begin{cases} \mathbf{u}^{(i)}(0) = 0, \; i = 0, \ldots, 3, \; D\_{0+}^{14/3} \mathbf{u}(0) = 0, \\\ D\_{0+}^{17/8} \mathbf{u}(1) = \frac{91}{6} \int\_{0}^{1} D\_{0+}^{3/2} \mathbf{v}(t) \, dt + \frac{4}{5} D\_{0+}^{16/7} \mathbf{v}\left(\frac{2}{3}\right) + \mathbf{c}\_{0\prime} \\\ \mathbf{v}^{(i)}(0) = 0, \; i = 0, \ldots, 4, \; D\_{0+}^{11/2} \mathbf{v}(0) = 0, \; D\_{0+}^{19/6} \mathbf{v}(1) = \frac{10}{13} D\_{0+}^{3/7} \mathbf{u}\left(\frac{8}{11}\right) + \mathbf{c}\_{0\prime} \end{cases} \tag{20}$$

We obtain here Δ<sup>1</sup> ≈ 40.01662964, Δ<sup>2</sup> ≈ 0.49478575, and Δ ≈ 452.46647281 > 0. Therefore, assumptions (*K*1), (*K*2), and (*K*4) are satisfied. In addition, we deduce

<sup>g</sup>1(*t*, *<sup>ζ</sup>*) = <sup>1</sup> Γ(14/3) - *t* 11/3(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24 <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>ζ</sup>*)11/3, 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t* 11/3(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> 1, <sup>g</sup>11(*τ*, *<sup>ζ</sup>*) = <sup>1</sup> Γ(89/21) - *<sup>τ</sup>*68/21(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>ζ</sup>*)68/21, 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*68/21(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> 1, <sup>g</sup>2(*t*, *<sup>ζ</sup>*) = <sup>1</sup> Γ(11/2) - *t* 9/2(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>ζ</sup>*)9/2, 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t* 9/2(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> 1, <sup>g</sup>21(*τ*, *<sup>ζ</sup>*) = <sup>1</sup> 6 - *<sup>τ</sup>*3(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>ζ</sup>*)3, 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*3(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> 1, <sup>g</sup>22(*τ*, *<sup>ζ</sup>*) = <sup>1</sup> Γ(45/14) - *<sup>τ</sup>*31/14(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>ζ</sup>*)31/14, 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*31/14(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> 1, <sup>G</sup>1(*t*, *<sup>ζ</sup>*) = <sup>g</sup>1(*t*, *<sup>ζ</sup>*) + <sup>10</sup>Δ1*<sup>t</sup>* 11/3 <sup>13</sup><sup>Δ</sup> <sup>g</sup><sup>11</sup> <sup>8</sup> <sup>11</sup>, *<sup>ζ</sup>* , <sup>G</sup>2(*t*, *<sup>ζ</sup>*) = *<sup>t</sup>* 11/3Γ(11/2) ΔΓ(7/3) 91 6 <sup>1</sup> 0 g21(*τ*, *ζ*) *dτ* + 4 5 <sup>g</sup>22 <sup>2</sup> 3 , *ζ* , <sup>G</sup>3(*t*, *<sup>ζ</sup>*) = <sup>10</sup>*<sup>t</sup>* 9/2Γ(14/3) <sup>13</sup>ΔΓ(61/24) <sup>g</sup><sup>11</sup> <sup>8</sup> <sup>11</sup>, *<sup>ζ</sup>* , <sup>G</sup>4(*t*, *<sup>ζ</sup>*) = <sup>g</sup>2(*t*, *<sup>ζ</sup>*) + *<sup>t</sup>* 9/2Δ<sup>2</sup> Δ 91 6 <sup>1</sup> 0 g21(*τ*, *ζ*) *dτ* + 4 5 <sup>g</sup>22 <sup>2</sup> 3 , *ζ* , <sup>h</sup>1(*ζ*) = <sup>1</sup> Γ(14/3) (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24 <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)17/8 , <sup>h</sup>2(*ζ*) = <sup>1</sup> Γ(11/2) (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)19/6 ,

for all *t*, *τ*, *ζ* ∈ [0, 1]. In addition, we find

J1(*ζ*) = ⎧ ⎨ ⎩ <sup>h</sup>1(*ζ*) + <sup>10</sup>Δ<sup>1</sup> 13ΔΓ(89/21) <sup>8</sup> <sup>11</sup> 68/21(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24 <sup>−</sup> <sup>8</sup> <sup>11</sup> − *ζ* 68/21 , 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; <sup>8</sup> 11 , <sup>h</sup>1(*ζ*) + <sup>10</sup>Δ<sup>1</sup> 13ΔΓ(89/21) <sup>8</sup> <sup>11</sup> 68/21(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24, <sup>8</sup> <sup>11</sup> ≤ *ζ* ≤ 1, J2(*ζ*) = ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ Γ(11/2) ΔΓ(7/3) 91 <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> <sup>91</sup> <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)<sup>4</sup> <sup>+</sup> <sup>4</sup> 5Γ(45/14) × <sup>2</sup> 3 31/14(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> <sup>2</sup> <sup>3</sup> − *ζ* 31/14 , 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; <sup>2</sup> 3 , Γ(11/2) ΔΓ(7/3) 91 <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> <sup>91</sup> <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)<sup>4</sup> <sup>+</sup> <sup>4</sup> 5Γ(45/14) <sup>×</sup> <sup>2</sup> 3 31/14(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 , <sup>2</sup> <sup>3</sup> ≤ *ζ* ≤ 1, J3(*ζ*) = ⎧ ⎨ ⎩ 10Γ(14/3) 13ΔΓ(61/24)Γ(89/21) <sup>8</sup> <sup>11</sup> 68/21(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24 <sup>−</sup> <sup>8</sup> <sup>11</sup> − *ζ* 68/21 , 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; <sup>8</sup> 11 , 10Γ(14/3) 13ΔΓ(61/24)Γ(89/21) <sup>8</sup> <sup>11</sup> 68/21(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)37/24, <sup>8</sup> <sup>11</sup> ≤ *ζ* ≤ 1, J4(*ζ*) = ⎧ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ <sup>h</sup>2(*ζ*) + <sup>Δ</sup><sup>2</sup> Δ 91 <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> <sup>91</sup> <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)<sup>4</sup> <sup>+</sup> <sup>4</sup> 5Γ(45/14) × <sup>2</sup> 3 31/14(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> <sup>2</sup> <sup>3</sup> − *ζ* 31/14 , 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; <sup>2</sup> 3 , <sup>h</sup>2(*ζ*) + <sup>Δ</sup><sup>2</sup> Δ 91 <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 <sup>−</sup> <sup>91</sup> <sup>144</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)<sup>4</sup> <sup>+</sup> <sup>4</sup> 5Γ(45/14) <sup>×</sup> <sup>2</sup> 3 31/14(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)4/3 , <sup>2</sup> <sup>3</sup> ≤ *ζ* ≤ 1.

We also obtain Ξ<sup>1</sup> = 1 and Ξ<sup>2</sup> = 1. After some computations, we find

$$\begin{aligned} P\_1 &:= \frac{2^{61/12}}{\Gamma(7/4)} \left( \int\_0^1 \mathfrak{J}\_1(\zeta) \xi^{9/61} \, d\zeta \right)^{61/12} \approx 4.11609161 \times 10^{-9} \, \\\ P\_2 &:= \frac{2^{51/8}}{\Gamma(7/5)} \left( \int\_0^1 \mathfrak{J}\_2(\zeta) \xi^{16/255} \, d\zeta \right)^{51/8} \approx 3.11233481 \times 10^{-10} \, \end{aligned}$$

$$\begin{aligned} P\_3 &:= \frac{2^{61/12}}{\Gamma(7/4)} \left( \int\_0^1 \mathfrak{J}\_3(\zeta) \zeta^{9/61} \, d\zeta \right)^{61/12} \approx 1.39796164 \times 10^{-18}, \\\ P\_4 &:= \frac{2^{51/8}}{\Gamma(7/5)} \left( \int\_0^1 \mathfrak{J}\_4(\zeta) \zeta^{16/25} \, d\zeta \right)^{51/8} \approx 1.16007238 \times 10^{-13} \end{aligned}$$

and so *<sup>L</sup>* <sup>=</sup> max{*Pi*, *<sup>i</sup>* <sup>=</sup> 1, ... , 4} <sup>=</sup> *<sup>P</sup>*1. We choose <sup>e</sup><sup>0</sup> <sup>=</sup> 10, *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> <sup>31</sup> <sup>6</sup> , *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> <sup>13</sup> <sup>2</sup> , and if we select *ω*<sup>1</sup> < <sup>1</sup> *<sup>L</sup>* <sup>10</sup>−1/12 and *<sup>ω</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>1</sup> *<sup>L</sup>* <sup>10</sup>−1/8, then we deduce that <sup>f</sup>(*z*) <sup>&</sup>lt; <sup>10</sup>61/12 *<sup>L</sup>* and g(*z*) < 1051/8 *<sup>L</sup>* for all *<sup>z</sup>* <sup>∈</sup> [0, 10]. For example, if *<sup>ω</sup>*<sup>1</sup> <sup>≤</sup> 2.0053 <sup>×</sup> <sup>10</sup><sup>8</sup> and *<sup>ω</sup>*<sup>2</sup> <sup>≤</sup> 1.8218 <sup>×</sup> 108, then the above conditions for f and g are satisfied. Therefore, assumption (*K*3) is also satisfied. By Theorem 1, we conclude that there exist positive constants c<sup>1</sup> and d<sup>1</sup> such that for any c<sup>0</sup> ∈ (0,c1] and d<sup>0</sup> ∈ (0, d1], problem (19) and (20) has at least one positive solution (u(*t*), v(*t*)), *t* ∈ [0, 1]. By Theorem 2, we deduce that there exist positive constants c<sup>2</sup> and d<sup>2</sup> such that for any c<sup>0</sup> ≥ c<sup>2</sup> and d<sup>0</sup> ≥ d2, problem (19) and (20) has no positive solution.

#### **5. Conclusions**

In this paper, we studied the system of coupled Riemann–Liouville fractional differential Equation (1) with 1-Laplacian and 2-Laplacian operators, subject to the nonlocal coupled boundary conditions (2), which contain fractional derivatives of various orders, Riemann–Stieltjes integrals, and two positive parameters c<sup>0</sup> and d0. Under some assumptions for the nonlinearities f and g of system (1), we established intervals for the parameters c<sup>0</sup> and d<sup>0</sup> such that our problem (1) and (2) has at least one positive solution. First, we made a change of unknown functions such that the new boundary conditions have no positive parameters. By using the corresponding Green functions, the new boundary value problem was then written equivalently as a system of integral equations (namely the system (15)). We associated to this integral system an operator (S), and we proved the existence of at least one fixed point for it by applying the Schauder fixed point theorem. Intervals for parameters c<sup>0</sup> and d<sup>0</sup> were also given such that problem (1) and (2) has no positive solution. Finally, we presented an example to illustrate our main results.

**Author Contributions:** Conceptualization, R.L.; formal analysis, J.H., R.L. and A.T.; methodology, J.H., R.L. and A.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Study on Infinitely Many Solutions for a Class of Fredholm Fractional Integro-Differential System**

**Dongping Li 1, Yankai Li 2,\* and Fangqi Chen 3,4**

	- Qingdao 266590, China

**Abstract:** This paper deals with a class of nonlinear fractional Sturm–Liouville boundary value problems. Each sub equation in the system is a fractional partial equation including the second kinds of Fredholm integral equation and the *p*-Laplacian operator, simultaneously. Infinitely many solutions are derived due to perfect involvements of fractional calculus theory and variational methods with some simpler and more easily verified assumptions.

**Keywords:** fractional integro-differential equation; Sturm–Liouville boundary condition; variational method

**MSC:** 26A33; 34B15; 35A15

#### **1. Introduction**

Nano/microactuators, as an indispensable portion of nano/microelectromechanical systems, are always subject to different inherent nonlinear forces. Many studies show that an integro-differential equation is generated in the modeling process of the nano/microactuator governing equation owing to axial forces ([1–3]). In [4,5], the following nanoactuator beam equation augmented to boundary conditions and containing an integro-differential expression, was discussed

$$\begin{cases} \frac{d^4f}{dt^4} - (\mu \int\_0^1 (\frac{df}{dt})^2 dt + \mathcal{L}) \frac{d^2f}{dt^2} + \frac{\theta}{f'^3} + \frac{\kappa}{(r+f)^2} + \frac{\mathbf{s}}{f} = \mathbf{0}, \ t \in [0, 1],\\ f(0) = f(1) = 0, \ f'(0) = f'(1) = 0, \end{cases} \tag{1}$$

where *f* and *t* denote the deflection and length of the beam, respectively. *μ*, *L*, *κ* and *r* denote some inherent nonlinear forces. Actually, in practical engineering applications, actuators are constructed by the billions for chipsets, therefore, developing more effective and accurate strategies for the study of nano/microactuator structures is of great significance.

Furthermore, it is often not appropriate to establish models with delayed behaviors by ordinary differential equations or partial differential equations, while integral equations are ideal tools. Moreover, fractional calculus operators are convolution operators (For details, please refer to the definitions of fractional integral and differential operators in [6], in which the definitions involving convolution integrals.), because they are nonlocal and have full-memory function, and those characteristics can be well used to describe various phenomena and complex processes involving delay and global correlations. For this reason, fractional calculus has been extensively applied in interdisciplinary fields such as fluid and viscoelastic mechanics, control theory, signal and image processing, electricity, physical, etc., (see [7–9]). Therefore, matching fractional calculus operators and integro-differential equations is ideal to complete the mathematical modeling of practical problems. Taking

**Citation:** Li, D.; Li, Y.; Chen, F. Study on Infinitely Many Solutions for a Class of Fredholm Fractional Integro-Differential System. *Fractal Fract.* **2022**, *6*, 467. https://doi.org/ 10.3390/fractalfract6090467

Academic Editor: Rodica Luca

Received: 8 July 2022 Accepted: 22 August 2022 Published: 26 August 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

into account the effect of a full-memory system, the integer derivatives in Equation (1) can be substituted for fractional ones. Inspired by this fact in Equation (1), Shivanian [10] introduced the following overdetermined Fredholm fractional integro-differential equations

$$\begin{cases} \ \_tD\_T^{a\_i}(a\_j(t)\_0 D\_1^{a\_j} u\_j(t)) = \lambda F\_{\mathbf{u}\_j}(t, u\_1(t), \dots, u\_{\mathbf{m}}(t)) + \int\_0^T k\_j(t, s) u\_j(s) ds, \ t \in (0, T), \ j = 1, 2, \dots, m, \\\ u\_j(t) = \int\_0^T k\_j(t, s) u\_j(s) ds, \ t \in (0, T), \ j = 1, 2, \dots, m, \\\ u\_j(0) = u\_j(T) = 0, \ j = 1, 2, \dots, m, \end{cases} \tag{2}$$

where *<sup>α</sup><sup>j</sup>* <sup>∈</sup> (0, 1], *aj*(*t*) <sup>∈</sup> *<sup>L</sup>*∞[0, *<sup>T</sup>*], *<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>m</sup>*. The existence of at least three weak solutions was obtained through the three critical points theorem.

Committed to fully considering more general systems, this paper studies a class of nonlinear Fredholm fractional integro-differential equations with *p*-Laplacian operator and Sturm–Liouville boundary conditions as below

$$\begin{cases} \, \_lD\_T^{\gamma\_l}(k\_j(t)\,\Phi\_p(\_{l}^{\text{C}}D\_t^{\gamma\_l}z\_j(t))) + l\_j(t)\,\Phi\_p(z\_j(t)) \\ = \lambda f\_{\bar{z}\_j}(t, z\_1(t), \dots, z\_m(t)) + \int\_0^T g\_j(t, s)\Phi\_p(z\_j(s))ds, \; t \in [0, T], \; j = 1, 2, \dots, m, \\\ z\_j(t) = \int\_0^T g\_j(t, s)\Phi\_p(z\_j(s))ds, \; t \in [0, T], \; j = 1, 2, \dots, m, \\\ c\_j k\_j(0)\Phi\_p(z\_j(0)) - c\_{jt}^\prime D\_T^{\gamma\_{j-1}}(k\_j(0)\Phi\_p(\_{l}^{\text{C}}D\_t^{\gamma\_{j-1}}z\_j(0))) = 0, \; j = 1, 2, \dots, m, \\\ d\_j k\_j(T)\Phi\_p(z\_j(T)) + d\_{jt}^\prime D\_T^{\gamma\_{j-1}}(k\_j(T)\Phi\_p(\_{l}^{\text{C}}D\_t^{\gamma\_{j}}z\_j(T))) = 0, \; j = 1, 2, \dots, m, \end{cases} (3)$$

where *cj*, *c j* , *dj* and *d <sup>j</sup>* are positive constants, *<sup>λ</sup>* <sup>∈</sup> (0, <sup>+</sup>∞) is a parameter, *kj*, *lj* <sup>∈</sup> *<sup>L</sup>*∞[0, *<sup>T</sup>*] with *kj* <sup>=</sup> *ess*inf[0,*T*] *kj*(*t*) <sup>&</sup>gt; 0 and *lj* <sup>=</sup> *ess*inf[0,*T*] *lj*(*t*) <sup>≥</sup> 0, *<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>m</sup>*. For 1 <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>, Φ*p*(*s*) = |*s*| *<sup>p</sup>*−2*s*(*<sup>s</sup>* <sup>=</sup> <sup>0</sup>), <sup>Φ</sup>*p*(0) = 0, *<sup>f</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> <sup>R</sup>*<sup>m</sup>* <sup>→</sup> <sup>R</sup> satisfies *<sup>f</sup>*(·, *<sup>z</sup>*1(*t*), ... , *zm*(*t*)) <sup>∈</sup> *<sup>C</sup>*[0, *<sup>T</sup>*] and *<sup>f</sup>*(*t*, ·, ... , ·) <sup>∈</sup> *<sup>C</sup>*1[R*m*], *gj*(·, ·) <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*], [0, *<sup>T</sup>*]). *<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup>* and *tDγ<sup>j</sup> <sup>T</sup>* denote the left Caputo fractional derivative and right Riemann–Liouville fractional derivative of order *γj*, respectively, which are defined by Kilbas et al. in [6]

$$\,\_{l}D\_{T}^{\gamma\_{j}}u(t) = (-1)^{n} \frac{d^{n}}{dt^{n}} \,\_{l}D\_{T}^{\gamma\_{j}-n}u(t) = \frac{(-1)^{n}}{\Gamma(n-\gamma\_{j})} \frac{d^{n}}{dt^{n}} \int\_{t}^{T} (\zeta - t)^{n-\gamma\_{j}-1} u(\zeta) d\zeta,\tag{4}$$

$$\, \_0^\mathbb{C} D\_t^{\gamma\_j} u(t) = \_0 D\_t^{\gamma\_j - n} u^{(n)}(t) = \frac{1}{\Gamma(n - \gamma\_j)} \int\_0^t (t - \zeta)^{n - \gamma\_j - 1} u^{(n)}(\zeta) d\zeta. \tag{5}$$

for <sup>∀</sup>*u*(*t*) <sup>∈</sup> *AC*([0, *<sup>T</sup>*], <sup>R</sup>), *<sup>n</sup>* <sup>−</sup> <sup>1</sup> <sup>≤</sup> *<sup>γ</sup><sup>j</sup>* <sup>&</sup>lt; *<sup>n</sup>*, *<sup>n</sup>* <sup>∈</sup> N.

We emphasize that this paper extends previous results in several directions, which are listed as follows: (i) In recent years, a large number of existence results for fractional differential equations have been acquired by variational methods and critical point theory ([11–14]). However, not many research works are available in related references to handle fractional integro-differential equations, let alone involving the *p*-Laplacian operator and Sturm–Liouville boundary conditions. (ii) It is not hard to see that Equation (3) can turn into the Dirichlet boundary value problem Equation (2) under *p* = 2, *c <sup>j</sup>* = *d <sup>j</sup>* = 0, *lj*(*t*) ≡ 0, *j* = 1, 2, ... , *m*, which means that Equation (2) is a special case of Equation (3). Furthermore, since the *p*-Laplacian operator is considered with 1 < *p* < ∞ in the paper, the linear differential operator *tD<sup>γ</sup> T C* <sup>0</sup> *<sup>D</sup><sup>γ</sup> <sup>t</sup>* is extended to the nonlinear differential operator *tD<sup>γ</sup> T*Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup><sup>γ</sup> <sup>t</sup>* ). In short, the form of Equation (3) is more generalized, as well as the boundary value conditions. (iii) Infinitely many solutions are obtained in this paper with some simpler and more easily verified assumptions. Hence, our work improves and replenishes some existing results form the literature.

#### **2. Preliminaries**

Assume *<sup>H</sup>* is a Banach space and F ∈ *<sup>C</sup>*1(*H*, <sup>R</sup>). Functional <sup>F</sup> satisfies the Palais–Smale condition if each sequence {*zk*}<sup>∞</sup> *<sup>k</sup>*=<sup>1</sup> ⊂ *H* such that {F(*zk*)} is bounded and lim *<sup>k</sup>*→<sup>∞</sup> <sup>F</sup> (*zk*) = 0 possesses strongly convergent subsequence in *H*.

**Theorem 1** ([15])**.** *Let <sup>H</sup> be an infinite-dimensional Banach space,* F ∈ *<sup>C</sup>*1(*H*, <sup>R</sup>) *is an even functional and satisfies the Palais–Smale condition. Assume that:*


Then, F has infinitely many critical points.

**Definition 1.** *Let* 1 < *p* < ∞*,* <sup>1</sup> *<sup>p</sup>* < *γ<sup>j</sup>* ≤ 1*, j* = 1, 2, ... , *m. Define the fractional derivative space <sup>H</sup>* = <sup>Π</sup>*j*=*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *<sup>H</sup>γj*,*<sup>p</sup> with the weighted norm*

$$\|\|Z\|\|\_{H} = \sum\_{j=1}^{j=m} \|z\_j\|\_{\left(\gamma\_j, p\right)^\*} \, z\_j \in H^{\gamma\_j, p}, Z = \left(z\_1, \dots, z\_m\right) \in H,\tag{6}$$

*where*

$$H^{\gamma\_j, p} = \{ z\_j \in A\mathcal{C}([0, T], \mathbb{R}) : \prescript{\subset}{0}{D}\_t^{\gamma\_j} z\_j(t) \in L^p([0, T], \mathbb{R}) \}$$

*as the closure of C*∞([0, *T*], R) *endowed with the norm*

$$\|\|z\_j\|\|\_{\left(\gamma\_j,p\right)} := \left(\int\_0^T |\,z\_j(t)\,|\,^p d t + \int\_0^T |\,^{\mathbb{C}}\_0 D\_t^{\gamma\_j} z\_j(t)\,|\,^p dt\right)^{\frac{1}{p}}, \forall z\_j \in H^{\gamma\_j,p}.\tag{7}$$

*Hγj*,*<sup>p</sup> is a reflexive and separable Banach space [16]. Therefore, H also is a reflexive and separable Banach space.*

**Lemma 1** ([13])**.** *For any zj*(*t*) <sup>∈</sup> *<sup>H</sup>γj*,*p,* <sup>1</sup> <sup>&</sup>lt; *<sup>p</sup>*, *<sup>q</sup>* <sup>&</sup>lt; <sup>∞</sup> *with* <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>1</sup> *<sup>q</sup>* = 1*, there exists a constant <sup>W</sup>*(*γj*,*p*) <sup>=</sup> max - *<sup>T</sup>γj*<sup>−</sup> <sup>1</sup> *p* Γ(*γj*)((*γj*−1)*q*+1) 1 *q* , 1) + \* 2*p*−<sup>1</sup> *<sup>T</sup>* max - 1, *Tγj* Γ(*γj*+1) *p*)+ <sup>1</sup> *p such that zj* ∞ ≤ *W*(*γj*,*p*) *zj* (*γj*,*p*)*, j* = 1, 2, . . . , *m.*

Taking into account Lemma 1, one has

$$\|z\_{j}\|\_{\infty} \le \frac{\mathcal{W}\_{\{\gamma\_{j},p\}}}{(\min\{\hat{k}\_{j},\hat{l}\_{j}\})^{\frac{1}{p}}} \left(\int\_{0}^{T} l\_{j}(t) \mid z\_{j}(t) \mid ^{p} dt + \int\_{0}^{T} k\_{j}(t) \mid ^{\mathbb{C}}\_{0} D\_{t}^{\gamma\_{j}} z\_{j}(t) \mid ^{p} dt\right)^{\frac{1}{p}}, \forall \, z\_{j}(t) \in H^{\gamma\_{j},p}, \tag{8}$$

*j* = 1, 2, . . . , *m*. In order to describe it more easily for the further analysis, denote

$$\mathcal{W}\_{\hat{l}} = \frac{\mathcal{W}\_{(\gamma\_{\hat{l}}, p)}}{(\min\{\hat{k}\_{\hat{l}}, \hat{l}\_{\hat{l}}\})^{\frac{1}{p}}} \prime \hat{\mathcal{W}} = \max\_{1 \le j \le m} \{\mathcal{W}\_{\hat{l}}\}. \tag{9}$$

Obviously, the norm defined by (7) is equivalent to

$$\|\|z\_{j}\|\|\_{\left(\gamma\_{j},p\right)} = \left(\int\_{0}^{T} l\_{j}(t) \mid z\_{j}(t) \mid \prescript{p}{}{d}t + \int\_{0}^{T} k\_{j}(t) \mid \prescript{c}{0}D\_{t}^{\gamma\_{j}} z\_{j}(t) \mid \prescript{p}{}{d}t\right)^{\frac{1}{p}}, \ j = 1,2,\ldots,m. \tag{10}$$

We work with the norm (10) hereinafter.

**Lemma 2** ([17])**.** *Let* <sup>1</sup> <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>*, <sup>γ</sup><sup>j</sup>* <sup>∈</sup> ( <sup>1</sup> *<sup>p</sup>* , 1]*, j* = 1, 2, ... , *m. Suppose that any sequence* {*zk*,*j*} *converges to zj in Hγj*,*<sup>p</sup> weakly. Then, zk*,*<sup>j</sup>* <sup>→</sup> *zj in C*([0, *<sup>T</sup>*]) *as k* <sup>→</sup> <sup>∞</sup>*.*

**Lemma 3** ([18])**.** *Let Hj be any finite-dimensional subspace of Hγj*,*p, j* = 1, 2, ... , *m. There exists a constant ζ*<sup>0</sup> > 0 *such that meas*{*t* ∈ [0, *T*] :| *zj*(*t*) |≥ *ζ*<sup>0</sup> *zj* (*γj*,*p*)} ≥ *ζ*0*,* ∀*zj*(*t*) ∈ *Hj* \ {0}*.*

**Lemma 4** ([6])**.** *Let <sup>γ</sup>* <sup>&</sup>gt; <sup>0</sup>*, <sup>p</sup>* <sup>≥</sup> <sup>1</sup>*, <sup>q</sup>* <sup>≥</sup> <sup>1</sup> *and* <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>1</sup> *<sup>q</sup>* ≤ 1 + *γ (p* = 1, *q* = 1 *in the case when* <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>1</sup> *<sup>q</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>γ</sup>). If <sup>z</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>p*([*a*, *<sup>b</sup>*]) *and <sup>z</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>q*([*a*, *<sup>b</sup>*])*, then,* % *<sup>b</sup> <sup>a</sup>* (*aD*−*<sup>γ</sup> <sup>t</sup> z*1(*t*))*z*2(*t*)*dt* = % *b <sup>a</sup> <sup>z</sup>*1(*t*)(*tD*−*<sup>γ</sup> <sup>b</sup> z*2(*t*))*dt.*

**Lemma 5.** *It is said Z* = (*z*1, ... , *zm*) ∈ *H is a weak solution of Equations (3), if the following equation holds*

$$\sum\_{j=1}^{m} \left\{ \int\_{0}^{T} k\_{j}(t) \Phi\_{p}(\hat{\mathbf{j}}) \hat{D}\_{1}^{\gamma\_{j}} z\_{j}(t) \zeta\_{0}^{\gamma\_{j}} D\_{1}^{\gamma\_{j}} y\_{j}(t) + l\_{j}(t) \Phi\_{p}(z\_{j}(t)) y\_{j}(t) dt + \frac{c\_{j}}{d\_{j}} k\_{l}(0) \Phi\_{p}(z\_{j}(0)) y\_{j}(0) + \frac{d\_{j}}{d\_{j}^{\gamma}} k\_{l}(T) \Phi\_{p}(z\_{j}(T)) y\_{j}(T) \right\} $$

$$= \sum\_{j=1}^{m} \left\{ \int\_{0}^{T} \int\_{0}^{T} g\_{j}(t,s) \Phi\_{p}(z\_{j}(s)) y\_{j}(t) ds dt + \lambda \int\_{0}^{T} f\_{\bar{z}\_{j}}(t, z\_{1}(t), \dots, z\_{\mathfrak{w}}(t)) y\_{j}(t) dt \right\} \forall Y = (y\_{1}, \dots, y\_{\mathfrak{w}}) \in H. \tag{11}$$

**Proof.** Consider (4) and (5), the boundary conditions in Equation (3) and Lemma 4 yield:

 *<sup>T</sup>* 0 *tDγ<sup>j</sup> <sup>T</sup>* (*kj*(*t*)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(*t*)))*yj*(*t*)*dt* = − *<sup>T</sup>* 0 *yj*(*t*)*d*[*tDγj*−<sup>1</sup> *<sup>T</sup>* (*kj*(*t*)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(*t*)))] <sup>=</sup>*tDγj*−<sup>1</sup> *T kj*(0)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(0)) *yj*(0) <sup>−</sup> *tDγj*−<sup>1</sup> *T kj*(*T*)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(*T*)) *yj*(*T*) + *<sup>T</sup>* 0 *tDγj*−<sup>1</sup> *<sup>T</sup>* (*kj*(*t*)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(*t*)))*y j* (*t*)*dt* (12) =*cj c j kj*(0)Φ*p*(*zj*(0))*yj*(0) + *dj d j kj*(*T*)Φ*p*(*zj*(*T*))*yj*(*T*) + *<sup>T</sup>* 0 *kj*(*t*)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(*t*))*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> yj*(*t*)*dt*.

Substituting *yj*(*t*) into Equation (3) and integrating on both sides from 0 to *T*, then summing from *j* = 1 to *j* = *m* and combining with (12), we can obtain Equation (11). The proof is completed.

$$\textbf{Remark 1.}\text{ For any } z\_{j} \in H^{\gamma\_{j}; p} \subset \mathbb{C}([0, T]), j = 1, 2, \dots, m,\text{ from Equation (3) we have}$$

$$\_{t}D\_{T}^{\gamma\_{j}}(k\_{j}(t)\Phi\_{p}(^{\gamma}\_{0}D\_{t}^{\gamma\_{j}}z\_{j}(t))) + l\_{j}(t)\Phi\_{p}(z\_{j}(t)) = \lambda f\_{z\_{j}}(t, z\_{1}(t), \dots, z\_{m}(t)) + \int\_{0}^{T} g\_{j}(t, s)\Phi\_{p}(z\_{j}(s))ds, t \in [0, T],$$

$$\text{because } f(t, \cdot, \cdot, \dots, \cdot) \in \mathbb{C}^{1}[\mathbb{R}^{m}], z\_{j}(t) = \int\_{0}^{T} g\_{j}(t, s)\Phi\_{p}(z\_{j}(s))ds \in H^{\gamma\_{j}; p} \text{ and}$$

$$\square$$

$${}\_{t}D\_{T}^{\gamma\_{j}}(k\_{j}(t)\Phi\_{p}({}\_{0}^{\mathbb{C}}D\_{t}^{\gamma\_{j}}z\_{j}(t)))=\left({}\_{t}D\_{T}^{\gamma\_{j}-1}(k\_{j}(t)\Phi\_{p}({}\_{0}^{\mathbb{C}}D\_{t}^{\gamma\_{j}}z\_{j}(t)))\right)',$$

*one gets*

$$\,\_tD\_T^{\gamma\_j-1}(k\_j(t)\Phi\_p(^{\subset}\_0D\_t^{\gamma\_j}z\_j(t))) \in AC([0,T]).$$

*Hence, the terms tDγj*−<sup>1</sup> *<sup>T</sup>* (*kj*(0)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(0))) *and tDγj*−<sup>1</sup> *<sup>T</sup>* (*kj*(*T*)Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zj*(*T*))) *exist in this paper.*

Consider the functional <sup>F</sup> : *<sup>H</sup>* <sup>→</sup> <sup>R</sup> with

$$\begin{split} \mathcal{F}(\boldsymbol{Z}) &:= \frac{1}{p} \sum\_{j=1}^{j-m} \int\_{0}^{T} k\_{j}(t) \left[ \left. \left\langle \boldsymbol{0} \right| \boldsymbol{I}\_{j}^{\boldsymbol{\eta}} \boldsymbol{z}\_{j}(t) \right| \boldsymbol{r} \right. \\ & \left. - \sum\_{j=1}^{j-m} \int\_{0}^{T} G\_{j}(\boldsymbol{z}\_{j}(t)) dt - \lambda \int\_{0}^{T} f(t, \boldsymbol{z}\_{1}(t), \dots, \boldsymbol{z}\_{m}(t)) dt \right. \\ & \left. - \frac{1}{p} \sum\_{j=1}^{j-m} ||\boldsymbol{z}\_{j}|| \right|\_{\boldsymbol{\gamma}(p,p)}^{p} + \sum\_{j=1}^{j-m} \left[ \frac{c\_{j}}{p\kappa\_{j}} k\_{j}(0) \mid \boldsymbol{z}\_{j}(0) \mid \boldsymbol{r} \right. + \frac{d\_{j}}{p d\_{j}} k\_{j}(T) \mid \boldsymbol{z}\_{j}(T) \mid \boldsymbol{r} \right] \\ & - \sum\_{j=1}^{j-m} \int\_{0}^{T} G\_{j}(\boldsymbol{z}\_{j}(t)) dt - \lambda \int\_{0}^{T} f(t, \boldsymbol{z}\_{1}(t), \dots, \boldsymbol{z}\_{m}(t)) dt \end{split} \tag{13}$$

where *Gj*(*zj*(*t*)) = <sup>1</sup> 2 % *T* <sup>0</sup> *gj*(*t*,*s*)Φ*p*(*zj*(*s*))*zj*(*t*)*ds*, *t* ∈ (0, *T*), *j* = 1, 2, ... , *m*. Owing to *zj*(*t*) = % *T* <sup>0</sup> *gj*(*t*,*s*)Φ*p*(*zj*(*s*))*ds*, *j* = 1, 2, . . . , *m*, the Gateaux derivative of ˆ *Gj* is

$$\begin{split} G\_j^t(z\_j)(y\_j) &= \lim\_{h \to 0} \frac{G\_j(z\_j + hy\_j) - G\_j(z\_j)}{h} \\ &= \lim\_{h \to 0} \frac{\frac{1}{2} \int\_0^T g\_j^T(t,s) \Phi\_p(z\_j(s) + hy\_j(s))(z\_j(t) + hy\_j(t)) - g\_j(t,s) \Phi\_p(z\_j(s))z\_j(t)ds}{h} \\ &= \lim\_{h \to 0} \frac{\frac{1}{2} h^2 g\_j^2(t) + hz\_j(t)y\_j(t)}{h} = z\_j(t)y\_j(t) = \int\_0^T g\_j(t,s) \Phi\_p(z\_j(s))y\_j(t)ds, j = 1,2,\dots,m. \end{split} \tag{14}$$

Then, combining the continuity of *<sup>f</sup>* and (14), we can see that F ∈ *<sup>C</sup>*1(*H*, <sup>R</sup>) and

$$\begin{split} \mathcal{F}'(\mathbf{Z})(\mathbf{Y}) &= \sum\_{j=1}^{j=n} \left\{ \int\_{0}^{T} k\_{j}(t) \Phi\_{p}(\prescript{\gamma}{}{D}\_{t}^{\gamma} z\_{j}(t)) \prescript{\gamma}{}{D}\_{t}^{\gamma} y\_{j}(t) + l\_{j}(t) \Phi\_{p}(z\_{j}(t)) y\_{j}(t) dt + \frac{c\_{j}}{\mathcal{c}\_{j}^{\gamma}} k\_{j}(0) \Phi\_{p}(z\_{j}(0)) y\_{j}(0) \\ &+ \frac{d\_{j}}{d\_{j}^{\gamma}} k\_{j}(T) \Phi\_{p}(z\_{j}(T)) y\_{j}(T) - \int\_{0}^{T} \int\_{0}^{T} g\_{j}(t,s) \Phi\_{p}(z\_{j}(s)) y\_{j}(t) ds dt - \lambda \int\_{0}^{T} f\_{\overline{z}\_{j}}(t, Z(t)) y\_{j}(t) dt \right\}, \forall \mathbf{Z}, \boldsymbol{Y} \in H. \end{split} \tag{15}$$

Notice that, the critical point of F is the weak solution of Equation (3).

#### **3. Main Results**

First, some hypotheses related to nonlinearity *f* are given, which play important roles in the remaining discussion.

$$\begin{aligned} (H\_0) \lim\_{\forall \boldsymbol{j}; |\boldsymbol{z}\_j| \to \infty} \frac{f(t, \boldsymbol{Z}(t))}{\sum\_{j=1}^{j=m} |z\_j|^p} &= \infty \text{ uniformly for } t \in [0, T], \boldsymbol{Z}(t) = (z\_1(t), \dots, z\_m(t)) \in \mathbb{R}^m; \\ (H\_1) \ 0 \le f(t, \boldsymbol{Z}(t)) &= o(\sum\_{j=1}^{j=m} |\boldsymbol{z}\_j|^p) \text{ as } \sum\_{j=1}^{j=m} |\boldsymbol{z}\_j| \to 0 \text{ uniformly for } t \in [0, T]; \end{aligned}$$

$$\begin{cases} (H\_2) \text{ For any } Z(t) = (z\_1(t), \dots, z\_m(t)) \in \mathbb{R}^m, f(t, Z(t)) = \sum\_{j=1}^{j=m} \frac{\eta\_j}{p} \mid z\_j \mid^p - f(t, Z(t)) \text{ with } 0 \le t \le m\\ f(t, 0) \equiv 0, \text{and} \end{cases}$$

$$\begin{aligned} \min\_{1 \le j \le m} \{\eta\_j\} &> \frac{1}{\lambda \zeta\_0^{p+1}} (\frac{3}{2} + p \sum\_{j=1}^{j=m} [\frac{c\_j}{p c\_j'} k\_j(0) + \frac{d\_j}{p d\_j'} k\_j(T)] \mathcal{W}\_j^p), \\\sum\_{j=1}^{j=m} (\frac{\eta\_j}{p} + \frac{\beta\_j}{2\lambda}) \mid z\_j \mid^{\omega\_j} &\le J(t, Z(t)) \le \sum\_{j=1}^{j=m} \delta\_j \mid z\_j \mid^{\omega\_j} \end{aligned}$$

where *ω<sup>j</sup>* ∈ (0, *p*), *δ<sup>j</sup>* > 0, *ζ*<sup>0</sup> > 0 is a constant and *β* is introduced thereinafter, *<sup>j</sup>* <sup>=</sup> 1, 2, . . . , *<sup>m</sup>*.

**Lemma 6.** F *satisfies the Palais–Smale condition under* (*H*0)*.*

**Proof.** Suppose that sequence {F(*Zk*)}*k*∈<sup>N</sup> is bounded and lim *<sup>k</sup>*→<sup>∞</sup> <sup>F</sup> (*Zk*) = 0, *Zk*(*t*) = (*zk*,1(*t*), ... , *zk*,*m*(*t*)). We claim that {*Zk*}*k*∈<sup>N</sup> is bounded in *<sup>H</sup>*. Indeed, assume ∀*j* : *zk*,*j* (*γj*,*p*) → ∞(*k* → ∞). From (*H*0), for any *L* > 0, there exists *k*<sup>0</sup> ∈ N such that

$$\frac{f(t, Z\_k(t))}{\sum\_{j=1}^{j=m} \|z\_{k,j}\|\_{\left(\gamma\_j, p\right)}^p} \ge L\_\prime \,\forall \, k > k\_{0\prime} \,\, t \in [0, T]. \tag{16}$$

For any fixed *k*<sup>∗</sup> ∈ N with *k*<sup>∗</sup> > *k*0, from the integral mean value theorem, there exists *ξ*(*k*∗) ∈ (0, 1] such that

$$\int\_{0}^{T} f(t, Z\_{k\_\*}(t))dt = Tf(\xi(k\_\*)T, Z\_{k\_\*}(\xi(k\_\*)T)).\tag{17}$$

Combining (16) and (17) yields

$$\frac{\int\_{0}^{T} f(t, Z\_{k\_{\ast}}(t))dt}{\sum\_{j=1}^{j=m} ||z\_{k\_{\ast},j}||\_{\left(\gamma\_{j},p\right)}^{p}} = \frac{T f(\check{\xi}(k\_{\ast})T, Z\_{k\_{\ast}}(\check{\xi}(k\_{\ast})T))}{\sum\_{j=1}^{j=m} ||z\_{k\_{\ast},j}||\_{\left(\gamma\_{j},p\right)}^{p}} \geq \frac{T L \sum\_{j=1}^{j=m} ||z\_{k\_{\ast},j}||\_{\left(\gamma\_{j},p\right)}^{p}}{\sum\_{j=1}^{j=m} ||z\_{k\_{\ast},j}||\_{\left(\gamma\_{j},p\right)}^{p}} = T L.$$

Hence, we can get

$$\frac{\int\_{0}^{T} f(t, Z\_{k}(t))dt}{\sum\_{j=1}^{j=m} ||z\_{k,j}||\_{\left(\gamma\_{j}, p\right)}^{p}} \ge TL\_{\prime} \,\forall \, k > k\_{0\prime} \,\, t \in [0, T]. \tag{18}$$

In view of (8), (9), (13) and (18) we have

$$\frac{\mathcal{F}(Z\_{k}(t))}{\sum\_{j=1}^{j=m} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{p}} = \frac{1}{p} + \frac{\sum\_{j=1}^{j=m} \left[\frac{c\_{j}}{p c\_{j}} k\_{j}(0) \mid z\_{k,j}(0) \mid {p}^{p} + \frac{d\_{j}}{p d\_{j}} k\_{j}(T) \mid z\_{k,j}(T) \mid {p}^{p}\right]}{\sum\_{j=1}^{j=m} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{p}}$$

$$-\frac{\sum\_{j=1}^{j=m} \int\_{0}^{T} G\_{j}(z\_{k,j}(t))dt + \lambda \int\_{0}^{T} f(t, Z\_{k}(t))dt}{\sum\_{j=1}^{j=m} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{p}}$$

$$\leq \frac{1}{p} + \frac{\sum\_{j=1}^{j=m} \frac{c\_{j}}{{p c\_{j}}} k\_{j}(0) + \frac{d\_{j}}{{p d\_{j}}} k\_{j}(T) \|W\_{j}^{p}\|\_{\left(\gamma\_{j},p\right)}^{p}}{\sum\_{j=1}^{j=m} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{p}} - \lambda TL$$

$$\leq \frac{1}{p} + \sum\_{j=1}^{j=m} \left[\frac{c\_{j}}{{p c\_{j}}} k\_{j}(0) + \frac{d\_{j}}{{p d\_{j}}} k\_{j}(T) \|W\_{j}^{p} - \lambda TL.\right]. \tag{19}$$

Choose *L* large enough such that <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>∑</sup>*j*=*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> [ *cj pc j kj*(0) + *dj pd j kj*(*T*)]*W<sup>p</sup> <sup>j</sup>* − *λTL* < −1, then combining (19) yields that <sup>F</sup>(*Zk*(*t*)) ≤ − <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) , which means that F(*Zk*(*t*)) → −∞ as *zk*,*j* (*γj*,*p*) → ∞, ∀*j* = 1, 2, ... , *m*. It contradicts that {F(*Zk*)} is bounded. Hence, {*Zk*} is bounded in *H*. Because of the reflexivity of *H*, we get that *Zk Z*<sup>∗</sup> in *H* (up to subsequences). From Lemma 2, we have *Zk* → *Z*<sup>∗</sup> uniformly in *C*([0, *T*] *<sup>m</sup>*) and *Lp*([0, *T*] *<sup>m</sup>*). Then,

$$\begin{cases} \begin{array}{l} (\mathcal{F}'(\mathcal{Z}\_{k}) - \mathcal{F}'(\mathcal{Z}^\*))(\mathcal{Z}\_{k} - \mathcal{Z}^\*) \to 0, \, k \to \infty, \\ \int\_{0}^{T} (f\_{\boldsymbol{z}\_{j}}(t, \mathcal{Z}\_{k}(t)) - f\_{\boldsymbol{z}\_{j}}(t, \mathcal{Z}^\*(t)))(z\_{k,j}(t) - z\_{j}^\*(t))dt \to 0, \, k \to \infty, j = 1, 2, \ldots, m, \\ \int\_{0}^{T} |z\_{k,j}(t) - z\_{j}^\*(t)|^{2} \, dt \to 0, z\_{k,j}(0) - z\_{j}^\*(0) \to 0, z\_{k,j}(T)) - z\_{j}^\*(T) \to 0, \, k \to \infty, j = 1, 2, \ldots, m. \end{cases} \end{cases} \tag{20}$$

From (15), we obtain that

(F (*Zk*) − F (*Z*∗))(*Zk* − *Z*∗) = F (*Zk*)(*Zk* − *Z*∗) − F (*Z*∗)(*Zk* − *Z*∗) = *j*=*m* ∑ *j*=1 - *<sup>T</sup>* 0 *kj*(*t*) Φ*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> zk*,*j*(*t*)) <sup>−</sup> <sup>Φ</sup>*p*(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup> z*<sup>∗</sup> *<sup>j</sup>* (*t*)) *C* <sup>0</sup> *<sup>D</sup>γ<sup>j</sup> <sup>t</sup>* (*zk*,*j*(*t*) − *z*<sup>∗</sup> *<sup>j</sup>* (*t*)) + *lj*(*t*) Φ*p*(*zk*,*j*(*t*)) − Φ*p*(*z*<sup>∗</sup> *<sup>j</sup>* (*t*)) (*zk*,*j*(*t*) − *z*<sup>∗</sup> *<sup>j</sup>* (*t*))*dt* (21) − *<sup>T</sup>* 0 *<sup>T</sup>* 0 *gj*(*t*,*s*) Φ*p*(*zk*,*j*(*s*)) − Φ*p*(*z*<sup>∗</sup> *<sup>j</sup>* (*s*)) (*zk*,*j*(*t*) − *z*<sup>∗</sup> *<sup>j</sup>* (*t*))*dsdt* <sup>+</sup> *cj c j kj*(0) Φ*p*(*zk*,*j*(0)) − Φ*p*(*z*<sup>∗</sup> *<sup>j</sup>* (0)) (*zk*,*j*(0) − *z*<sup>∗</sup> *<sup>j</sup>* (0)) + *dj d j kj*(*T*) Φ*p*(*zk*,*j*(*T*)) − Φ*p*(*z*<sup>∗</sup> *<sup>j</sup>* (*T*)) (*zk*,*j*(*T*) − *z*<sup>∗</sup> *<sup>j</sup>* (*T*)) − *λ <sup>T</sup>* 0 (*fzj*(*t*, *Zk*(*t*)) − *fzj*(*t*, *Z*∗(*t*)))(*zk*,*j*(*t*) − *z*<sup>∗</sup> *<sup>j</sup>* (*t*))*dt*) ;

moreover,

$$\int\_{0}^{T} \int\_{0}^{T} g\_{j}(t, \mathbf{s}) \Big( \Phi\_{p}(\mathbf{z}\_{k,j}(\mathbf{s})) - \Phi\_{p}(\mathbf{z}\_{j}^{\*}(\mathbf{s})) \Big) (\mathbf{z}\_{k,j}(t) - \mathbf{z}\_{j}^{\*}(t)) d\mathbf{s} dt = \int\_{0}^{T} |\operatorname{\boldsymbol{\boldsymbol{z}}}\_{k,j}(t) - \boldsymbol{z}\_{j}^{\*}(t)|^{2} \, dt. \tag{22}$$

Denote

$$\begin{aligned} \Psi\_{k,j}(\gamma\_{j},p) &= \int\_{0}^{T} k\_{j}(t) \left( \Phi\_{p}(\prescript{C}{}{0}D\_{t}^{\gamma\_{j}}z\_{k,j}(t)) - \Phi\_{p}(\prescript{C}{}{0}D\_{t}^{\gamma\_{i}}z\_{j}^{\*}(t)) \right) \prescript{C}{}{D}\_{t}^{\gamma\_{j}}(z\_{k,j}(t) - z\_{j}^{\*}(t))dt, \\ \Psi\_{k,j}(p) &= \int\_{0}^{T} l\_{j}(t) \left( \Phi\_{p}(z\_{k,j}(t)) - \Phi\_{p}(z\_{j}^{\*}(t)) \right) (z\_{k,j}(t) - z\_{j}^{\*}(t))dt, \end{aligned}$$

combining (20), (21) and (22), we obtain <sup>∑</sup>*j*=*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> {Ψ*k*,*j*(*γj*, *p*) + Ψ*k*,*j*(*p*)} → 0 as *k* → ∞. As in the discussion of Θ(*α*, *p*), Θ(*p*) in [19], we can get

$$\mathbb{P}\_{k,\boldsymbol{j}}(\gamma\_{\boldsymbol{j}\boldsymbol{\nu}}\boldsymbol{p}) + \mathbb{P}\_{k,\boldsymbol{j}}(\boldsymbol{p}) \geq \begin{cases} \ \boldsymbol{e}\_{\boldsymbol{j}} ||\boldsymbol{z}\_{k,\boldsymbol{j}} - \boldsymbol{z}\_{\boldsymbol{j}}^{\*}||\_{\gamma\_{\boldsymbol{j}},\boldsymbol{p}\boldsymbol{\nu}}^{p} \ \boldsymbol{p} \geq 2, \\\ \boldsymbol{e}\_{\boldsymbol{j}}' ||\boldsymbol{z}\_{k,\boldsymbol{j}} - \boldsymbol{z}\_{\boldsymbol{j}}^{\*}||\_{\left(\gamma\_{\boldsymbol{j}},\boldsymbol{p}\right)}^{2} \left(||\boldsymbol{z}\_{k,\boldsymbol{j}}||\_{L^{p}}^{p} + ||\boldsymbol{z}\_{\boldsymbol{j}}^{\*}||\_{L^{p}}^{p}\right)^{\frac{p-2}{p}}, \ \boldsymbol{1} < p < 2, \end{cases}$$

where *ej*,*e <sup>j</sup>* are constants, *j* = 1, 2, ... , *m*. Based on the above discussion, we can obtain *zk*,*<sup>j</sup>* − *z*<sup>∗</sup> *j* (*γj*,*p*) → 0, *j* = 1, 2, ... , *m*, for all 1 < *p* < ∞. Hence, the Palais–Smale condition holds.

**Theorem 2.** *Assume that* (*H*0) *and* (*H*1) *hold and f*(*t*, *Z*) = *f*(*t*, −*Z*)*. Then, Equation (3) has infinitely many solutions with* <sup>1</sup> *T pW <sup>p</sup>* <sup>−</sup> *<sup>β</sup>* <sup>&</sup>gt; <sup>0</sup> *and* <sup>0</sup> <sup>&</sup>lt; *<sup>λ</sup>* <sup>&</sup>lt; <sup>∞</sup>*.*

**Proof.** Due to *f*(*t*, *Z*) = *f*(*t*, −*Z*), it is easy to verify that F is even. Obviously, F(0) = 0. Taking into account (*H*1) that, for any *ε* > 0, there exists *r*(*ε*) such that

$$f(t, Z(t)) \le \varepsilon \sum\_{j=1}^{j=m} |\lfloor z\_j \rfloor|^p \text{ } \forall t \in [0, T] \text{ } \sum\_{j=1}^{j=m} |\lfloor z\_j \rfloor| \le r(\varepsilon). \tag{23}$$

Further, *gj*(·, ·) ∈ *C*([0, *T*], [0, *T*]) means that the kernel *gj* is bounded by, say *βj*, i.e., | *gj*(*t*,*s*) |≤ *βj*, and

$$\mathcal{G}\_{\rangle}(\boldsymbol{z}\_{\boldsymbol{j}}(t)) = \frac{1}{2} \int\_{0}^{T} \boldsymbol{g}\_{\boldsymbol{j}}(t, \boldsymbol{s}) \boldsymbol{\Phi}\_{\boldsymbol{p}}(\boldsymbol{z}\_{\boldsymbol{j}}(\boldsymbol{s})) \boldsymbol{z}\_{\boldsymbol{j}}(t) d\boldsymbol{s} \leq \frac{\beta\_{\boldsymbol{j}}}{2} \boldsymbol{z}\_{\boldsymbol{j}}(t) \parallel \boldsymbol{z}\_{\boldsymbol{j}} \parallel \boldsymbol{z}\_{\boldsymbol{\alpha}} \parallel \boldsymbol{\varepsilon}^{p-1} \leq \frac{\beta\_{\boldsymbol{j}}}{2} \parallel \boldsymbol{z}\_{\boldsymbol{j}} \parallel \boldsymbol{\varepsilon}^{p}, j = 1, 2, \dots, m. \tag{24}$$

Let *τ* = *<sup>r</sup> <sup>W</sup>* . For any *<sup>Z</sup>* <sup>∈</sup> <sup>Υ</sup>*τ*, one has *Z <sup>H</sup>* <sup>=</sup> <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zj* (*γj*,*p*) <sup>≤</sup> *<sup>r</sup> <sup>W</sup>* . Then,

$$\frac{r}{\widehat{W}} \ge \sum\_{j=1}^{j=m} ||z\_j||\_{\left(\gamma\_j, p\right)} \ge \sum\_{j=1}^{j=m} \frac{1}{W\_j} ||z\_j||\_{\infty} \ge \frac{1}{\widehat{W}} \sum\_{j=1}^{j=m} ||z\_j||\_{\infty} \tag{25}$$

which means that <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zj* <sup>∞</sup> ≤ *r*(*ε*). At this point, from (13), (23) and (24) we can see

$$\begin{split} \mathcal{F}(Z) &\geq \frac{1}{p} \sum\_{j=1}^{j=m} \|z\_{j}\|\_{\left(\gamma\_{j},p\right)}^{p} - \sum\_{j=1}^{j=m} \int\_{0}^{T} \frac{\beta\_{j}}{2} \|\left|\left.z\_{j}\right|\right\|\_{\infty}^{p} dt - \lambda \int\_{0}^{T} \varepsilon \sum\_{j=1}^{j=m} \left|\left.z\_{j}\right|^{p} dt \\ &\geq \frac{1}{p} \sum\_{j=1}^{j=m} \|\left|z\_{j}\right|\right|\_{\left(\gamma\_{j},p\right)}^{p} - \sum\_{j=1}^{j=m} \left(\frac{T\beta\_{j}}{2} + \lambda \varepsilon T\right) \mathcal{W}\_{j}^{p} \|\left|z\_{j}\right|\right|\_{\left(\gamma\_{j},p\right)}^{p} \\ &\geq \left|\frac{1}{p} - \left(\frac{T\widehat{\beta}}{2} + \lambda \varepsilon T\right) \widehat{\mathcal{W}}^{p}\right| \frac{1}{m^{p}} \left(\sum\_{j=1}^{j=m} ||z\_{j}||\_{\left(\gamma\_{j},p\right)}\right)^{p} \\ &= \left|\frac{1}{p} - \left(\frac{T\widehat{\beta}}{2} + \lambda \varepsilon T\right) \widehat{\mathcal{W}}^{p}\right| \frac{1}{m^{p}} \|Z\|\|\_{H^{p}}^{p} \,\forall Z \in \overline{\mathbf{Y}\_{\tau}}. \end{split} \tag{26}$$

where *β* <sup>=</sup> max 1≤*j*≤*m* {*βj*}. Choose *<sup>ε</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup>*<sup>λ</sup>* ( <sup>1</sup> *T pW <sup>p</sup>* <sup>−</sup> *<sup>β</sup>* ), from (26), we get

$$\|\mathcal{F}(Z)\| \ge \frac{1}{2^p m^p} \|Z\|\_{H}^p \ge 0. \tag{27}$$

Hence, <sup>Υ</sup>*<sup>τ</sup>* ⊂ {*<sup>Z</sup>* <sup>∈</sup> *<sup>H</sup>* | F(*Z*) <sup>≥</sup> <sup>0</sup>} and <sup>F</sup>(*Z*) <sup>≥</sup> <sup>1</sup> 2*pm<sup>p</sup> Z p <sup>H</sup>*, ∀*Z* ∈ *∂*Υ*τ*. Therefore, the condition (*i*) in Theorem 1 holds.

For any finite-dimensional space *<sup>H</sup>*<sup>0</sup> <sup>⊂</sup> *<sup>H</sup>*, we claim that *<sup>H</sup>* <sup>=</sup> *<sup>H</sup>*<sup>0</sup> <sup>5</sup>{*Z* ∈ *H* | F(*Z*) ≥ <sup>0</sup>} is bounded. Assume that there exists at least a sequence {*Zk*} ⊂ *<sup>H</sup>* such that *Zk <sup>H</sup>* → ∞ as *k* → ∞. From F(*Zk*) ≥ 0 and (19), we obtain

$$0 \le \frac{\mathcal{F}(Z\_k(t))}{\sum\_{j=1}^{j=m} ||z\_{k,j}||\_{\left(\gamma\_j, p\right)}^p} \le \frac{1}{p} + \sum\_{j=1}^{j=m} [\frac{c\_j}{pc\_j'}k\_j(0) + \frac{d\_j}{pd\_j'}k\_j(T)]\mathcal{W}\_j^p - \lambda T L.$$

Since *<sup>L</sup>* is arbitrary, we draw a contradiction. Therefore, *<sup>H</sup>* <sup>=</sup> *<sup>H</sup>*<sup>0</sup> <sup>5</sup>{*Z* ∈ *H* | F(*Z*) ≥ 0} is bounded. Based on Theorem 1, functional F has infinitely many critical points, which means that Equation (3) has infinitely many solutions in *H*.

**Theorem 3.** *Assume that* (*H*2) *holds and J*(*t*, *Z*) = *J*(*t*, −*Z*)*. Then, Equation (3) has infinitely many solutions with* <sup>∑</sup>*j*=*<sup>m</sup> j*=1 1 *p* − *<sup>β</sup>jT* <sup>2</sup> + *λTη<sup>j</sup> p W<sup>p</sup> <sup>j</sup>* > 0*.*

**Proof.** Suppose that the sequence {F(*Zk*)}*k*∈<sup>N</sup> is bounded and lim *<sup>k</sup>*→<sup>∞</sup> <sup>F</sup> (*Zk*) = 0, *Zk*(*t*) = (*zk*,1(*t*), ... , *zk*,*m*(*t*)). In what follows, we prove that F satisfies the Palais–Smale condition. Indeed, assume ∀*j* : *zk*,*j* (*γj*,*p*) → ∞(*k* → ∞), from (13), (24), (*H*2) and (8), we have

$$\begin{split} \frac{1}{p} \sum\_{j=1}^{j=m} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{p} \leq & \mathcal{F}(Z\_{k}) + \sum\_{j=1}^{j=m} \int\_{0}^{T} G\_{j}(z\_{j}(t))dt + \lambda \int\_{0}^{T} f(t,z\_{1}(t),\ldots,z\_{m}(t))dt \\ \leq & \mathcal{F}(Z\_{k}) + \sum\_{j=1}^{j=m} \int\_{0}^{T} \frac{\beta\_{j}}{2} \mid z\_{k,j}\parallel^{p} dt + \lambda \sum\_{j=1}^{j=m} \int\_{0}^{T} \frac{\eta\_{j}}{p} \mid z\_{k,j}\parallel^{p} - (\frac{\eta\_{j}}{p} + \frac{\beta\_{j}}{2\lambda}) \mid z\_{k,j}\parallel^{\omega\_{j}} dt \\ \leq & \mathcal{F}(Z\_{k}) + \sum\_{j=1}^{j=m} \left(\frac{\beta\_{j}T}{2} + \frac{\lambda T \eta\_{j}}{p}\right) \mathcal{W}\_{j}^{p} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{p} + \lambda T \sum\_{j=1}^{j=m} \left(\frac{\eta\_{j}}{p} + \frac{\beta\_{j}}{2\lambda}\right) \mathcal{W}\_{j}^{\omega\_{j}} \|z\_{k,j}\|\_{\left(\gamma\_{j},p\right)}^{\omega\_{j}}. \end{split} \tag{28}$$

namely

$$\sum\_{j=1}^{j=m} \left[ \frac{1}{p} - \left( \frac{\mathcal{\beta}\_j T}{2} + \frac{\lambda T \eta\_j}{p} \right) \mathcal{W}\_j^p \right] \|\boldsymbol{z}\_{k,j}\|\_{\left(\gamma\_j, p\right)}^p - \lambda T \sum\_{j=1}^{j=m} \left( \frac{\eta\_j}{p} + \frac{\mathcal{\beta}\_j}{2\lambda} \right) \mathcal{W}\_j^{\omega\gamma} \|\boldsymbol{z}\_{k,j}\|\_{\left(\gamma\_j, p\right)}^{\omega\gamma} \le \mathcal{F}(\boldsymbol{Z}\_k). \tag{29}$$

Recall that <sup>∑</sup>*j*=*<sup>m</sup> j*=1 1 *p* − *<sup>β</sup>jT* <sup>2</sup> + *λTη<sup>j</sup> p W<sup>p</sup> <sup>j</sup>* > 0, *ω<sup>j</sup>* ∈ (0, *p*) and {F(*Zk*)} is bounded, we get a contradiction. Hence, {*Zk*} is bounded on *H*. The rest of the proof for the Palais–Smale condition is similar to that of Lemma 6, so we do not repeat it.

Let *<sup>τ</sup>* <sup>∈</sup> (0, <sup>1</sup> *<sup>W</sup>* ). For any *<sup>Z</sup>* <sup>∈</sup> <sup>Υ</sup>*τ* , one has *Z <sup>H</sup>* <sup>=</sup> <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zj* (*γj*,*p*) <sup>≤</sup> *<sup>τ</sup>* <sup>&</sup>lt; <sup>1</sup> *<sup>W</sup>* . A similar analysis with (25) yields <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zj* <sup>∞</sup> < 1. From (28), we get

F(*Z*) ≥ 1 *p j*=*m* ∑ *j*=1 *zj p* (*γj*,*p*) − *j*=*m* ∑ *j*=1 *<sup>T</sup>* 0 *βj* <sup>2</sup> <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup> dt* <sup>−</sup> *<sup>λ</sup> j*=*m* ∑ *j*=1 *<sup>T</sup>* 0 *ηj <sup>p</sup>* <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup>* <sup>−</sup>( *ηj p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup>*λ*) <sup>|</sup> *zj* <sup>|</sup> *<sup>ω</sup><sup>j</sup> dt* <sup>=</sup> <sup>1</sup> *p j*=*m* ∑ *j*=1 *zj p* (*γj*,*p*) − *j*=*m* ∑ *j*=1 \* *<sup>T</sup>* 0 *βj* <sup>2</sup> <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup> dt* + *λ <sup>T</sup>* 0 *ηj <sup>p</sup>* <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup>* <sup>−</sup>( *ηj p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup>*λ*) <sup>|</sup> *zj* <sup>|</sup> *<sup>ω</sup><sup>j</sup> dt*<sup>+</sup> <sup>=</sup> <sup>1</sup> *p j*=*m* ∑ *j*=1 *zj p* (*γj*,*p*) − *j*=*m* ∑ *j*=1 \* *<sup>T</sup>* 0 ( *βj* <sup>2</sup> <sup>+</sup> *λη<sup>j</sup> <sup>p</sup>* ) <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup>* <sup>−</sup>( *λη<sup>j</sup> p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup> ) <sup>|</sup> *zj* <sup>|</sup> *<sup>ω</sup><sup>j</sup> dt*<sup>+</sup> <sup>=</sup> <sup>1</sup> *p j*=*m* ∑ *j*=1 *zj p* (*γj*,*p*) <sup>+</sup> *j*=*m* ∑ *j*=1 *<sup>T</sup>* 0 ( *λη<sup>j</sup> p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup> ) <sup>|</sup> *zj* <sup>|</sup> *<sup>ω</sup><sup>j</sup>* <sup>−</sup>( *λη<sup>j</sup> p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup> ) <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup> dt* ≥ 1 *p j*=*m* ∑ *j*=1 *zj p* (*γj*,*p*) <sup>+</sup> *j*=*m* ∑ *j*=1 *<sup>T</sup>* 0 ( *λη<sup>j</sup> p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup> ) <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup>* <sup>−</sup>( *λη<sup>j</sup> p* <sup>+</sup> *<sup>β</sup><sup>j</sup>* <sup>2</sup> ) <sup>|</sup> *zj* <sup>|</sup> *<sup>p</sup> dt* <sup>=</sup> <sup>1</sup> *p j*=*m* ∑ *j*=1 *zj p* (*γj*,*p*) ≥ 1 *pm<sup>p</sup>* ( *j*=*m* ∑ *j*=1 *zj* (*γj*,*p*))*<sup>p</sup>* <sup>=</sup> <sup>1</sup> *pm<sup>p</sup> Z p <sup>H</sup>* ≥ 0, ∀ *Z* ∈ Υ*τ* .

$$\text{Clearly, } \overline{\Upsilon}\_{\tau'} \subset \{ Z \in H \mid \mathcal{F}(Z) \ge 0 \} \text{ and } \mathcal{F}(Z) \ge \underset{\ldots}{\underset{\ldots}{\sim}} \parallel Z \vert \vert\_{H'}^p \lor Z \in \partial \Upsilon\_{\tau'}.$$

For any finite-dimensional space *H* <sup>0</sup> <sup>⊂</sup> *<sup>H</sup>*, we claim that *<sup>H</sup>* <sup>=</sup> *<sup>H</sup>* 0 <sup>5</sup>{*Z* ∈ *H* | F(*Z*) ≥ <sup>0</sup>} is bounded. Assume that there exists at least a sequence {*Zk*} ⊂ *<sup>H</sup>* such that *Zk <sup>H</sup>* → ∞ as *k* → ∞. Then, according to (19), (*H*2) and Lemma 3 we obtain

<sup>0</sup> <sup>≤</sup> <sup>F</sup>(*Zk*(*t*)) <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) ≤ 1 *p* + *j*=*m* ∑ *j*=1 [ *cj pc j kj*(0) + *dj pd j kj*(*T*)]*W<sup>p</sup> <sup>j</sup>* <sup>−</sup> *<sup>λ</sup>* % *T* <sup>0</sup> <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *ηj <sup>p</sup>* | *zk*,*<sup>j</sup>* | *<sup>p</sup>* <sup>−</sup>*J*(*t*, *Zk*(*t*))*dt* <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) (30) ≤ 1 *p* + *j*=*m* ∑ *j*=1 [ *cj pc j kj*(0) + *dj pd j kj*(*T*)]*W<sup>p</sup> j* − <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *λη<sup>j</sup> p* % Ω*zk*,*<sup>j</sup> ζ p* 0 *zk*,*j p* (*γj*,*p*) *dt* <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) + *λ* % *T* <sup>0</sup> <sup>∑</sup>*j*=*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *δ<sup>j</sup>* | *zk*,*<sup>j</sup>* | *<sup>ω</sup><sup>j</sup> dt* <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) ≤ 1 *p* + *j*=*m* ∑ *j*=1 [ *cj pc j kj*(0) + *dj pd j kj*(*T*)]*W<sup>p</sup> <sup>j</sup>* <sup>−</sup> *λζ <sup>p</sup>*+<sup>1</sup> 0 *p* min 1≤*j*≤*m* {*ηj*} + *<sup>λ</sup><sup>T</sup>* <sup>∑</sup>*j*=*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *<sup>δ</sup>jW<sup>ω</sup><sup>j</sup> j zk*,*j ωj* (*γj*,*p*) <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) ,

where Ω*zk*,*<sup>j</sup>* = {*t* ∈ [0, *T*] :| *zk*,*j*(*t*) |≥ *ζ*<sup>0</sup> *zk*,*j* (*γj*,*p*)} and *meas*{Ω*zk*,*<sup>j</sup>* } ≥ *ζ*0. Since min1≤*j*≤*m*{*ηj*} <sup>&</sup>gt; <sup>1</sup> *λζ <sup>p</sup>*+<sup>1</sup> 0 ( 3 <sup>2</sup> <sup>+</sup> *<sup>p</sup>* <sup>∑</sup>*j*=*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> [ *cj pc j kj*(0) + *dj pd j kj*(*T*)]*W<sup>p</sup> <sup>j</sup>* ), then

$$\frac{1}{p} + \sum\_{j=1}^{j=m} [\frac{c\_j}{p c\_j'} k\_j(0) + \frac{d\_j}{p d\_j'} k\_j(T)] \mathcal{W}\_j^p - \frac{\lambda \zeta\_0^{p+1}}{p} \min\_{1 \le j \le m} \{\eta\_j\} < -\frac{1}{2p},\tag{31}$$

based on *ω<sup>j</sup>* ∈ (0, *p*) and *Zk <sup>H</sup>* → ∞ as *k* → ∞, we get

$$\frac{\lambda T \sum\_{j=1}^{j=m} \delta\_j \mathsf{W}\_j^{\omega\_j} ||z\_{k,j}||\_{\left(\gamma\_j, p\right)}^{\omega\_j}}{\sum\_{j=1}^{j=m} ||z\_{k,j}||\_{\left(\gamma\_j, p\right)}^p} \to 0, k \to \infty. \tag{32}$$

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Combining (31) and (32), we obtain that 0 <sup>≤</sup> <sup>F</sup>(*Zk* (*t*)) <sup>∑</sup>*j*=*<sup>m</sup> j*=1 *zk*,*j p* (*γj*,*p*) <sup>&</sup>lt; <sup>−</sup> <sup>1</sup> <sup>2</sup>*<sup>p</sup>* as *k* → ∞, which

draws a contradiction. Hence, *<sup>H</sup>* is bounded. Based on Theorem 1, functional <sup>F</sup> has infinitely many critical points, which means that Equation (3) has infinitely many solutions in *H*.

**Example 1.** *Focus on the following Fredholm fractional partial integro-differential equations with m* = 3 *and p* = 4*:*

*tD*0.5 <sup>1</sup> ((*<sup>t</sup>* <sup>+</sup> <sup>1</sup>)Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.5 *<sup>t</sup> <sup>z</sup>*1(*t*))) + ( <sup>1</sup> <sup>2</sup> <sup>+</sup> *<sup>t</sup>*)Φ4(*z*1(*t*)) = *Dz*<sup>1</sup> *<sup>f</sup>*(*t*, *<sup>z</sup>*1(*t*), *<sup>z</sup>*2(*t*), *<sup>z</sup>*3(*t*)) + % <sup>1</sup> <sup>0</sup> <sup>10</sup>−5*<sup>t</sup>* sin(*s*)Φ4(*z*1(*s*))*ds*, *<sup>t</sup>* <sup>∈</sup> [0, 1], *<sup>z</sup>*1(*t*) = % <sup>1</sup> <sup>0</sup> <sup>10</sup>−5*<sup>t</sup>* sin(*s*)Φ4(*z*1(*s*))*ds*, *<sup>t</sup>* <sup>∈</sup> [0, 1], *tD*0.6 <sup>1</sup> ((*t* <sup>2</sup> + 1)Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.6 *<sup>t</sup> <sup>z</sup>*2(*t*))) + ( <sup>1</sup> <sup>3</sup> + *t* <sup>2</sup>)Φ4(*z*2(*t*)) = *Dz*<sup>2</sup> *<sup>f</sup>*(*t*, *<sup>z</sup>*1(*t*), *<sup>z</sup>*2(*t*), *<sup>z</sup>*3(*t*)) + % <sup>1</sup> <sup>0</sup> <sup>10</sup>−5*<sup>t</sup>* <sup>2</sup> sin(*s*)Φ4(*z*2(*s*))*ds*, *<sup>t</sup>* <sup>∈</sup> [0, 1], *<sup>z</sup>*2(*t*) = % <sup>1</sup> <sup>0</sup> <sup>10</sup>−5*<sup>t</sup>* <sup>2</sup> sin(*s*)Φ4(*z*2(*s*))*ds*, *<sup>t</sup>* <sup>∈</sup> [0, 1], *tD*0.75 <sup>1</sup> ((*t* <sup>3</sup> + 1)Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.75 *<sup>t</sup> <sup>z</sup>*3(*t*))) + ( <sup>1</sup> <sup>4</sup> + *t* <sup>3</sup>)Φ4(*z*3(*t*)) = *Dz*<sup>3</sup> *<sup>f</sup>*(*t*, *<sup>z</sup>*1(*t*), *<sup>z</sup>*2(*t*), *<sup>z</sup>*3(*t*)) + % <sup>1</sup> <sup>0</sup> <sup>10</sup>−5*<sup>t</sup>* <sup>3</sup> sin(*s*)Φ4(*z*3(*s*))*ds*, *<sup>t</sup>* <sup>∈</sup> [0, 1], *<sup>z</sup>*3(*t*) = % <sup>1</sup> <sup>0</sup> <sup>10</sup>−5*<sup>t</sup>* <sup>3</sup> sin(*s*)Φ4(*z*3(*s*))*ds*, *<sup>t</sup>* <sup>∈</sup> [0, 1], <sup>Φ</sup>4(*z*1(0)) <sup>−</sup> *tD*−0.5 <sup>1</sup> (Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.5 *<sup>t</sup> <sup>z</sup>*1(0))) = 0, <sup>Φ</sup>4(*z*1(1)) + *tD*−0.5 <sup>1</sup> (Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.5 *<sup>t</sup> z*1(1))) = 0, <sup>Φ</sup>4(*z*2(0)) <sup>−</sup> *tD*−0.4 <sup>1</sup> (Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.6 *<sup>t</sup> <sup>z</sup>*2(0))) = 0, <sup>Φ</sup>4(*z*2(1)) + *tD*−0.4 <sup>1</sup> (Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.6 *<sup>t</sup> z*2(1))) = 0, <sup>Φ</sup>4(*z*3(0)) <sup>−</sup> *tD*−0.25 <sup>1</sup> (Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.75 *<sup>t</sup> <sup>z</sup>*3(0))) = 0, <sup>Φ</sup>4(*z*3(1)) + *tD*−0.25 <sup>1</sup> (Φ4(*<sup>C</sup>* <sup>0</sup> *<sup>D</sup>*0.75 *<sup>t</sup> z*3(1))) = 0, (33)

$$where \ c\_{\dot{\jmath}} = c\_{\dot{\jmath}'} = 1, d\_{\dot{\jmath}} = d\_{\dot{\jmath}'} = \frac{1}{2}, \dot{\jmath} = 1, 2, 3.$$

$$f(t, z\_1, z\_2, z\_3) = (1+t) \begin{cases} \left(z\_1^4 + z\_2^4 + z\_3^4\right)^2 , z\_1^4 + z\_2^4 + z\_3^4 \le 1, \\\ 2(z\_1^4 + z\_2^4 + z\_3^4)^2 - (z\_1^4 + z\_2^4 + z\_3^4)^{\frac{1}{2}} , z\_1^4 + z\_2^4 + z\_3^4 > 1. \end{cases}$$

It is easy to verify that *f* is continuous with respect to *t* and continuously differentiable with respect to *z*1, *z*<sup>2</sup> and *z*<sup>3</sup> (see Figures 1 and 2) and satisfies (*H*0) and (*H*1). Obviously, *k*1(0) = *k*2(0) = *k*3(0) = 1, *k*1(1) = *k*2(1) = *k*3(1) = 2, *β* <sup>=</sup> <sup>10</sup>−5. By direct calculation we have *k*<sup>1</sup> <sup>=</sup> *k*<sup>2</sup> <sup>=</sup> *k*<sup>3</sup> <sup>=</sup> 1, *l*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> ,*l*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>3</sup> ,*l*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>4</sup> , and

$$W\_{(0.5A)} = \max\left\{\frac{1}{\Gamma(0.5)[(-\frac{1}{2})\frac{4}{3}+1]^{\frac{3}{4}}}, 1\right\} + \left[8\max\left\{1, \left(\frac{1}{\Gamma(1.5)}\right)^{4}\right\}\right]^{\frac{1}{4}} = 3.184,$$

$$W\_{(0.6.4)} = \max\left\{\frac{1}{\Gamma(0.6)[(-\frac{2}{5})\frac{4}{3}+1]^{\frac{3}{4}}}, 1\right\} + \left[8\max\left\{1, \left(\frac{1}{\Gamma(1.6)}\right)^{4}\right\}\right]^{\frac{1}{4}} = 3.072,$$

$$W\_{(0.75A)} = \max\left\{\frac{1}{\Gamma(0.75)[(-\frac{1}{4})\frac{4}{3}+1]^{\frac{3}{4}}}, 1\right\} + \left[8\max\left\{1, \left(\frac{1}{\Gamma(1.75)}\right)^{4}\right\}\right]^{\frac{1}{4}} = 2.936,$$

then

$$\frac{W\_{(0.5A)}^4}{\min\{\hat{k}\_1,\hat{l}\_1\}} = 206, \frac{W\_{(0.6,4)}^4}{\min\{\hat{k}\_2,\hat{l}\_2\}} = 267, \frac{W\_{(0.75,4)}^4}{\min\{\hat{k}\_3,\hat{l}\_3\}} = 297.$$

namely, *<sup>W</sup>* <sup>=</sup> 297, <sup>1</sup> *pW* <sup>=</sup> 8.4 <sup>×</sup> <sup>10</sup>−5, then <sup>1</sup> *pW* <sup>−</sup> *<sup>β</sup>* <sup>&</sup>gt; 0. Hence, from Theorem 2 we can see that Equation (33) has infinitely many solutions.

**Figure 1.** the contour-plot of Equation (33) for *t* = 0.

**Figure 2.** the contour-plot of Equation (33) for *t* = 1.

**Author Contributions:** Conceptualization, Y.L.; Investigation, D.L.; Writing—original draft, D.L.; Writing—review and editing, Y.L. and F.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China grant numbers 12101481, 62103327, 11872201; Young Talent Fund of Association for Science and Technology in Shaanxi, China grant number 20220529; Young Talent Fund of Association for Science and Technology in Xi'an, China grant number 095920221344.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the editor and reviewers greatly for their precious comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Existence and Approximate Controllability of Mild Solutions for Fractional Evolution Systems of Sobolev-Type**

**Yue Liang**

Center for Quantitative Biology, College of Science, Gansu Agricultural University, Lanzhou 730070, China; liangy@gsau.edu.cn; Tel.: +86-139-1915-6640

**Abstract:** This paper investigates the existence and approximate controllability of Riemann–Liouville fractional evolution systems of Sobolev-type in abstract spaces. At first, a group of sufficient conditions is established for the existence of mild solutions without the compactness of operator semigroup. Then the approximate controllability is studied under the assumption that the corresponding linear system is approximate controllability. The proof is based on the fixed point theory and the method of operator semigroup. An example is given as an application of the obtained results.

**Keywords:** fractional evolution systems; approximate controllability; Sobolev operator; compactness; Schauder fixed point theorem

**MSC:** 26A33; 93B05

#### **1. Introduction**

Let *X* be a Hilbert space, whose norm is denoted by · . We consider the fractional evolution equation of sobolev type with the Riemann–Liouville derivative of the form

$$\begin{cases} \, \, ^L D\_t^a(Ex(t)) = Ax(t) + f(t, x(t)) + Bu(t), \quad t \in \mathcal{J}' := (0, b], \\\, \, \, \, \, \, ^{1-a}\_t(Ex(t))|\_{t=0} + g(\mathbf{x}) = \mathbf{x}\_{0\prime} \end{cases} \tag{1}$$

where *LD<sup>α</sup> <sup>t</sup>* is the Riemann–Liouville fractional derivative operator of order *α* ∈ (0, 1), *I* 1−*α t* is the fractional integral operator of order 1 − *α*, *A* : *D*(*A*) ⊂ *X* → *X* and *E* : *D*(*E*) ⊂ *X* → *X* are linear operators, *B* is a linear bounded operator from U to X; here *U* is another Hilbert space, the control function *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>U</sup>*) for *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1, *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>X</sup>*, *<sup>f</sup>* is the nonlinear function and *g* represents the nonlocal function which satisfies specific conditions.

Fractional differential equations, including of the Caputo type and Riemann–Liouville type, have been proved to be crucial tools in portraying the hereditary and memory property of various materials and processes. In 2011, Du et al. [1] pointed out that Riemann– Liouville fractional derivatives are more suitable to describe certain characteristics of viscoelastic materials than Caputo ones. Therefore, it is significant to study Riemann– Liouville fractional differential systems. In 2013, Zhou et al. [2], applying the Laplace transform technique and probability density functions, presented a suitable concept of mild solutions of Riemann–Liouville fractional evolution equations, and proved the existence of mild solutions for the fractional Cauchy problems under the cases that the *C*0-semigroup is compact or noncompact. For the existence of mild solutions of fractional evolution equations, we refer to [3–8] and the references therein. In these papers, the compactness of operator semigroup or the measure of non-compactness conditions on nonlinearity are required. Sometimes, in order to obtain the uniqueness of mild solutions, the Lipschitz condition is also assumed.

In recent years, the controllability of fractional evolution equations has gained considerable attention. Generally speaking, the controllability of fractional evolution equations in

**Citation:** Liang, Y. Existence and Approximate Controllability of Mild Solutions for Fractional Evolution Systems of Sobolev-Type. *Fractal Fract.* **2022**, *6*, 56. https://doi.org/ 10.3390/fractalfract6020056

Academic Editor: Rodica Luca

Received: 5 January 2022 Accepted: 18 January 2022 Published: 22 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

abstract spaces includes two cases: the exact controllability and the approximate controllability. When we study the exact controllability of fractional evolution systems in abstract spaces, we assume that the control operator has a bounded inverse operator in a quotient space. However, if the state space is infinite dimensional and the operator semigroup is compact, the inverse of the control operator may not exist, see [9]. Hence, the assumptions for the exact controllability are too strong. Contrasting with the exact controllability, approximate controllability is more suitable to describe the natural phenomena. There are many research works focusing on the approximate controllability of fractional evolution systems, see [10–12] and the references therein. In [10], Chang et al. investigated the approximate controllability of fractional differential systems of Sobolev type in Banach spaces under the assumption that the resolvent operators, generated by the linear part, are compact. Sakthivel et al. [11] studied the approximate controllability of nonlinear fractional stochastic evolution systems when the linear part generates a compact semigroup. Recently, In [12], Yang demonstrated the existence and approximate controllability of mild solutions for *α* ∈ (1, 2)-order fractional evolution equations of Sobolev type when the pair (*A*, *E*) generates a compact resolvent family.

Inspired by the above mentioned papers, the aim of this work is to investigate the existence and approximate controllability of Riemann–Liouville fractional evolution system (1) in Hilbert space *X*. By using the Schauder fixed point theorem and the operator semigroup theory, we first prove the existence of mild solutions of the considered system without the compactness of operator semigroup and the measure of non-compactness conditions on nonlinearity. Then the approximate controllability is studied under the assumption that the corresponding linear system is approximate controllability. It is emphasized that the compactness of the operator semigroup and the Lipschitz continuity of nonlinearity are deleted in our work. The redundant assumptions on the linear operator *E*, such as the conditions [*C*1] and [*C*4] of [13], are removed in this paper.

#### **2. Preliminaries**

Let *J* = [0, *b*] and *C*(*J*, *X*) be the continuous function space. Denote by

$$\mathcal{C}\_{1-\mathfrak{a}}(f, X) := \{ \mathfrak{x} : \cdot^{1-\mathfrak{a}} \mathfrak{x}(\cdot) \in \mathcal{C}(f, X) \}.$$

Then *C*1−*α*(*J*, *X*) is a Banach space endowed with the norm *x <sup>C</sup>*1−*<sup>α</sup>* = sup *t* <sup>1</sup>−*<sup>α</sup> x*(*t*) .

*t*∈*J* At first, for any *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>X</sup>*) with *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1, we consider the following linear fractional initial value problem

$$\begin{cases} \, \, ^L D\_t^\alpha (\text{Ex}(t)) = A\text{x}(t) + h(t), \quad t \in \mathbb{J}',\\\, \, \, \, I\_t^{1-\alpha} (\text{Ex}(t))|\_{t=0} + \text{g}(\text{x}) = \text{x}\_0. \end{cases} \tag{2}$$

Throughout this paper, we suppose the following assumptions on *A* and *E*.

(*A*1) The linear operator *A* is densely defined and closed.

(*A*2) *D*(*E*) ⊂ *D*(*A*) and *E* is bijective.

(*A*3) The linear operator *<sup>E</sup>*−<sup>1</sup> : *<sup>X</sup>* <sup>→</sup> *<sup>D</sup>*(*E*) <sup>⊂</sup> *<sup>X</sup>* is compact.

By (*A*1)–(*A*3), the linear operator *AE*−<sup>1</sup> : *<sup>X</sup>* <sup>→</sup> *<sup>X</sup>* is bounded due to the closed graph theorem. Hence, *AE*−<sup>1</sup> generates a *<sup>C</sup>*0-semigroup *<sup>T</sup>*(*t*)(*<sup>t</sup>* <sup>≥</sup> <sup>0</sup>), which is expressed by *<sup>T</sup>*(*t*) = *<sup>e</sup>AE*−1*<sup>t</sup>* for *<sup>t</sup>* <sup>≥</sup> 0. We suppose that *<sup>M</sup>* :<sup>=</sup> sup *t*≥0 *T*(*t*) < +∞.

**Remark 1.** *Contrasting with [13], we delete the redundant conditions* [*C*1] *and* [*C*4] *of [13] in our paper. Hence, the results obtained in this work extends the results of [13].*

Applying the Riemann–Liouville fractional integral operator on both sides of (2), we obtain

$$E\mathfrak{x}(t) = \frac{t^{\mathfrak{a}-1}}{\Gamma(\mathfrak{a})} I\_t^{1-\mathfrak{a}}(E\mathfrak{x}(t))|\_{t=0} + I\_t^{\mathfrak{a}} A\mathfrak{x}(t) + I\_t^{\mathfrak{a}} h(t)$$

$$= \frac{t^{\mathfrak{a}-1}}{\Gamma(\mathfrak{a})} [\mathfrak{x}\_0 - \mathfrak{g}(\mathfrak{x})] + \frac{1}{\Gamma(\mathfrak{a})} \int\_0^t (t-s)^{\mathfrak{a}-1} \left[ A\mathfrak{x}(s) + h(s) \right] ds.$$

Let *λ* > 0. Taking the Laplace transform

$$
\widehat{\mathfrak{x}}(\lambda) = \int\_0^\infty e^{-\lambda t} \mathfrak{x}(t) dt
$$

and

$$
\widehat{h}(\lambda) = \int\_0^\infty e^{-\lambda t} h(t) dt
$$

on both sides of the above equality, we can obtain

$$\begin{aligned} E\hat{x}(t) &= \frac{1}{\lambda^a} [\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] + \frac{1}{\lambda^a} AE^{-1} E\hat{x}(\lambda) + \frac{1}{\lambda^a} \hat{h}(\lambda) \\ &= \left(\lambda^a I - AE^{-1}\right)^{-1} [\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] + (\lambda^a I - AE^{-1})^{-1} \hat{h}(\lambda) \\ &= \int\_0^\infty e^{-\lambda^a s} T(s) [\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] ds + \int\_0^\infty e^{-\lambda^a s} T(s) \hat{h}(\lambda) ds \end{aligned}$$

where (*λ<sup>α</sup> <sup>I</sup>* <sup>−</sup> *AE*−1)−<sup>1</sup> <sup>=</sup> % <sup>∞</sup> <sup>0</sup> *<sup>e</sup>*−*λ<sup>α</sup>sT*(*s*)*ds*. Consider the one-side stable probability density function

$$\zeta\_{\theta}(\theta) = \frac{1}{\pi} \sum\_{n=1}^{\infty} (-1)^{n-1} \theta^{-an-1} \frac{\Gamma(n\alpha + 1)}{n!} \sin(n\pi\alpha), \; \theta \in (0, +\infty),$$

whose Laplace transform is given by

$$\int\_0^\infty e^{-\lambda \theta} \tilde{\zeta}\_\alpha(\theta) d\theta = e^{-\lambda^\alpha}, \quad \alpha \in (0,1).$$

A similar argument as in [2] shows that

$$\begin{aligned} E\widehat{\mathfrak{x}}(\lambda) &= \int\_0^\infty e^{-\lambda t} \int\_0^\infty a\theta \xi\_a(\theta) T(t^a \theta) t^{a-1} [\mathfrak{x}\_0 - \mathfrak{g}(\mathfrak{x})] d\theta dt \\ &+ \int\_0^\infty e^{-\lambda t} \int\_0^t \int\_0^\infty a\theta \xi\_a(\theta) T((t-s)^a \theta) (t-s)^{a-1} h(s) d\theta ds dt, \end{aligned}$$

where *ξα*(*θ*) = <sup>1</sup> *<sup>α</sup> <sup>θ</sup>*−1<sup>−</sup> <sup>1</sup> *<sup>α</sup> α*(*θ*<sup>−</sup> <sup>1</sup> *<sup>α</sup>* ). This fact implies that

$$\begin{aligned} \left(E\mathbf{x}(t)\right) &= \int\_0^\infty a\theta \xi\_a^\mathbf{x}(\theta) T(t^a\theta) t^{a-1} [\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] d\theta \\ &+ \int\_0^t \int\_0^\infty a\theta \xi\_a^\mathbf{z}(\theta) T((t-s)^a\theta) (t-s)^{a-1} h(s) d\theta ds. \end{aligned}$$

Thus, we obtain

$$\mathbf{x}(t) = t^{a-1} T\_E(t)[\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] + \int\_0^t (t - s)^{a-1} T\_E(t - s) h(s) ds\_\prime$$

where

$$T\_E(t) = E^{-1} \int\_0^\infty \alpha \theta \xi\_\alpha^\*(\theta) T(t^\alpha \theta) d\theta.$$

**Remark 2.** *When E* = *I*, *I* : *X* → *X is the identity operator, we have*

$$T\_I(t) = \int\_0^\infty a\theta \xi\_\alpha(\theta) T(t^\alpha \theta) d\theta, \ t \ge 0.$$

*Therefore, TE*(*t*) = *<sup>E</sup>*−<sup>1</sup>*TI*(*t*) *for all t* <sup>≥</sup> <sup>0</sup>*.*

From the above arguments, we introduce the definition of mild solution of the system (1) as follows.

**Definition 1.** *For each <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>U</sup>*), *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1, *a function <sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*1−*α*(*J*, *<sup>X</sup>*) *is called a mild solution of the system (1) if I*1−*<sup>α</sup> <sup>t</sup>* (*Ex*(*t*))|*t*=<sup>0</sup> <sup>+</sup> *<sup>g</sup>*(*x*) = *<sup>x</sup>*<sup>0</sup> *and*

$$\mathbf{x}(t) = t^{a-1} T\_E(t)[\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] + \int\_0^t (t-s)^{a-1} T\_E(t-s)[f(s, \mathbf{x}(s)) + Bu(s)]ds, \ t \in J'. \tag{3}$$

For the operator family {*TE*(*t*)}*t*≥0, we have the following lemma.

**Lemma 1.** *Let the assumptions* (*A*1)*–*(*A*3) *hold. Then* {*TE*(*t*)}*t*≥<sup>0</sup> *has the following properties:* (*i*) *For fixed t* ≥ 0*, TE*(*t*) *is a linear and bounded operator, i.e., for any x* ∈ *X,*

$$||T\_E(t)x|| \le \frac{M||E^{-1}||}{\Gamma(\alpha)}||x||.$$

(*ii*) {*TE*(*t*)}*t*≥<sup>0</sup> *is continuous in the uniform operator topology for t* ≥ 0*.* (*iii*) {*TE*(*t*)}*t*≥<sup>0</sup> *is compact.*

**Proof.** From Proposition 3.1 of [2] and Remark 2, it is easy to verify that (*i*) holds. By virtue of the definition of the operator *T*(*t*)(*t* ≥ 0) and the Lebesgue dominated convergence theorem, we can deduce (*ii*). Next, we prove (*iii*). For any *r* > 0, *x* ∈ *X* with *x* ≤ *r*, we have

$$\begin{aligned} \|T\_I(t)\mathfrak{x}\| &\leq \|\mathfrak{a}M\int\_0^\infty \theta \tilde{\xi}\_a(\theta)d\theta\|\|\mathfrak{x}\|\\ &\leq \|\frac{\mathfrak{a}M}{\Gamma(\mathfrak{a}+1)}\|\mathfrak{x}\|\\ &\leq \frac{Mr}{\Gamma(\mathfrak{a})}. \end{aligned}$$

This fact means that *TI*(*t*) maps bounded subset of *X* into the bounded set. Then *E*−<sup>1</sup>*TI*(*t*) maps the bounded subset of *X* into relatively compact set due to the compactness of *E*−1. Thus, {*TE*(*t*)}*t*≥<sup>0</sup> is compact.

**Definition 2.** *Let Kf*(*b*) = {*x*(*b*) : *<sup>x</sup> be a mild solution of the system* (1) *for some <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>U</sup>*)}*. If Kf*(*b*) = *X, the system* (1) *is said to be approximate controllability on J.*

We consider the linear fractional control system corresponding to (1) in the form

$$\begin{cases} \, \, ^L D\_t^\alpha (\text{Ex}(t)) = A\text{x}(t) + Bu(t), \quad t \in \mathcal{I}',\\ \, \, \, \, I\_t^{1-\alpha} (\text{Ex}(t)) = \text{x}\_0. \end{cases} \tag{4}$$

Define two operators Π*<sup>b</sup>* <sup>0</sup> and *<sup>R</sup>*(, <sup>Π</sup>*<sup>b</sup>* <sup>0</sup>) by

$$
\Pi\_0^b = \int\_0^b (b-s)^{\alpha-1} T\_E(b-s) B B^\* T\_E^\*(b-s) ds,
$$

$$
R\left(\varepsilon, \Pi\_0^b\right) = \left(\varepsilon I + \Pi\_0^b\right)^{-1}, \quad \varepsilon > 0,
$$

where *B*∗ and *T*∗ *<sup>E</sup>*(*t*) denote the adjoint operators of *<sup>B</sup>* and *TE*(*t*), respectively. Then, <sup>Π</sup>*<sup>b</sup>* <sup>0</sup> is a linear operator. From [14], we obtain the following result.

**Lemma 2.** *The following conditions are equivalent:*


#### **3. Existence and Approximate Controllability**

In order to study the approximate controllability of the fractional control system (1), we first investigate the existence of solutions for the following integral system

$$\begin{cases} \mathbf{x}(t) = t^{a-1} T\_E(t)[\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] + \int\_0^t (t - s)^{a-1} T\_E(t - s)[f(\mathbf{s}, \mathbf{x}(s)) + Bu(\mathbf{s}; \mathbf{x})] ds, \quad t \in \mathcal{I}', \\\ u(t; \mathbf{x}) = \mathcal{B}^\* T\_E^\*(b - t)\mathcal{R}(\mathbf{c}, \Pi\_0^b)\mathcal{P}(\mathbf{x}), \\\ \mathcal{P}(\mathbf{x}) = \mathbf{x}\_b - b^{a-1} T\_E(b)(\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})) - \int\_0^b (b - s)^{a-1} T\_E(b - s)f(\mathbf{s}, \mathbf{x}(s)) ds, \end{cases} (5)$$

where *xb* is an arbitrary element in *X* which is different from *x*0. By Definition 1, the mild solution of the system (1) is equivalent to the solution of the integral system (5) for *u*(·; *x*) ∈ *Lp*(*J*, *X*).

For this purpose, we make the following assumptions.

(*A*4) *f* : *J* × *X* → *X* satisfies the following conditions.

(*i*) For each *x* ∈ *X*, *f*(·, *x*) : *J* → *X* is strongly measurable, and for every *t* ∈ *J*, *f*(·, *x*) : *X* → *X* is continuous.

(*ii*) For any *<sup>r</sup>* <sup>&</sup>gt; 0, there is a function *<sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, <sup>R</sup>+), *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1 such that

$$\|f(t, \mathbf{x})\| \le \phi(t)$$

for any *t* ∈ *J* and *x* ∈ *X* with *x* ≤ *r*.

(*A*5) *g* : *C*1−*α*(*J*, *X*) → *X* is continuous and maps bounded subset of *C*1−*α*(*J*, *X*) into the bounded set.

(*A*6) *B* : *U* → *X* is a bounded linear operator, i.e., ∃ *MB* > 0 such that *B* ≤ *MB*.

(*A*7) *R*(, Π*<sup>b</sup>* 0) <sup>≤</sup> <sup>1</sup> for all > 0.

For any *r* > 0, let *Br* = *x* ∈ *C*1−*α*(*J*, *X*) : *x <sup>C</sup>*1−*<sup>α</sup>* ≤ *r* . Then *Br* is a nonempty bounded, closed and convex subset of *C*1−*α*(*J*, *X*). By the assumption (*A*5) we know that there exists a constant *M*<sup>1</sup> > 0 such that *g*(*x*) ≤ *M*<sup>1</sup> for any *x* ∈ *Br*. From the assumption (*A*6) we deduce that *Bu* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>X</sup>*) for any *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>X</sup>*) with *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1.

**Lemma 3.** *For any* F ∈ *<sup>L</sup>p*(*J*, *<sup>X</sup>*)*, the operator* <sup>ℵ</sup> : *<sup>L</sup>p*(*J*, *<sup>X</sup>*) <sup>→</sup> *<sup>C</sup>*(*J*, *<sup>X</sup>*)*, defined by*

$$(\aleph \mathcal{F})(\cdot) = \cdot^{1-\alpha} \int\_0^\cdot (\cdot - s)^{\alpha - 1} T\_E(\cdot - s) \mathcal{F}(s) ds \,\omega$$

*is compact.*

**Proof.** Denote by

$$(\aleph\_0 \mathcal{F})(t) = t^{1-\mathfrak{a}} \int\_0^t (t-s)^{\mathfrak{a}-1} T\_I(t-s) \mathcal{F}(s) ds.$$

It follows from Lemma 1 that

$$\left\|(\aleph\_0 \mathcal{F})(t)\right\| \leq \quad \frac{M}{\Gamma(\alpha)} (\frac{bp-b}{p\alpha-1})^{1-\frac{1}{p}} \|\mathcal{F}\|\_{L^p}.$$

So, owing to the compactness of *E*−1, we conclude that the set

$$\{ (\aleph \mathcal{F})(t) = E^{-1}(\aleph\_0 \mathcal{F})(t) : \mathcal{F} \in L^p(\mathcal{J}, X), t \in \mathcal{J} \}$$

is relatively compact in *X*.

Next, we will prove that the set {ℵF : F ∈ *<sup>L</sup>p*(*J*, *<sup>X</sup>*)} is equi-continuous in *<sup>C</sup>*(*J*, *<sup>X</sup>*). For *t*1, *t*<sup>2</sup> ∈ *J* with 0 ≤ *t*<sup>1</sup> < *t*<sup>2</sup> < *b*, we have

$$\begin{split} \| \| (\aleph\_{\mathcal{F}})(t\_2) - (\aleph\_{\mathcal{F}})(t\_1) \| &\leq \quad \| (t\_2^{1-a} - t\_1^{1-a}) \int\_0^{t\_2} (t\_2 - s)^{a-1} T\_{\mathcal{E}}(t\_2 - s) \mathcal{F}(s) ds \| \\ &\quad + \quad t\_1^{1-a} \| \int\_0^{t\_1} [(t\_2 - s)^{a-1} - (t\_1 - s)^{a-1}] T\_{\mathcal{E}}(t\_2 - s) \mathcal{F}(s) ds \| \\ &\quad + \quad t\_1^{1-a} \| \int\_0^{t\_1} (t\_1 - s)^{a-1} [T\_{\mathcal{E}}(t\_2 - s) - T\_{\mathcal{E}}(t\_1 - s)] \mathcal{F}(s) ds \| \\ &\quad + \quad t\_1^{1-a} \| \int\_{t\_1}^{t\_2} (t\_2 - s)^{a-1} T\_{\mathcal{E}}(t\_2 - s) \mathcal{F}(s) ds \| \\ &= \quad \sum\_{i=1}^4 I\_i . \end{split}$$

Obviously, if *t*<sup>2</sup> − *t*<sup>1</sup> → 0, we have

$$\begin{aligned} I\_1 &=& \| (t\_2^{1-\alpha} - t\_1^{1-\alpha}) \int\_0^{t\_2} (t\_2 - s)^{\alpha - 1} T\_E(t\_2 - s) \mathcal{F}(s) ds \| \\ &\le& \frac{M \| |E^{-1}| }{\Gamma(\alpha)} (\frac{p-1}{p\alpha - 1})^{1 - \frac{1}{p}} \| \mathcal{F} \|\_{L^p} (t\_2 - t\_1)^{1 - \alpha} \\ &\to& 0, \end{aligned}$$

$$\begin{aligned} I\_2 &=& t\_1^{1-a} \| \int\_0^{t\_1} [(t\_2 - s)^{a-1} - (t\_1 - s)^{a-1}] T\_E(t\_2 - s) \mathcal{F}(s) ds \| \\ &\le& \frac{M \| E^{-1} \| b^{1-a}}{\Gamma(a)} \int\_0^{t\_1} [(t\_2 - s)^{a-1} - (t\_1 - s)^{a-1}] \mathcal{F}(s) ds \\ &\to& 0 \end{aligned}$$

and

$$\begin{aligned} I\_4 &=& \|t\_1^{1-\alpha}\| \int\_{t\_1}^{t\_2} (t\_2 - s)^{\alpha - 1} T\_E(t\_2 - s) \mathcal{F}(s) ds \| \\ &\le \quad \frac{M \|E^{-1}\| \|b^{1-\alpha}(\frac{p-1}{p\alpha - 1})^{1 - \frac{1}{p}}\|\mathcal{F}\|\_{L^p}(t\_2 - t\_1)^{\frac{p\alpha - 1}{p}} \\ &\to \quad 0. \end{aligned}$$

Since *TE*(*t*) is continuous in the uniform operator topology for *t* ≥ 0, we obtain that

$$\begin{aligned} I\_3 &= \|t\_1^{1-\alpha}\| \int\_0^{t\_1} (t\_1 - s)^{\alpha - 1} [T\_E(t\_2 - s) - T\_E(t\_1 - s)] \mathcal{F}(s) ds\| \\ &\le \sup\_{s \in [0, t\_1]} \|T\_E(t\_2 - s) - T\_E(t\_1 - s)\| |(\frac{bp - b}{p\alpha - 1})^{1 - \frac{1}{p}}| \|\mathcal{F}\|\_{L^p} \\ &\to \quad 0 \end{aligned}$$

as *t*<sup>2</sup> − *t*<sup>1</sup> → 0. Consequently, we have

$$\|(\aleph \mathcal{F})(t\_2) - (\aleph \mathcal{F})(t\_1)\| \to 0 \ (t\_2 - t\_1 \to 0).$$

This fact yields that the set {ℵF : F ∈ *<sup>L</sup>p*(*J*, *<sup>X</sup>*)} is equi-continuous in *<sup>C</sup>*(*J*, *<sup>X</sup>*). According to the Ascoli–Arzela theorem, the set {ℵF : F ∈ *<sup>L</sup>p*(*J*, *<sup>X</sup>*)} is relatively compact in *C*(*J*, *X*).

**Theorem 1.** *Let the assumptions* (*A*1)*–*(*A*7) *hold. Then, the system* (1) *has at least one mild solution on J.*

**Proof.** For any > 0, let *r* > 0 be large enough such that

$$r \geq N^\* \|\mathbf{x}\_{\theta}\| + \frac{M\|\mathbb{E}^{-1}\|}{\Gamma(a)} (N^\* b^{a-1} + 1)(\|\mathbf{x}\_0\| + M\_1) + \frac{M\|\mathbb{E}^{-1}\|}{\Gamma(a)} (\frac{bp-b}{pa-1})^{1-\frac{1}{p}} (\|\boldsymbol{\phi}\|\_{L^p} (N^\* + 1), \tag{6}$$

where *N*<sup>∗</sup> = *<sup>b</sup>* ( *MMB <sup>E</sup>*−<sup>1</sup> <sup>Γ</sup>(*α*) )2( *<sup>p</sup>*−<sup>1</sup> *<sup>p</sup>α*−<sup>1</sup> ) <sup>1</sup><sup>−</sup> <sup>1</sup> *<sup>p</sup>* . Define an operator Φ : *Br* → *C*1−*α*(*J*, *X*) by

$$(\Phi \mathbf{x})(t) = t^{a-1} T\_E(t) [\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})] + \int\_0^t (t - s)^{a-1} T\_E(t - s) [f(s, \mathbf{x}(s)) + Bu(s; \mathbf{x})] ds$$

where

$$u(s; \mathbf{x}) = B^\* T\_E^\*(b - s) R(\varepsilon, \Pi\_0^b) \mathcal{P}(\mathbf{x}),$$

$$\mathcal{P}(\mathbf{x}) = \mathbf{x}\_b - b^{\mu - 1} T\_E(b) (\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})) - \int\_0^b (b - s)^{\mu - 1} T\_E(b - s) f(s, \mathbf{x}(s)) ds.$$

$$\begin{array}{cccc} \text{Step 1. We want the } \Phi^{\circ}(\alpha) \text{ for } \alpha \text{ or } \beta\\ \text{Step 1. We will prove } \Phi: B\_{\mathcal{I}} \to B\_{\mathcal{I}}. \end{array}$$

For any > 0, by assumptions (*A*4)–(*A*7) and Lemma 1, we have

$$||\mu(t; \boldsymbol{x})|| \le \frac{MM\_B||E^{-1}||}{\epsilon \Gamma(\alpha)}||\mathcal{P}(\boldsymbol{x})||, \quad \boldsymbol{x} \in \mathcal{B}\_{r\_\star} \ t \in \mathcal{J}'$$

and

$$||\mathcal{P}(\mathbf{x})|| \le ||\mathbf{x}\_b|| + \frac{M||E^{-1}||b^{a-1}}{\Gamma(a)}(M\_1 + ||\mathbf{x}\_0||) + \frac{M||E^{-1}||}{\Gamma(a)}(\frac{bp-b}{pa-1})^{1-\frac{1}{p}}||\phi||\_{L^{p}}, \quad \mathbf{x} \in B\_{\mathbf{r}}.$$

Together this fact with (6), for any > 0, we have

*t* <sup>1</sup>−*<sup>α</sup>* (Φ*x*)(*t*) ≤ *TE*(*t*)[*x*<sup>0</sup> − *g*(*x*)] + *t* <sup>1</sup>−*<sup>α</sup> <sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*α*−<sup>1</sup>*TE*(*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)[ *<sup>f</sup>*(*s*, *<sup>x</sup>*(*s*)) + *Bu*(*s*)]*ds* <sup>≤</sup> *<sup>M</sup> <sup>E</sup>*−<sup>1</sup> <sup>Γ</sup>(*α*) ( *x*0 <sup>+</sup> *<sup>M</sup>*1) + *<sup>b</sup>*1−*<sup>α</sup> <sup>M</sup> <sup>E</sup>*−<sup>1</sup> Γ(*α*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*α*−1(*φ*(*s*) + *MB u*(*s*) )*ds* <sup>≤</sup> *<sup>M</sup> <sup>E</sup>*−<sup>1</sup> <sup>Γ</sup>(*α*) ( *x*0 <sup>+</sup> *<sup>M</sup>*1) + *<sup>M</sup> <sup>E</sup>*−<sup>1</sup> <sup>Γ</sup>(*α*) ( *bp* − *b pα* − 1 ) <sup>1</sup><sup>−</sup> <sup>1</sup> *<sup>p</sup>* ( *φ <sup>L</sup><sup>p</sup>* + *MB u <sup>L</sup><sup>p</sup>* ) ≤ *N*∗ *xb* + *M <sup>E</sup>*−<sup>1</sup> <sup>Γ</sup>(*α*) (*N*∗*bα*−<sup>1</sup> <sup>+</sup> <sup>1</sup>)( *x*0 + *M*1) + *M <sup>E</sup>*−<sup>1</sup> <sup>Γ</sup>(*α*) ( *bp* − *b pα* − 1 ) <sup>1</sup><sup>−</sup> <sup>1</sup> *p φ <sup>L</sup><sup>p</sup>* (*N*<sup>∗</sup> + 1) ≤ *r*.

Thus, Φ*x <sup>C</sup>*1−*<sup>α</sup>* = sup *t*∈*J t* <sup>1</sup>−*<sup>α</sup>* (Φ*x*)(*t*) ≤ *r*, which implies Φ : *Br* → *Br*. Step 2. Φ : *Br* → *Br* is continuous.

Let {*xn*} ⊂ *Br* with *xn* → *x* as *n* → ∞. From the continuity of *f* and *g*, we have

$$f(t, \mathfrak{x}\_n(t)) \to f(t, \mathfrak{x}(t)), \quad t \in J$$

and

$$
\mathcal{g}(\mathfrak{x}\_n) \to \mathcal{g}(\overline{\mathfrak{x}}),
$$

as *n* → ∞. Since

$$\left| \| (t - s)^{a - 1} [f(s, \chi\_n(s)) - f(s, \overline{\chi}(s))] \| \right| \le 2(t - s)^{a - 1} \phi(s) \in L^1(f, \mathbb{R}^+),$$

it follows from the Lebesgue dominated convergence theorem that

$$\begin{aligned} &\quad t^{1-a} \| (\boldsymbol{\Phi} \mathbf{x}\_{\boldsymbol{n}})(t) - (\boldsymbol{\Phi} \overline{\boldsymbol{x}})(t) \| \\ &\leq \quad \| T\_{\boldsymbol{E}}(t) (\boldsymbol{g}(\mathbf{x}\_{\boldsymbol{n}}) - \boldsymbol{g}(\overline{\boldsymbol{x}})) \| + b^{1-a} \frac{M \| \boldsymbol{E}^{-1} \|}{\Gamma(a)} \int\_{0}^{t} (t - s)^{a - 1} \| f(\boldsymbol{s}, \mathbf{x}\_{\boldsymbol{n}}(s)) - f(\boldsymbol{s}, \overline{\mathbf{x}}(s)) \| ds \\ &\rightarrow \quad 0 \quad (n \to \infty). \end{aligned}$$

Hence,

$$\|\Phi \mathbf{x}\_n - \Phi \overline{\mathbf{x}}\|\_{\mathcal{C}\_{1-\alpha}} \to 0$$

as *n* → ∞ and Φ : *Br* → *Br* is continuous.

Step 3. The set {Φ*x* : *x* ∈ *Br*} is relatively compact in *C*1−*α*(*J*, *X*).

In order to prove the relative compactness of {Φ*x* : *x* ∈ *Br*} in *C*1−*α*(*J*, *X*), we prove that the set {·1−*α*Φ*x*(·) : *<sup>x</sup>* <sup>∈</sup> *Br*} is relatively compact in *<sup>C</sup>*(*J*, *<sup>X</sup>*).

Denote by

$$(\Phi\_1 \mathfrak{x})(t) = T\_E(t)(\mathfrak{x}\_0 - \mathfrak{g}(\mathfrak{x})), \quad t \in J$$

and

$$(\Phi\_2 \mathfrak{x})(t) = t^{1-a} \int\_0^t (t-s)^{a-1} T\_E(t-s) [f(s, \mathfrak{x}(s)) + Bu(s)] ds, \quad t \in \mathbb{J}.$$

Then for any *t* ∈ *J*, we have

$$t^{1-\alpha} \Phi \mathfrak{x}(t) = (\Phi\_1 \mathfrak{x})(t) + (\Phi\_2 \mathfrak{x})(t).$$

It is sufficient to prove that {Φ1*x* : *x* ∈ *Br*} and {Φ2*x* : *x* ∈ *Br*} are relatively compact in *C*(*J*, *X*).

For any *x* ∈ *Br* and *t* ∈ *J*, by virtue of

$$\left\|\left|T\_I(t)(\mathbf{x}\_0 - \mathbf{g}(\mathbf{x}))\right|\right\| \le \frac{M}{\Gamma(\alpha)} (\left\|\mathbf{x}\_0\right\| + M\_1)\_{\nu}$$

we obtain that {(Φ1*x*)(*t*) : *x* ∈ *Br*, *t* ∈ *J*} is relatively compact in *X* owing to the compactness of *<sup>E</sup>*−1. It is obvious that the set {Φ1*<sup>x</sup>* : *<sup>x</sup>* <sup>∈</sup> *Br*} is equi-continuous in *<sup>C</sup>*(*J*, *<sup>X</sup>*) because *TE*(*t*) is continuous in the uniform operator topology for *t* ≥ 0. Hence, it follows from the Ascoli–Arzela theorem that the set {Φ1*x* : *x* ∈ *Br*} is relatively compact in *C*(*J*, *X*).

By assumptions (*A*4) and (*A*6), we know that

$$f(t, \mathfrak{x}(t)) + Bu(t) \in L^p(f, \mathbb{X}).$$

By Lemma 3, the set {Φ2*x* : *x* ∈ *Br*} is relatively compact in *C*(*J*, *X*). Consequently, the set {Φ*x* : *x* ∈ *Br*} is relatively compact in *C*1−*α*(*J*, *X*).

Hence, Φ is completely continuous in *C*1−*α*(*J*, *X*). By the Schauder fixed point theorem, Φ has at least one fixed point in *Br*, which is the mild solution of the system (1).

**Remark 3.** *In [15], Lian et al. proved the existence of mild solutions of fractional evolution equations under the assumption that the nonlocal function g is continuous, uniformly bounded and satisfies some other conditions. In [2], Zhou et al. investigated the existence of mild solutions of fractional evolution equations when the nonlocal function g is Lipschitz continuous or completely continuous. In our Theorem 1, we only assume that the nonlocal function g is continuous and maps bounded subset into bounded set, without the Lipschitz continuity and the complete continuity and any other extra conditions we obtain the existence of mild solutions of the fractional evolution Equation (1). Hence, Theorem 1 greatly extends the main results in [2,15].*

If the assumptions (*A*4) and (*A*5) are replaced by the following conditions: (*A*4) *f* : *J* × *X* → *X* satisfies the following conditions.

(*i*) For each *x* ∈ *X*, *f*(·, *x*) : *J* → *X* is strongly measurable, and for every *t* ∈ *J*, *f*(·, *x*) : *X* → *X* is continuous.

(*ii*) There exists a function *<sup>ψ</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, <sup>R</sup>+), *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1 and a constant *<sup>ρ</sup>* <sup>&</sup>gt; 0 such that

$$\|f(t, \mathbf{x})\| \le \psi(t) + \rho t^{1-\alpha} \|\mathbf{x}\|\_{\prime} \quad t \in J\_{\prime} \ \mathbf{x} \in X.$$

(*A*5) *g* : *C*1−*α*(*J*, *X*) → *X* is continuous and there exists a constant *M*<sup>2</sup> > 0 such that *g*(*x*) ≤ *M*<sup>2</sup> for any *x* ∈ *C*1−*α*(*J*, *X*).

then by Theorem 1 we can obtain the following existence theorem.

**Theorem 2.** *Let the assumptions* (*A*1)*–*(*A*3),(*A*4) ,(*A*5) ,(*A*6) *and* (*A*7) *hold. Then the system* (1) *has at least one mild solution in C*1−*α*(*J*, *X*)*.*

**Proof.** It is clear that (*A*5) ⇒ (*A*5) and (*A*4) ⇒ (*A*4) with *φ*(·) = *ψ*(·) + *rρ*· <sup>1</sup>−*<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>X</sup>*) for any *<sup>r</sup>* <sup>&</sup>gt; 0 and *<sup>x</sup>* <sup>∈</sup> *Br*. Therefore, by Theorem 1 we can prove that the system (1) has a mild solution *x* ∈ *C*1−*α*(*J*, *X*).

Now, we state and prove the approximate controllability of the fractional control system (1).

**Theorem 3.** *Let the conditions* (*A*1)*–*(*A*3),(*A*4),(*A*5) *and* (*A*6) *be satisfied, where*

(*A*4) *f* : *J* × *X* → *X satisfies the following conditions.* (*i*) *For each x* ∈ *X*, *f*(·, *x*) : *J* → *X is strongly measurable, and for every t* ∈ *J*, *f*(·, *x*) :

*X* → *X is continuous.*

(*ii*) *There exist a function <sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>p*(*J*, <sup>R</sup>+) *with p<sup>α</sup>* <sup>&</sup>gt; <sup>1</sup> *such that*

$$\|f(t, \mathfrak{x})\| \le \varrho(t), \quad \forall t \in J, \ \mathfrak{x} \in X.$$

*In addition, the linear fractional control system* (4) *is approximately controllable on J. Then the fractional control system (1) is approximately controllable on J.*

**Proof.** It is clear that (*A*4) ⇒ (*A*4) and (*A*5) ⇒ (*A*5). By Lemma 2 we know that the condition (*H*7) holds. It follows from Theorem 1 that the system (1) has a mild solution *x* ∈ *C*1−*α*(*J*, *X*) for every > 0, which is expressed by

$$\begin{split} \mathbf{x}\_{\varepsilon}(t) &= \quad t^{a-1} T\_E(t) \left[ \mathbf{x}\_0 - \mathbf{g}(\mathbf{x}) \right] + \int\_0^t (t-s)^{a-1} T\_E(t-s) f(s, \mathbf{x}\_{\varepsilon}(s)) ds \\ &+ \quad \int\_0^t (t-s)^{a-1} T\_E(t-s) B B^\* T\_E^\*(b-s) R(\varepsilon, \Pi\_0^b) \left[ \mathbf{x}\_b - b^{a-1} T\_E(b) (\mathbf{x}\_0 - \mathbf{g}(\mathbf{x})) \right] ds \\ &- \quad \int\_0^b (b-\theta)^{a-1} T\_E(b-\theta) f(\theta, \mathbf{x}\_{\varepsilon}(\theta)) d\theta \big] ds. \end{split}$$

In view of *<sup>I</sup>* <sup>−</sup> <sup>Π</sup>*<sup>b</sup>* <sup>0</sup>(*<sup>I</sup>* + <sup>Π</sup>*<sup>b</sup>* <sup>0</sup>)−<sup>1</sup> = *R*(, <sup>Π</sup>*<sup>b</sup>* <sup>0</sup>), we have

$$\mathfrak{x}\_{\epsilon}(b) = \mathfrak{x}\_{b} - \epsilon R(\epsilon, \Pi\_{0}^{b}) p(\mathfrak{x}\_{\epsilon})\_{\prime}$$

where

$$p(\mathbf{x}\_{\varepsilon}) = \mathbf{x}\_{b} - b^{a-1} T\_E(b) (\mathbf{x}\_0 - \mathbf{g}(\mathbf{x}\_{\varepsilon})) - \int\_0^b (b-s)^{a-1} T\_E(b-s) f(s, \mathbf{x}\_{\varepsilon}(s)) ds.$$

By the assumption (*A*5) , we have

$$||b^{\alpha-1}(\mathfrak{x}\_0 - \mathfrak{g}(\mathfrak{x}\_\epsilon))|| \le b^{\alpha-1}(||\mathfrak{x}\_0|| + M\_2).$$

Then the set {*bα*−<sup>1</sup>*TE*(*b*)(*x*<sup>0</sup> <sup>−</sup> *<sup>g</sup>*(*x*))} is relatively compact since *TE*(*b*) is a compact operator. There exists a subsequence of {*bα*−<sup>1</sup>*TE*(*b*)(*x*<sup>0</sup> <sup>−</sup> *<sup>g</sup>*(*x*))}, still denoted by itself, and a function *g*∗ such that

$$b^{\mathfrak{a}-1}T\_E(b)(\mathfrak{x}\_0 - \mathfrak{g}(\mathfrak{x}\_\mathfrak{e})) \to \mathfrak{g}^\* \quad (\mathfrak{e} \to 0^+).$$

By means of (*A*4) we have

$$\|\|f(\cdot,\mathbf{x}\_{\varepsilon}(\cdot))\|\|\_{L^p} = \left(\int\_0^b \|\|f(\mathbf{s},\mathbf{x}\_{\varepsilon}(\mathbf{s}))\|\|^p d\mathbf{s}\right)^{\frac{1}{p}} \le \|\|\boldsymbol{\varrho}\|\|\_{L^p}.$$

Hence, the set { *<sup>f</sup>*(·, *<sup>x</sup>*(·))} is bounded in *<sup>L</sup>p*(*J*, *<sup>X</sup>*). So there is a subsequence, still denoted by { *<sup>f</sup>*(·, *<sup>x</sup>*(·))}, converges weakly to some *<sup>f</sup>* <sup>∗</sup>(·) <sup>∈</sup> *<sup>L</sup>p*(*J*, *<sup>X</sup>*), that is,

$$f(\mathbf{s}, \mathfrak{x}\_{\varepsilon}(\mathbf{s})) \stackrel{w}{\longrightarrow} f^\*(\mathbf{s}), \quad a.e. \,\mathbf{s} \in \mathcal{J}$$

as → 0. By Lemma 3 and the Lebesgue dominated convergence theorem, we can obtain

$$\int\_0^b (b-s)^{a-1} T\_E(b-s) f(s, x\_\varepsilon(s)) ds \to \int\_0^b (b-s)^{a-1} T\_E(b-s) f^\*(s) ds$$

as → 0. Denote by

$$h = \mathfrak{x}\_b - \mathfrak{g}^\* - \int\_0^b (b - s)^{a - 1} T\_E(b - s) f^\*(s) ds.$$

Then by the definition of *p*(*x*), we obtain that

$$p(\mathfrak{x}\_{\varepsilon}) \to h \text{ ( $\varepsilon \to 0$ )}.$$

Consequently, we have

$$\begin{array}{rcl} \left||\boldsymbol{\chi\_{\varepsilon}}(\boldsymbol{b}) - \boldsymbol{\chi\_{b}}\right|| &=& \left||\boldsymbol{\varepsilon}\mathcal{R}(\boldsymbol{\varepsilon}, \boldsymbol{\Pi}\_{0}^{b})\boldsymbol{p}(\boldsymbol{x\_{\varepsilon}})\right|| \\ &=& \left||\boldsymbol{\varepsilon}\mathcal{R}(\boldsymbol{\varepsilon}, \boldsymbol{\Pi}\_{0}^{b})(\boldsymbol{p}(\boldsymbol{x\_{\varepsilon}}) - \boldsymbol{h})\right|| + \left|\boldsymbol{\varepsilon}\mathcal{R}(\boldsymbol{\varepsilon}, \boldsymbol{\Pi}\_{0}^{b})\boldsymbol{h}\right|| \\ &\to & 0 \quad (\boldsymbol{\varepsilon} \to \boldsymbol{0}). \end{array}$$

By Definition 2, the fractional control system (1) is approximately controllable on *J*.

#### **4. An Example**

Consider the Sobolev-type partial differential equation with Riemann-Liouville fractional derivatives

$$\begin{cases} \, ^L D\_t^{\frac{3}{2}}[(I - \frac{\partial^2}{\partial y^2})x(t, y)] = \frac{\partial^2}{\partial y^2}x(t, y) + \frac{e^{-3t}\sqrt{\sin x(t, y)}}{3 + |x(t, y)|} + u(t), \quad (t, y) \in (0, 1] \times [0, \pi], \\\ x(t, 0) = x(t, \pi) = 0, \quad t \in [0, 1], \\\ \left. I\_{0^+}^{1 - a}[(I - \frac{\partial^2}{\partial y^2})x(t, y)]|\_{t = 0} + \sum\_{i = 1}^m c\_i \sqrt[3]{\sin(t^{1 - a}x(t, y)) + 7} \right. \\\ \left. \left. \frac{1}{2}x(t, \frac{1}{2}x(t, \frac{1}{2}x(t, y)) + 7} \right] \right|\_{t = 0} = x\_0(y), \end{cases} (7)$$

where *ci* > 0, *i* = 1, 2, ··· , *m* are given positive constants.

Let *<sup>X</sup>* <sup>=</sup> *<sup>U</sup>* :<sup>=</sup> *<sup>L</sup>*2[0, *<sup>π</sup>*]. Denote *<sup>D</sup>*(*A*) = *<sup>D</sup>*(*E*) :<sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>* : *<sup>x</sup>*, *<sup>x</sup>* are absolutely continuous, *x* ∈ *X* and *x*(*t*, 0) = *x*(*t*, *π*) = 0}. We define two operators *A* : *D*(*A*) ⊂ *X* → *X* and *E* : *D*(*E*) ⊂ *X* → *X* by

$$A\mathfrak{x} = \frac{\partial^2}{\partial y^2}\mathfrak{x}, \quad \mathfrak{x} \in D(A); \quad E\mathfrak{x} = (I - \frac{\partial^2}{\partial y^2})\mathfrak{x}, \quad \mathfrak{x} \in D(E).$$

Let *en*(*y*) = <sup>6</sup> <sup>2</sup> *<sup>π</sup>* sin *ny*, *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> be the orthonormal set of eigenvectors of *<sup>A</sup>*. By [4,16], we have

*Ax* <sup>=</sup> <sup>−</sup>Σ<sup>∞</sup> *<sup>n</sup>*=1*n*2*x*,*enen*, *<sup>x</sup>* <sup>∈</sup> *<sup>D</sup>*(*A*)

and

$$E\mathfrak{x} = \Sigma\_{n=1}^{\infty} (1 + n^2) \langle \mathfrak{x}, e\_n \rangle e\_{n\prime} \quad \mathfrak{x} \in D(E).$$

This implies, for any *x* ∈ *H*, that

$$E^{-1}\mathfrak{x} = \Sigma\_{n=1}^{\infty} \frac{1}{1+n^2} \langle \mathfrak{x}, e\_n \rangle e\_{n\prime}$$

$$AE^{-1}\mathfrak{x} = \Sigma\_{n=1}^{\infty} \frac{-n^2}{1+n^2} \langle \mathfrak{x}, e\_n \rangle e\_n$$

and

$$T(t)\mathfrak{x} = \Sigma\_{n=1}^{\infty} e^{\frac{-n^2}{1+n^2}t} \langle \mathfrak{x}, e\_n \rangle e\_{n\nu}$$

where *T*(*t*)*x* = *eAE*−1*<sup>t</sup> <sup>x</sup>*, *<sup>t</sup>* <sup>≥</sup> 0. Then *<sup>E</sup>*−<sup>1</sup> is a linear operator which is compact and *<sup>E</sup>*−<sup>1</sup> ≤ 1. Hence,

$$T\_E(t) = \frac{3}{4} \int\_0^\infty E^{-1} \theta \xi\_{\frac{3}{4}}(\theta) T(t^{\frac{3}{4}}\theta) d\theta$$

with

$$||T\_E(t)\mathfrak{x}|| \le \frac{1}{\Gamma(\frac{3}{4})}||\mathfrak{x}||\_{\prime}$$

where

$$\xi\_{\frac{3}{4}}(\theta) = \frac{1}{\pi} \sum\_{n=1}^{\infty} (-1)^{n-1} \theta^{-\frac{3}{4}n-1} \frac{\Gamma(\frac{3}{4}n+1)}{n!} \sin(\frac{3}{4}n\pi), \; \theta \in (0, +\infty).$$

Let *x*(*t*)(*y*) = *x*(*t*, *y*). Denote

$$(f(t, \mathfrak{x}(t))(y) = \frac{e^{-3t} \sqrt{\sin \mathfrak{x}(t, y)}}{3 + |\mathfrak{x}(t, y)|}$$

and

$$g(\boldsymbol{x})(\boldsymbol{y}) = \sum\_{i=1}^{m} c\_i \sqrt[3]{\sin(t^{1-\alpha}\boldsymbol{x}(t,\boldsymbol{y})) + 7}.$$

Then the problem (7) can be rewritten as the abstract control system (1). Moreover, the assumptions (*A*1)–(*A*6) are fulfilled with *f*(*t*, *x*) *<sup>X</sup>* = <sup>1</sup> <sup>3</sup> and *g*(*x*) *<sup>X</sup>* <sup>≤</sup> <sup>2</sup> *<sup>m</sup>* ∑ *i*=1 *ci*. If the linear system corresponding to (7) is approximately controllable on [0, 1], then by Theorem 3, the fractional partial differential equation of (7) is approximately controllable on [0, 1].

#### **5. Conclusions**

In this paper, with the aid of the compactness of the operator *E*−1, we prove the existence of mild solutions of the fractional evolution system (1) without the compactness of operator semigroup. The Lipschitz continuity and the compactness of the nonlocal function *g* are not needed in our main results. Under the assumption that the associate linear control system (4) is approximately controllable, the approximate controllability of the fractional evolution system (1) is also studied.

**Funding:** The research is supported by the National Natural Science Function of China (No. 11701457).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

#### **Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


## *Article* **New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators**

**Daliang Zhao \* and Yongyang Liu**

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China; 2020020448@stu.sdnu.edu.cn

**\*** Correspondence: dlzhao928@sdnu.edu.cn

**Abstract:** This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of *C*0-semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results.

**Keywords:** approximate controllability; resolvent operator; delay; nonlocal conditions; parameter

#### **1. Introduction**

This manuscript mainly investigates the sufficient conditions of the approximate controllability of some fractional control systems as below:

$$\begin{cases} \ ^{\mathbb{C}}D^{\beta}\mathbf{x}(t) = A\mathbf{x}(t) + f(t, \mathbf{x}\_{l}) + Bu(t), \ t \in I := [0, a], \\\ \mathbf{x}(t) = \phi(t), \ t \in [-b, 0], \end{cases} \tag{1}$$

and

$$\begin{cases} \, \, ^C D^\beta \mathbf{x}(t) = A \mathbf{x}(t) + f(t, \mathbf{x}\_l) + Bu(t), \; t \in I := [0, a], \\\, \mathbf{x}(t) + \lambda \mathbf{g}\_l(\mathbf{x}) = \boldsymbol{\phi}(t), \; t \in [-b, 0], \end{cases} \tag{2}$$

where *CD<sup>β</sup>* means the Caputo derivative with order <sup>1</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>β</sup>* <sup>≤</sup> 1. *<sup>X</sup>* and *<sup>U</sup>* are Banach spaces. Linear operator *A* : D ⊂ *X* → *X* is unbounded with dense domain D. The delay term *xt* is explained in Equation (5). The control *<sup>u</sup>* takes values in *<sup>L</sup>*2(*I*; *<sup>U</sup>*). For any *<sup>t</sup>* <sup>∈</sup> [−*b*, 0], the non-local term *gt* : *C*([−*b*, *a*]; *X*) → *X* satisfies some given conditions. *λ* is a parameter. Let *<sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*). *<sup>B</sup>* : *<sup>L</sup>*2(*I*; *<sup>U</sup>*) <sup>→</sup> *<sup>L</sup>*2(*I*; <sup>D</sup>) is a bounded linear operator. *<sup>f</sup>* is a non-linearity that will be specified later.

Fractional differential systems and evolution systems have been studied extensively owing to its widespread backgrounds of some scientific and engineering realms, such as signal processing, finance, anomalous diffusion phenomena, heat conduction, etc. We refer readers to [1–4] for further detailed information. On the other side, controllability has gained a lot of importance and interest, and it plays a significant role in the description of various dynamical problems [5–8]. It is known to all that the fractional evolution system is closely related to time. In this regard, it has something in common with the controllability problem. Therefore, the controllability of some kinds of fractional evolution systems has become an important research hotspot. For example, exact controllability and approximate controllability are two mainstream research directions and they have important differences from the viewpoint of mathematics. Exact controllability can steer the control system to

**Citation:** Zhao, D.; Liu, Y. New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators. *Fractal Fract.* **2022**, *6*, 424. https://doi.org/10.3390/ fractalfract6080424

Academic Editor: Rodica Luca

Received: 7 July 2022 Accepted: 28 July 2022 Published: 30 July 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

any given final time point. The control operator is usually assumed to be reversible. Then, the controllability problem is transformed into a fixed point problem [9–13]. Furthermore, an induced inverse of the control operator is not necessarily true in infinite-dimensional space. In consideration of these strong assumptions, more and more scholars begin to study the approximate controllability in various abstract spaces, which means that it can steer the control system to an any small neighborhood of final time point [14–20]. In addition, controllability of fractional evolution systems also has important applications in the research areas of logical control networks or Boolean networks.

For instance, S. Ji [16] and F. Ge et al. [17] studied the approximate controllability of fractional semi-linear non-local evolution systems and fractional differential systems with impulsive conditions via approximating method under the assumption that *A* generated a *C*0-semigroup, respectively. Moreover, the approximate controllability of some other fractional systems, such as stochastic equations, neutral equations, etc., have also been deeply investigated (one can see [18–20] for more details). However, approximate controllability of the linear systems correspondence to the considered systems is necessary in this method. Therefore, some other approaches, such as the iterative method, are used to solve the approximate controllability problems for some evolution systems. For example, H. Zhou [21] obtained a sufficient condition of the approximate controllability for certain first-order evolution equations by utilizing iterative approach and the theory of strongly continuous semigroup. Authors in [22] dealt with the approximate controllability of some evolution systems with fractional order without delay by using iterative method. The properties of *C*0-semigroup are also included. By applying the same method, [23] also derived some appropriate controllability conclusions for some fractional differential equations with no delay effects.

It is noted that the results of approximate controllability discussed above are based on the *C*0-semigroup together with some associated characteristic solution operators [24]. However, in many cases, infinitesimal generator *A* may not be able to generate a *C*0 semigroup, but it can generate a resolvent operator instead [25]. On the other side, a resolvent operator can degenerate into a *C*0-semigroup when the integral kernel is equal to 1, that is, a resolvent operator covers a *C*0-semigroup as a special case. Of course, this can also be explained by the subordinate principle [26].

In comparision with results in [27] considering the influence of delay, we shall study the approximate controllability for some fractional control systems on the supposition that *A* is an infinitesimal generator of a differentiable resolvent operator rather than a *C*0-semigroup; we shall consider a control problem with variable delay, not fixed delay by contrast; the function *φ*(*t*) is supposed to be integrable rather than continuous. Under these generalized conditions, the difficulty mainly lies in how to overcome the obstacles caused by the variable delay and how to make use of the differentiability of resolvent operators. We solve this problem by means of a new special complete space we introduced and the theory of differentiable resolvent operator developed in [25].

Motivated by the aforementioned discussions, we shall establish a set of new approximate controllability results for systems (1) and (2) by using iterative method. As far as we know, the approximate controllability for the fractional evolution equations with finite variable delay and with non-local conditions and a parameter under the hypothesis that *A* generate a differentiable resolvent operator is still an untreated topic in the existing literature. Therefore, it is necessary to make further investigations to fill the gap in this regard.

Summarily, different from the above discussed papers, some highlights of the manuscript are presented as follows. (i) The approximate controllability of considered systems is studied on the supposition that the resolvent operator is differentiable, rather than utilizing the theory of *C*0-semigroup together with the properties of associated characteristic solution operators; (ii) The delay-induced-difficulty is overcome by introducing a special complete integrable space since we generalize the delay term from continuity to integrability compared with some other papers; (iii) The system (2) discussed in this manuscript is

provided with some more generalized nonlocal conditions compared with many related papers [5,9,11,16,17] (*λ* = 1, *t* = 0).

This manuscript is arranged as below. In the next part, we include some necessary preparations for the main controllability results. InSection 3, some existence results of the mild solution of the considered systems are obtained. In Section 4, we investigate the approximate controllability for the fractional delay control systems, and the case with non-local conditions and a parameter is discussed in Section 5. An instance is proposed in Section 6 to illustrate our abstract conclusions.

#### **2. Preparations**

Let *X* be a Banach space with norm *x* , *x* ∈ *X*. The linear operator *A* : D ⊂ *X* → *X* is closed and unbounded, in which D means the domain of *A* equipped with graph norm *x* <sup>D</sup> = *x* + *Ax* . *C*(*I*; *X*) stands for the space with all the continuous functions mapping *I* into *X* equipped with the sup-norm *x <sup>C</sup>*, *L*2(*I*; *X*) stands for the space of all Bochner integrable functions mapping *I* into *X* equipped with the norm *x <sup>L</sup>*2(*I*;*X*) = *<sup>a</sup>* <sup>2</sup>*dt*1/2 , and *Cβ*(*I*; *X*) denotes the space of all the *β*-Hölder continuous func-

0 *x*(*t*) tions mapping *I* into *X* provided with the norm *x <sup>C</sup>β*(*I*;*X*) = *x <sup>C</sup>*(*I*;*X*) + [|*x*|] *<sup>C</sup>β*(*I*;*X*), where

$$\|\|\mathbf{x}\|\|\_{\mathcal{C}^{\beta}(I;\mathcal{X})} = \sup\_{t,\mathbf{s}\in I, t\neq\mathbf{s}} \frac{\|\|\mathbf{x}(t) - \mathbf{x}(s)\|\|}{(t-s)^{\beta}}.$$

In the next discussion, the following equation

$$\mathbf{x}(t) = \frac{1}{\Gamma(\beta)} \int\_0^t \frac{A\mathbf{x}(s)}{(t-s)^{1-\beta}} ds, \ t \ge 0,\tag{3}$$

is assumed to possess an resolvent operator {R(*t*)}*t*≥<sup>0</sup> on *X*.

**Definition 1** ([28])**.** *The fractional integral of order β* > 0 *with the lower limit zero is written as*

$$I\_{0^{+}}^{\\\\\beta} \mathfrak{x}(t) = \frac{1}{\Gamma(\beta)} \int\_{0}^{t} (t-s)^{\beta-1} \mathfrak{x}(s) ds, \ t > 0,$$

*where* Γ *denotes the Gamma function.*

**Definition 2** ([28])**.** *The fractional derivative of the function <sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*((0, <sup>+</sup>∞); <sup>R</sup>) *in the Caputo sense can be defined by*

$${}^{C}D\_{0^{+}}^{\\\\\beta}x(t) = \frac{1}{\Gamma(n-\beta)} \int\_{0}^{t} \frac{x^{(n)}(s)}{(t-s)^{\beta-n+1}}ds, \; t > 0, \; t$$

*where n* = [*β*] + 1, [*β*] *represents the integer part of the positive constant β.*

**Definition 3** ([25])**.** *Suppose a set of operators* {R(*t*)}*t*≥<sup>0</sup> *to be bounded and linear on space X. If it fulfills hypotheses as below:*

*(i)* <sup>R</sup>(*t*) *is strongly continuous on* <sup>R</sup><sup>+</sup> *and* <sup>R</sup>(0) = <sup>I</sup>*; (ii)* R(*t*)D⊂D; *for each x* ∈ D, *t* ≥ 0, *it satisfies A*R(*t*)*x* = R(*t*)*Ax; (iii) The following equality can be established*

$$\Re(t)x = x + \frac{1}{\Gamma(\beta)} \int\_0^t \frac{Ax(s)}{(t-s)^{1-\beta}} ds \lambda$$

then we define it as a resolvent operator of Equation (3).

**Definition 4** ([25])**.** *A resolvent operator* R(*t*) *of Equation (3) is known as differentiable, if it satisfies* <sup>R</sup>(·)*<sup>x</sup>* <sup>∈</sup> *<sup>W</sup>*1,1 *loc* (R+; *<sup>X</sup>*)*,* <sup>∀</sup>*<sup>x</sup>* ∈ D*. In addition, for* <sup>∀</sup>*<sup>x</sup>* ∈ D*, there exists a function <sup>ω</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*(R+) *satisfying*

$$\|\dot{\Re}(t)\mathbf{x}\| \le \omega(t) \|\mathbf{x}\|\_{\mathcal{D}} \text{ a.e. on } \mathbb{R}\_+. $$

Consider the following equality

$$w(t) = w(t) + \frac{1}{\Gamma(\beta)} \int\_0^t \frac{A\mathbf{x}(s)}{(t-s)^{1-\beta}} ds, \ t \in I,\tag{4}$$

where *<sup>w</sup>* <sup>∈</sup> *<sup>L</sup>*1(*I*; *<sup>X</sup>*).

**Definition 5** ([25])**.** *A function x* ∈ *C*(*I*; *X*) *is said to be a mild solution of equality Equation (4) if it satisfies <sup>t</sup>* 0 *x*(*s*) (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)1−*<sup>β</sup> ds* ∈ D *and*

$$\varkappa(t) = w(t) + \frac{1}{\Gamma(\beta)} A \int\_0^t \frac{\varkappa(s)}{(t-s)^{1-\beta}} ds, \; \forall t \in I.$$

The following result provides another equivalent form of mild solution for Equation (4).

**Lemma 1** ([25])**.** *If the resolvent operator* R(*t*) *of Equation (4) is differentiable, then for w* ∈ *C*(*I*; D)*, the following function*

$$\mathbf{x}(t) = \int\_0^t \dot{\Re}(t - s) w(s) ds + w(t), \ t \in I\_\nu$$

*is called a mild solution of Equation (4).*

To end this section, the set *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*) is proposed which stands for a space of all the integrable functions mapping [−*b*, 0] into *X* equipped with norm · *<sup>L</sup>*1[−*b*,0] <sup>=</sup> <sup>0</sup> −*b* · (*t*) *dt*. Obviously, it is complete. Considering Equation (1), for any *x* ∈ *C*(*I*; *X*), *t* ∈ *I*, let

$$\chi\_t(\theta) = \begin{cases} \ x(t+\theta), & t+\theta \ge 0, \\\ \phi(t+\theta), & t+\theta \le 0, \end{cases} \tag{5}$$

for any *θ* ∈ [−*b*, 0], where *φ*(*t*) denotes the function mentioned in Equation (1). Obviously, we can check that *xt* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*).

On the basis of Equation (5), we give the following result.

**Lemma 2.** *Assume that xn* → *x*<sup>0</sup> (*n* → +∞) *for xn*, *x*<sup>0</sup> ∈ *C*(*I*; *X*)*. Then, for any t* ∈ *I, one can derive that* (*xn*)*<sup>t</sup>* <sup>→</sup> (*x*0)*<sup>t</sup>* (*<sup>n</sup>* <sup>→</sup> <sup>+</sup>∞) *for* (*xn*)*t*,(*x*0)*<sup>t</sup>* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*)*.*

**Proof.** In view of (5), we can easily derive

$$\|(\mathfrak{x}\_n)\_t - (\mathfrak{x}\_0)\_t\|\_{L^1[-b,0]} = \begin{cases} \int\_0^t ||\mathfrak{x}\_n(s) - \mathfrak{x}\_0(s)|| ds, & t \le b, \\\int\_{t-b}^t ||\mathfrak{x}\_n(s) - \mathfrak{x}\_0(s)|| ds, & t \ge b, \end{cases}$$

which indicates that

$$\|\| (\mathbf{x}\_n)\_t - (\mathbf{x}\_0)\_t \|\|\_{L^1[-b,0]} \le b \|\mathbf{x}\_n - \mathbf{x}\_0 \|\|\_{\mathbb{C}'} \tag{6}$$

for any *t* ∈ *I*.

#### **3. Existence Results**

This part establishes the existence results of mild solution of Equation (1). Now, assume resolvent operator {R(*t*)}*t*≥<sup>0</sup> to be differentiable. Let *ω<sup>A</sup>* be the function mentioned in Definition 4.

From Definition 1 and Definition 5, we can obtain

**Definition 6.** *For any <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), *a function <sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*(*I*; *<sup>X</sup>*) *is called a mild solution of Equation (1) on I, provided that*

$$\mathbf{x}(t) = \phi(0) + \frac{1}{\Gamma(\beta)} A \int\_0^t \frac{\mathbf{x}(s)}{(t-s)^{1-\beta}} ds + \frac{1}{\Gamma(\beta)} \int\_0^t \frac{f(s, \mathbf{x}\_s)}{(t-s)^{1-\beta}} ds + \frac{1}{\Gamma(\beta)} \int\_0^t \frac{Bu(s)}{(t-s)^{1-\beta}} ds \,\omega$$

*where <sup>t</sup>* 0 *x*(*s*) (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)1−*<sup>β</sup> ds* ∈ D*,* <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*, *and xs is defined by Equation (5).*

In the next content, we will need the following assumptions.

**Hypothesis 1** (**H1**)**.** *<sup>f</sup> is a continuous function from <sup>I</sup>* <sup>×</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*) *into* <sup>D</sup> *and <sup>φ</sup>*(0) ∈ D*. There is a real number β*<sup>1</sup> ∈ (0, *β*) *and a function m* ∈ *L* 1 *<sup>β</sup>*<sup>1</sup> (*I*; <sup>R</sup>+) *satisfying f*(*t*, *x*) <sup>D</sup> ≤ *<sup>m</sup>*(*t*) *for any t* <sup>∈</sup> *I and x* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*)*.*

**Hypothesis 2** (**H2**)**.** *For any x*, *<sup>y</sup>* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*), *there exists a constant L* <sup>&</sup>gt; <sup>0</sup> *satisfying*

$$||f(t,x) - f(t,y)||\_{\mathcal{D}} \le L||x - y||\_{L^1[-b,0]}.$$

*For simplicity, we denote*

$$\mathfrak{F}\_x(t) = \frac{1}{\Gamma(\beta)} \int\_0^t \frac{f(s, \chi\_s)}{(t - s)^{1 - \beta}} ds, \; \mathfrak{B}\_\mu(t) = \frac{1}{\Gamma(\beta)} \int\_0^t \frac{Bu(s)}{(t - s)^{1 - \beta}} ds, \; \vartheta = \frac{\beta - 1}{1 - \beta\_1}.$$

*From Lemma 1 and Definition 6, we can derive the mild solution of Equation (1) on I of another expression as follows.*

**Definition 7.** *For any <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), *a function <sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*(*I*; *<sup>X</sup>*) *is called a mild solution of Equation (1) on I, provided that*

$$\mathbf{x}(t) = \boldsymbol{\phi}(0) + \mathfrak{F}\_{\boldsymbol{x}}(t) + \mathfrak{B}\_{\boldsymbol{u}}(t) + \int\_{0}^{t} \dot{\mathfrak{R}}(t-s)(\boldsymbol{\phi}(0) + \mathfrak{F}\_{\boldsymbol{x}}(s) + \mathfrak{B}\_{\boldsymbol{u}}(s))ds.$$

**Remark 1.** *It follows from Definition 1 that the classical solution of system Equation (1) is a convolution equation. Hence, it is natural to apply Laplace transform on it to express an appropriate formula for the mild solution representation of the considered system. For this purpose, we suppose that resolvent operator* R(*t*) *is exponentially bounded. By utilizing the theory of the Laplace transform and inverse Laplace transform, the mild solution of Equation (1) could be defined by*

$$\mathbf{x}(t) = \begin{cases} \Re(t)\phi(0) + \int\_0^t \mathcal{K}\_\odot(t-s)f(s, \mathbf{x}\_s)ds + \int\_0^t \mathcal{K}\_\odot(t-s)Bu(s)ds, & t \in I = [0, a]\_\prime, \\\ \phi(t), \quad t \in [-b, 0]\_\prime \end{cases}$$

*where* K(*t*) = *<sup>d</sup> dt*(*<sup>I</sup> β* <sup>0</sup><sup>+</sup>R(*t*)) *and xs is defined by Equation (5).*

**Lemma 3.** *(i) If hypothesis (H1) holds, then for arbitrarily given x* ∈ *C*(*I*; *X*)*, we have* F*<sup>x</sup>* ∈ *<sup>C</sup>β*−*β*<sup>1</sup> (*I*; <sup>D</sup>), *and*

$$\mathbb{E}\left[|\mathfrak{F}\_{\boldsymbol{x}}|\right]\_{\mathcal{C}^{\beta-\beta\_{1}}(I;\mathcal{D})} \leq \frac{2||m||\_{L^{\frac{1}{\beta\_{1}}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_{1}}}.$$

*(ii) For any u* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), *we have* <sup>B</sup>*<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>β*<sup>−</sup> <sup>1</sup> <sup>2</sup> (*I*; D), *and*

$$\left[\left|\mathfrak{B}\_{\boldsymbol{u}}\right|\right]\_{\mathcal{C}^{\boldsymbol{\beta}-\frac{1}{2}}\left(I;\mathcal{D}\right)} \leq \frac{2\left\|\boldsymbol{B}\boldsymbol{u}\right\|\_{L^{2}\left(I;\mathcal{D}\right)}}{\Gamma\left(\boldsymbol{\beta}\right)\left(2\boldsymbol{\beta}-1\right)^{\frac{1}{2}}}.$$

**Proof.** (i) For arbitrarily given *x* ∈ *C*(*I*; *X*), ∀*t* ∈ [0, *a*), ∀*h* > 0 satisfying *t* + *h* ∈ [0, *a*], by using Hölder inequality, one can derive

 F*x*(*t* + *h*) − F*x*(*t*) D ≤ 1 Γ(*β*) *<sup>t</sup>* 0 [(*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> <sup>−</sup> (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−1] *f*(*s*, *xs*) <sup>D</sup>*ds* + 1 Γ(*β*) *<sup>t</sup>*+*<sup>h</sup> t* (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> *f*(*s*, *xs*) <sup>D</sup>*ds* ≤ 1 Γ(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> <sup>−</sup> (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> <sup>1</sup> <sup>1</sup>−*β*<sup>1</sup> *ds*1−*β*<sup>1</sup> *m L* 1 *β*1 + 1 Γ(*β*) *<sup>t</sup>*+*<sup>h</sup> t* (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> <sup>1</sup> <sup>1</sup>−*β*<sup>1</sup> *ds*1−*β*<sup>1</sup> *m L* 1 *β*1 ≤ 1 Γ(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*<sup>ϑ</sup>* <sup>−</sup> (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>* <sup>−</sup> *<sup>s</sup>*)*<sup>ϑ</sup> ds*1−*β*<sup>1</sup> *m L* 1 *β*1 + 1 Γ(*β*) *<sup>t</sup>*+*<sup>h</sup> t* (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>* <sup>−</sup> *<sup>s</sup>*)*ϑds*1−*β*<sup>1</sup> *m L* 1 *β*1 ≤ *m L* 1 *β*1 Γ(*β*)(1 + *ϑ*)1−*β*<sup>1</sup> (*t* <sup>1</sup>+*<sup>ϑ</sup>* <sup>−</sup> (*<sup>t</sup>* <sup>+</sup> *<sup>h</sup>*)1+*<sup>ϑ</sup>* <sup>+</sup> *<sup>h</sup>*1+*ϑ*)1−*β*<sup>1</sup> <sup>+</sup> *m L* 1 *β*1 Γ(*β*)(1 + *ϑ*)1−*β*<sup>1</sup> *h*(1+*ϑ*)(1−*β*1) ≤ 2 *m L* 1 *β*1 Γ(*β*)(1 + *ϑ*)1−*β*<sup>1</sup> *h*(1+*ϑ*)(1−*β*1) = 2 *m L* 1 *β*1 Γ(*β*)(1 + *ϑ*)1−*β*<sup>1</sup> *hβ*−*β*<sup>1</sup> ,

which indicates that [|F*x*|] *<sup>C</sup>β*−*β*<sup>1</sup> (*I*;D) ≤ 2 *m L* 1 *β*1 <sup>Γ</sup>(*β*)(<sup>1</sup> <sup>+</sup> *<sup>ϑ</sup>*)1−*β*<sup>1</sup> and <sup>F</sup>*<sup>x</sup>* <sup>∈</sup> *<sup>C</sup>β*−*β*<sup>1</sup> (*I*; <sup>D</sup>). (ii) In the light of the proof for (i), it can be obtained similarly.

**Lemma 4.** *(i) If Hypotheses (H1) and (H2) hold, then for* ∀*x*, *y* ∈ *C*(*I*; *X*)*,*

$$\|\|\mathfrak{F}\_{\mathbf{x}}(t) - \mathfrak{F}\_{\mathcal{Y}}(t)\|\|\_{\mathcal{D}} \le \frac{La^{\beta}b}{\Gamma(\beta + 1)} \|\mathbf{x} - \mathbf{y}\|\_{\mathcal{C}'} \,\forall t \in I\_{\prime}$$

*and*

$$\|\|\mathfrak{F}\_{\mathbf{x}}(t)\|\|\_{\mathcal{D}} \le \frac{a^{\beta-\beta\_1}||m||\_{\frac{1}{L^{\frac{1}{\beta\_1}}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_1}}, \forall t \in I.$$

*(ii) For any u*, *<sup>v</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*)*,*

$$\|\mathfrak{B}\_{\mathfrak{U}}(t) - \mathfrak{B}\_{\upsilon}(t)\|\_{\mathcal{D}} \le \frac{1}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta - 1}}{2\beta - 1}} \|Bu - Bv\|\_{L^2(I; \mathcal{D})'} \,\,\forall t \in I, \ell$$

*and*

$$\|\mathfrak{B}\_{\mathfrak{u}}(t)\|\_{\mathcal{D}} \le \frac{1}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta-1}}{2\beta-1}} \|Bu\|\_{L^2(I;\mathcal{D})'} \,\,\forall t \in I.$$

**Proof.** (i) In view of Lemma 2, we can obtain

$$\begin{array}{rcl} \|\mathfrak{F}\_{\mathbf{x}}(t) - \mathfrak{F}\_{\mathbf{y}}(t)\|\|\_{\mathcal{D}} & \leq & \frac{1}{\Gamma(\beta)} \int\_{0}^{t} (t-s)^{\beta-1} \|f(s, \mathbf{x}\_{s}) - f(s, \mathbf{y}\_{s})\|\|\_{\mathcal{D}} ds \\ & \leq & \frac{L}{\Gamma(\beta)} \int\_{0}^{t} (t-s)^{\beta-1} \|\mathbf{x}\_{s} - \mathbf{y}\_{s}\|\_{L[-b,0]} ds \\ & \leq & \frac{Lb}{\Gamma(\beta)} \int\_{0}^{t} (t-s)^{\beta-1} \|\mathbf{x} - \mathbf{y}\|\_{\mathcal{C}} ds \\ & = & \frac{La^{\beta}b}{\Gamma(\beta+1)} \|\mathbf{x} - \mathbf{y}\|\_{\mathcal{C}} \quad \forall t \in I. \end{array}$$

In addition,

$$\begin{array}{llll} \|\mathfrak{F}\_{\mathbf{x}}(t)\|\!|\mathcal{D} &\leq&\frac{1}{\Gamma(\beta)}\int\_{0}^{t}(t-s)^{\beta-1}\|f(s,\mathbf{x}\_{s})\|\!|\mathcal{D}ds\\ &\leq&\frac{1}{\Gamma(\beta)}\left(\int\_{0}^{t}[(t-s)^{\beta-1}]^{\frac{1}{1-\beta\_{1}}}ds\right)^{1-\beta\_{1}}||m||\_{L^{\frac{1}{\beta\_{1}}}}\\ &\leq&\frac{t^{(1+\theta)(1-\beta\_{1})}}{\Gamma(\beta)(1+\theta)^{1-\beta\_{1}}}||m||\_{L^{\frac{1}{\beta\_{1}}}}\\ &\leq&\frac{a^{\beta-\beta\_{1}}||m||\_{L^{\frac{1}{\beta\_{1}}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_{1}}},\ \forall t\in I. \end{array}$$

(ii) Obviously, we can obtain that

$$\begin{split} \|\mathfrak{B}\_{\boldsymbol{u}}(t) - \mathfrak{B}\_{\boldsymbol{v}}(t)\|\_{\mathcal{D}} &\leq \quad \frac{1}{\Gamma(\boldsymbol{\beta})} \int\_{0}^{t} (t-s)^{\boldsymbol{\beta}-1} \|Bu(s) - B\boldsymbol{v}(s)\|\_{\mathcal{D}} ds \\ &\leq \quad \frac{1}{\Gamma(\boldsymbol{\beta})} \left( \int\_{0}^{t} [(t-s)^{\boldsymbol{\beta}-1}]^{2} ds \right)^{\frac{1}{2}} \|\mathfrak{B}\_{\boldsymbol{u}} - \mathfrak{B}\_{\boldsymbol{v}}\|\_{L^{2}(I;\mathcal{D})} \\ &= \quad \frac{1}{\Gamma(\boldsymbol{\beta})} \sqrt{\frac{a^{2\boldsymbol{\beta}-1}}{2\boldsymbol{\beta}-1}} \|\mathfrak{B}\_{\boldsymbol{u}} - \mathfrak{B}\_{\boldsymbol{v}}\|\_{L^{2}(I;\mathcal{D})^{\*}} \quad \forall t \in I. \end{split}$$

Similarly, we can obtain

$$\|\mathfrak{B}\_{\mathfrak{U}}(t)\|\_{\mathcal{D}} \le \frac{1}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta-1}}{2\beta-1}} \|Bu\|\_{L^2(I;\mathcal{D})'} \,\,\forall t \in I.$$

**Theorem 1.** *If the Hypotheses (H1) and (H2) hold, then for any given control <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), *fractional evolution system Equation (1) has an unique mild solution on I, provided that*

$$\frac{La^{\beta}b(1+||\omega\_{A}||\_{L^{1}(I)})}{\Gamma(\beta+1)} < 1. \tag{7}$$

**Proof.** In view of Definition 7, for any *t* ∈ *I*, define an operator Ψ : *C*(*I*; *X*) → *C*(*I*; *X*) as below

$$
\phi(\Psi \mathbf{x})(t) = \phi(0) + \mathfrak{F}\_{\mathbf{x}}(t) + \mathfrak{B}\_{\mathbf{u}}(t) + \int\_{0}^{t} \dot{\mathfrak{R}}(t-s)(\phi(0) + \mathfrak{F}\_{\mathbf{x}}(s) + \mathfrak{B}\_{\mathbf{u}}(s))ds.\tag{8}
$$

Evidently, we only need to consider the fixed point of Ψ.

**Step 1**. Ψ maps *C*(*I*; *X*) into *C*(*I*; *X*).

For every *x* ∈ *C*(*I*; *X*), 0 < *t* < *t* + *h* ≤ *a*, we have

$$\begin{array}{rcl} (\mathsf{F}\mathsf{x})(t+h) - (\mathsf{F}\mathsf{x})(t) &=& \mathfrak{F}\_{\mathsf{X}}(t+h) - \mathfrak{F}\_{\mathsf{X}}(t) + \mathfrak{B}\_{\mathsf{u}}(t+h) - \mathfrak{B}\_{\mathsf{u}}(t) \\ &+& \int\_{0}^{t+h} \mathfrak{R}(t+h-s)\phi(0)ds - \int\_{0}^{t} \mathfrak{R}(t-s)\phi(0)ds \\ &+& \int\_{0}^{t+h} \mathfrak{R}(t+h-s)\mathfrak{F}\_{\mathsf{X}}(s)ds - \int\_{0}^{t} \mathfrak{R}(t-s)\mathfrak{F}\_{\mathsf{X}}(s)ds \\ &+& \int\_{0}^{t+h} \mathfrak{R}(t+h-s)\mathfrak{B}\_{\mathsf{u}}(s)ds - \int\_{0}^{t} \mathfrak{R}(t-s)\mathfrak{B}\_{\mathsf{u}}(s)ds \\ &=& \sum\_{i=1}^{5} \mathsf{Y}\_{i\mathsf{u}} \end{array}$$

where

$$\begin{split} \mathsf{Y}\_{1} &= \mathsf{F}\_{\mathsf{X}}(t+h) - \mathsf{F}\_{\mathsf{X}}(t), \\ \mathsf{Y}\_{2} &= \mathsf{B}\_{\mathsf{u}}(t+h) - \mathsf{B}\_{\mathsf{u}}(t), \\ \mathsf{Y}\_{3} &= \int\_{0}^{t+h} \mathsf{R}(t+h-s)\phi(0)ds - \int\_{0}^{t} \mathsf{R}(t-s)\phi(0)ds, \\ \mathsf{Y}\_{4} &= \int\_{0}^{t+h} \mathsf{R}(t+h-s)\mathfrak{F}\_{\mathsf{x}}(s)ds - \int\_{0}^{t} \mathsf{R}(t-s)\mathfrak{F}\_{\mathsf{x}}(s)ds, \\ \mathsf{Y}\_{5} &= \int\_{0}^{t+h} \mathsf{R}(t+h-s)\mathfrak{B}\_{\mathsf{u}}(s)ds - \int\_{0}^{t} \mathsf{R}(t-s)\mathfrak{B}\_{\mathsf{u}}(s)ds. \\ \text{By Lemma 3, we can obtain} \end{split}$$

$$||\Upsilon\_1|| \le \frac{2||m||\_{L^{\frac{1}{\beta\_1}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_1}} h^{\beta-\beta\_1} \to 0, \text{ as } h \to 0,$$

and

$$\|\|\mathbf{Y}\_2\|\|\leq \frac{2||Bu||\_{L^2(I;\mathcal{D})}}{\Gamma(\beta)(2\beta-1)^{\frac{1}{2}}}h^{\beta-\frac{1}{2}}\to 0,\text{ as }h\to 0.$$

Notice that

$$\begin{aligned} \Upsilon\_3 &= \int\_0^h \dot{\Re}(t+h-s)\phi(0)ds + \int\_h^{t+h} \dot{\Re}(t+h-s)\phi(0)ds - \int\_0^t \dot{\Re}(t-s)\phi(0)ds\\ &= \int\_0^h \dot{\Re}(t+h-s)\phi(0)ds. \end{aligned}$$

Then, we have

$$\|\|\mathbf{Y}\_3\|\| \le \|\phi(\mathbf{0})\|\|\_{\mathcal{D}} \int\_0^h \omega\_A(t+h-s)ds \to 0, \text{ as } h \to 0.$$

In addition, since

$$\begin{array}{rcl} \Upsilon\_{4} &=& \int\_{0}^{h} \dot{\mathfrak{R}}(t+h-s)\mathfrak{F}\_{\boldsymbol{x}}(s)ds + \int\_{h}^{t+h} \dot{\mathfrak{R}}(t+h-s)\mathfrak{F}\_{\boldsymbol{x}}(s)ds - \int\_{0}^{t} \dot{\mathfrak{R}}(t-s)\mathfrak{F}\_{\boldsymbol{x}}(s)ds \\ &=& \int\_{0}^{h} \dot{\mathfrak{R}}(t+h-s)\mathfrak{F}\_{\boldsymbol{x}}(s)ds + \int\_{0}^{t} \dot{\mathfrak{R}}(s)\mathfrak{F}\_{\boldsymbol{x}}(t+h-s)ds - \int\_{0}^{t} \dot{\mathfrak{R}}(s)\mathfrak{F}\_{\boldsymbol{x}}(t-s)ds, \end{array}$$

we thus can derive from Definition 4, Lemma 3 and Lemma 4 that

$$\begin{split} \|\|\mathbf{Y}\_{4}\|\| &\leq \int\_{0}^{h} \|\dot{\Re}(t+h-s)\widetilde{\mathbf{x}}\_{x}(s)\|\| ds + \int\_{0}^{t} \|\dot{\Re}(s)(\widetilde{\mathbf{x}}\_{x}(t-s+h)-\widetilde{\mathbf{x}}\_{x}(t-s))\|\| ds \\ &\leq \int\_{0}^{h} \omega\_{A}(t+h-s) \|\|\widetilde{\mathbf{x}}\_{x}(s)\|\|\_{\mathscr{D}} ds + \int\_{0}^{t} \omega\_{A}(s) \|\|\widetilde{\mathbf{x}}\_{x}\|\|\_{\mathscr{C}^{\beta-\frac{1}{2}}(I,\mathscr{D})} h^{\beta-\widetilde{\rho}\_{1}} ds \\ &\leq \frac{\|\beta^{-\beta\_{1}}\|\|\mathbf{m}\|\|\_{\mathscr{L}^{\frac{1}{\beta\_{1}}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_{1}}} \int\_{0}^{h} \omega\_{A}(t+h-s) ds + \frac{2\|m\|\_{\mathscr{L}^{\frac{1}{\beta\_{1}}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_{1}}} \int\_{0}^{t} \omega\_{A}(s) ds \\ &\leq \frac{\|m\|\_{\mathscr{L}^{\frac{1}{\beta\_{1}}}}}{\Gamma(\beta)(1+\theta)^{1-\beta\_{1}}} \Big( a^{\beta-\beta\_{1}} \int\_{0}^{h} \omega\_{A}(t+h-s) ds + 2h^{\beta-\beta\_{1}} \|\omega\_{A}\|\_{\mathscr{L}^{1}(I)} \Big) \\ &\to 0, \text{ as } h \to 0. \end{split}$$

It is not difficult to have

$$\begin{split} \Upsilon\_{5} &= \int\_{0}^{h} \dot{\mathfrak{R}}(t+h-s)\mathfrak{B}\_{\mathfrak{u}}(s)ds + \int\_{h}^{t+h} \dot{\mathfrak{R}}(t+h-s)\mathfrak{B}\_{\mathfrak{u}}(s)ds - \int\_{0}^{t} \dot{\mathfrak{R}}(t-s)\mathfrak{B}\_{\mathfrak{u}}(s)ds \\ &= \int\_{0}^{h} \dot{\mathfrak{R}}(t+h-s)\mathfrak{B}\_{\mathfrak{u}}(s)ds + \int\_{0}^{t} \dot{\mathfrak{R}}(s)\mathfrak{B}\_{\mathfrak{u}}(t+h-s)ds - \int\_{0}^{t} \dot{\mathfrak{R}}(s)\mathfrak{B}\_{\mathfrak{u}}(t-s)ds, \end{split}$$

which together with Lemma 3 and Lemma 4 implies

$$\begin{split} \|\|\mathbf{Y}\_{5}\|\| &\leq \int\_{0}^{h} \|\dot{\mathfrak{R}}(t+h-s)\mathfrak{B}\_{u}(s)\|\| ds + \int\_{0}^{t} \|\dot{\mathfrak{R}}(s)(\mathfrak{B}\_{u}(t-s+h) - \mathfrak{B}\_{u}(t-s))\|\| ds \\ &\leq \int\_{0}^{h} \omega\_{A}(t+h-s) \|\|\mathfrak{B}\_{u}(s)\|\|\varphi ds + \int\_{0}^{t} \omega\_{A}(s) \|\|\mathfrak{B}\_{u}\|\|\_{\mathscr{C}^{\theta-\frac{1}{2}}(\varPi)} h^{\theta-\frac{1}{2}} ds \\ &\leq \frac{1}{\Gamma(\theta)} \sqrt{\frac{a^{2\theta-1}}{2\theta-1}} \|\|Bu\|\_{L^{2}(\varPi)} \int\_{0}^{h} \omega\_{A}(t+h-s) ds \\ &+ \frac{2\|\|Bu\|\|\_{L^{2}(\varPi)}\theta^{1-\frac{1}{2}}}{\Gamma(\theta)(2\theta-1)^{\frac{1}{2}}} \int\_{0}^{t} \omega\_{A}(s) ds \\ &\leq \frac{\|\|Bu\|\|\_{L^{2}(\varPi)}}{\Gamma(\theta)(2\theta-1)^{\frac{1}{2}}} \left(a^{\theta-\frac{1}{2}} \int\_{0}^{h} \omega\_{A}(t+h-s) ds + 2h^{\theta-\frac{1}{2}} \|\omega\_{A}\|\_{L^{1}(I)}\right) \\ &\to 0, \text{ as } h \to 0. \end{split}$$

Hence, (Ψ*x*)(*t* + *h*) − (Ψ*x*)(*t*) → 0, *h* → 0, which indicates that Ψ*x* ∈ *C*(*I*; *X*), ∀*x* ∈ *C*(*I*; *X*).

**Step 2.** Ψ is contractive on *C*(*I*; *X*). In fact, Lemma 2 indicates that

 (Ψ*x*)(*t*) − (Ψ*y*)(*t*) ≤ F*x*(*t*) − F*y*(*t*) <sup>D</sup> + *<sup>t</sup>* 0 *ωA*(*t* − *s*) F*x*(*s*) − F*y*(*s*) <sup>D</sup>*ds* ≤ 1 Γ(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> *f*(*s*, *xs*) − *f*(*s*, *ys*) <sup>D</sup>*ds* + 1 Γ(*β*) *<sup>t</sup>* 0 *ωA*(*t* − *s*) *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>τ</sup>*)*β*−<sup>1</sup> *f*(*τ*, *xτ*) − *f*(*τ*, *yτ*) <sup>D</sup>*dτ ds* ≤ *L* Γ(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)*β*−<sup>1</sup> *xs* − *ys <sup>L</sup>*1[−*b*,0] *ds* + *L* Γ(*β*) *<sup>t</sup>* 0 *ωA*(*t* − *s*) *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>τ</sup>*)*β*−<sup>1</sup> *x<sup>τ</sup>* − *y<sup>τ</sup> <sup>L</sup>*1[−*b*,0] *dτ ds* ≤ *Laβb* Γ(*β* + 1) *x* − *y <sup>C</sup>* + *Laβ<sup>b</sup> ωA L*1(*I*) Γ(*β* + 1) *x* − *y C* <sup>=</sup> *Laβb*(<sup>1</sup> <sup>+</sup> *ωA <sup>L</sup>*1(*I*)) Γ(*β* + 1) *x* − *y <sup>C</sup>*, ∀*t* ∈ *I*,

which shows that

$$||\Psi\mathfrak{x} - \Psi\mathfrak{y}||\_{\mathbb{C}} \le \frac{La^{\beta}b(1 + ||\omega\_A||\_{L^1(I)})}{\Gamma(\beta + 1)}||\mathfrak{x} - \mathfrak{y}||\_{\mathbb{C}}.$$

Hence, Ψ is contractive on *C*(*I*; *X*) due to the Hypothesis (1). By utilizing the Banach's fixed point theorem, we find that Ψ has a unique fixed point on *C*(*I*; *X*).

#### **4. Main Results**

This part gives the results of approximate controllability of Equation (1). Let us show the next definitions which is critical to our work.

**Definition 8.** *The set <sup>K</sup>*(*a*, *<sup>f</sup>*) = {*x*(*a*; *<sup>u</sup>*) : *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*)} *is said to be the reachable set of Equation (1) at final point a, where x*(*t*; *u*) *is the state value of Equation (1) at time point t corresponding to control <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*). *If <sup>K</sup>*(*a*, *<sup>f</sup>*) = *<sup>X</sup>*, *we call that Equation (1) is approximately controllable on I, where K*(*a*, *f*) *stands for the closure of K*(*a*, *f*)*.*

Denote Nemytskii operator <sup>F</sup> : *<sup>C</sup>*(*I*; *<sup>X</sup>*) <sup>→</sup> *<sup>L</sup>*2(*I*; <sup>D</sup>) corresponding to the non-linearity *f* by

$$\mathcal{F}\mathfrak{x}(t) = f(t, \mathfrak{x}\_t), \ t \in I\_\nu$$

and define the continuous operator <sup>P</sup> : *<sup>L</sup>*2(*I*; <sup>D</sup>) <sup>→</sup> *<sup>X</sup>* by

$$\mathcal{P}y = \frac{1}{\Gamma(\beta)} \int\_0^a \frac{y(t)}{(a-t)^{1-\beta}} dt + \frac{1}{\Gamma(\beta)} \int\_0^a \dot{\Re}(a-t) \left( \int\_0^t \frac{y(s)}{(t-s)^{1-\beta}} ds \right) dt, \ y \in \mathcal{L}^2(I; \mathcal{D}). \tag{9}$$

It is not difficult to see that the approximate controllability of Equation (1) on *I* is equivalent to that the set *K*(*a*, *f*) is dense on *X*. That is to say, we can obtain an equivalent definition as below.

**Definition 9.** *System (1) is said to be approximately controllable on I, provided that for any ε* > 0 *and any final value <sup>ξ</sup>* <sup>∈</sup> *X, there exists a control term u<sup>ε</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*) *satisfying*

$$\left\| \left| \mathcal{J} - \mathfrak{R}(a)\phi(0) - \mathcal{P}(\mathcal{F}\mathfrak{x}\_{\varepsilon}) - \mathcal{P}(Bu\_{\varepsilon}) \right| \right\| < \varepsilon\_{\varepsilon} $$

*where xε*(*t*) = *<sup>x</sup>*(*t*; *<sup>u</sup>ε*) *is a mild solution of Equation (1) corresponding to u<sup>ε</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*)*.*

In addition, following hypotheses to obtain our approximate controllability results are presented.

**Hypothesis 3** (**H3**)**.** *For arbitrarily given <sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>ψ</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; <sup>D</sup>)*, there is a function <sup>u</sup>* <sup>∈</sup> *L*2(*I*; *U*) *satisfying*

$$||\mathcal{P}\psi - \mathcal{P}(Bu)|| < \varepsilon,$$

*and*

$$||Bu||\_{L^{2}(I;\mathcal{D})} < \mu ||\psi||\_{L^{2}(I;\mathcal{D})'} $$

*where μ* > 0 *is a real number independent of ψ.*

**Hypothesis 4** (**H4**)**.** *Under Equation (7), the following inequality holds*

$$\frac{1}{2}\mu \text{L}a^{\frac{1}{2}}b\left(1-\frac{\text{L}a^{\beta}b(1+||\omega\_{A}||\_{L^{1}(I)})}{\Gamma(\beta+1)}\right)^{-1}\frac{1+||\omega\_{A}||\_{L^{1}(I)}}{\Gamma(\beta)}\sqrt{\frac{a^{2\beta-1}}{2\beta-1}}<1.$$

*Next, to demonstrate our main result, we still need a lemma as below.*

**Lemma 5.** *If the Hypotheses (H1) and (H2) hold, then for any mild solutions of Equation (1), the following result holds*

$$\|\|\mathbf{x}\_1 - \mathbf{x}\_2\|\|\_{\mathbb{C}} \le \left(1 - \frac{La^{\theta}b(1 + \|\|\omega\_A\|\|\_{L^1(I)})}{\Gamma(\beta + 1)}\right)^{-1} \frac{1 + \|\|\omega\_A\|\|\_{L^1(I)}}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta - 1}}{2\beta - 1}} \|Bu\_1 - Bu\_2\|\|\_{L^2(I; \mathcal{D})'},$$

*for any u*1, *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*)*.*

**Proof.** The mild solution *xi*(*t*) = *x*(*t*; *ui*) (*i* = 1, 2) of system (1) corresponding to *ui* (*i* = 1, 2) satisfy

$$\mathbf{x}\_{i}(t) = \boldsymbol{\phi}(0) + \mathfrak{F}\_{\mathbf{x}\_{i}}(t) + \mathfrak{B}\_{\boldsymbol{u}\_{i}}(t) + \int\_{0}^{t} \dot{\mathfrak{R}}(t-s)(\boldsymbol{\phi}(0) + \mathfrak{F}\_{\mathbf{x}\_{i}}(s) + \mathfrak{B}\_{\boldsymbol{u}\_{i}}(s))ds, \ \forall t \in I.$$

From Lemma 4, one can obtain

 *x*1(*t*) − *x*2(*t*) ≤ F*x*<sup>1</sup> (*t*) − F*x*<sup>2</sup> (*t*) + B*u*<sup>1</sup> (*t*) − B*u*<sup>2</sup> (*t*) + *<sup>t</sup>* 0 <sup>R</sup>˙ (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)(F*x*<sup>1</sup> (*s*) <sup>−</sup> <sup>F</sup>*x*<sup>2</sup> (*s*)) *ds* + *<sup>t</sup>* 0 <sup>R</sup>˙ (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)(B*u*<sup>1</sup> (*t*) <sup>−</sup> <sup>B</sup>*u*<sup>2</sup> (*t*)) *ds* ≤ F*x*<sup>1</sup> (*t*) − F*x*<sup>2</sup> (*t*) <sup>D</sup> + B*u*<sup>1</sup> (*t*) − B*u*<sup>2</sup> (*t*) D + *<sup>t</sup>* 0 *ωA*(*t* − *s*) F*x*<sup>1</sup> (*s*) − F*x*<sup>2</sup> (*s*) <sup>D</sup>*ds* + *<sup>t</sup>* 0 *ωA*(*t* − *s*) B*u*<sup>1</sup> (*s*) − B*u*<sup>2</sup> (*s*) <sup>D</sup>*ds* ≤ *Laβb*(<sup>1</sup> <sup>+</sup> *ωA <sup>L</sup>*1(*I*)) Γ(*β* + 1) *x*<sup>1</sup> − *x*<sup>2</sup> *<sup>C</sup>* + 1 + *ωA L*1(*I*) Γ(*β*) 7 *a*2*β*−<sup>1</sup> 2*β* − 1 *Bu*<sup>1</sup> − *Bu*<sup>2</sup> *<sup>L</sup>*2(*I*;D), ∀*<sup>t</sup>* ∈ *<sup>I</sup>*,

which implies that

$$\|\|\mathbf{x}\_1 - \mathbf{x}\_2\|\|\_{\mathbb{C}} \le \left(1 - \frac{La^6b(1 + \|\|\boldsymbol{\omega}\_A\|\|\_{L^1(I)})}{\Gamma(\beta + 1)}\right)^{-1} \frac{1 + \|\|\boldsymbol{\omega}\_A\|\|\_{L^1(I)}}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta - 1}}{2\beta - 1}} \|Bu\_1 - Bu\_2\|\|\_{L^2(I; \mathcal{D})}.$$
 
$$\Box$$

**Theorem 2.** *If the Hypotheses (H1)–(H4) hold, then system (1) is approximately controllable on I*.

**Proof.** It is only needed to prove that D ⊂ *K*(*a*, *f*) due to the fact that D is dense, i.e., for <sup>∀</sup>*<sup>ε</sup>* <sup>&</sup>gt; 0 and *<sup>ξ</sup>* ∈ D, there is a control term *<sup>u</sup><sup>ε</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*) satisfying

$$\left\| \left| \boldsymbol{\xi} - \mathcal{R}(\boldsymbol{a})\boldsymbol{\phi}(\boldsymbol{0}) - \mathcal{P}(\mathcal{F}\mathbf{x}\_{\varepsilon}) - \mathcal{P}(\mathcal{B}\boldsymbol{u}\_{\varepsilon}) \right| \right\| < \varepsilon. \tag{10}$$

It follows from the Definition 3 that R(*a*)*φ*(0) ∈ D for *φ*(0) ∈ D, which indicates that *<sup>ξ</sup>* <sup>−</sup> <sup>R</sup>(*a*)*φ*(0) ∈ D. Then, it can be see that there exists some *<sup>ψ</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; <sup>D</sup>), such that <sup>P</sup>*<sup>ψ</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>−</sup> <sup>R</sup>(*a*)*φ*(0). Next, we are to show that there is a control *<sup>u</sup><sup>ε</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*) satisfying (4.2). Actually, for <sup>∀</sup>*<sup>ε</sup>* <sup>&</sup>gt; 0 and *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), in view of (H3), we can find a function *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), such that

$$\left\| \left| \boldsymbol{\xi} - \Re(\boldsymbol{a})\phi(0) - \mathcal{P}(\mathcal{F}\boldsymbol{x}\_{1}) - \mathcal{P}(\boldsymbol{B}\boldsymbol{u}\_{2}) \right| \right\| < \frac{\varepsilon}{2^{2}}\lambda$$

where *<sup>x</sup>*1(*t*) = *<sup>x</sup>*(*t*; *<sup>u</sup>*1), *<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*. Further, for *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), we can find a function *<sup>v</sup>*<sup>2</sup> <sup>∈</sup> *L*2(*I*; *U*) by (H3) again, such that

$$||\mathcal{P}(\mathcal{F}\mathbf{x}\_2 - \mathcal{F}\mathbf{x}\_1) - \mathcal{P}(Bv\_2)|| < \frac{\varepsilon}{2^{3}}\epsilon$$

where *x*2(*t*) = *x*(*t*; *u*2), *t* ∈ *I*. Then, from Lemma 5, we derive

$$\begin{array}{l} & \|Bv\_{2}\|\_{L^{2}(I;\mathcal{D})} \\ & \leq \quad \mu\|\mathcal{F}\mathbf{x}\_{2}-\mathcal{F}\mathbf{x}\_{1}\|\_{L^{2}(I;\mathcal{D})} \\ & \leq \quad \mu La^{\frac{1}{2}}b\|\mathbf{x}\_{2}-\mathbf{x}\_{1}\|\_{\mathcal{C}} \\ & \leq \quad \mu La^{\frac{1}{2}}b\left(1-\frac{La^{\theta}b(1+\|\omega\_{A}\|\_{L^{1}(I)})}{\Gamma(\beta+1)}\right)^{-1}\frac{1+\|\omega\_{A}\|\_{L^{1}(I)}}{\Gamma(\beta)}\sqrt{\frac{a^{2\beta-1}}{2\beta-1}}\|Bu\_{1}-Bu\_{2}\|\_{L^{2}(I;\mathcal{D})}. \end{array}$$

Next, define *<sup>u</sup>*<sup>3</sup> <sup>=</sup> *<sup>u</sup>*<sup>2</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), and, thus, it has

$$\begin{array}{ll} & \left||\mathfrak{f} - \mathfrak{R}(a)\phi(0) - \mathcal{P}(\mathcal{F}\mathfrak{x}\_{2}) - \mathcal{P}(Bu\_{3})\right|| \\ & \leq & \left||\mathfrak{f} - \mathfrak{R}(a)\phi(0) - \mathcal{P}(\mathcal{F}\mathfrak{x}\_{1}) - \mathcal{P}(Bu\_{2})\right|| + \left||\mathcal{P}(\mathcal{B}v\_{2}) - \mathcal{P}(\mathcal{F}\mathfrak{x}\_{2} - \mathcal{F}\mathfrak{x}\_{1})\right|| \\ & \leq & \left(\frac{1}{2^{2}} + \frac{1}{2^{3}}\right)\varepsilon. \end{array}$$

Utilizing induction, it is not hard to find a sequence {*un* : *<sup>n</sup>* <sup>≥</sup> <sup>1</sup>} ⊂ *<sup>L</sup>*2(*I*; *<sup>U</sup>*) satisfying

$$\left\| \left| \xi - \Re(a)\phi(0) - \mathcal{P}(\mathcal{F}\mathbf{x}\_n) - \mathcal{P}(Bu\_{n+1}) \right| \right\| < \left( \frac{1}{2^2} + \frac{1}{2^3} + \dots + \frac{1}{2^{n+1}} \right) \varepsilon,\tag{11}$$

where *xn*(*t*) = *x*(*t*; *un*), *t* ∈ *I*, and

$$\begin{aligned} &\quad \|Bu\_{n+1} - Bu\_{n}\|\_{L^{2}(I;\mathcal{D})} \\ &\leq \quad \mu La^{\frac{1}{2}}b \left(1 - \frac{La^{\theta}b(1 + \|\omega\_{A}\|\_{L^{1}(I)})}{\Gamma(\beta + 1)}\right)^{-1} \frac{1 + \|\omega\_{A}\|\_{L^{1}(I)}}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta - 1}}{2\beta - 1}} \|Bu\_{n} - Bu\_{n-1}\|\_{L^{2}(I;\mathcal{D})}. \end{aligned}$$

From Hypothesis (H4), we know that {*Bun* : *<sup>n</sup>* <sup>≥</sup> <sup>1</sup>} is a Cauchy sequence on *<sup>L</sup>*2(*I*; <sup>D</sup>), and, thus, there exists a function *<sup>u</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; <sup>D</sup>) satisfying

$$\lim\_{n \to \infty} B\mu\_n = \mu^\* \text{ in } L^2(I; \mathcal{D}).$$

Hence, for every *ε* > 0, we can obtain a number *N* > 0 satisfying

$$\left\|\left|\mathcal{P}(Bu\_{N+1}) - \mathcal{P}(Bu\_N)\right|\right\| < \frac{\varepsilon}{2}.\tag{12}$$

Then, from Equations (11) and (12), it is easy to deduce

$$\begin{array}{l} \left\| \left| \mathfrak{F} - \mathfrak{R}(a)\phi(0) - \mathcal{P}(\mathcal{F}\mathfrak{x}\_{N}) - \mathcal{P}(Bu\_{N}) \right| \right\| \\ \leq \left\| \left| \mathfrak{F} - \mathfrak{R}(a)\phi(0) - \mathcal{P}(\mathcal{F}\mathfrak{x}\_{N}) - \mathcal{P}(Bu\_{N+1}) \right| \right\| + \left\| \mathcal{P}(Bu\_{N+1}) - \mathcal{P}(Bu\_{N}) \right\| \\ \leq \left( \frac{1}{2^{2}} + \frac{1}{2^{3}} + \dots + \frac{1}{2^{N+1}} \right) \varepsilon + \frac{\varepsilon}{2} < \varepsilon, \end{array}$$

where *xN*(*t*) = *x*(*t*; *uN*), *t* ∈ *I*. Consequently, the fractional evolution system (1) is approximately controllable on *I*.

#### **5. Non-Local Conditions**

The practical usefulness and significance of non-local conditions in the field of technology and mechanical engineering have been demonstrated [5,9,11]. It has been proved that the non-local initial condition can provide more accurate descriptions than the classical initial conditions. Therefore, we concern the following system involving non-local conditions and a parameter as below:

$$\begin{cases} \ ^C D^\beta \mathfrak{x}(t) = A\mathfrak{x}(t) + f(t, \mathfrak{x}\_t) + B\mathfrak{u}(t), \ t \in I := [0, a]\_\prime, \\\ \mathfrak{x}(t) + \lambda \mathfrak{g}\_t(\mathfrak{x}) = \mathfrak{g}(t), \ t \in [-b, 0]. \end{cases}$$

Firstly, we present the following hypothesis about the non-local conditions.

**Hypothesis 5** (**H5**)**.** *gt* : *C*([−*b*, *a*]; *X*) → D*, for any t* ∈ [−*b*, 0]*; (i) For* ∀*x*, *y* ∈ *C*(*I*; *X*)*, there has a number l* > 0 *satisfying*

$$\|\|g\_t(\mathbf{x}) - g\_t(y)\|\|\_{\mathcal{D}} \le l \|\|\mathbf{x} - y\|\|\_{\mathcal{D}}.$$

*(ii) The non-local term gt*(*x*) *is continuous in t* ∈ [−*b*, 0] *for all x* ∈ *C*([−*b*, *a*]; *X*)*, and there has a constant C* > 0 *satisfying gt*(*x*) <sup>D</sup> ≤ *C*.

*Next, for* ∀*x* ∈ *C*(*I*; *X*) *and t* ∈ *I, let*

$$\chi\_t(\theta) = \begin{cases} \ x(t+\theta), & t+\theta \ge 0, \\\ \phi(t+\theta) - \lambda g\_{t+\theta}(\mathbf{x}), & t+\theta \le 0, \end{cases} \tag{13}$$

*for* <sup>∀</sup>*<sup>θ</sup>* <sup>∈</sup> [−*b*, 0]*. Obviously, we can check that xt* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*). *On the basis of Equation (13) and (H5), we have the following result similar to Lemma 2.*

**Lemma 6.** *Assume that xn* → *x*<sup>0</sup> (*n* → +∞) *for xn*, *x*<sup>0</sup> ∈ *C*(*I*; *X*)*. Then, for any t* ∈ *I, one can derive that* (*xn*)*<sup>t</sup>* <sup>→</sup> (*x*0)*<sup>t</sup>* (*<sup>n</sup>* <sup>→</sup> <sup>+</sup>∞) *for* (*xn*)*t*,(*x*0)*<sup>t</sup>* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*)*, and satisfies*

$$\| (\mathfrak{x}\_{\mathsf{n}})\_{t} - (\mathfrak{x}\_{\mathsf{0}})\_{t} \|\_{L^{1}[-b,0]} \le (|\lambda|l+1)b||\mathfrak{x}\_{\mathsf{n}} - \mathfrak{x}\_{\mathsf{0}}||\_{\mathsf{C}'} \ t \in I.$$

**Proof.** In accordance with Equation (13) and condition (H5), we can draw the inequalities as below:

$$\begin{array}{rcl} \| (\mathbf{x}\_{\boldsymbol{n}})\_{t} - (\mathbf{x}\_{0})\_{t} \| \|\_{L^{1}[-b,0]} &=& \int\_{-b}^{0} \| (\mathbf{x}\_{\boldsymbol{n}})\_{t} (\boldsymbol{\theta}) - (\mathbf{x}\_{0})\_{t} (\boldsymbol{\theta}) \| d\boldsymbol{\theta} \\ &=& \int\_{t-b}^{0} |\boldsymbol{\lambda}| \| \| \mathbf{g}\_{s} (\mathbf{x}\_{\boldsymbol{n}}) - \mathbf{g}\_{s} (\mathbf{x}\_{0}) \| d\mathbf{s} + \int\_{0}^{t} \| \mathbf{x}\_{\boldsymbol{n}} (\mathbf{s}) - \mathbf{x}\_{0} (\mathbf{s}) \| d\mathbf{s} \\ &\leq & \| \boldsymbol{\lambda}| \| b \| \| \mathbf{x}\_{\boldsymbol{n}} - \mathbf{x}\_{0} \| \_{\complement} + b \| \mathbf{x}\_{\boldsymbol{n}} - \mathbf{x}\_{0} \| \|\_{\complement} \\ &=& (\| \boldsymbol{\lambda}| l + 1) b \| \| \mathbf{x}\_{\boldsymbol{n}} - \mathbf{x}\_{0} \| \|\_{\complement} \ t \leq b \,\end{array}$$

and

$$\begin{aligned} \| |(\mathfrak{x}\_{\mathfrak{n}})\_t - (\mathfrak{x}\_0)\_t | \|\_{L^1[-b,0]} &= \int\_{-b}^0 ||(\mathfrak{x}\_{\mathfrak{n}})\_t(\theta) - (\mathfrak{x}\_0)\_t(\theta)|| d\theta \\ &= \int\_{\substack{t=b\\b \le |\mathfrak{x}\_{\mathfrak{n}} - \mathfrak{x}\_0|}}^{t} ||\mathfrak{x}\_{\mathfrak{n}}(s) - \mathfrak{x}\_0(s)|| ds \\ &\le \| b \|\mathfrak{x}\_{\mathfrak{n}} - \mathfrak{x}\_0 \|\_{\mathcal{C}\mathcal{I}} \quad t \ge b, \end{aligned}$$

which imply that

$$\|(\mathfrak{x}\_n)\_t - (\mathfrak{x}\_0)\_t\|\_{L^1[-b,0]} \le (|\lambda|l+1)b||\mathfrak{x}\_n - \mathfrak{x}\_0||\_{\mathbb{C}'} $$

for any *t* ∈ *I*.

**Definition 10.** *(i) For any <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), *a function <sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*(*I*; *<sup>X</sup>*) *is called a mild solution of Equation (2) on I, provided that*

$$\mathbf{x}(t) = \boldsymbol{\phi}(0) - \lambda \mathbf{g}\_0(\mathbf{x}) + \mathfrak{F}\_\mathbf{x}(t) + \mathfrak{B}\_\mathbf{u}(t) + \int\_0^t \dot{\mathfrak{R}}(t-s)(\boldsymbol{\phi}(0) - \lambda \mathbf{g}\_0(\mathbf{x}) + \mathfrak{F}\_\mathbf{x}(s) + \mathfrak{B}\_\mathbf{u}(s))ds, \ t \in I.$$

*(ii) System (2) is said to be approximately controllable on I, provided that for any ε* > 0 *and any final value <sup>ξ</sup>* <sup>∈</sup> *X, there exists a control term u<sup>ε</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*) *satisfying*

$$\left\| \left| \boldsymbol{\xi} - \Re(\boldsymbol{a}) \left( \boldsymbol{\phi}(\boldsymbol{0}) - \lambda \boldsymbol{g}\_{0}(\boldsymbol{x}\_{\varepsilon}) \right) - \mathcal{P} (\mathcal{F} \boldsymbol{x}\_{\varepsilon}) - \mathcal{P} (\boldsymbol{B} \boldsymbol{u}\_{\varepsilon}) \right\| < \varepsilon\_{\varepsilon} $$

where *<sup>x</sup>ε*(*t*) = *<sup>x</sup>*(*t*; *<sup>u</sup>ε*) is a mild solution of Equation (2) corresponding to *<sup>u</sup><sup>ε</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*).

**Theorem 3.** *In accordance with the proof steps of Theorem 1, one finds that if the Hypotheses (H1)–(H2) hold, then for any given control <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*), *system (2) has an unique mild solution on I, provided that*

$$(1 + \|\omega\_A\|\_{L^1(I)}) \left( |\lambda| l + \frac{L a^\beta (|\lambda| l + 1) b}{\Gamma(\beta + 1)} \right) < 1. \tag{14}$$

Under the condition Equation (14), we further suppose the following hypothesis:

**Hypothesis 6** (**H6**)**.** *The following inequality holds*

$$\mu L a^{\frac{1}{2}} (|\lambda|l+1)b \left(1 - (1 + \|\omega\_A\|\_{L^1(I)}) \left(|\lambda|l + \frac{L a^{\beta} (|\lambda|l+1)b}{\Gamma(\beta+1)}\right)\right)^{-1} \frac{1 + \|\omega\_A\|\_{L^1(I)}}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta-1}}{2\beta-1}} < 1.$$

In addition, to obtain the non-local results, we still need a lemma as below.

**Lemma 7.** *If the hypotheses (H1)–(H2) hold, then for any mild solutions of system (2), the following result holds*

$$\|\|\mathbf{x}\_1 - \mathbf{x}\_2\|\|\_{\mathbb{C}} \le \left(1 - (1 + \|\omega\_A\|\_{L^1(I)})\left(|\lambda|l + \frac{La^\beta(|\lambda|l+1)b}{\Gamma(\beta+1)}\right)\right)^{-1} \frac{1 + \|\omega\_A\|\_{L^1(I)}}{\Gamma(\beta)} \sqrt{\frac{a^{2\beta-1}}{2\beta-1}} \|Bu\_1 - Bu\_2\|\_{L^2(I;\mathcal{D})'},$$
  $for \ a \,\, u\_1 \,\, u\_2 \in I^2(I;II)$ 

*for any u*1, *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>*2(*I*; *<sup>U</sup>*)*.*

By means of iterative method utilized in Theorem 2 similarly, we now can obtain the main controllability result of the non-local case:

**Theorem 4.** *If the Hypotheses (H1)–(H3) and (H5) hold, then system (2) is approximately controllable on I*.

**Remark 2.** *Usually, the non-local condition can be given as follows*

$$\lambda g\_t(\mathbf{x}) = \lambda \sum\_{i=1}^{q} l\_i \mathbf{x}(t + \iota\_i), \ t \in [-b, 0]\_q$$

*where li* (*i* = 1, ···, *q*) *are some real numbers;* 0 < *ι*<sup>1</sup> < *ι*<sup>2</sup> < ··· < *ι<sup>q</sup>* ≤ *a*. *When λ* = 1 *and at time t* = 0*, it is evident that*

$$g\_0(\mathbf{x}) = g(\mathbf{x}) = \sum\_{i=1}^{q} l\_i \mathbf{x}(\iota\_i),$$

*which is exactly the case in [5,9,11,16,17].*

#### **6. Applications**

Evolutionary fractional behavior has widespread backgrounds of some practical fields of science and engineering. For example, in an electrical circuit, the voltage produced by some non-linear device can be expressed by the non-linear term *f* in the evolution systems; some related resistances can be represented by *A*; and linear operator *B* can denote some inductances. On the other hand, non-local conditions are more extensive in practical applications because they usually includes many other conditions, such as conditions of initial value, multipoint average, and periodic, etc. In this part, we consider the following fractional non-local delayed evolution systems

⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ *∂* 3 4 *∂t* 3 4 *<sup>x</sup>*(*t*, *<sup>ξ</sup>*) = *<sup>∂</sup>*<sup>2</sup> *∂ξ*<sup>2</sup> *<sup>x</sup>*(*t*, *<sup>ξ</sup>*) + (*t*)*e*−*<sup>t</sup>* 1 + *e*2*<sup>t</sup> <sup>t</sup> t*−*b* (*t* − *s*) sin(*x*(*s*, *ξ*))*ds* + *Bu*(*t*, *ξ*), (*t*, *ξ*) ∈ [0, *a*] × (0, *π*), *x*(*t*, 0) = *x*(*t*, *π*) = 0, *t* ∈ [0, *a*], *x*(*t*, *ξ*) + *λ m* ∑ *j*=1 *kj* sin(*x*(*ς<sup>j</sup>* + *t*, *ξ*)) = *φ*(*t*, *ξ*), (*t*, *ξ*) ∈ [−*b*, 0] × [0, *π*], *ς<sup>j</sup>* ∈ [0, *a*], (15) where <sup>∈</sup> *<sup>C</sup>*([0, *<sup>a</sup>*]; <sup>R</sup>), <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*(R+), and *<sup>φ</sup>* <sup>∈</sup> *<sup>C</sup>*2,1([−*b*, 0] <sup>×</sup> [0, *<sup>π</sup>*]; <sup>R</sup>). *<sup>φ</sup>*(*t*, 0) = *<sup>φ</sup>*(*t*, *<sup>π</sup>*) = 0, ∀*t* ∈ [−*b*, 0]. Let *<sup>X</sup>* <sup>=</sup> *<sup>U</sup>* <sup>=</sup> *<sup>L</sup>*2([0, *<sup>π</sup>*]), *Ax* <sup>=</sup> *<sup>x</sup>* for *<sup>x</sup>* ∈ D, where D = {*x* ∈ *X* : *x*, *x are absolutely continuous*, *x* ∈ *X*, *x*(0) = *x*(*π*) = 0}.

Evidently, *A* is an infinitesimal generator of a semigroup {*T*(*t*)}*t*≥<sup>0</sup> satisfying

$$T(t)\mathfrak{x} = \sum\_{n=1}^{\infty} e^{-n^2 t} \langle \mathfrak{x}, \delta\_n \rangle \delta\_{n\nu} \ x \in X.$$

In view of subordinate principle (Chapter 3, [26]), we know that *A* is also an infinitesimal generator of a continuous differentiable bounded linear operators family {R(*t*)}*t*≥<sup>0</sup> satisfying R(0) = I, and

$$\Re(t) = \int\_0^\infty \eta\_{t,\emptyset}(s) T(s) ds, \ t > 0,$$

where *ηt*,*β*(*s*) = *t* <sup>−</sup>*β*Φ*β*(*st*−*β*), and

$$\Phi\_{\beta}(y) = \sum\_{n=0}^{\infty} \frac{(-y)^n}{n! \Gamma(-\beta n + 1 - \beta)} = \frac{1}{2\pi i} \int\_{\mathcal{H}} \zeta^{\beta - 1} \exp(\zeta - y\zeta^{\beta}) d\zeta, \ 0 < \beta < 1,$$

where H is a contour which encircles the origin once counterclockwise.

For each *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>a</sup>*]; *<sup>U</sup>*), one has

$$
\mu(t) = \sum\_{n=0}^{\infty} \mu\_n(t)\delta\_{n\prime} \ u\_n(t) = \langle \mu(t), \delta\_n \rangle.
$$

Then, an operator *B* can be defined by

$$B\underline{u} = \sum\_{n=1}^{\infty} \overline{u}\_n \delta\_{n\lambda}$$

where

$$\overline{u}\_{\hbar}(t) = \begin{cases} \begin{array}{ll} 0, & 0 \le t < a - \frac{a}{n^2}, \\\\ u\_{\hbar}(t), & a - \frac{a}{n^2} \le t \le a, \end{array} \end{cases}$$

for every *n* = 1, 2, ···. This ensures that *B* is a bounded linear operator. In addition, the operator P in Equation (9) is exactly the case of the operator in [29] when *B* = I and *t* = *a*. Furthermore, denote by

$$\begin{aligned} \beta &= \frac{3}{4} \in (\frac{1}{2}, 1], \\ ^C D^{\frac{3}{4}} \mathfrak{x}(t)(\tilde{\mathfrak{z}}) &= \frac{\partial^{\frac{3}{4}}}{\partial t^{\frac{3}{4}}} \mathfrak{x}(t, \tilde{\mathfrak{z}}), \\ \mathfrak{x}(t)(\tilde{\mathfrak{z}}) &= \mathfrak{x}(t, \tilde{\mathfrak{z}}), \\ B\mathfrak{u}(t)(\tilde{\mathfrak{z}}) &= Bu(t, \tilde{\mathfrak{z}}), \\ \phi(t)(\tilde{\mathfrak{z}}) &= \phi(t, \tilde{\mathfrak{z}}), \end{aligned}$$

$$\begin{aligned} g\_t(\mathbf{x})(\xi) &= \sum\_{j=1}^m k\_j \sin(\mathfrak{x}(\xi\_j + t, \xi)), \\ f(t, \mathbf{x}\_t)(\xi) &= \frac{\mathcal{O}(t)e^{-t}}{1 + e^{2t}} \int\_{t-b}^t \boldsymbol{\varrho}(t-s) \sin(\mathfrak{x}(s, \xi)) ds. \\ \text{Hence, Equation (15) can be regarded as} \end{aligned}$$

$$\begin{cases} \ ^{\mathbb{C}}D^{\beta}\mathbf{x}(t) = A\mathbf{x}(t) + f(t, \mathbf{x}\_{t}) + Bu(t), \ t \in [0, a]\_{\prime},\\ \mathbf{x}(t) + \lambda g\_{t}(\mathbf{x}) = \phi(t), \ t \in [-b, 0]\_{\prime} \end{cases}$$

In addition, it can be checked that *f* , *B*, *gt*, *φ* satisfy all assumptions in Theorem 4. Therefore, system (15) is approximately controllable on [0, *a*]. In addition, it is well known to all that the prospect of digital signal processing (DSP) is widespread and developmental, and digital filters play a significant role in it. Therefore, in this part, we also present the filter pattern of the system we studied which is given in Figure 1.

**Figure 1.** Filter system.

For any time *t*, the resultant values of samples *xt* and *f*(*t*) are produced and transferred to the integrators *I*<sup>1</sup> and *I*2, where the signals are integrated over time 0 to *t*. The signals of resultant values of *B* and *ux*(*t*) are integrated in integrators *I*<sup>3</sup> and *I*4. Integrators *I*<sup>1</sup> and *I*<sup>3</sup> are entered into summer network-1; Integrators *I*<sup>2</sup> and *I*<sup>4</sup> are entered into summer network-2. Inputs *φ*(*t*) and *λgt*(*x*) at time *t* = 0 are added up in the summer network-3 and summer network-4. The integral for the product of <sup>R</sup>˙ (*<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*) and the signals in summer network-4 over time 0 to *t* is performed in integrators *I*5. At last, move the above outputs and integrators *I*<sup>5</sup> to summer network-5, and, thus, the final outputs *x*(*t*) is derived, which is bounded and approximately controllable.

#### **7. Conclusions**

In this manuscript, some approximate controllability results of fractional delay systems with non-local conditions and a parameter are derived by using an iterative method. We substitute for the theory of *C*0-semigroup and its associated characteristic solution operators by utilizing differentiability properties about resolvent operator. A special complete space is used to assist in solving the disturbance due to delay effects. Then, the current results seem to be more general and generalize some recent analogous outcomes, e.g., [21–23,27].

By means of iterative method, some further new study can be devoted to the approximate controllability of fractional impulsive systems as below:

$$\begin{cases} \ ^C D^\beta \boldsymbol{x}(t) = A \boldsymbol{x}(t) + f(t, \boldsymbol{x}\_t, \boldsymbol{Q}\_{\boldsymbol{x}}(t)) + Bu(t), \ a.e. \ t \in I = [0, a]\_\prime\\ \Delta \boldsymbol{x}(t\_i) = \boldsymbol{x}(t\_i^+) - \boldsymbol{x}(t\_i^-) = I\_i(\boldsymbol{x}(t\_i^-)), \ i = 1, 2, \cdots, m\_\prime\\ \boldsymbol{x}(t) + \lambda g\_t(\boldsymbol{x}) = \boldsymbol{\phi}(t), \ t \in [-b, 0]\_\prime \end{cases}$$

where *Qx*(*t*) = *<sup>t</sup>* 0 *q*(*t*,*s*, *xs*)*ds*, *q* : Λ × *L*([−*b*, 0]; *X*) → *X* and Λ = {(*t*,*s*) ∈ *I* × *I* : *s* ≤ *t*}. The impulsive items *Ii* (*i* = 1, 2, ···, *m*) are given functions that satisfy some appropriate hypotheses. *<sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*). The main tools we are about to use here can be the theory of differentiable resolvent operators or analytic resolvent operators [25,30,31]. Furthermore, evolutionary fractional behavior is more accurately captured by variable-order fractional calculus. To this end, extending the present results to the more generalized variable-order fractional system will be an interesting problem.

**Author Contributions:** Investigation, D.Z.; writing-original draft, D.Z.; writing-review and editing, Y.L.; software, Y.L.; conception of the work, D.Z.; Validation, D.Z.; Revising, D.Z. and Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China under grant number 62073204.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Topological Structure of the Solution Sets for Impulsive Fractional Neutral Differential Inclusions with Delay and Generated by a Non-Compact Demi Group**

**Zainab Alsheekhhussain 1,\* , Ahmed Gamal Ibrahim <sup>2</sup> and Akbar Ali <sup>1</sup>**


**Abstract:** In this paper, we give an affirmative answer to a question about the sufficient conditions which ensure that the set of mild solutions for a fractional impulsive neutral differential inclusion with state-dependent delay, generated by a non-compact semi-group, are not empty compact and an *Rδ*-set. This means that the solution set may not be a singleton, but it has the same homology group as a one-point space from the point of view of algebraic topology. In fact, we demonstrate that the solution set is an intersection of a decreasing sequence of non-empty compact and contractible sets. Up to now, proving that the solution set for fractional impulsive neutral semilinear differential inclusions in the presence of impulses and delay and generated by a non-compact semigroup is an *Rδ*-set has not been considered in the literature. Since fractional differential equations have many applications in various fields such as physics and engineering, the aim of our work is important. Two illustrative examples are given to clarify the wide applicability of our results.

**Keywords:** impulsive fractional differential inclusions; neutral differential inclusions; mild solutions; contractible sets; *Rδ*-set

#### **1. Introduction**

Impulsive differential equations and inclusions describe phenomena in which states are changing rapidly at certain moments. In [1–8], the authors examined whether a mild solution for different types of impulsive differential inclusions exist.

The study of neutral differential equations appears in many applied mathematical sciences, such as viscoelasticity and equations that describe the distribution of heat. The structure of neutral equations involve derivatives related to delay beside the function. Neutral differential equations and inclusions were studied in [9–12]. These papers examined the mild solutions and controllability of the system.

Because the set of mild solutions for a differential inclusion having the same initial point may not be a singleton, many authors are interested in investigating the structure of this set in a topological point of view. An important aspect of such structure is the *Rδ*-property, which means that the homology group of the set of mild solutions is the same as a one-point space. We list some studies in which the authors demonstrated the solution sets satisfying *Rδ*-property: Gabor [13] considered impulsive semilinear differential inclusions with finite delay on the half-line of order one generated by a non-compact semi-group; Djebali et al. [14] worked on impulsive differential inclusions on unbounded domains; Zhou et al. [15] studied the neutral evolution inclusions of order one generated by a non-compact semi-group; Zhou et al. [16] considered fractional stochastic evolution inclusions generated by a compact semi-group; Zhao et al. [17] studied a stochastic differential equation of Sobolev-type which is semilinear with Poisson jumps of order

**Citation:** Alsheekhhussain, Z.; Ibrahim, A.G.; Ali, A. Topological Structure of the Solution Sets for Impulsive Fractional Neutral Differential Inclusions with Delay and Generated by a Non-Compact Demi Group. *Fractal Fract.* **2022**, *6*, 188. https://doi.org/10.3390/ fractalfract6040188

Academic Editor: Rodica Luca

Received: 12 March 2022 Accepted: 24 March 2022 Published: 28 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

*α* ∈ (1, 2); Beddani [18] examined a differential inclusion involving Riemann–Liouville fractional derivatives; Wang et al. [19] worked on semilinear fractional differential inclusions with non-instantaneous impulses; Ouahab et al. [20] considered fractional inclusions that are non-local and have impulses at different times; Zaine [21] studied weighted fractional differential equations. Recently, Zhang et al. [22] proved that the set of *C*0-solutions for impulsive evolution inclusions of order one is an *Rδ*-set and generated by *m*–dissipative operator. Wang et al. [23] proved that the solution for evolution equations that have nonlinear delay and multivalued perturbation on a non-compact interval is an *Rδ*-set.

In [6,24–26], the authors studied different kinds of fractional differential inclusions, and, in all cases, they showed that the set of solutions is a compact set. For more work related to this, the reader can consult the book in [27] about the topological properties for evolution inclusions.

However, up to now, proving that the solution set for fractional impulsive neutral semilinear differential inclusions involving delay and generated by a non-compact semigroup is an *Rδ*-set has not been considered in the literature. Thus, this topic is new and interesting and, hence, the question whether there exists a solution set carrying an *Rδ*-structure remains unsolved for fractional differential inclusions when there are impulses, delay (finite or infinite) and the operator families generated by the linear part lack compactness. Therefore, our main goal is to give an affirmative answer to this question. In fact, we study a neutral fractional impulsive differential inclusion with delay which is generated by a non-compact semigroup, and we show that the set of solutions is non-empty and equal to an intersection of a decreasing sequence of sets each of which is non-empty compact and has a homotopy equivalent to a point.

Let *α* ∈ (0, 1), *r* > 0, *J* = [0, *b*], *T* = {Υ(*η*) : *η* ≥ 0} a semigroup on *E*, which is Banach space, and *<sup>A</sup>* the infinitesimal generator of *<sup>T</sup>*. Let *<sup>F</sup>* : *<sup>J</sup>* <sup>×</sup> <sup>Θ</sup> <sup>→</sup> <sup>2</sup>*<sup>E</sup>* − {*φ*} be a multifunction, *h* : *J* × Θ → *E*, 0 = *η*<sup>0</sup> < *η*<sup>1</sup> < ··· < *η<sup>m</sup>* < *ηm*+<sup>1</sup> = *b*, and *ψ* ∈ Θ be given. For every *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*, let <sup>κ</sup>(*η*) : H → <sup>Θ</sup>, (κ(*η*)*x*)(*θ*) = *<sup>x</sup>*(*<sup>η</sup>* <sup>+</sup> *<sup>θ</sup>*); *<sup>θ</sup>* <sup>∈</sup> [−*r*, 0]; where <sup>Θ</sup> and <sup>H</sup> are defined later.

The present paper shows the solution set of a fractional neutral impulsive semilinear differential inclusion with delay having details as follows:

$$\begin{cases} \, ^cD\_{0,\eta}^a[\mathbf{x}(\eta) - h(\eta, \varkappa(\eta)\mathbf{x})] \in A\mathbf{x}(\eta) + \mathcal{F}(\eta, \varkappa(\eta)\mathbf{x}), \ a.e. \ \eta \in [0, b] - \{\eta\_1, \dots, \eta\_{|\mathbf{m}|}\},\\\ I\_i \mathbf{x}(\eta\_i^-) = \mathbf{x}(\eta\_i^-) - \mathbf{x}(\eta\_i^+), i = 1, \dots, m, \\\ \mathbf{x}(\eta) = \boldsymbol{\psi}(\eta), \boldsymbol{\eta} \in [-r, 0], \end{cases} \tag{1}$$

is not empty, compact and an *<sup>R</sup>δ*-set, where *Ii* : *<sup>E</sup>* −→ *<sup>E</sup>*, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>m</sup>*, and *<sup>x</sup>*(*η*<sup>+</sup> *<sup>i</sup>* ), *x*(*η*<sup>−</sup> *i* ) are the limits of the function *<sup>x</sup>* evaluated at *<sup>η</sup><sup>i</sup>* from the right and the left. Furthermore, *cD <sup>α</sup>* 0,*<sup>η</sup>* denotes the Caputo derivative that has order *α* ∈ (0, 1) and lower limit at zero [28].

In the following points, we clarify the originality, importance and the main contributions of this article:


7. Our technique can be used to derive suitable conditions, which implies that the solution set is an *Rδ*-set for the problems studied in [13–23] when they contain impulses and delay.

In order to clarify the difficulties encountered to achieve our aim, we point to the normed space P C([−*r*, *b*], *E*], which consists of piecewise continuous bounded functions defined on [−*r*, *b*] with a finite number of discontinuity points and is left continuous at the discontinuity points, and is not necessarily complete. Moreover, unlike the Banach spaces *C*([−*r*, *b*], *E*) and *PC*(*J*, *E*), the Hausdorff measure of noncompactness on P C([−*r*, *b*], *E*] is not specific. Thus, when the problem involves delay and impulses, we cannot consider P C([−*r*, *b*], *E*] as the space of solutions. To overcome these difficulties, a complete metric space *H* is introduced as the space of mild solutions (see the next section). In addition, the function *<sup>η</sup>* <sup>→</sup> <sup>κ</sup>(*η*)*x*; *<sup>x</sup>* <sup>∈</sup> *<sup>H</sup>* is not necessarily measurable (see Remark 1, and so, a norm different from the uniform convergence norm is introduced (see Equation (2) below).

For recent contributions on neutral differential inclusions of fractional order, Burqan et al. [29] give a numerical approach in solving fractional neutral pantograph equations via the ARA integral transform. Ma et al. [30] studied the controllability for a neutral differential inclusion with Hilfer derivative, and Etmad et al. [31] investigated a neutral fractional differential inclusion of Katugampola-type involving both retarded and advanced arguments. For more recent papers we cite [32–34].

The sections of the paper are organized as follows: We include some background materials in Section 2 as we need them in the main sections. Section 3 is assigned for proving that the solution set of Problem (1) is non-empty and compact. In Section 4, we show that this set is an *Rδ*-set in the complete metric space *H*. In Section 5, e give an example as an application of the obtained results. Sections 6 and 7 are the discussion and conclusion sections.

#### **2. Preliminaries and Notation**

In all the text we denote for the set of mild solutions for Problem 1 by Σ*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*] and by *<sup>L</sup>*1(*J*, *<sup>E</sup>*) to the quotient space consisting of *<sup>E</sup>*−valued Bohner integrable functions defined on *J* having the norm *f <sup>L</sup>*1(*J*,*E*) <sup>=</sup> % *<sup>b</sup>* 0 *f*(*θ*) *dθ*. Let *Pck*(*E*) = {*B* ⊆ *E* : *B* be non-empty, convex and compact}.

**Definition 1.** *(Ref. [35]) Let h* : *J* → *E,* {Υ(*η*) : *η* ≥ 0} *a C*0−*semigroup and A be the infinitesimal generator of it. A continuous function x* : *J* → *E is called a mild solution for the problem:*

$$\begin{cases} \,^cD \,^\kappa z(\eta) = Az(\eta) + h(\eta), \eta \in J\_\prime \\ z(0) = z\_0 \in E\_\prime \end{cases}$$

*if*

$$z(\eta) = \mathfrak{K}\_1(\eta)z\_0 + \int\_0^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_2(\eta - \tau)h(\tau)d\tau, \eta \in \mathfrak{J}\_2$$

*where* <sup>K</sup>1(*η*) = % <sup>∞</sup> <sup>0</sup> *ξα*(*θ*)Υ(*<sup>η</sup> αθ*)*dθ*,K2(*η*) = *<sup>α</sup>* % <sup>∞</sup> <sup>0</sup> *θξα*(*θ*)Υ(*ηαθ*)*dθ*,

*ξα*(*θ*) = <sup>1</sup> *<sup>α</sup> <sup>θ</sup>*−1<sup>−</sup> <sup>1</sup> *<sup>α</sup> <sup>w</sup>α*(*θ*<sup>−</sup> <sup>1</sup> *<sup>α</sup>* ) <sup>≥</sup> 0, *<sup>w</sup>α*(*θ*) = <sup>1</sup> *<sup>π</sup>* <sup>∑</sup><sup>∞</sup> *n*=1(−1)*n*−1*θ*−*αn*−<sup>1</sup> <sup>Γ</sup>(*<sup>n</sup> <sup>α</sup>*+1) *<sup>n</sup>*! sin(*nπα*), *θ* ∈ (0, ∞) *and* % <sup>∞</sup> <sup>0</sup> *ξα*(*θ*)*dθ* = 1*.*

**Lemma 1.** *(Ref. [35] (lemma 3.1)) The properties stated below are held:*


$$\lim\_{\eta \to \tau} ||\mathfrak{K}\_1(\eta)\mathfrak{x} - \mathfrak{K}\_1(\tau)\mathfrak{x}|| = 0,\\
\text{and}\\
\lim\_{\eta \to \tau} ||\mathfrak{K}\_2(\eta)\mathfrak{x} - \mathfrak{K}\_2(\tau)\mathfrak{x}|| = 0.$$

Consider the spaces:

1. The normed space

Θ : = {*x* : [−*r*, 0] → *E*, where *x* is discontinuous at finite number of points *<sup>τ</sup>* <sup>=</sup> 0, and all the limits *<sup>x</sup>*(*τ*+) and *<sup>x</sup>*(*τ*−) are less than <sup>∞</sup>}

endowed with the norm:

$$||\mathfrak{x}||\_{\oplus} := \int\_{-r}^{0} ||\mathfrak{x}(\mathfrak{r})|| d\mathfrak{r}.\tag{2}$$

2. The Banach space

$$\begin{aligned} \text{PC}(f, E) & \quad = \{ u : f \to E : u\_{|I\_i} \in \text{C}(f\_{i\nu} E), i = 0, 1, 2, \dots, m, \text{ and } u(\eta\_i^+), i = 1, 2, \dots, m \}, \\ u(\eta\_i) &= \quad u(\eta\_i^-) \text{ are finite for every } i = 1, 2, \dots, m \} \end{aligned}$$

where *J*<sup>0</sup> = [0, *η*1], *Ji* = (*ηi*, *ηi*+1], *i* = 1, 2, . . . , *m*, and ||*v*||*PC*(*J*;*E*) = *τupη*∈*J*||*v*(*η*)||. 3. The complete metric space

$$H = \{ \mathbf{x} : [-r, b] \to E : \text{ where } \mathbf{x} \text{ is continuous at } \eta = 0, \mathbf{x}\_{|\_{[-r, 0]}} = \psi\_r \mathbf{x}\_{|\_{l\_l}} \in \text{PC } (f, E) \}\_l$$

where the metric function is given by:

$$d\_H(\mathfrak{x}, \mathfrak{y}) = \pi \mathfrak{u} p\_{\eta \in I} ||\mathfrak{x}(\eta) - \mathfrak{y}(\eta)||.$$

4. The Banach space

$$\mathcal{H} := \{ \mathbf{x} : [-r, b] \to E \text{ where } \mathbf{x}(\eta) = \mathbf{0}, \forall \eta \in [-r, \mathbf{0}]. \mathbf{x}\_{|\_{l\_i}} \in \text{PC } (f, E) \}$$

together with the norm ||*x*||H <sup>=</sup> *<sup>τ</sup>upη*∈*J*||*x*(*η*)|| <sup>+</sup> ||*x*|[−*r*,0] ||<sup>Θ</sup> = *τupη*∈*J*||*x*(*η*)||.

The Hausdorff measure of noncompactness on a Banach space *PC*(*J*, *E*) is given by

$$\chi\_{\text{PC}}(B) := \max\_{i=0,1,2,\dots,m} \chi\_i(B\_{\mid \cdot \overline{f\_i}})\_{\prime \prime}$$

where *B* is a bounded subset of *PC*(*J*, *E*) and *χ<sup>i</sup>* is the Hausdorff measure of noncompactness on the Banach space *C*(*Ji*, *E*) and

$$B\_{|\overline{l\_i}} := \{ \mathbf{x}^\* : \overline{l\_i} \to E : \mathbf{x}^\*(\eta) = \mathbf{x}(\eta), \eta \in I\_{\overline{l}} \text{ and } \mathbf{x}^\*(\eta\_{\overline{l}}) = \mathbf{x}(\eta\_{\overline{l}}^+), \mathbf{x} \in B \}.$$

The Hausdorff measure of noncompactness on H is defined by:

$$\chi\_{\mathcal{H}}(B) = \max\_{i=0,1,2,\dots,m} \chi\_i(B\_{\mid \cdot \overline{f\_i}})\_{\prime \cdot}$$

where *B* is a bounded subset of H.

**Remark 1.** *Since the function <sup>η</sup>* <sup>→</sup> <sup>κ</sup>(*η*)*x*; *<sup>x</sup>* <sup>∈</sup> *<sup>H</sup> is not necessarily measurable*, *we do not consider the uniform convergence norm to be the norm defined on the space* Θ *(see Example 3.1, [36]). Therefore, the multivalued superposition operator*

$$\mathfrak{a}\mathfrak{x} \to \operatorname{S}^1\_{F(\cdot,\varkappa(\cdot)\mathfrak{x})} = \{ f \in L^1(f,E) : f(\eta) \in F(\eta,\varkappa(\eta)\mathfrak{x}), a.e., \eta \in I \}$$

*would not be well defined. Therefore, we consider a norm defined by Equation* (2)*.*

**Definition 2.** *A function x* ∈ *H is said to be a mild solution for* (1) *if*

$$
\overline{\mathfrak{X}}(\eta) = \begin{cases}
\psi(\eta), \eta \in [-r, 0], \\
\mathfrak{K}\_{1}(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\mathfrak{x}}) \\
+ \int\_{0}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\mathfrak{x}}) d\tau \\
+ \int\_{0}^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_{2}(\eta - \tau) f(\tau) d\tau \\
+ \sum\_{0 < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\overline{\mathfrak{x}}(\eta\_{i}^{-}))\_{\prime} \eta \in \mathcal{J} = [0, b],
\end{cases} \tag{3}
$$

where *<sup>f</sup>* <sup>∈</sup> *<sup>S</sup>*<sup>1</sup> *F*(.,κ(.)*x*)

We assume the following conditions:

(*HA*) *A* is the infinitesimal generator of *T*, 0 is an element of the resolvent of *A*, *ρ*(*A*) and sup*η*≥<sup>0</sup> ||Υ(*η*)|| ≤ *<sup>M</sup>*, where *<sup>M</sup>* <sup>≥</sup> 1.

(*HF*) *F* : *J* × Θ → *Pck*(*E*) where:

.

(*HF*1) For any *z* ∈ Θ, the multifunction *η* −→ *F*(*η*, *z*) has a measurable selection, and for *η* ∈ *J*, *a*.*e*., the multifunction *z* −→ *F*(*η*, *z*) is upper semicontinuous.

(*HF*2) There exists a *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>P*(*I*, <sup>R</sup>+)(*<sup>P</sup>* <sup>&</sup>gt; <sup>1</sup> *<sup>α</sup>* ) satisfying

> *F*(*η*, *z*) ≤ *ϕ*(*η*) (1 + *z* <sup>Θ</sup>), ∀*z* ∈ Θ and for *a*.*e*. *η* ∈ *J*.

(*HF*3) There is a *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>P*([0, *<sup>b</sup>*], *<sup>E</sup>*), *<sup>p</sup>* <sup>&</sup>gt; <sup>1</sup> *<sup>α</sup>* such that, for any *D* ⊂ Θ that is bounded, we have

$$\chi\_E(F(\eta, D)) \le \beta(\eta) \sup\_{\theta \in [-r, 0]} \chi\_E\{z(\theta) : z \in D\}, a.e. \text{ for } \eta \in \mathfrak{J}. \tag{4}$$

(*H I*) For any *i* = 1, ... , *m*, the function *Ii* : *E* → *E* is continuous, and there are *σ<sup>i</sup>* > 0 and *ς<sup>i</sup>* > 0 satisfying ||*Ii*(*x*)|| ≤ *σi*||*x*||, and for any bounded subset *D* ⊆ *E*,

$$
\chi\_E(I\_!(D)) \le \varsigma\_i \chi\_E(I\_!(D)).
$$

**Lemma 2.** *(Ref. [37]) Under condition* (*HA*), *for any <sup>γ</sup>* <sup>∈</sup> (0, 1)*, the fractional power <sup>A</sup><sup>γ</sup> can be defined, and it is linear and closed on its domain D*(*Aγ*)*. In addition, the following properties are satisfied:*

*(i) D*(*Aγ*) *is a Banach space with the norm*

$$||\mathfrak{x}||\_{\mathfrak{T}} = ||A^{\gamma}\mathfrak{x}||.$$


$$||A^{\gamma}\Upsilon(\eta)|| \le \frac{\mathcal{C}\_{\gamma}}{\eta^{\gamma}}.\tag{5}$$


$$A\mathfrak{K}\_2(\eta)\mathfrak{x} = A^{1-\gamma}\mathfrak{K}\_2(\eta)A^{\gamma}\mathfrak{x}, \eta \in \mathfrak{J},\tag{6}$$

*and*

$$||A^{\gamma} \mathfrak{K}\_2(\eta)|| \le \frac{a \mathbb{C}\_{\gamma} \Gamma(2-\gamma)}{\eta^{a\gamma} \Gamma(1+a(1-\gamma))}, \eta \in (0,b]. \tag{7}$$

We need the next lemmas in order to prove our main results.

**Lemma 3.** *Assume W* ⊆ *E to be bounded, closed and convex,* Φ<sup>1</sup> : *W* → *E is a single-valued function,* Φ<sup>2</sup> : *W* → *Pck*(*E*) *is a multifunction, and for any x* ∈ *W*, Φ1(*x*) + *y* ∈ *W*, ∀*y* ∈ Φ2(*x*). *Suppose that*


Then, the fixed point set of Φ<sup>1</sup> + Φ<sup>2</sup> is not empty. Moreover, the set of fixed points for Φ<sup>1</sup> + Φ<sup>2</sup> is compact if it is bounded.

**Proof.** Φ<sup>1</sup> is continuous on *W* since it is a contraction and, hence, it follows by the closeness of Φ2, that the multifunction *R* = Φ<sup>1</sup> + Φ<sup>2</sup> is closed. We show that *R* is *χE*−condensing, where *χ<sup>E</sup>* is the Hausdorff measure of noncompactness on *E*. Let *Z* be a bounded set of *W*. Since Φ<sup>1</sup> is a contraction with the contraction constant *k*, we get *μE*(Φ1(*Z*)) ≤ *kμE*(*Z*) ≤ 2*kχE*(*Z*) < *χE*(*Z*), where *μ<sup>E</sup>* is the Kuratowski measure of noncompactness on *E*. Because Φ<sup>2</sup> is compact, *χE*(Φ2(*Z*)) = 0. Therefore,

$$\begin{aligned} \chi\_E(R(Z)) &= \quad \chi\_E(\Phi\_1(Z)) + \chi\_E(\Phi\_2(Z)) \\ &= \quad \chi\_E(\Phi\_1(Z)) \le \mu\_E(\Phi\_1(Z)) \\ &< \quad \chi\_E(Z). \end{aligned}$$

This means that *R* is *χE*−condensing. By Proposition 3.5.1 in [38], the fixed point set of Φ<sup>1</sup> + Φ<sup>2</sup> is not empty. The second part follows from Proposition 3.5.1 in [38].

#### **3. The Compactness of Σ***<sup>F</sup> <sup>ψ</sup>***[***−r***,** *b***]**

In this section, we show that the set of mild solutions for Problem 1 is nonempty and compact.

For any *x* ∈ H with *x*(0) = *ψ*(0), let *x* ∈ *H* be defined by

$$\mathfrak{X}(\eta) := \begin{cases} \; \psi(\eta), \eta \in [-r, 0], \\ \; x(\eta), \eta \in (0, b]. \end{cases} \tag{8}$$

**Lemma 4.** *For any <sup>x</sup>* <sup>∈</sup> *H, the function <sup>η</sup>* <sup>→</sup> <sup>κ</sup>(*η*)*x is continuous from J to* <sup>Θ</sup>*.*

**Proof.** Assume *η*, *τ* ∈ *J*, *η* ≤ *τ*. Then,

$$||\varkappa(\eta)\overline{\varkappa} - \varkappa(\tau)\overline{\varkappa}||\_{\Theta} = \int\_{-r}^{0} ||\overline{\varkappa}(\eta + \theta) - \overline{\varkappa}(\tau + \theta)||d\theta.$$

Because *x* is continuous on [−*r*, *b*] except for a finite number of points, it follows that lim*η*→*<sup>τ</sup>* ||*x*(*η* + *θ*) − *x*(*τ* + *θ*)|| = 0, a.e. Since *x* ∈ *H*, *limη*→*<sup>τ</sup>* % 0 <sup>−</sup>*<sup>r</sup>* ||*x*(*<sup>η</sup>* <sup>+</sup> *<sup>θ</sup>*) <sup>−</sup> *<sup>x</sup>*(*<sup>τ</sup>* <sup>+</sup> *<sup>θ</sup>*)||*d<sup>θ</sup>* <sup>=</sup> 0, and the proof is completed.

**Theorem 1.** *Assume that* (*HA*) *and* (*HF*) *are held and that* {Υ(*η*) : *η* ≥ 0} *is equicontinuous. Assume also that the following conditions are satisfied.*

(*Hh*) The function *h* : *J* × Θ → *E* is continuous and there exists a *γ* ∈ (0, 1) satisfying *<sup>h</sup>*(*η*, *<sup>u</sup>*) <sup>∈</sup> *<sup>D</sup>*(*Aγ*), <sup>∀</sup>(*η*, *<sup>u</sup>*) <sup>∈</sup> *<sup>J</sup>* <sup>×</sup> <sup>Θ</sup> and


$$d\_1 ||A^{-\gamma}|| + \frac{d\_1 b^{\alpha \gamma} \mathbb{C}\_{1-\gamma} \Gamma(1+\gamma)}{\gamma \Gamma(1+\alpha \gamma)} < \frac{1}{2r'} \tag{9}$$

$$||A^{\gamma}h(\eta,\mu)|| \le d\_2(1+||u||\_{\Theta}), \forall (\eta,u) \in f \times \Theta,\tag{10}$$

and

$$||A^{\gamma}h(\eta, u\_1) - A^{\gamma}h(\eta, u\_2)|| \le d\_1 ||u\_1 - u\_2||\_{\Theta^{\gamma}} \forall \eta \in J. \tag{11}$$

Then, Σ*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*] is not empty and a compact subset of *H* provided that

$$||A^{-\gamma}||d\_2r + d\_2\frac{\mathbb{C}\_{1-\gamma}\Gamma(1+\gamma)b^{a\gamma}}{\Gamma(1+a\gamma)\gamma}r + \frac{M}{\Gamma(a)}\Delta||\varphi||\_{L^P\_{(l,\mathbb{R}^+)}}r + \sigma M < 1,\tag{12}$$

and

$$\frac{4\Delta M}{\Gamma(\alpha)}||\beta||\_{L^P(I,\mathbb{R}^+)} + 2M\sum\_{i=1}^{i=m} \zeta\_i < \frac{1}{2} \tag{13}$$

where *<sup>σ</sup>* <sup>=</sup> <sup>∑</sup>*i*=*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>σ</sup><sup>i</sup>* and <sup>Δ</sup> = ( *<sup>P</sup>*−<sup>1</sup> *<sup>α</sup>P*−<sup>1</sup> ) *P*−1 *<sup>P</sup> bα*<sup>−</sup> <sup>1</sup> *P* .

**Proof.** A multioperator Φ : H → *P*(H) is defined as the following: let *x* ∈ H, hence, as a consequence of (*HF*1), the multifunction *<sup>η</sup>* −→ *<sup>F</sup>*(*η*,κ(*η*)*x*) admits a measurable selection which, by (*HF*2), belongs to *S*<sup>1</sup> *F*(.,κ(.)*x*) , and, therefore, *y* ∈ Φ(*x*) can be defined by

$$y(\eta) = \begin{cases} 0, \eta \in [-r, 0], \\ \mathfrak{K}\_1(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\pi}) \\ + \int\_0^{\eta} (\eta - \tau)^{a-1} A \mathfrak{K}\_2(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\pi}) d\tau \\ + \int\_0^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_2(\eta - \tau) f(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{K}\_1(\eta - \eta\_i) I\_i(\overline{\pi}(\eta\_i^{-})), \eta \in \mathfrak{J}\_i \end{cases} \tag{14}$$

where *<sup>f</sup>* <sup>∈</sup> *<sup>S</sup>*<sup>1</sup> *<sup>F</sup>*(.,κ(.)*x*) and *x* is defined by (8).

We show that a point *<sup>x</sup>* is a fixed point for <sup>Φ</sup> if and only if *<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*]. Assume *x* is a fixed point to Φ. Hence,

$$\mathbf{x}(\eta) = \begin{cases} \begin{array}{l} 0, \eta \in [-r, 0], \\ \mathfrak{K}\_{1}(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\pi}) \\ + \int\_{0}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\pi}) d\tau \\ + \int\_{0}^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_{2}(\eta - \tau) f(\tau) d\tau \\ + \sum\_{0 < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\overline{\pi}(\eta\_{i}^{-})), \eta \in \mathbb{J}. \end{array} \end{cases}$$

Therefore,

$$
\overline{\boldsymbol{\pi}}(\boldsymbol{\eta}) = \begin{cases}
\begin{array}{c}
\psi(\boldsymbol{\eta}),\boldsymbol{\eta} \in [-r,0], \\
\mathfrak{K}\_{1}(\boldsymbol{\eta})[\boldsymbol{\psi}(\boldsymbol{0}) - h(\boldsymbol{0},\boldsymbol{\psi})] + h(\boldsymbol{\eta},\boldsymbol{\varkappa}(\boldsymbol{\eta})\overline{\boldsymbol{\pi}}) \\
+ \int\_{0}^{\boldsymbol{\eta}} (\boldsymbol{\eta} - \boldsymbol{\tau})^{a-1} A \mathfrak{K}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) h(\boldsymbol{\tau},\boldsymbol{\varkappa}(\boldsymbol{\tau})\overline{\boldsymbol{\pi}}) d\boldsymbol{\pi} \\
+ \int\_{0}^{\boldsymbol{\eta}} (\boldsymbol{\eta} - \boldsymbol{\tau})^{a-1} \mathfrak{K}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) f(\boldsymbol{\tau}) d\boldsymbol{\pi} \\
+ \sum\_{0 < \boldsymbol{\eta}\_{i} < \boldsymbol{\eta}} \mathfrak{K}\_{1}(\boldsymbol{\eta} - \boldsymbol{\eta}\_{i}) I\_{i}(\overline{\boldsymbol{\pi}}(\boldsymbol{\eta}\_{i}^{-})), \boldsymbol{\eta} \in \mathcal{J},
\end{array}
$$

which means that *x* satisfies (3), and, thus, it is a mild solution for problem (1). In a similar way, it can be seen that if *x* satisfies (3), then *x* is a fixed point for Φ. Let Φ<sup>1</sup> : H→H and Φ<sup>2</sup> : Φ<sup>2</sup> → *P*(H) be such that

$$\Phi\_1(\mathbf{x})(\eta) = \begin{cases} 0, \eta \in [-r, 0], \\ \Re\_1(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\mathbf{x}}) \\ + \int\_0^{\eta} (\eta - \tau)^{\alpha - 1} A \pounds\_2(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\mathbf{x}}) d\tau, \eta \in \mathfrak{J}, \end{cases} \tag{15}$$

and a function *y* ∈ Φ2(*x*) if and only if

$$y(\eta) = \begin{cases} 0, \eta \in [-r, 0], \\ + \int\_0^\eta (\eta - \tau)^{a-1} \mathfrak{K}\_2(\eta - \tau) f(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{K}\_1(\eta - \eta\_i) I\_i(\overline{\mathfrak{X}}(\eta\_i^{-})), \eta \in \mathfrak{J}, \end{cases} \tag{16}$$

where *<sup>f</sup>* <sup>∈</sup> *<sup>S</sup>*<sup>1</sup> *F*(.,κ(.)*x*) . Notice that <sup>Φ</sup> <sup>=</sup> <sup>Φ</sup><sup>1</sup> <sup>+</sup> <sup>Φ</sup>2. Let *<sup>ξ</sup>* <sup>=</sup> sup*θ*∈[−*r*,0] ||*ψ*(*θ*)||,

$$\begin{aligned} \omega &= \, \_M\mathrm{M}\left[\zeta + ||A^{-\gamma}||d\_2(1+r\xi)\right] \\ &+ (1+r\xi)\left[||A^{-\gamma}||d\_2 + d\_2 \frac{\mathrm{C}\_{1-\gamma}\Gamma(1+\gamma)b^{a\gamma}}{\Gamma(1+a\gamma)\gamma} + \frac{M}{\Gamma(a)}\Delta||\varphi||\_{L^P\_{(l,\mathbb{R}^+)}}\right] \end{aligned}$$

and *υ* be a positive real number satisfying

$$\upsilon > \frac{\omega}{1 - \left[||A^{-\gamma}||d\_2r + d\_2\frac{\mathbb{C}\_{1-\gamma}\Gamma(1+\gamma)b^{\mu\gamma}}{\Gamma(1+a\gamma)\gamma}r + \frac{M}{\Gamma(a)}\Delta||\varrho||\_{L^P\_{(f,\mathbb{R}^+)}}r + \sigma M\right]}.\tag{17}$$

Put *<sup>B</sup><sup>υ</sup>* = {*<sup>u</sup>* ∈ H : ||*u*||H ≤ *<sup>ν</sup>*}. Due to (12), *<sup>υ</sup>* is well defined. The rest of the proof is divided in the following steps:

Step 1. This step shows that Φ(*Bν*) ⊆ *Bν*. Let *x* ∈ *B<sup>υ</sup>* and *y* ∈ Φ(*x*). There exists *<sup>f</sup>* <sup>∈</sup> *<sup>S</sup>*<sup>1</sup> *<sup>F</sup>*(.,κ(.)*x*) where

$$y(\eta) = \begin{cases} 0, \eta \in [-r, 0], \\ \Re\_1(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\pi}) \\ + \int\_0^{\eta} (\eta - \tau)^{\alpha - 1} A \mathfrak{K}\_2(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\pi}) d\tau \\ + \int\_0^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_2(\eta - \tau) f(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{K}\_1(\eta - \eta\_i) I\_i(\overline{\pi}(\eta\_i^{-})), \eta \in \mathcal{I}. \end{cases}$$

Let *η* ∈ *J*. For every *x* ∈ H, we get

$$||\varkappa(\eta)\overline{\varkappa}||\_{\Theta} = \int\_{-r}^{0} ||\overline{\varkappa}(\eta+\theta)||d\theta \le r(\xi+\upsilon)\_{\prime}$$

which implies that (*HF*2), || *<sup>f</sup>*(*τ*)|| ≤ *<sup>ϕ</sup>*(*τ*)(<sup>1</sup> <sup>+</sup> ||κ(*η*)*x*||Θ) <sup>≤</sup> *<sup>r</sup>*(*<sup>ξ</sup>* <sup>+</sup> *<sup>υ</sup>*); *<sup>a</sup>*.*e*.*<sup>τ</sup>* <sup>∈</sup> *<sup>J</sup>*. So, by (ii) of Lemma 1, and the Holder inequality, it follows that

$$\begin{aligned} &\quad ||\int\_0^\eta (\eta-\tau)^{\alpha-1} \mathfrak{K}\_2(\eta-\tau) f(\tau) d\tau|| \\ &\leq \quad \frac{M}{\Gamma(\alpha)} (1+r(\xi+\upsilon)) \int\_0^\eta (\eta-\tau)^{\alpha-1} \varrho(\tau) d\tau \\ &\leq \quad \frac{M}{\Gamma(\alpha)} \Delta ||\varrho||\_{L^p\_{(\downarrow,\mathbb{R}^+)}} (1+r(\xi+\upsilon)). \end{aligned}$$

Then, from (6), (7), (10) and (*H I*), one has, for *η* ∈ *J*,


This equation with (12) leads to

$$\begin{split} ||y||\_{\mathcal{H}} &\leq \quad \mathcal{M}\left[\check{\xi} + ||A^{-\gamma}||d\_{2}(1+r\check{\xi})\right] \\ &+ (1+r\check{\xi}) [||A^{-\gamma}||d\_{2} + d\_{2}\frac{\mathsf{C}\_{1-\gamma}\Gamma(1+\gamma)b^{\mathsf{a}\gamma}}{\Gamma(1+\mathsf{a}\gamma)\gamma} + \frac{\mathsf{M}}{\Gamma(\mathsf{a})}\Delta||\varphi||\_{L^{p}\_{(\J,\mathbb{R}^{+})}} \, ] \\ &+ \upsilon[||A^{-\gamma}||d\_{2}\tau + d\_{2}\frac{\mathsf{C}\_{1-\gamma}\Gamma(1+\gamma)b^{\mathsf{a}\gamma}}{\Gamma(1+\mathsf{a}\gamma)\gamma}\tau + \frac{\mathsf{M}}{\Gamma(\mathsf{a})}\Delta||\varphi||\_{L^{p}\_{(\J,\mathbb{R}^{+})}}\, \tau + \sigma\mathsf{M}] \\ &<\,\,\,\nu. \end{split}$$

Then, Φ(*Bυ*) ⊆ *Bυ*.

Step 2. Φ<sup>1</sup> is a contraction with a contraction constant *k* < <sup>1</sup> 2 .

Let *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>B</sup><sup>υ</sup>* and *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*. Then, ||κ(*η*)*<sup>u</sup>* <sup>−</sup> <sup>κ</sup>(*η*)*v*||<sup>Θ</sup> <sup>=</sup> % <sup>0</sup> <sup>−</sup>*<sup>r</sup>* ||*u*(*<sup>η</sup>* <sup>+</sup> *<sup>θ</sup>*) <sup>−</sup> *<sup>v</sup>*(*<sup>η</sup>* <sup>+</sup> *<sup>θ</sup>*)||*d<sup>θ</sup>* <sup>≤</sup> *r*||*u* − *v*||H. From (6), (7) and (11), for every *u*, *v* ∈ *B<sup>υ</sup>* and any *η* ∈ *J*, we have that


which yields with (9) that Φ<sup>1</sup> is a contraction with a contraction constant *k* < <sup>1</sup> 2 .

Step 3. Φ<sup>2</sup> has a closed graph and Φ2(*x*); *x* ∈ *B<sup>υ</sup>* is compact.

Assume (*xn*)*n*≥<sup>1</sup> and (*yn*)*n*≥<sup>1</sup> are sequences in *B<sup>υ</sup>* where *xn* → *x*, *yn* → *y* and *yn* ∈ Φ2(*xn*); *n* ≥ 1. Then,

$$y\_n(\eta) = \begin{cases} 0, \eta \in [-r, 0], \\ + \int\_0^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_2(\eta - \tau) f\_n(\tau) d\tau \\ + \sum\_{0 < \eta\_k < \eta} \mathfrak{K}\_1(\eta - \eta\_k) I\_i(\ge\_n (\eta\_k^{-})), \eta \in I\_i \end{cases} \tag{18}$$

where *fn* <sup>∈</sup> *<sup>τ</sup>*<sup>1</sup> *F*(.,κ(.)*xn*) . Using (*HF*2), it yields that

$$||f\_n(\eta)|| \le \varrho(\eta)(1 + r(\upsilon + \xi)), \text{a.e.} \eta \in f.$$

So, (*fn*)*n*≥<sup>1</sup> is bounded in *<sup>L</sup>P*(*J*, *<sup>E</sup>*) and, hence, there exists a subsequence of { *fn*}<sup>∞</sup> *<sup>n</sup>*=1. We denote them by (*fn*)*n*≥1, where *fn* −→ *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>P*(*J*, *<sup>E</sup>*). From Mazur's Lemma, there exists a sequence of convex combination, {*zn*}<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> of { *fn*}<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> that converges almost everywhere to *f* . Note that by (*HF*2), again, for any *η* ∈ *J*, *τ* ∈ (0, *η*] and any *n* ≥ 1,

$$||(\eta - \tau)^{a-1} f\_n(\tau)|| \le |\eta - \tau|^{|a-1|} \rho(\tau) (1 + r(\upsilon + \xi)) \in L^P((0, \eta], \mathbb{R}^+).$$

Set

$$\tilde{y}\_n(\eta) = \begin{cases} 0, \eta \in [-r, 0]\_\prime \\ + \int\_0^\eta (\eta - \tau)^{n-1} \mathfrak{K}\_2(\eta - \tau) z\_n(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{K}\_1(\eta - \eta\_i) I\_i(\mathfrak{x}\_n(\eta\_i^{-})) \, \eta \in \mathfrak{J}. \end{cases} \tag{19}$$

Note that by (18), *<sup>y</sup><sup>n</sup>*(*η*) <sup>→</sup> *<sup>y</sup>*(*η*), *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*. Moreover, since <sup>κ</sup>(*η*)*xn* <sup>→</sup> <sup>κ</sup>(*η*)*x*; *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*, *<sup>F</sup>*(*η*, .); *<sup>a</sup>*.*e*. *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>* is upper semicontinuous, it yields *<sup>f</sup>*(*η*) <sup>∈</sup> *<sup>F</sup>*(*η*,κ(*η*)*x*), *<sup>a</sup>*.*e*. Therefore, from the continuity of K2(*η* − *τ*); *τ* ∈ [0, *η*], *Ii* (*i* = 1, 2, ...), and by taking the limit of (19) as *n* → ∞ , one gets *y* ∈ Φ2(*x*).

To prove that the values of Φ<sup>2</sup> are compact, assume *x* ∈ H and *yn* ∈ Φ2(*x*), *n* ≥ 1. Using similar arguments to the above, we get that {*yn* : *n* ≥ 1} has a convergent subsequence (*<sup>y</sup>*)*n*≥1. So, <sup>Φ</sup>2(*x*) is relatively compact. Since the graph of <sup>Φ</sup><sup>2</sup> is closed its values are closed and, hence, Φ2(*x*) is relatively compact in H.

Step 4. We claim that the subsets *<sup>Z</sup>*|*Ji* (*<sup>i</sup>* <sup>=</sup> 0, 1, . . . , *<sup>m</sup>*) are equicontinuous, where

$$Z\_{|\overline{\boldsymbol{l}}|} = \{ \boldsymbol{y}^\* \in \mathbb{C}(\overline{f}\_{\boldsymbol{l}}, \boldsymbol{E}) : \boldsymbol{y}^\*(\boldsymbol{\eta}) = \boldsymbol{y}(\boldsymbol{\eta}), \boldsymbol{\eta} \in (\eta\_{\boldsymbol{l}}, \eta\_{\boldsymbol{i+1}}], \boldsymbol{y}^\*(\boldsymbol{\eta}\_{\boldsymbol{i}}) = \boldsymbol{y}(\eta\_{\boldsymbol{i}}^+), \boldsymbol{y} \in \Phi\_2(\boldsymbol{x}), \boldsymbol{x} \in B\_{\boldsymbol{\upsilon}}\}.$$

Assume *<sup>y</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>Z</sup>*|*Ji* . Then, there exists *<sup>x</sup>* <sup>∈</sup> *<sup>B</sup><sup>υ</sup>* and *<sup>f</sup>* <sup>∈</sup> *<sup>S</sup>*<sup>1</sup> *<sup>F</sup>*(.,κ(.)*x*) , where, for *η* ∈ *Ji*,

$$\begin{aligned} y^\*(\eta) &= \int\_0^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_2(\eta - \tau) f(\tau) d\tau \\ &+ \sum\_{0 < \eta\_k < \eta} \mathfrak{K}\_1(\eta - \eta\_k) I\_k(\overline{\mathfrak{x}}(\eta\_k^{-}))\_{\eta} \end{aligned}$$

and *y*∗(*ηi*) = *y*(*η*<sup>+</sup> *<sup>i</sup>* ).

Case 1. Let *η*1, *η*<sup>2</sup> (*η*<sup>1</sup> < *η*2) be two points in (*ηi*, *ηi*+1]. Then,

 *y*∗(*η*2) − *y*∗(*η*1) ≤ || *<sup>η</sup>*<sup>2</sup> 0 (*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1K2(*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*f*(*τ*)*d<sup>τ</sup>* − *<sup>η</sup>*<sup>1</sup> 0 (*η*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1K2(*η*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*f*(*τ*)|| +|| ∑ 0<*ηk*<*η*<sup>2</sup> K1(*η*<sup>2</sup> − *ηk*)*Ik*(*x*(*η*<sup>−</sup> *<sup>k</sup>* )) − ∑ 0<*ηi*<*η*<sup>1</sup> K1(*η*<sup>1</sup> − *ηk*)*Ik*(*x*(*η*<sup>−</sup> *<sup>k</sup>* ))|| ≤ || *<sup>η</sup>*<sup>2</sup> *η*1 (*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1K2(*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*f*(*τ*)*dτ*|| + *<sup>η</sup>*<sup>1</sup> 0 <sup>|</sup>(*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−<sup>1</sup> <sup>−</sup> (*η*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1| ||K2(*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*f*(*τ*)||*d<sup>τ</sup>* <sup>+</sup>|| *<sup>η</sup>*<sup>1</sup> 0 (*η*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*α*−1||K2(*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*f*(*τ*) <sup>−</sup> <sup>K</sup>2(*η*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*f*(*τ*)|| *<sup>d</sup><sup>τ</sup>* . + ∑ 0<*ηk*<*η*<sup>2</sup> ||K1(*η*<sup>2</sup> − *ηk*) − K1(*η*<sup>1</sup> − *ηk*)|| ||*Ii*(*x*(*η*<sup>−</sup> *<sup>i</sup>* ))|| = *i*=4 ∑ *i*=1 *Ii*.

The hypothesis (*HF*2) implies || *f*(*η*)|| ≤ *ϕ*(*η*) (1 + *r*(*υ* + *ξ*)), *a*.*e*.*η* ∈ *J*, and, hence, by Lemma 1, we get

$$\begin{split} \lim\_{\eta\_{2}\to\eta\_{1}} I\_{1} &= \lim\_{\eta\_{2}\to\eta\_{1}} ||\int\_{\eta\_{1}}^{\eta\_{2}} (\eta\_{2}-\tau)^{a-1} \mathfrak{K}\_{2}(\eta\_{2}-\tau) f(\tau) d\tau|| \\ &\leq \frac{M(1+r(\upsilon+\xi))}{\Gamma(a)} \lim\_{\eta\_{2}\to\eta\_{1}} \int\_{\eta\_{1}}^{\eta\_{2}} (\eta\_{2}-\tau)^{a-1} \varrho(\tau) d\tau \\ &= \frac{M(1+r(\upsilon+\xi))}{\Gamma(a)} ||\varrho||\_{L^{P}([J\mathbb{R}^{+}]} \lim\_{\eta\_{2}\to\eta\_{1}} (\int\_{\eta\_{1}}^{\eta\_{2}} (\eta\_{2}-\tau)^{\frac{P(\kappa-1)}{P-1}} d\tau) \frac{^{P-1}}{^{P}} = 0. \end{split}$$

For *I*2, we have

$$\begin{split} \lim\_{\eta\_{2}\to\eta\_{1}} I\_{2} &\leq \lim\_{\eta\_{2}\to\eta\_{1}} \int\_{0}^{\eta\_{1}} |(\eta\_{2}-\tau)^{a-1} - (\eta\_{1}-\tau)^{a-1}| \, ||\mathfrak{K}\_{2}(\eta\_{2}-\tau)f(\tau)|| |d\tau| \\ &= \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(a)} \lim\_{\eta\_{2}\to\eta\_{1}} \int\_{0}^{\eta\_{1}} |(\eta\_{2}-\tau)^{a-1} - (\eta\_{1}-\tau)^{a-1}| \, \rho(\tau)d\tau. \end{split}$$

Note that *ω* = *<sup>α</sup>*−<sup>1</sup> <sup>1</sup><sup>−</sup> <sup>1</sup> *P* <sup>∈</sup> (−1, 0), then, for *<sup>τ</sup>* <sup>&</sup>lt; *<sup>η</sup>*<sup>1</sup> , we have (*η*<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*<sup>ω</sup>* <sup>≥</sup> (*η*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*)*ω*. As an application of Lemma 3 in [8] and considering *<sup>P</sup>*−<sup>1</sup> *<sup>P</sup>* ∈ (0, 1), we get

$$|\left[\left(\eta\_1-\tau\right)^{\overline{\omega}}\right]^{1-\frac{1}{\overline{p}}}-\left[\left(\eta\_2-\tau\right)^{\overline{\omega}}\right]^{\frac{p-1}{\overline{p}}}|\;\;\leq\left[\left(\eta\_1-\tau\right)^{\overline{\omega}}-\left(\eta-\tau\right)^{\overline{\omega}}\right]^{\frac{p-1}{\overline{p}}}.$$

Then,

$$|(\eta\_1 - \tau)^{a-1} - (\eta\_2 - \tau)^{a-1}| \le \left[ (\eta\_1 - \tau)^{\overline{\omega}} - (\eta\_2 - \tau)^{\overline{\omega}} \right]^{\frac{p-1}{p}}.$$

This leads to

$$|(\eta - \tau)^{a-1} - (\eta + \lambda - \tau)^{a-1}|^{\frac{p-1}{p}} \le \left[ (\eta - \tau)^{\overline{\omega}} - (\eta + \lambda - \tau)^{\overline{\omega}} \right].$$

Therefore,

$$\begin{split} &\quad \lim\_{\eta\_{2}\to\eta\_{1}} I\_{2} \\ &\leq \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(\alpha)} \lim\_{\eta\_{2}\to\eta\_{1}} \int\_{0}^{\eta\_{1}} |(\eta\_{2}-\tau)^{\alpha-1} - (\eta\_{1}-\tau)^{\alpha-1}| \varrho(\tau) d\tau \\ &\leq \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(\alpha)} \lim\_{\eta\_{2}\to\eta\_{1}} \left[\int\_{0}^{\eta\_{1}} |(\eta\_{2}-\tau)^{\alpha-1} - (\eta\_{1}-\tau)^{\alpha-1}|^{\frac{p}{p-1}} d\tau\right]^{\frac{p-1}{p}} ||\varrho||\_{L^{p}\_{(j,\mathbb{R}^{+})}} \\ &\leq \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(\alpha)} \lim\_{\eta\_{2}\to\eta\_{1}} \left[\int\_{0}^{\eta\_{1}} [(\eta\_{2}-\tau)^{\overline{\alpha}} - (\eta\_{1}-\tau)^{\overline{\alpha}}] d\tau\right]^{\frac{p-1}{p}} ||\varrho||\_{L^{p}\_{(j,\mathbb{R}^{+})}} \\ &\leq \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(\alpha)} \lim\_{\eta\_{2}\to\eta\_{1}} \left[\frac{1}{\omega+1} [\eta\_{2}^{\overline{\alpha}+1} - (\eta\_{2}-\eta\_{1})^{\overline{\alpha}+1} - \eta\_{1}^{\overline{\alpha}+1}] \xrightarrow{\frac{p-1}{T}} ||\varrho||\_{L^{p}\_{(j,\mathbb{R}^{+})}} \\ &= \quad 0. \end{split}$$

For *I*3,

$$\lim\_{\eta\_2 \to \eta\_1} I\_3 \le \lim\_{\eta\_2 \to \eta\_1} ||\int\_0^{\eta\_1} (\eta\_1 - \tau)^{a-1} ||\mathfrak{A}\_2(\eta\_2 - \tau) f(\tau) - \mathfrak{A}\_2(\eta\_1 - \tau) f(\tau)|| \, d\tau.$$

Observe that for every *τ* ∈ [0, *η*],

$$\begin{aligned} & \quad (\eta\_1 - \tau)^{\alpha - 1} ||\mathcal{K}\_{\alpha}(\eta\_2 - \tau)f(\tau) - \mathcal{K}\_{\alpha}(\eta\_1 - \tau)f(\tau)|| \\ & \le \quad \frac{2M(\nu + 1)}{\Gamma(\alpha)}(\eta\_1 - \tau)^{\alpha - 1}\rho(\tau) \in L^P(f, \mathbb{R}^+). \end{aligned}$$

Moreover, since {*η*(*η*) : *η* > 0} is equicontinuous, and, using the Lebesgue-dominated convergence theorem, one gets

$$\begin{split} \lim\_{\eta\_{2}\to\eta\_{1}} I\_{3} &\leq \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(a)} \lim\_{\eta\_{2}\to\eta\_{1}} \int\_{0}^{\eta\_{1}} (\eta\_{1}-\tau)^{a-1} ||\mathfrak{A}\_{2}(\eta\_{2}-\tau)-\mathfrak{A}\_{2}(\eta\_{1}-\tau)||\varrho(\tau)d\tau \\ &= \quad \frac{M(1+r(\upsilon+\xi))}{\Gamma(a)} \int\_{0}^{\eta\_{1}} \int\_{0}^{\infty} \theta(\eta\_{1}-\tau)^{a-1}\zeta\_{a}(\theta) \times \\ &\quad \Big[\lim\_{\eta\_{2}\to\eta\_{1}} ||(Y((\eta\_{2}-\tau)^{a}\theta)-Y(\eta\_{1}-\tau)^{a}\theta)|| \, | \, d\theta \, \rho(\tau)d\tau \\ &= \quad 0. \end{split}$$

For *I*4,

$$\lim\_{\eta\_2 \to \eta\_1} I\_4 \le \sigma \upsilon \lim\_{\eta\_2 \to \eta\_1} \sum\_{0 < \eta\_k < \eta\_2} ||\mathfrak{K}\_1(\eta\_2 - \eta\_k) - \mathfrak{K}\_1(\eta\_1 - \eta\_k)|| = 0.$$

Case 2. *η* = *η<sup>i</sup>* , *i* = 1, ... , *m*. Assume *δ* > 0, *η<sup>i</sup>* + *δ* ∈ (*ηi*, *ηi*+1] and *λ* > 0 where *η<sup>i</sup>* < *λ* < *η<sup>i</sup>* + *δ* ≤ *ηi*+1. Hence, as above, it can be shown that

$$\|y^\*(\eta\_i + \delta) - y^\*(\eta\_i)\| = \lim\_{\lambda \to \eta\_i^+} \|y(\eta\_i + \delta) - y(\lambda)\| = 0.$$

Then, *<sup>Z</sup>*|*Ji* (*<sup>i</sup>* <sup>=</sup> 0, 1, . . . , *<sup>m</sup>*) are equicontinuous.

Step 5. Set *B*<sup>1</sup> = *conv*Φ(*Bυ*) and *Bn* = *conv*Φ(*Bn*−1), *n* ≥ 2. Then, the sequence (*Bn*), *n* ≥ 1 is a decreasing sequence of not empty, closed and bounded subsets of H. So, the set *B* = 5 *n*≥1 *Bn* is bounded, closed, convex and Φ(*B*) ⊂ *B*. Next, we show that *B* is compact.

According to the generalized Cantor's intersection property, we only need to prove that

$$\lim\_{n \to \infty} \chi\_{\mathcal{H}}(B\_n) = 0,\tag{20}$$

where *<sup>χ</sup>*<sup>H</sup> is the Hausdorff measure of noncompactness on <sup>H</sup>. Assume *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> and *<sup>n</sup>* <sup>≥</sup> 1 are fixed. From the fact that Φ<sup>1</sup> is a contraction with a contraction constant *k* < <sup>1</sup> <sup>2</sup> , it follows that

$$\begin{aligned} &\chi\_{\mathcal{H}}\Phi(\mathcal{B}\_{n-1})\\ &\leq \quad \chi\_{\mathcal{H}}\Phi\_{1}(\mathcal{B}\_{n-1}) + \chi\_{\mathcal{H}}\Phi\_{2}(\mathcal{B}\_{n-1})\\ &\leq \quad \frac{1}{2}\chi\_{\mathcal{H}}(\mathcal{B}\_{n-1}) + \chi\_{\mathcal{H}}\Phi\_{2}(\mathcal{B}\_{n-1}).\end{aligned} \tag{21}$$

Let *<sup>ε</sup>* > 0. Using Lemma 5 in [39], there is a (*yk*)*k*≥<sup>1</sup> in <sup>Φ</sup>2(*Bn*−1) with

$$
\chi\_{\mathcal{H}} \Phi\_2(B\_{n-1}) \le 2 \chi\_{\mathcal{H}} \{ y\_k : k \ge 1 \} + \varepsilon.
$$

From the fact that the subsets *<sup>Z</sup>*|*Ji* (*<sup>i</sup>* <sup>=</sup> 0, 1, . . . , *<sup>m</sup>*) are equicontinuous, one obtains

$$\begin{aligned} &\chi\_{\mathcal{H}}\Phi\_2(\mathcal{B}\_{n-1}) \\ &\le \ &2\chi\_{\mathcal{H}}\{y\_k:k\ge 1\}+\varepsilon \\ &\le \ &2\sup\_{\eta\in[0,b]}\chi\_E\{y\_k(\eta):k\ge 1\}+\varepsilon. \tag{22} \end{aligned}$$

Now, let *xk* ∈ *Bn*−<sup>1</sup> and *yk* ∈ Φ2(*xk*), *k* ≥ 1. Then, for every *k* ≥ 1, there is a *fk* <sup>∈</sup> *<sup>τ</sup>*<sup>1</sup> *<sup>F</sup>*(.,κ(*η*)*xk* ) such that, for any *<sup>η</sup>* ∈ *<sup>J</sup>*,

$$y\_k(\eta) = \begin{cases} 0, \eta \in [-r, 0], \\ \int\_0^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{R}\_2(\eta - \tau) f\_k(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{R}\_1(\eta - \eta\_i) I\_i(\overline{\mathfrak{x}}\_k(\eta\_i^{-})), \eta \in \mathcal{J}. \end{cases}$$

Note that the assumption (*H I*) implies that for *η* ∈ *J*,

$$\begin{split} \chi\_{\mathbb{E}} \left\{ \sum\_{0 < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\overline{\mathfrak{x}}\_{k}(\eta\_{i}^{-})) \right. \left. : \quad k \geq 1 \right\} \\ & \leq \ \ M \sum\_{i=1}^{i=m} \xi\_{i} \, \chi\_{\mathbb{E}} \{ \overline{\mathfrak{x}}\_{k}(\eta\_{i}^{-}) \} : k \geq 1 \right\} \\ & \leq \ \ M \sum\_{i=1}^{i=m} \xi\_{i} \, \chi\_{\mathbb{E}} \{ \mathfrak{x}\_{k}(\eta\_{i}^{-}) \} : k \geq 1 \right\} \\ & \leq \ \ M \, \chi\_{\mathcal{H}}(\mathcal{B}\_{n-1}) \sum\_{i=1}^{i=m} \xi\_{i} \, . \end{split} \tag{23}$$

Moreover, from (4) , we have that for *a*.*e*.*τ* ∈ *J*,

$$\begin{split} \chi\_{\mathbb{E}}\{f\_{k}(\tau) \quad : \quad k \ge 1\} &\leq \chi\{F(\tau, \varkappa(\tau)\overline{x}\_{k}) : k \ge 1\} \\ &\leq \quad \beta(\tau) \sup\_{\theta \in [-r, 0]} \chi\{\overline{x}\_{k}(\tau + \theta) : k \ge 1\} \\ &\leq \quad \beta(\tau) \sup\_{\delta \in [-r, \tau]} \chi\{\overline{x}\_{k}(\delta) : k \ge 1\} \\ &\leq \quad \beta(\tau) \sup\_{\delta \in [0, \tau]} \chi\{x\_{k}(\delta) : k \ge 1\} \\ &\leq \quad \beta(\tau)\chi\_{\mathcal{H}}(B\_{n-1}) = \gamma(\eta). \end{split}$$

Again, by (*HF*2)∗, for every *k* ≥ 1, and for almost *η* ∈ *J*, || *fk*(*η*)|| ≤ *ϕ*(*η*) (1 + *r*(*υ* + *ξ*)) and, hence, { *fk* : *k* ≥ 1} is integrably bounded. As a consequence of Lemma 4 in [40], there is a compact set *K* ⊆ *E*, a measurable set *J* ⊂ *J* having a measure less than and {*z <sup>k</sup>*} ⊂ *<sup>L</sup>P*(*J*, *<sup>E</sup>*) such that for every *<sup>τ</sup>* <sup>∈</sup> *<sup>J</sup>*, {*z <sup>k</sup>* (*τ*) : *k* ≥ 1} ⊆ *K* and

$$||f\_k(\tau) - z\_k^\epsilon(\tau)|| < 2\gamma(\tau) + \epsilon \text{ for all } k \ge 1 \text{ and all } \tau \in f - f\_\epsilon. \tag{25}$$

Then, by (24) and (25) and Minkowski's inequality, it follows that for *k* ≥ 1,

$$\begin{split} & \quad ||\int\_{I-I\_{\varepsilon}} (\eta - \tau)^{\kappa - 1} \mathfrak{s}\_{2} (\eta - \tau) (f\_{k}(\tau) - z\_{k}^{\varepsilon}(\tau)) d\tau|| \\ & \leq \quad \frac{\Lambda M}{\Gamma(\alpha)} ||f\_{k} - z\_{k}^{\varepsilon}||\_{L^{p}(I\_{0} - I\_{\varepsilon}, \mathbb{R}^{+})} (\int\_{I-I\_{\varepsilon}} (\eta - \tau)^{\frac{(\kappa - 1)\mathbb{P}}{P-1}} d\tau)^{\frac{p-1}{P}} \\ & \leq \quad \frac{\Lambda M}{\Gamma(\alpha)} ||f\_{k} - z\_{k}^{\varepsilon}||\_{L^{p}(I\_{0} - I\_{\varepsilon}, \mathbb{R}^{+})} \\ & \leq \quad \frac{\Lambda M}{\Gamma(\alpha)} (2||\gamma||\_{L^{p}(I - I\_{\varepsilon}, \mathbb{R}^{+})} + \epsilon b^{\frac{1}{p}}) \\ & = \quad \frac{\Lambda M}{\Gamma(\alpha)} \left( 2||\beta||\_{L^{p}(I\_{\varepsilon}, \mathbb{R}^{+})} \chi \eta \left( \mathcal{B}\_{n-1} \right) + \epsilon b^{\frac{1}{p}} \right), \end{split} \tag{26}$$

and

$$\begin{split} & \quad ||\int\_{I\_{\varepsilon}} (\eta - \tau)^{a-1} \mathfrak{z}\_{2} (\eta - \tau) f\_{k}(\tau) d\tau|| \\ & \leq \quad \frac{M}{\Gamma(a)} (1 + r(\upsilon + \xi)) \int\_{I\_{\varepsilon}} (\eta - \tau)^{a-1} \varrho(\tau) d\tau \\ & \leq \quad \frac{M}{\Gamma(a)} (1 + r(\upsilon + \xi)) ||\varrho||\_{L^{P}(I\_{\varepsilon}, \mathbb{R}^{+})} (\int\_{I\_{\varepsilon}} (\eta - \tau)^{\frac{(a-1)P}{P-1}} d\tau)^{\frac{P-1}{P}}. \end{split} \tag{27}$$

Moreover, from the fact that {*z <sup>k</sup>* (*τ*) : *k* ≥ 1}; *τ* ∈ *J* is contained in a compact subset, we get

$$\propto \{ \int\_{J-J\_{\varepsilon}} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_2(\eta - \tau) z\_k^{\varepsilon}(\tau) d\tau : k \ge 1 \} = 0.$$

Combining this relation with (26) and (27), it follows that

$$\begin{split} \chi\{\int\_{0}^{\eta} (\eta-\tau)^{a-1} \mathfrak{K}\_{2}(\eta-\tau) f\_{k}(\tau) d\tau \quad : \quad k \geq 1\} \\ &\leq \quad \frac{\Delta M}{\Gamma(a)} (2||\beta||\_{L^{p}(\mathfrak{J},\mathbb{R}^{+})} \chi\_{\mathcal{H}}(B\_{n-1}) + \varepsilon b^{\frac{1}{p}}) \\ &\quad + \frac{(1+r(\upsilon+\xi)M)}{\Gamma(a)}||\varphi||\_{L^{p}(\mathfrak{J}\_{\mathbb{C}},\mathbb{R}^{+})} \Delta\_{\varepsilon} \end{split} \tag{28}$$

where Δ = (% *J* (*η* − *τ*) (*α*−1)*P <sup>P</sup>*−<sup>1</sup> *dτ*) *P*−1 *<sup>P</sup>* . Using the fact that *ε* is chosen arbitrary, relation (28) becomes

$$\begin{aligned} \left\{ \chi \{ \int\_0^\eta (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_2(\eta - \tau) f\_k(\tau) d\tau \quad : \quad k \ge 1 \right\} \\ &\leq \quad \frac{2\Delta M}{\Gamma(\alpha)} ||\beta||\_{L^p(I, \mathbb{R}^+)} \chi\_{\mathcal{H}}(\mathcal{B}\_{n-1}). \end{aligned}$$

Using the above inequality and (21)–(23), in addition to the fact that *ε* is arbitrary, it follows that

$$\chi\_{\mathcal{H}}(B\_n) \le \left(\frac{4\Delta M}{\Gamma(a)} \, \vert \, \vert \mathcal{J} \vert \vert\_{L^P(I, \mathbb{R}^+)} + 2M \sum\_{i=1}^{i=m} \zeta\_i + \frac{1}{2} \right) \chi\_{\mathcal{H}}(B\_{n-1}) .$$

This leads to

$$\chi\_{\mathcal{H}}(\mathcal{B}\_n) \le (\frac{4\Delta M}{\Gamma(a)} \, ||\beta||\_{L^P(I, \mathbb{R}^+)} + M \sum\_{i=1}^{i=m} \varsigma\_i + \frac{1}{2})^{n-1} \chi\_{\mathcal{H}}(\mathcal{B}\_1), \,\forall n \ge 1.$$

The above inequality holds for any natural number *n*, and by (13) together with taking the limit as *n* → ∞, we get (20). Then, *B* is not empty and a compact subset of H. So, Φ : *B* → *Pck*(*B*) is completely continuous. By applying Lemma 3, we conclude that the fixed points set of Φ is not an empty subset of H. Furthermore, by arguing as in Step 1, we can prove that the set of fixed points of Φ is bounded and, hence, by Lemma 3, it is compact in <sup>H</sup>. Therefore, the set <sup>Σ</sup>*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*] is not empty and a compact subset of *H*.

#### **4. The Structure Topological of Σ***<sup>F</sup> <sup>ψ</sup>***[***−r***,** *b***]**

In the section we prove that Σ*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*] is an *Rδ*-set

**Definition 3** ([41])**.** *A topological space X, which is homotopy equivalent to a point, is called contractible. In other words, there is a continuous map h* : [0, 1] × *X* → *X*, *h*(0, .*x*) = *x and h*(1, *x*) = *x*<sup>0</sup> ∈ *X*.

**Lemma 5** ([41])**.** *Let A* ⊆ *X, where A is not empty and X is a complete metric space. Then, A is said to be Rδ-set if and only if it is an intersection of a decreasing sequence* {*An*} *of contractible sets and χX*(*An*) → 0, *as n* → ∞.

Now, consider the multi-valued function *<sup>F</sup>* : *<sup>J</sup>* <sup>×</sup> <sup>Θ</sup> <sup>→</sup> *Pck*(*E*) that is given by:

$$\widetilde{F}\left(\eta, u\right) := \begin{cases} |F(\eta, u)\_{\prime}| |u|| < \upsilon\_{\prime} \\ |F(\eta, \frac{\upsilon u}{||u||})\_{\prime}| |u|| \ge \upsilon\_{\prime} \end{cases}$$

where *<sup>υ</sup>* is defined by (17). Since *<sup>F</sup>* <sup>=</sup> *<sup>F</sup>* on *<sup>D</sup>υ*, the set of solutions consisting of mild solutions for Problem (1) is equal to the set of solutions consisting of mild solutions for the problem:

$$\begin{cases} \ ^cD\_{0,\eta}^a[\mathbf{x}(\eta) - h(\eta, \mathbf{x}(\eta)\mathbf{x})] \in Ax(\eta) + \widetilde{F}(\eta, \mathbf{x}(\eta)\mathbf{x}), \ a.e. \ \eta \in [0, b] - \{\eta\_1, \dots, \eta\_m\}, \\\ I\_i(\mathbf{x}(\eta\_i^-)) = x(\eta\_i^-) - x(\eta\_i^+), i = 1, \dots, m, \\\ x(\eta) = \psi(\eta), \eta \in [-r, 0]. \end{cases}$$

Obviously, *<sup>F</sup>* verifies (*HF*1) and, for *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*, *<sup>a</sup>*.*e*.,

$$||\widetilde{F}(\eta, u)|| \le \begin{cases} \varrho(\eta)(1 + ||u||) \le \varrho(\eta)(1 + r(\widetilde{\xi} + \upsilon)) = \zeta(\eta), ||u|| < \upsilon, \\\varrho(\eta)(1 + ||\frac{\upsilon u}{||u||}||) = \varrho(\eta)(1 + r(\widetilde{\xi} + \upsilon)) = \zeta(\eta), ||u|| \ge \upsilon. \end{cases}$$

Then, we can assume that *F* verifies the next condition: (*HF*2)<sup>∗</sup> There exists a function *<sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>P*(*I*, <sup>R</sup>+)(*<sup>P</sup>* <sup>&</sup>gt; <sup>1</sup> *<sup>α</sup>* ), where for every *z* ∈ Θ,

$$\|\|F(\eta, z)\|\| \le \mathcal{J}(\eta), \ a.e. \ \eta \in \mathcal{J}.$$

We recall the next Lemma. For its proof, we refer the reader to the second step in the proof of Theorem 3.5 in [13].

**Lemma 6.** *Assume that* (*HF*1) *and* (*HF*2)<sup>∗</sup> *are satisfied. Then, there exists a sequence of multifunctions* {*Fi*}<sup>∞</sup> *<sup>i</sup>*=<sup>1</sup> *with Fi* : *J* × Θ → *Pck*(*E*) *such that:*


**Remark 2.** *(Ref. [19]) The property (iv) in Lemma 6 implies that, for almost η* ∈ *J, gi*(*η*, .)*, i* ≥ 1 *is continuous.*

Assume Σ*Fi <sup>ψ</sup>* [−*r*, *b*] is the mild solutions set of the following fractional neutral impulsive semilinear differential inclusions with delay:

$$\begin{cases} \ ^cD\_{0,\eta}^a[\mathbf{x}(\eta) - h(\eta, \varkappa(\eta)\mathbf{x})] \in A\mathbf{x}(\eta) + F\_i(\eta, \varkappa(\eta)\mathbf{x}), \ a.e.\ \eta \in [0, b] - \{\eta\_1, \dots, \eta\_{\parallel n}\},\\\ I\_i(\mathbf{x}(\eta\_i^-)) = \mathbf{x}(\eta\_i^-) - \mathbf{x}(\eta\_i^+), i = 1, \dots, m, \\\ \mathbf{x}(\eta) = \boldsymbol{\psi}(\eta), \boldsymbol{\eta} \in [-r, 0]. \end{cases} \tag{29}$$

**Theorem 2.** *Assume that the conditions in Theorem 1 after substituting* (*HF*2) *by* (*HF*2)∗ *are held. Then, there exists <sup>N</sup>*<sup>0</sup> <sup>∈</sup> <sup>N</sup> *such that, for <sup>i</sup>* <sup>≥</sup> *<sup>N</sup>*0*, the set* <sup>Σ</sup>*Fi <sup>ψ</sup>* [−*r*, *b*] *is compact and not empty in H*.

**Proof.** Let *i* be a fixed natural number. We define a multioperator Φ*<sup>i</sup>* : H → *P*(H) as the following : *y* ∈ Φ*i*(*x*) if and only if

$$y(\eta) = \begin{cases} 0, \eta \in [-r, 0]\_{\prime} \\ \mathfrak{K}\_{1}(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\pi}) \\ + \int\_{0}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\pi}) d\tau \\ + \int\_{0}^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_{2}(\eta - \tau) f(\tau) d\tau \\ + \sum\_{0 < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\overline{\pi}(\eta\_{i}^{-}))\_{i} \eta \in \mathbb{J}\_{i} \end{cases}$$

where *<sup>f</sup>* <sup>∈</sup> *<sup>τ</sup>*<sup>1</sup> *Fi*(.,κ(.)*x*) . Due to Lemma 5, *Fi* verifies (*F*1), (*F*2)∗. As a result of Theorem 1, Φ*<sup>i</sup>* is closed , Φ*i*(*Bυ*) ⊆ *B<sup>υ</sup>* and Φ*i*(*Bυ*) is equicontinuous. Set *B*1,*<sup>i</sup>* = *conv*Φ*i*(*Bυ*) and *Bn*,*<sup>i</sup>* = *conv*Φ*i*(*Bn*−1,*i*), *<sup>n</sup>* ≥ 2. As in Theorem 1, the sequence (*Bn*,*i*), *<sup>n</sup>* ≥ 1 is a decreasing sequence of non-empty, closed and bounded subsets of H. We show that

$$\lim\_{n \to \infty} \mathcal{X}\_{\mathbb{C}([-r, b], E)}(B\_{n, i}) = 0. \tag{30}$$

Let *ε* > 0. Choose a natural number *N*<sup>0</sup> with 31−*N*<sup>0</sup> < *<sup>ε</sup>* 2||*β*||*LP*(*J*, <sup>R</sup>+) and let *i* > *N*<sup>0</sup> be a fixed natural number. Using a similar argument as the one used in the proof of Theorem 1, one gets

$$\begin{aligned} & \chi\_{\mathcal{H}}(B\_{\eta,i}) \\ & \le \sup\_{\eta \in I} \chi\_E\{y\_k(\eta) : k \ge 1\} + \frac{\varepsilon}{2}, \end{aligned}$$

where

$$y\_k(\eta) = \begin{cases} 0, \eta \in [-r, 0] \\ \int\_0^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_2(\eta - \tau) f\_k(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{K}\_1(\eta - \eta\_i) I\_i(\overline{\mathfrak{x}}(\eta\_i^{-}))\_{\prime} \eta \in f\_k \end{cases}$$

and *fk* <sup>∈</sup> *<sup>τ</sup>*<sup>1</sup> *Fi*(.,κ(*η*)*xk* ) . Next, due to Remark 4.2 in [7], it follows that for any bounded subset *D* ⊂ Θ,

$$\chi\_E(\mathcal{F}\_i(\eta, D)) \le \beta(\eta) [\sup\_{\theta \in [-r, \eta]} \chi\_E\{z(\theta) : z \in D\} + \mathfrak{J}^{1-i}].\tag{31}$$

Then, it yields from (ii) in Lemma 5 and (31), for *a*.*e*.*τ* ∈ *J*,

$$\begin{split} \chi\_{E}(\{f\_{k}(\tau) \quad:\ k \geq 1\} \\ &\leq \chi\_{E}\{F\_{i}(\tau,\varkappa(\tau)\mathbf{x}\_{k}):k \geq 1\} \\ &\leq \beta(\tau)[\sup\_{\theta \in [-r,0]} \chi\_{E}\{\mathbf{x}\_{k}(\tau+\theta):k \geq 1\} + 3^{1-N\_{0}}] \\ &\leq \beta(\tau)[\sup\_{\delta \in [-r,\tau]} \chi\_{E}\{\mathbf{x}\_{k}(\delta):k \geq 1\} + 3^{1-N\_{0}}] \\ &\leq \beta(\tau)[\sup\_{\theta \in [0,\tau]} \chi\_{E}\{\mathbf{x}\_{k}(\delta):k \geq 1\} + 3^{1-N\_{0}}] \\ &\leq \beta(\tau)\chi\_{\mathcal{H}}(\mathcal{B}\_{n-1,i}) + \beta(\tau)3^{1-N\_{0}} = \overline{\gamma}(\tau). \end{split}$$

As in (28) but by using (32) instead of (24), we get

$$\begin{split} \chi\{\int\_{0}^{\eta} (\eta-\tau)^{a-1} \mathfrak{K}\_{2}(\eta-\tau) f\_{k}(\tau) d\tau \quad :\quad k \geq 1\} \\ &\leq \quad \frac{\Delta M}{\Gamma(a)} (2||\beta||\_{L^{P}(I,\mathbb{R}^{+})} \chi\_{\mathcal{H}}(B\_{n-1}) + \varepsilon b^{\frac{1}{P}}) + \frac{\varepsilon}{2} \\ &\quad + \frac{M}{\Gamma(a)} (1+r\upsilon+r\xi) \times \\ ||\phi||\_{L^{P}(I,\mathbb{R}^{+})} (\int\_{I\_{\xi}} (\eta-\tau)^{\frac{P}{P-1}} d\tau)^{\frac{P-1}{P}}. \end{split}$$

Similarly, as in the proof of Theorem 1, we confirm the validity of (30). Therefore, by the generalized Cantor's intersection property, the set *Bi* is not empty and compact in H. As in Theorem 1, the fixed points set of the multivalued function Φ*<sup>i</sup>* : *Bi* → *Pck*(*Bi*) is not empty and a compact subset in <sup>H</sup>. Consequently, the set <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*] is not empty and a compact subset of *H*.

**Theorem 3.** *Under the conditions of Theorem 2,* ∑*<sup>F</sup> <sup>ψ</sup>*[−*r*, *<sup>b</sup>*] = <sup>∩</sup><sup>∞</sup> *<sup>n</sup>*=*N*<sup>0</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*]. **Proof.** In view of (iii) in Lemma 8, it can be seen that ∑*<sup>F</sup> <sup>ψ</sup>*[−*r*, *<sup>b</sup>*] ⊆ ∩<sup>∞</sup> *<sup>n</sup>*=*N*<sup>0</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*]. Let *<sup>x</sup>* ∈ ∩<sup>∞</sup> *<sup>n</sup>*=*N*<sup>0</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *<sup>b</sup>*]. Then, there is <sup>f</sup>*<sup>n</sup>* <sup>∈</sup> *<sup>τ</sup>*<sup>1</sup> *Fn*(.,κ(.)*x*) , *n* ≥ *N*<sup>0</sup> such that

$$\overline{\mathbf{x}}(\eta) = \begin{cases} \psi(\eta), \eta \in [-r, 0], \\ \mathcal{R}\_1(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\mathbf{x}}) \\ + \int\_0^{\eta} (\eta - \tau)^{a-1} A \mathfrak{A}\_2(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\mathbf{x}}) d\tau \\ + \int\_0^{\eta} (\eta - \tau)^{a-1} \mathfrak{A}\_2(\eta - \tau) \mathfrak{f}\_n(\tau) d\tau \\ + \sum\_{0 < \eta\_i < \eta} \mathfrak{A}\_1(\eta - \eta\_i) I\_i(\overline{\mathfrak{x}}(\eta\_i^{-})), \eta \in \mathcal{I}. \end{cases} \tag{33}$$

It follows from (*HF*2)∗ that

$$||f\_n(\eta)|| \le \zeta(\eta), \text{for } a.e.\eta \in \mathcal{J}.$$

This means that the sequence (f*n*)*n*≥<sup>1</sup> is weakly relatively compact in *<sup>L</sup>P*(*J*, *<sup>E</sup>*), so we can assume <sup>f</sup>*<sup>n</sup> <sup>f</sup>* weakly, where *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>P*(*J*, <sup>R</sup>+). As in the proof of Theorem 1, there is a sequence of convex combinations (*zn*)*n*≥<sup>1</sup> of (f*n*)*n*≥<sup>1</sup> that converges almost everywhere to *f* . Note that

$$\overline{\mathfrak{X}}(\eta) = \begin{cases} \psi(\eta), \eta \in [-r, 0], \\ \mathfrak{K}\_{1}(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\mathfrak{x}}) \\ \quad + \int\_{0}^{\eta} (\eta - \tau)^{\alpha - 1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\mathfrak{x}}) d\tau \\ \quad + \int\_{0}^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_{2}(\eta - \tau) z\_{n}(\tau) d\tau \\ \quad + \sum\_{0 < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\overline{\mathfrak{x}}(\eta\_{i}^{-})), \eta \in \mathfrak{J}\_{i} \end{cases} \tag{34}$$

and *zn*(*η*) <sup>∈</sup> *Fn* (*η*,κ(*η*)*x*), *<sup>n</sup>* <sup>≥</sup> 1. It yields, from (ii) of Lemma 8, that for almost *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*,

$$z\_n(\eta) \in \overline{\operatorname{co}} \mathcal{F}(\eta, \{ y \in \Theta : ||y - \varkappa(\eta)\overline{x}|| \le \mathfrak{Z}^{1-n} \}), n \ge 1, \iota$$

which implies that *<sup>f</sup>*(*η*) <sup>∈</sup> *<sup>F</sup>*(*η*,κ(*η*)*x*), for *<sup>a</sup>*.*e*. *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*. Moreover, using the fact that K2(*η*)(*η* > 0) is continuous, and taking the limit as *n* → ∞ in (34), one gets

$$
\overline{\boldsymbol{\pi}}(\boldsymbol{\eta}) = \begin{cases}
\begin{array}{c}
\psi(\boldsymbol{\eta}),\boldsymbol{\eta} \in [-r,0], \\
\hline
\mathfrak{K}\_{1}(\boldsymbol{\eta})[\psi(\boldsymbol{0}) - h(\boldsymbol{0},\boldsymbol{\psi})] + h(\boldsymbol{\eta},\boldsymbol{\varkappa}(\boldsymbol{\eta})\overline{\boldsymbol{\pi}}) \\
+ \int\_{0}^{\boldsymbol{\eta}} (\boldsymbol{\eta} - \boldsymbol{\tau})^{a-1} A \mathfrak{K}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) h(\boldsymbol{\tau},\boldsymbol{\varkappa}(\boldsymbol{\tau})\overline{\boldsymbol{\pi}}) d\boldsymbol{\pi} \\
+ \int\_{0}^{\boldsymbol{\eta}} (\boldsymbol{\eta} - \boldsymbol{\tau})^{a-1} \mathfrak{K}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) f(\boldsymbol{\tau}) d\boldsymbol{\pi} \\
+ \sum\_{0 < \boldsymbol{\eta}\_{i} < \boldsymbol{\eta}} \mathfrak{K}\_{1}(\boldsymbol{\eta} - \boldsymbol{\eta}\_{i}) I\_{i}(\overline{\boldsymbol{\varpi}}(\boldsymbol{\eta}\_{i}^{-})), \boldsymbol{\eta} \in \mathcal{J}.
\end{array}
$$

This means that *<sup>x</sup>* <sup>∈</sup> <sup>∑</sup>*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*].

To prove our main results, we need the next lemma.

**Lemma 7** ([19], Lemma 4.5)**.** *Assume that* (*X*, *d*) *and* (*Y*, *ρ*) *are two metric spaces. Then, if f* : (*M*, *d*) → (*Y*, *ρ*) *is locally Lipschitz, then it is Lipschitz on all subsets of X that are compact.*

**Theorem 4.** *Under the assumptions of Theorem 2*, *the set* ∑*<sup>F</sup> <sup>ψ</sup>*[−*r*, *b*] *is an Rδ-set in H provided that rd*1||*A*−*γ*|| <sup>&</sup>lt; 1.

**Proof.** Using Lemma 4 and Theorems 1–3, we only need to prove that ∑*Fn <sup>ψ</sup>* [−*r*, *b*], where *<sup>n</sup>* <sup>≥</sup> *<sup>N</sup>*<sup>0</sup> is contractible. Assume that *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> and *<sup>n</sup>* <sup>≥</sup> *<sup>N</sup>*<sup>0</sup> . Consider the following fractional neutral impulsive semilinear:

$$\begin{cases} \,^cD\_{0,\eta}^a\left[\mathbf{x}(\eta) - h(\eta, \varkappa(\eta)\mathbf{x})\right] = A\mathbf{x}(\eta) + g\_n(\eta, \varkappa(\eta)\mathbf{x}), \; a.e.\ \eta \in [0, b] - \{\eta\_1, \dots, \eta\_{\left[m\right]}\},\\\ I\_i(\mathbf{x}(\eta\_i^{-})) = \mathbf{x}(\eta\_i^{-}) - \mathbf{x}(\eta\_i^{+}), i = 1, \dots, m, \\\ \mathbf{x}(\eta) = \boldsymbol{\varphi}(\eta), \boldsymbol{\eta} \in [-r, 0]. \end{cases} \tag{35}$$

Using Lemma 6 and Remark 3, *gn*(., *u*) is measurable, and for *η* ∈ *J*, *a*.*e*., *gn*(*η*, .) is continuous. Since the multi-valued *F* satisfies (*F*2)∗and (*F*3), then, following the arguments employed in the proof of Theorem 2, the fractional differential Equation (35) has a mild solution *<sup>y</sup>* <sup>∈</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*] satisfying the following integral equation:

$$
\overline{\boldsymbol{y}}(\boldsymbol{\eta}) = \begin{cases}
\boldsymbol{\psi}(\boldsymbol{\eta}), \boldsymbol{\eta} \in [-r, 0], \\
\mathscr{R}\_{1}(\boldsymbol{\eta})[\boldsymbol{\psi}(\boldsymbol{0}) - h(\boldsymbol{0}, \boldsymbol{\psi})] + h(\boldsymbol{\eta}, \varkappa(\boldsymbol{\eta})\overline{\boldsymbol{y}}) \\
+ \int\_{0}^{\boldsymbol{\eta}} (\boldsymbol{\eta} - \boldsymbol{\tau})^{\alpha - 1} A \mathscr{R}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) h(\boldsymbol{\tau}, \varkappa(\boldsymbol{\tau})\overline{\boldsymbol{y}}) d\boldsymbol{\tau} \\
+ \int\_{0}^{\boldsymbol{\eta}} (\boldsymbol{\eta} - \boldsymbol{\tau})^{\alpha - 1} \mathscr{R}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) \mathscr{g}\_{\alpha}(\boldsymbol{\eta}, \varkappa(\boldsymbol{\eta})\overline{\boldsymbol{y}}) d\boldsymbol{\tau} \\
+ \sum\_{0 < \boldsymbol{\eta}\_{i} < \boldsymbol{\eta}} \mathscr{R}\_{i}(\boldsymbol{\eta} - \boldsymbol{\eta}\_{i}) I\_{i}(\overline{\boldsymbol{y}}(\boldsymbol{\eta}\_{i}^{-})), \boldsymbol{\eta} \in \overline{\boldsymbol{y}}.
\end{cases} \tag{36}
$$

Next, we show that the solution is unique. Assume that *<sup>x</sup>* <sup>∈</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*] is another mild solution for (35). Then,

$$\overline{\mathfrak{X}}(\eta) = \begin{cases} \psi(\eta), \eta \in [-r, 0], \\ \mathfrak{K}\_{1}(\eta)[\psi(0) - h(0, \psi)] + h(\eta, \varkappa(\eta)\overline{\mathfrak{x}}) \\ \quad + \int\_{0}^{\eta} (\eta - \tau)^{\alpha - 1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{\mathfrak{x}}) d\tau \\ \quad + \int\_{0}^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_{2}(\eta - \tau) g\_{\alpha}(\eta, \varkappa(\eta)\overline{\mathfrak{x}}) d\tau \\ \quad + \sum\_{0 < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\overline{\mathfrak{x}}(\eta\_{i}^{-})), \eta \in \mathfrak{f}. \end{cases} \tag{37}$$

Let *η* ∈ [0, *η*1] be fixed. Due to (6), (7), (11) (36) and (37), it yields


Now, from Lemma 5, the function *<sup>τ</sup>* <sup>→</sup> <sup>κ</sup>(*τ*)*<sup>x</sup>* is continuous from [0, *<sup>η</sup>*1] to <sup>Θ</sup> and, hence, the subset *Zx* <sup>=</sup> {κ(*τ*)*<sup>x</sup>* : *<sup>τ</sup>* <sup>∈</sup> [0, *<sup>η</sup>*1]} is compact in <sup>Θ</sup>. Similarly, the set *Zy* <sup>=</sup> {κ(*τ*)*<sup>y</sup>* : *<sup>τ</sup>* <sup>∈</sup> [0, *<sup>η</sup>*1]} is compact in <sup>Θ</sup> and, therefore, the set *Zx*,*<sup>y</sup>* <sup>=</sup> *Zx* <sup>∪</sup> *Zy* is compact in <sup>Θ</sup>, and consequently, [0, *η*1] × *Zx*,*<sup>y</sup>* is compact in [0, *η*1] × Θ. Thus, by (iv) in Lemma 6 and Lemma 7, there exists *cη*<sup>1</sup> > 0 , for which the estimate

$$||\mathfrak{g}\_{\boldsymbol{\eta}}(\boldsymbol{\pi},\boldsymbol{\varkappa}(\boldsymbol{\pi})\overline{\boldsymbol{y}}) - \mathfrak{g}\_{\boldsymbol{\eta}}(\boldsymbol{\pi},\boldsymbol{\varkappa}(\boldsymbol{\pi})\overline{\boldsymbol{x}})|| \leq c\_{\eta\_{1}}||\boldsymbol{\varkappa}(\boldsymbol{\pi})\overline{\boldsymbol{y}} - \boldsymbol{\varkappa}(\boldsymbol{\pi})\overline{\boldsymbol{x}}||\_{\boldsymbol{\Theta}^{\boldsymbol{\prime}}}$$

holds for *τ* ∈ *J*. Therefore, from (38), it yields

$$\begin{split} &||\overline{\boldsymbol{x}}(\eta)-\overline{\boldsymbol{y}}(\eta)|| \\ \leq & d\_{1}||A^{-\gamma}|| \,||\varkappa(\eta)\overline{\boldsymbol{y}}-\varkappa(\eta)\overline{\boldsymbol{x}}||\_{\Theta} \\ &+ d\_{1}||A^{-\gamma}||\frac{a\mathbb{C}\_{1-\gamma}\Gamma(1+\gamma)}{\Gamma(1+a\gamma)}\int\_{0}^{\eta}(\eta-\tau)^{a\gamma-1}||\varkappa(\tau)\overline{\boldsymbol{y}}-\varkappa(\tau)\overline{\boldsymbol{x}}||\_{\Theta}d\tau \\ &+ \frac{Mc\_{\eta\_{1}}}{\Gamma(a)}\int\_{0}^{\eta}(\eta-\tau)^{a-1}||\varkappa(\tau)\overline{\boldsymbol{y}}-\varkappa(\tau)\overline{\boldsymbol{x}}||\_{\Theta}d\tau. \end{split}$$

Note that when *τ* ∈ [0, *η*], we have

$$\begin{aligned} ||\varkappa(\tau)\overline{y} - \varkappa(\tau)\overline{\varkappa}||\_{\Theta} &= \int\_{-r}^{0} ||\overline{y}(\tau+\theta) - \overline{\varkappa}(\tau+\theta)|| d\theta \\ &\leq \quad r \sup\_{\delta \in [0,\tau]} ||\overline{y}(\delta) - \overline{\varkappa}(\delta)||. \end{aligned}$$

It yields

$$\begin{split} &||\overline{\boldsymbol{x}}(\eta) - \overline{\boldsymbol{y}}(\eta)|| \\ \leq & d\_{1}||A^{-\gamma}|| \,||\, \varkappa(\eta)\overline{\boldsymbol{y}} - \varkappa(\eta)\overline{\boldsymbol{x}}||\_{\Theta} \\ &+ r d\_{1}||A^{-\gamma}|| \frac{a \mathbb{C}\_{1-\gamma} \Gamma(1+\gamma)}{\Gamma(1+a\gamma)} \int\_{0}^{\eta} (\eta - \tau)^{a\gamma - 1} \sup\_{\delta \in [a,\tau]} ||\overline{\boldsymbol{y}}(\delta) - \overline{\boldsymbol{x}}(\delta)|| d\tau \\ &+ \frac{r Mc\_{\eta\_{1}}}{\Gamma(a)} \int\_{0}^{\eta} (\eta - \tau)^{a-1} \sup\_{\delta \in [0,\tau]} ||\overline{\boldsymbol{y}}(\delta) - \overline{\boldsymbol{x}}(\delta)|| d\tau. \end{split}$$

Since *x* and *y* are continuous on [0, *η*], there is *ρ* ∈ [0, *η*] with ||*x*(*ρ*) − *y*(*ρ*)|| = sup*δ*∈[0,*η*] ||*x*(*δ*) <sup>−</sup> *<sup>y</sup>*(*δ*)||. Then,

$$\begin{split} &\sup\_{\delta\in[0,\eta]} ||\overline{\pi}(\delta)-\overline{y}(\delta)|| = ||\overline{\pi}(\rho)-\overline{y}(\rho)|| \\ &\leq \quad d\_{1}||A^{-\gamma}|| ||\varkappa(\rho)\overline{y}-\varkappa(\rho)\overline{\varkappa}||\_{\Theta} \\ &\quad + rd\_{1}||A^{-\gamma}|| \frac{a\mathbb{C}\_{1-\gamma}\Gamma(1+\gamma)}{\Gamma(1+a\gamma)} \int\_{0}^{\rho} (\rho-\tau)^{a\gamma-1} \sup\_{\delta\in[0,\tau]} ||\overline{y}(\delta)-\overline{\varkappa}(\delta)||d\tau \\ &\quad + \frac{rMc\_{\eta\_{1}}}{\Gamma(a)} \int\_{0}^{\rho} (\rho-\tau)^{a-1} \sup\_{\delta\in[0,\tau]} ||\overline{y}(\delta)-\overline{\varkappa}(\delta)||d\tau \\ &\leq \quad rd\_{1}||A^{-\gamma}|| \sup\_{\delta\in[0,\eta]} ||\overline{\varkappa}(\delta)-\overline{y}(\delta)|| \\ &\quad + rd\_{1}||A^{-\gamma}|| \frac{a\mathbb{C}\_{1-\gamma}\Gamma(1+\gamma)}{\Gamma(1+a\gamma)} \int\_{a}^{\rho} (\rho-\tau)^{a\gamma-1} \sup\_{\delta\in[a,\tau]} ||\overline{y}(\delta)-\overline{\varkappa}(\delta)||d\tau \\ &\quad + \frac{rMc\_{\eta\_{1}}}{\Gamma(a)} \int\_{a}^{\rho} (\rho-\tau)^{a-1} \sup\_{\delta\in[a,\tau]} ||\overline{y}(\delta)-\overline{\varkappa}(\delta)||d\tau. \end{split}$$

Since *rd*1||*A*−*γ*|| <sup>&</sup>lt; 1, the last relations lead to

$$\begin{split} &\sup\_{\delta\in[0,\eta]} ||\overline{\mathfrak{x}}(\delta) - \overline{\mathfrak{y}}(\delta)|| \\ &\leq \quad \frac{1}{1 - rd\_{1}||A^{-\gamma}||} [\int\_{0}^{\rho} (\rho - \tau)^{a\gamma - 1} d\_{1} ||A^{-\gamma}|| \frac{r\mathcal{C}\_{1-\gamma}\Gamma(1+\gamma)}{\Gamma(1+a\gamma)}] \\ &+ \int\_{0}^{\rho} (\rho - \tau)^{a-1} \frac{r\mathcal{M}\mathcal{C}\_{V}}{\Gamma(\alpha)} \sup\_{\delta\in[0,\tau]} ||\overline{\mathfrak{y}}(\delta) - \overline{\mathfrak{x}}(\delta)|| d\tau. \end{split}$$

Using the generalized Gronwall inequality [42], one has sup*δ*∈[0,*η*] ||*x*(*δ*) <sup>−</sup> *<sup>y</sup>*(*δ*)|| <sup>=</sup> 0. Since *η* ∈ [0, *η*1] is arbitrary, we conclude that *x* = *y* on [0, *η*1].

Next, let *η* ∈ [*η*1, *η*2] be fixed. Note that *x*(*η*<sup>−</sup> <sup>1</sup> ) = *y*(*η*<sup>−</sup> <sup>1</sup> ). Then,

$$\begin{split} & ||\overline{\mathcal{Y}}(\eta) - \overline{\mathcal{X}}(\eta)|| \\ & \leq \quad ||h(\eta, \varkappa(\eta)\overline{\eta}) - h(\eta, \varkappa(\eta)\overline{\varkappa})||\_{\Theta} \\ & + ||\int\_{\eta\_{1}}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{sl}\_{2}(\eta - \tau) (h(\tau, \varkappa(\tau)\overline{y}) - h(\tau, \varkappa(\tau)\overline{\varkappa})) d\tau|| \\ & + ||\int\_{\eta\_{1}}^{\eta} (\eta - \tau)^{a-1} \mathfrak{sl}\_{2}(\eta - \tau) (g\_{n}(\tau, \varkappa(\tau)\overline{y}) - g\_{n}(\tau, \varkappa(\tau)\overline{\varkappa})) d\tau|| \\ & \leq \quad d\_{1} ||A^{-\gamma}|| ||\varkappa(\eta)\overline{y} - \varkappa(\eta)\overline{\varkappa}||\_{\Theta} \\ & + d\_{1} ||A^{-\gamma}|| \frac{a\mathfrak{C}\_{1-\gamma}\Gamma(1+\gamma)}{\Gamma(1+a\gamma)} \int\_{a}^{\eta} (\eta - \tau)^{a\gamma-1} ||\varkappa(\tau)\overline{y} - \varkappa(\tau)\overline{x}||\_{\Theta} d\tau \\ & + \frac{M}{\Gamma(a)} \int\_{\eta\_{1}}^{\eta} (\eta - \tau)^{a-1} ||g\_{n}(\tau, \varkappa(\tau)\overline{y}) - g\_{n}(\tau, \varkappa(\tau)\overline{x})|| |d\tau|. \end{split}$$

By repeating the arguments employed above, we get *x* = *y* on [*η*1, *η*2]. Continuing with the same processes, we arrive to *x* = *y* on *J*.

Next, we prove that ∑*Fn <sup>ψ</sup>* [−*r*, *b*] is homotopically equivalent to *y*. To this end, we define a continuous function *Zn* : [0, 1] <sup>×</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *<sup>b</sup>*] <sup>→</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *<sup>b</sup>*], where *Zn* (0, *<sup>x</sup>*) = *<sup>x</sup>* and (1, *<sup>x</sup>*) = *<sup>y</sup>*. Assume (*λ*, *<sup>x</sup>* ) <sup>∈</sup> [0, 1] <sup>×</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *<sup>b</sup>*] is fixed. Then, there exists a *<sup>f</sup>* <sup>∈</sup> *<sup>τ</sup>*<sup>1</sup> *Fn*(.,κ(.)*<sup>x</sup>*) such that

$$
\widetilde{\boldsymbol{x}}(\boldsymbol{\eta}) = \begin{cases}
\psi(\boldsymbol{\eta}), \boldsymbol{\eta} \in [-r, 0], \\
\mathfrak{K}\_{1}(\boldsymbol{\eta})[\boldsymbol{\psi}(0) - h(0, \boldsymbol{\psi})] + h(\boldsymbol{\eta}, \varkappa(\boldsymbol{\eta})\widetilde{\boldsymbol{x}}) \\
+ \int\_{0}^{\eta} (\boldsymbol{\eta} - \boldsymbol{\tau})^{\alpha - 1} A \mathfrak{K}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) h(\boldsymbol{\tau}, \varkappa(\boldsymbol{\tau})\widetilde{\boldsymbol{x}}) d\boldsymbol{\tau} \\
+ \int\_{0}^{\eta} (\boldsymbol{\eta} - \boldsymbol{\tau})^{\alpha - 1} \mathfrak{K}\_{2}(\boldsymbol{\eta} - \boldsymbol{\tau}) f(\boldsymbol{\tau}) d\boldsymbol{\tau} \\
+ \sum\_{0 < \boldsymbol{\eta}\_{i} < \eta} \mathfrak{K}\_{1}(\boldsymbol{\eta} - \boldsymbol{\eta}\_{i}) I\_{i}(\widetilde{\boldsymbol{x}}(\boldsymbol{\eta}\_{i}^{-})), \boldsymbol{\eta} \in \mathcal{J}.
\end{cases} \tag{39}
$$

Consider the partition { 0, <sup>1</sup> *<sup>m</sup>*+<sup>1</sup> , <sup>2</sup> *<sup>m</sup>*+<sup>1</sup> , ... , *<sup>m</sup>*+<sup>1</sup> *<sup>m</sup>*+<sup>1</sup> } for *<sup>J</sup>* = [0, 1]. We consider the following cases:

(i) *<sup>λ</sup>* <sup>∈</sup> [0, <sup>1</sup> *<sup>m</sup>*+<sup>1</sup> ]. Put *<sup>a</sup>*<sup>1</sup> *<sup>λ</sup>* = *ηm*+<sup>1</sup> − *λ* (*m* + 1)(*ηm*+<sup>1</sup> − *ηm*). The following fractional neutral differential inclusion is a result of the above discussion:

$$\begin{cases} \ ^cD\_{a^1,\eta}^{\alpha}[\mathbf{x}(\eta) - h(\eta, \varkappa(\eta)\mathbf{x})] = A\mathbf{x}(\eta) + \mathbf{g}\_{\imath}(\eta, \varkappa(\eta)\mathbf{x}), \ a.e. \ \eta \in [a\_{\lambda,1}, b]\_{\varkappa} \\\ \mathbf{x}(\eta) = \widetilde{\mathbf{x}}(\eta), \eta \in [-r, a\_{\lambda}^1]\_{\varkappa} \end{cases}$$

has a unique mild solution *x*<sup>1</sup> *<sup>λ</sup>* <sup>∈</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*] satisfying the next integral equation:

$$x\_{\lambda}^{1}(\eta) = \begin{cases} \begin{array}{l} \left\| \widetilde{\boldsymbol{x}}(\eta), \ \eta \in [-r, a\_{\lambda}^{1}]\_{\flat} \\ \mathfrak{s}\_{1}(\eta - a\_{\lambda}^{1})[\widetilde{\boldsymbol{x}}(a\_{\lambda}^{1}) - h(a\_{\lambda}^{1}, \varkappa(a\_{\lambda}^{1})\widetilde{\boldsymbol{x}}(a\_{\lambda}^{1})] \\ + h(\eta, \varkappa(\eta)\mathbbm{x}\_{\lambda}^{1}(\eta)) \\ + \int\_{a\_{\lambda}^{1}}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{s}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\mathbbm{x}\_{\lambda}^{1}(\eta)) d\tau \\ + \int\_{a\_{\lambda}^{1}}^{\eta} (\eta - \tau)^{a-1} \mathfrak{s}\_{2}(\eta - \tau) g\_{\mathfrak{n}}(\eta, \varkappa(\eta)\mathbbm{x}\_{\lambda}^{1}) d\tau, \eta \in [a\_{\lambda, 1}, b]. \end{array} \tag{40}$$

Note that *x*<sup>1</sup> <sup>0</sup>(*η*) = *<sup>x</sup>*(*η*); *<sup>η</sup>* <sup>∈</sup> [−*r*, *<sup>b</sup>*].

(ii) *<sup>λ</sup>* <sup>∈</sup> ( <sup>1</sup> *<sup>m</sup>*+<sup>1</sup> , <sup>2</sup> *<sup>m</sup>*+<sup>1</sup> ].Put *<sup>a</sup>*<sup>2</sup> *<sup>λ</sup>* <sup>=</sup> *<sup>η</sup><sup>m</sup>* <sup>−</sup> (*<sup>m</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>λ</sup>* <sup>−</sup> <sup>1</sup> *<sup>m</sup>*+<sup>1</sup> )(*η<sup>m</sup>* − *ηm*−1). Again, the following fractional neutral differential inclusion:

$$\begin{cases} \, \, ^cD\_{a\_\lambda^\eta \eta}^\alpha \left[ \mathfrak{x}(\eta) - h(\eta, \mathfrak{x}(\eta)\mathfrak{x}) \right] = A\mathfrak{x}(\eta) + g\_\mathfrak{u}(\eta, \mathfrak{x}(\eta)\mathfrak{x}), \, a.e. \, \eta \in [a\_{\lambda'}^2 b] - \{\eta\_m\}\_{\lambda'} \\\ I\_m(\mathfrak{x}(\eta\_m^-)) = \mathfrak{x}(\eta\_m^-) - \mathfrak{x}(\eta\_m^+), \\\ \mathfrak{x}(\eta) = \widetilde{\mathfrak{x}}(\eta), \eta \in [-r, a\_\lambda^2]. \end{cases}$$

has a unique mild solution *x*<sup>2</sup> *<sup>λ</sup>* <sup>∈</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*] and

$$
\mathbf{x}\_{\lambda}^{2}(\eta) = \begin{cases}
\begin{array}{l}
\quad \widetilde{\boldsymbol{\kappa}}(\eta), \ \eta \in [-r, a\_{\lambda}^{2}], \\
\quad \mathfrak{K}\_{1}(\eta - a\_{\lambda}^{2})[\widetilde{\boldsymbol{\kappa}}(a\_{\lambda}^{2}) - h(a\_{\lambda, 1}, \varkappa(a\_{\lambda}^{2})\widetilde{\boldsymbol{\kappa}}(a\_{\lambda}^{2})] \\
\quad + h(\eta, \varkappa(\eta)\chi\_{\lambda}^{2}(\eta)) \\
\quad + \int\_{a\_{\lambda}^{2}}^{\eta} (\eta - \tau)^{\alpha - 1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\chi\_{\lambda}^{2}(\eta)) d\tau \\
\quad + \int\_{a\_{\lambda}^{2}}^{\eta} (\eta - \tau)^{\alpha - 1} \mathfrak{K}\_{2}(\eta - \tau) g\_{\mathfrak{u}}(\eta, \varkappa(\eta)\chi\_{\lambda}^{2}) d\tau \\
\quad + \sum\_{a\_{\lambda}^{2} < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\varkappa\_{\lambda}^{2}(\eta\_{i}^{-})), \eta \in [a\_{\lambda}^{2}, b].
\end{cases}
$$

We continue up to *<sup>m</sup>* <sup>+</sup> <sup>1</sup>−step. That is *<sup>λ</sup>* <sup>∈</sup> ( *<sup>m</sup> <sup>m</sup>*+<sup>1</sup> , 1] and put *<sup>a</sup>m*+<sup>1</sup> *<sup>λ</sup>* = *η*<sup>1</sup> − (*m* + <sup>1</sup>)(*<sup>λ</sup>* <sup>−</sup> *<sup>m</sup> <sup>m</sup>*+<sup>1</sup> )*η*1. Let *<sup>x</sup>m*+<sup>1</sup> *<sup>λ</sup>* <sup>∈</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b* ] be the unique mild solution for the impulsive fractional neutral differential inclusion:

$$\begin{cases} \,^cD\_{a\_{\lambda}^{m+1},\eta}^{a}[\mathbf{x}(\eta) - h(\eta, \varkappa(\eta)\mathbf{x})] = A\mathbf{x}(\eta) + g\_{n}(\eta, \varkappa(\eta)\mathbf{x}), \; a.e.\; \eta \in [a\_{\lambda}^{m+1}, b] - \{\eta\_{1}, \eta\_{2}, \dots, \eta\_{m}\}, \\\ I\_{i}(\mathbf{x}(\eta\_{i}^{-})) = \mathbf{x}(\eta\_{i}^{-}) - \mathbf{x}(\eta\_{i}^{+}), i = 1, 2, \dots, m \\\ \mathbf{x}(\eta) = \widetilde{\mathbf{x}}(\eta), \eta \in [-r\_{\star} a\_{\lambda}^{m+1}]. \end{cases}$$

Then,

$$\mathbf{x}\_{\lambda}^{m+1}(\eta) = \begin{cases} \begin{array}{l} \widetilde{\boldsymbol{x}}(\eta), \ \eta \in [-r, a\_{\lambda}^{m+1}]\_{\prime} \\ \mathfrak{K}\_{1}(\eta)[\widetilde{\boldsymbol{x}}(a\_{\lambda}^{m+1}) - \mathfrak{h}(a\_{\lambda,1}, \varkappa(a\_{\lambda}^{m+1})) \widetilde{\boldsymbol{x}}(a\_{\lambda}^{m+1})] \\ + \mathfrak{h}(\eta, \varkappa(\eta) \boldsymbol{x}\_{\lambda}^{m+1}) \\ + \int\_{a\_{\lambda}^{m+1}}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{sl}\_{2}(\eta - \tau) \mathfrak{h}(\tau, \varkappa(\tau) \boldsymbol{x}\_{\lambda}^{m+1}(\eta)) d\tau \\ + \int\_{a\_{\lambda}^{m+1}}^{\eta} (\eta - \tau)^{a-1} \mathfrak{sl}\_{2}(\eta - \tau) \mathfrak{g}\_{\text{fl}}(\eta, \varkappa(\eta) \boldsymbol{x}\_{\lambda}^{m+1}) d\tau \\ + \sum\_{a\_{\lambda}^{m+1} < \eta\_{i} < \eta} \mathfrak{K}\_{1}(\eta - \eta\_{i}) I\_{i}(\boldsymbol{x}\_{\lambda}^{m+1}(\eta\_{i}^{-})), \eta \in [a\_{\lambda}^{m+1}, b]. \end{cases} \tag{41}$$

Note that *am*+<sup>1</sup> <sup>1</sup> <sup>=</sup> 0 and *<sup>x</sup>m*+<sup>1</sup> <sup>1</sup> <sup>=</sup> *<sup>y</sup>*. Now, we define *Zn* at (*λ*, *<sup>x</sup>*) as

$$Z\_{\mathfrak{n}}(\lambda,\hat{\mathfrak{x}}) = \begin{cases} \begin{array}{ll} \mathfrak{x}\_{\lambda'}^{1} & \text{if } \lambda \in [0, \frac{1}{m+1}], \\ \mathfrak{x}\_{\lambda'}^{2} \text{ if } \lambda \in (\frac{1}{m+1}, \frac{2}{m+1}], \\ \vdots \\ \vdots \\ \mathfrak{x}\_{\lambda}^{m+1}, \text{ if } \lambda \in (\frac{m}{m+1}, 1]. \end{array} \end{cases} \tag{42}$$

Therefore, *Zn*(0, *<sup>x</sup>*) = *<sup>x</sup>*<sup>1</sup> *<sup>λ</sup>* <sup>=</sup> *<sup>x</sup>* and *Zn*(1, *<sup>x</sup>*) = *<sup>x</sup>m*+<sup>1</sup> <sup>1</sup> = *y*.

It remains to clarify the continuity of *Zn*. Let (*λ*, *<sup>u</sup>* ),(, *<sup>v</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> <sup>∑</sup>*Fn <sup>ψ</sup>* [−*r*, *b*]. Let *<sup>λ</sup>* <sup>=</sup> <sup>=</sup> 0. Then, by (42), lim*u*→*<sup>v</sup> Zn*(*λ*, *<sup>u</sup>*) = lim*u*→*<sup>v</sup> <sup>u</sup>* <sup>=</sup> *<sup>v</sup>* <sup>=</sup> *Zn*(, *<sup>v</sup>*). Let *<sup>λ</sup>*, <sup>∈</sup> (0, <sup>1</sup> *<sup>m</sup>*+<sup>1</sup> ]. So, *Zn*(*λ*, *u*) = *u*<sup>1</sup> *<sup>λ</sup>* and *Zn*(*λ*, *<sup>v</sup>*) = *<sup>v</sup>*<sup>1</sup> *<sup>μ</sup>*, where

$$
\overline{u}\_{\lambda}^{1}(\eta) = \begin{cases}
\widetilde{\boldsymbol{x}}(\eta), \ \eta \in [-r, a\_{\lambda}^{1}]\_{\mathsf{T}} \\
\mathfrak{K}\_{1}(\eta - a\_{\lambda}^{1})[\widetilde{\boldsymbol{x}}(a\_{\lambda}^{1}) - h(a\_{\lambda}^{1}, \varkappa(a\_{\lambda}^{1})\widetilde{\boldsymbol{x}}(a\_{\lambda}^{1})] \\
+ h(\eta, \varkappa(\eta)\overline{u}\_{\lambda}^{1}(\eta)) \\
+ \int\_{a\_{\lambda}^{1}}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{K}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{u}\_{\lambda}^{1}(\eta)) d\tau \\
+ \int\_{a\_{\lambda}^{1}}^{\eta} (\eta - \tau)^{a-1} \mathfrak{K}\_{2}(\eta - \tau) g\_{\eta}(\eta, \varkappa(\eta)\overline{u}\_{\lambda}^{1}) d\tau, \eta \in [a\_{\lambda, 1}, b]\_{\mathsf{T}} \\
\end{cases} \tag{43}
$$

and

$$
\overline{v}\_{\mu}^{1}(\eta) = \begin{cases}
\widetilde{\boldsymbol{x}}(\eta), \ \eta \in [-r, a\_{\mu}^{1}], \\
\mathfrak{F}\_{1}(\eta - a\_{\lambda}^{1})[\widetilde{\boldsymbol{x}}(a\_{\lambda}^{1}) - h(a\_{\mu}^{1}, \varkappa(a\_{\mu}^{1})\widetilde{\boldsymbol{x}}(a\_{\mu}^{1})] \\
+ h(\eta, \varkappa(\eta)\overline{v}\_{\mu}^{1}(\eta)) \\
+ \int\_{a\_{\mu}^{1}}^{\eta} (\eta - \tau)^{a-1} A \mathfrak{A}\_{2}(\eta - \tau) h(\tau, \varkappa(\tau)\overline{v}\_{\mu}^{1}(\eta)) d\tau \\
+ \int\_{a\_{\mu}^{1}}^{\eta} (\eta - \tau)^{a-1} \mathfrak{A}\_{2}(\eta - \tau) g\_{\mu}(\eta, \varkappa(\eta)\overline{v}\_{\mu}^{1}) d\tau, \eta \in [a\_{\mu}^{1}, b],
\end{cases} \tag{44}
$$

*a*1 *<sup>λ</sup>* <sup>=</sup> *<sup>b</sup>* <sup>−</sup> *<sup>μ</sup>*(*<sup>m</sup>* <sup>+</sup>1)(*<sup>b</sup>* <sup>−</sup> *<sup>τ</sup>m*) and *<sup>a</sup>*<sup>1</sup> *<sup>μ</sup>* <sup>=</sup> *<sup>b</sup>* <sup>−</sup> *<sup>μ</sup>*(*<sup>m</sup>* <sup>+</sup>1)(*<sup>b</sup>* <sup>−</sup> *<sup>τ</sup>m*). Obviously, lim*λ*→*<sup>μ</sup> <sup>a</sup>*<sup>1</sup> *<sup>λ</sup>* = *<sup>a</sup>*<sup>1</sup> *μ* and, hence, by (43) and (44), and by arguing as above, we get

$$\lim\_{\substack{\lambda \to \mu \\ \mu \to \upsilon}} Z\_{\mathfrak{n}}(\lambda \,\mu) = Z\_{\mathfrak{n}}(\mu \,\upsilon)\_{\nu}$$

which implies the continuity of *Zn*(., .), when *<sup>λ</sup>* <sup>∈</sup> [0, <sup>1</sup> *<sup>m</sup>*+<sup>1</sup> ]. Similarly, we can show the continuity of *Zn* and consequently, ∑*Fn <sup>ψ</sup>* [−*r*, *b*] is contractible. This completes the proof.

#### **5. Example**

**Example 1.** *Assume that E* = *L*2([0, *π*], R), *J* = [0, 1], *r* = <sup>1</sup> <sup>2</sup> , *<sup>m</sup>* <sup>=</sup> 1, *<sup>η</sup>*<sup>0</sup> <sup>=</sup> <sup>0</sup> *and <sup>η</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2 , *<sup>η</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>*. For any <sup>x</sup>* : *<sup>J</sup>* <sup>→</sup> *<sup>E</sup>* <sup>=</sup> *<sup>L</sup>*2([0, *<sup>π</sup>*], <sup>R</sup>)*, we denote by <sup>x</sup>*(*η*, *<sup>ω</sup>*); *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*, *<sup>ω</sup>* <sup>∈</sup> [0, *<sup>π</sup>*] *the value of <sup>x</sup>*(*η*) *at <sup>ω</sup>. Let <sup>A</sup>* : *<sup>D</sup>*(*A*) <sup>⊆</sup> *<sup>L</sup>*2[0, *<sup>π</sup>*] <sup>→</sup> *<sup>L</sup>*2[0, *<sup>π</sup>*] *, Ax*(*η*, *<sup>ω</sup>*) :<sup>=</sup> <sup>−</sup> *<sup>∂</sup>*<sup>2</sup> *∂ω*<sup>2</sup> *<sup>x</sup>*(*η*, *<sup>ω</sup>*) *and domain <sup>A</sup> be defined as*

$$\begin{aligned} D(A) &=& \{ \mathbf{x} \in L^2[0, \pi] : \mathbf{x}, \mathbf{x'} \text{ are absolutely continuous, } \mathbf{x''} \in L^2[0, 1], \\ \mathbf{x}(\eta, 0) &=& \mathbf{x}(\eta, \pi) = 0 \}. \end{aligned}$$

Using [37], there is a compact analytic semi-group {Υ(*η*) : *η* ≥ 0} generated by *A* and

$$A\mathfrak{x} = \sum\_{n=1}^{\infty} n^2 < \mathfrak{x}\_{\prime} \ \mathfrak{x}\_{\hbar} > \mathfrak{x}\_{\hbar \prime} \mathfrak{x} \in D(A),\tag{45}$$

where *xn*(*y*) = <sup>√</sup>2 sin *ny*, *<sup>n</sup>* <sup>=</sup> 1, 2, ... is the orthonormal set of eigenvalues of *<sup>A</sup>*. In addition, for all *<sup>x</sup>* <sup>∈</sup> *<sup>L</sup>*2[0, 1], one gets

$$\Upsilon(\eta)(x) = \sum\_{n=1}^{\infty} e^{-n^2 \eta} < \infty, \ x\_n > \infty.$$

So, *<sup>M</sup>* <sup>=</sup> sup{||Υ(*η*)|| : *<sup>η</sup>* <sup>≥</sup> <sup>0</sup>} <sup>=</sup> 1. Furthermore, for each *<sup>x</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>π</sup>*], <sup>R</sup>),

$$\begin{aligned} A^{\frac{-1}{2}}\mathfrak{x} &= \sum\_{n=1}^{\infty} \frac{1}{n} < \mathfrak{x} \; \mathfrak{x}\_n > \mathfrak{x}\_n. \\\\ A^{\frac{1}{2}}\mathfrak{x} &= \sum\_{n=1}^{\infty} n < \mathfrak{x} \; \mathfrak{x}\_n > \mathfrak{x}\_{\mathfrak{H}}. \end{aligned}$$

and ||*<sup>A</sup>* <sup>−</sup><sup>1</sup> <sup>2</sup> || <sup>=</sup> 1. The domain of *<sup>A</sup>*<sup>1</sup> <sup>2</sup> is defined as

$$D(A^{\frac{1}{2}}) = \{ \mathfrak{x} \in L^2([0, \pi], \mathbb{R}) : \sum\_{n=1}^{\infty} n < \infty, \ \mathfrak{x}\_{\mathbb{R}} > \mathfrak{x}\_{\mathbb{R}} \in L^2([0, \pi], \mathbb{R}) \}.$$

Let *h* : *J* × Θ → *E* be such that

$$h(\eta, u) := A^{\frac{-1}{2}} \left( \int\_{-r}^{0} \lambda u(\theta) d\theta \right), \tag{46}$$

where *λ* > 0. We have

$$\begin{aligned} ||A^{\frac{1}{2}}h(\eta, u\_1) - A^{\frac{1}{2}}h(\eta, u\_2)||\_E & \leq \quad \lambda || \int\_{-r}^{0} (u\_1(\theta) - u\_2(\theta)) d\theta || \\ & \leq \quad \lambda \int\_{-r}^{0} ||u\_1(\theta) - u\_2(\theta)|| d\theta \\ & \leq \quad \lambda ||u\_1 - u\_2||\_{\Theta'} \end{aligned}$$

and

$$||A^{\gamma}h(\eta, u)|| \le \lambda \, ||\int\_{-r}^{0} (u(\theta)d\theta)|| \le \lambda ||u||\_{\Theta}.$$

Then, (10) and (11) are satisfied with *d*<sup>1</sup> = *d*<sup>2</sup> = *λ*.

Let Λ be a convex compact subset in *E*, sup{||*z*|| : *z* ∈ *Z*} = and *κ* > 0. Define *<sup>F</sup>* : *<sup>J</sup>* <sup>×</sup> <sup>Θ</sup> <sup>→</sup> <sup>2</sup>*L*2[0,*π*] by

$$F(\eta, u) := \frac{e^{-\kappa \eta} ||u||}{\varrho} \Lambda. \tag{47}$$

We have

$$||F(\eta, u)|| = \sup\{ |\langle \frac{e^{\kappa \eta} ||u||}{\varrho} z : z \in \Lambda \} \le e^{\kappa \eta} ; \eta \in \mathbb{J}.\}$$

Moreover, for any bounded subset *<sup>D</sup>* <sup>⊂</sup> <sup>Θ</sup>, we have *<sup>F</sup>*(*η*, *<sup>D</sup>*) <sup>⊆</sup> *<sup>ς</sup> <sup>e</sup>κη* Λ, where *ς* = sup{||*u*|| : *u* ∈ *D*} and, hence, *χE*(*F*(*η*, *D*)) = 0. Then, *F* satisfies (*HF*1),(*HF*2)∗and (*HF*3) with *<sup>ξ</sup>*(*η*) = *<sup>e</sup>*−*κη*, *<sup>β</sup>*(*η*) = 0; *<sup>η</sup>* <sup>∈</sup> *<sup>J</sup>*.

$$\text{Next, let}$$

$$I: E \to E, I\_l(\mathbf{x}) := \sigma \operatorname{proj}\_{\Lambda} \mathbf{x},\tag{48}$$

where *σ* is a positive number. Obviously, *I* verifies (*H I*) with *ς<sup>i</sup>* = 0 ; *i* = 1, 2, . . . .

Therefore, by applying Theorems 1 and 4, the set of solutions for the following fractional neutral impulsive semilinear differential inclusions with delay:

$$\begin{cases} \ ^cD\_{0,\eta}^{\alpha}[\mathbf{x}(\eta) - h(\eta, \varkappa(\eta)\mathbf{x})] \\ \in -\frac{\partial^2}{\partial\omega^2}\mathbf{x}(\eta, \omega) + \mathbf{F}(\eta, \varkappa(\eta)\mathbf{x}), \ a.e. \ \eta \in [0, 1] - \{\frac{1}{2}, 1\}, \\\ I\_i \mathbf{x}(\eta\_i^{-}, \omega) = \mathbf{x}(\eta\_i^{-}, \omega) - \mathbf{x}(\eta\_i^{+}, \omega), i = 1, 2, \omega \in [0, \pi], \\\ \mathbf{x}(\eta, \omega) = \boldsymbol{\psi}(\eta, \omega), \eta \in [-r, 0], \boldsymbol{\eta} \in [0, 1] - \{\frac{1}{2}, 1\}, \end{cases} \tag{49}$$

is a not empty, compact and an *Rδ*-set provided that

$$
\lambda \left( 1 + \frac{\mathbb{C}\_{1-\gamma} \Gamma \left( \frac{3}{2} \right)}{\Gamma \left( 1 + \frac{a}{2} \right)} \right) < 1,\tag{50}
$$

and

$$\frac{\lambda}{2} + 2\lambda \frac{\mathcal{C}\_{1-\gamma}\Gamma(\frac{3}{2})}{\Gamma(1+\frac{a}{2})} + \frac{1}{2\Gamma((a)}(\frac{P-1}{aP-1})^{\frac{P-1}{P}}||\xi||\_{L^{P}\_{(l,\mathbb{R}^+)}} + \sigma < 1,\tag{51}$$

where *F*, *h I* are given by (45)–(47). By choosing *λ* and *σ* small enough and *κ* large enough, we arrive to (50) and (51).

**Example 2.** *Let J*, *E*, *A*,*r*, *η*0, *η*1, *η*<sup>2</sup> Λ, *and be as in Example (1) and θ* ∈ [−*r*, 0] *be a fixed element.*

Let *h* : *J* × Θ → *E* be such that

$$h((\eta, \varkappa(\eta)\mathbf{x})(\omega) := \lambda \int\_0^\pi \mathbb{U}(\omega, y)\mathbf{x}(\theta + \eta)(\omega) dy; \omega \in [0, \pi]; \eta \in [0, 1], \tag{52}$$

where *<sup>λ</sup>* <sup>&</sup>gt; 0, *<sup>U</sup>* : [0, *<sup>π</sup>*] <sup>×</sup> [0, *<sup>π</sup>*] <sup>→</sup> <sup>R</sup> is measurable,% *<sup>π</sup>* 0 % *π* <sup>0</sup> *<sup>U</sup>*(*ω*, *<sup>y</sup>*)*dyd<sup>ω</sup>* <sup>&</sup>lt; <sup>∞</sup>, *<sup>∂</sup>U*(*ω*,*η*) *∂ω* is measurable, *U*(0, *y*) = *U*(*π*, *y*) = 0, ∀*y* ∈ [0, *π*] and ( % *π* 0 % *π* <sup>0</sup> ( *<sup>∂</sup>U*(*ω*,*η*) *∂ω* )2*dydω*) 1 <sup>2</sup> < ∞.

Next, let *<sup>F</sup>* : *<sup>J</sup>* <sup>×</sup> <sup>Θ</sup> <sup>→</sup> <sup>2</sup>*L*2[0,*π*] , *F*((*η*,κ(*η*)*x*)(*ω*) = *<sup>γ</sup>G*(*η*,*x*(*θ*+*η*)(*ω*))<sup>|</sup> Λ, where *γ* > 0, *G* : *<sup>J</sup>* <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> is a continuous function. Then, by choosing *<sup>λ</sup>* and *<sup>σ</sup>* small enough, one can show that *h* and *F* satisfy all assumptions of Theorems 2 (see [15,43]) and, hence, the set of mild solutions for the partial differential inclusions of impulsive neutral type with delay:

$$\begin{cases} \ ^cD\_{0,\eta}^{\alpha}[\chi(\eta,\omega)-\int\_0^{\pi} \mathcal{U}(\omega,y)\chi(\theta+\eta)(\omega)dy,\ ]\\ \in -\frac{\partial^2}{\partial\omega^2}\mathfrak{x}(\eta,\omega)+\frac{G(\eta,\mathfrak{x}(\theta+\eta)(\omega))|}{\varrho}\Lambda,\ \text{a.e.}\ \eta\in[0,1]-\{\frac{1}{2},1\},\\\ I\_i\mathfrak{x}(\eta\_i^{-},\omega)=\mathfrak{x}(\eta\_i^{-},\omega)-\mathfrak{x}(\eta\_i^{+},\omega), i=1,2,\omega\in[0,\pi],\\\ \mathfrak{x}(\eta,\omega)=\psi(\eta,\omega),\eta\in[-r,0],\eta\in[0,1]-\{\frac{1}{2},1\},\end{cases} \tag{53}$$

is an *Rδ*-set.

#### **6. Discussion**

The neutral differential equations and inclusions appear in many applied mathematical sciences such as viscoelasticity, and the equations describe the distribution of heat. Since the set of mild solutions for a differential inclusion having the same initial point may not be a singleton, many authors are interested to investigate the structure of this set in a topological point of view. An important aspect of such structure is the *Rδ*- property, which means that the homology group of the set of mild solutions is the same as a onepoint space. In the literature, there are many results on this subject but no result about the topological properties of the set of mild solutions for a fractional neutral differential inclusion generated by a non-compact semigroup in the presence of impulses and delay. As cited in the introduction, when the problem involves delay and impulses, we cannot consider the space P C([−*r*, *b*], *E*] as the space of solutions. To overcome these difficulties, a complete metric space *H* is introduced as the space of mild solutions. In addition, the function *<sup>η</sup>* <sup>→</sup> <sup>κ</sup>(*η*)*x*; *<sup>x</sup>* <sup>∈</sup> *<sup>H</sup>* is not necessarily measurable, therefore, a norm different from the uniform convergence norm is introduced on Θ (see Equation (2)).

#### **7. Conclusions**

During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering. For some of these applications, one can see [28] and the references therein. In this paper, we have given an affirmative answer for a basic question, which is whether there exists a solution set carrying an *Rδ*-structure when there are impulsive effects and delay on the system, the operator families generated by the linear part lack compactness and the order is fractional. More specifically,


**Author Contributions:** Funding acquisition, Z.A. and A.G.I.; investigation, Z.A. and A.G.I.; methodology, Z.A., A.G.I. and A.A.; writing—original draft, Z.A. and A.G.I.; writing—-review and editing, Z.A., A.G.I. and A.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This research has been funded by the Scientific Research Deanship at University of Ha'il—Saudi Arabia through project number RG-21 101.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators**

**Chandra Bose Sindhu Varun Bose and Ramalingam Udhayakumar \***

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India

**\*** Correspondence: udhayaram.v@gmail.com or udhayakumar.r@vit.ac.in

**Abstract:** This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli's fixed point theorem, we prove the primary results. In addition, the application is provided to demonstrate how the major results might be applied.

**Keywords:** Hilfer fractional system; neutral system; multi-valued maps; sectorial operators

**MSC:** 26A33; 34A08; 34K30; 47D09

#### **1. Introduction**

In modern mathematics, the fundamentals surrounding fractional computation and the fractional differential equation have taken center stage. The idea of fractional computation has now been put to the test in a wide variety of social, physical, signal, image processing, biological, control theory, engineering, etc., challenges. However, it has been demonstrated that fractional differential equations may be a valuable tool for describing a variety of situations. For many different types of realistic applications, fractional-order models are superior to integer-order models. The research articles [1–15] are concerned with the theory of fractional differential systems, and readers will find a number of fascinating findings about fractional dynamical systems. Please refer to [16–21] for more information.

Other fractional derivatives introduced by Hilfer [22] include the R-L derivative and Caputo fractional derivative. Many scholars have recently shown tremendous interest in this area, e.g., [23–25]; researchers have established their results with the help of Schauder's fixed point theorem. In [26–28], the authors worked on the existence and controllability of differential inclusions via the fixed point theorem approach. In references [29–31], the authors discussed the existence of a mild solution by using Martelli's fixed point theorem. As a result of these findings, we expand on the literature's earlier findings to a class of Hilfer fractional differential (*HFD*) systems in which the closed operator is almost sectorial.

In [32], M. Zhou, C. Li, and Y. Zhou studied the existence of mild solutions to Hilfer fractional differential equations with the order *λ* ∈ (0, 1) and type *ν* ∈ [0, 1] in the abstract sense, as follows:

$$\begin{aligned} ^HD\_{0^+}^{\lambda,\nu}y(t) &= Ay(t) + g(t, y(t)), \; t \in (0, T], \\ ^II\_{0^+}^{(1-\lambda)(1-\nu)}y(0) &= y\_{0\nu} \end{aligned}$$

here, *A* denotes the almost sectorial operator of the semigroup and the Schauder fixed point theorem is used.

**Citation:** Varun Bose, C.B.S.; Udhayakumar, R. Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators. *Fractal Fract.* **2022**, *6*, 532. https://doi.org/10.3390/ fractalfract6090532

Academic Editor: Rodica Luca

Received: 18 August 2022 Accepted: 16 September 2022 Published: 19 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In [33], Zhang and Zhou demonstrated the existence of fractional Cauchy problems using almost sectorial operators of the type,

$$\begin{aligned} \,^L D\_{0^+}^q \mathfrak{x}(t) &= A\mathfrak{x}(t) + f(t, \mathfrak{x}(t)) \, t \in [0, a]\_{\mathsf{V}},\\ \,^L I\_{0^+}^{(1-q)} \mathfrak{x}(0) &= \mathfrak{x}\_{0\prime} \end{aligned}$$

where *LD<sup>q</sup>* <sup>0</sup><sup>+</sup> is the *R* − *L* derivative of order *q*, 0 < *q* < 1, *I* (1−*q*) <sup>0</sup><sup>+</sup> is the *R* − *L* integral of order 1 − *q*, *A* is an almost sectorial operator on a complex Banach space. We refer the reader to [34–37] for information. These discoveries led us to extend past findings in the literature to Hilfer fractional Volterra–Fredholm integro-differential inclusions.

We will examine the following subject in the article: The almost sectorial operators are contained in the *HF* neutral integro-differential inclusion,

$$\begin{cases} D\_{0^{+}}^{\kappa,\varepsilon} \left[ \mathbf{y}(\mathfrak{z}) - \mathcal{N}(\mathfrak{z}, \mathbf{y}(\mathfrak{z})) \right] \in \mathsf{A}\mathbf{y}(\mathfrak{z}) + \mathcal{G} \left( \mathfrak{z}, \mathbf{y}(\mathfrak{z}), \int\_{0}^{\mathfrak{z}} e(\mathfrak{z}, \mathbf{s}, \mathbf{y}(\mathfrak{s})) d\mathfrak{s} \right), & \mathfrak{z} \in \mathcal{J}' = (0, d], \tag{2} \\\\ I\_{0^{+}}^{(1-\kappa)(1-\varepsilon)} \mathbf{y}(0) = \mathbf{y}\_{0^{+}} \end{cases} \tag{2}$$

where *Dκ*,*<sup>ε</sup>* <sup>0</sup><sup>+</sup> notates the *HFD* of order *κ*, 0 < *κ* < 1, type *ε*, 0 ≤ *ε* ≤ 1; and A is an almost sectorial operator of the analytic semigroup ' T(z),z ≥ 0 ( on Y. State y(·) takes the value in a Banach space Y with norm · . Let J = [0, *d*], N : J × *Y* be the appropriate function, <sup>G</sup> : J × *<sup>Y</sup>* <sup>×</sup> *<sup>Y</sup>* <sup>→</sup> <sup>2</sup>*Y*\{∅} be a non-empty, bounded, closed convex multi-valued map, N : J × *Y* → *Y* and *e* : J ×J × *Y* → *Y* are the appropriate functions.

This article is structured as follows: In Section 2, we present the fundamentals of fractional differential systems, semigroup, and closed linear operators. In Section 3, we present the existence of the required solution. In Section 4, we provide an application to demonstrate our main arguments and some inferences are established in the end.

#### **2. Preliminaries**

Here, we introduce some basic definitions, theorems, and lemmas that are applied to every part of the paper.

Let be the collection of all continuous functions from J to *Y*, where J = [0, *d*] and <sup>J</sup> = (0, *<sup>d</sup>*] with *<sup>d</sup>* <sup>&</sup>gt; 0. Take <sup>X</sup> <sup>=</sup> {<sup>y</sup> <sup>∈</sup> : limz→<sup>0</sup> <sup>z</sup>1−*ε*+*κε*−*κξ*y(z) exists and finite }, which is the Banach space and its norm on · <sup>X</sup> , defined as y <sup>X</sup> <sup>=</sup> supz∈I{z1−*ε*+*κε*−*κξ* y(z) }. Let <sup>y</sup>(z) = <sup>z</sup>−1+*ε*−*κε*+*κξu*(z), <sup>z</sup> <sup>∈</sup> (0, *<sup>d</sup>*] then, <sup>y</sup> ∈ X *iff y* <sup>∈</sup> *and* y <sup>X</sup> = *y* . Moreover, define *BP*(J ) = {y ∈ *such that* y ≤ *P*}.

**Definition 1** ([19])**.** *The left side of the R-L fractional integral of order κ with the lower limit d for function* <sup>G</sup> : [*d*, <sup>∞</sup>) <sup>→</sup> <sup>R</sup> *is presented by*

$$I\_{d^{+}}^{\\\kappa} \mathcal{G}(\mathfrak{z}) = \frac{1}{\Gamma(\kappa)} \int\_{d}^{\mathfrak{z}} \frac{\mathcal{G}(w)}{(\mathfrak{z} - w)^{1 - \kappa}} dw, \text{  $\mathfrak{z} > 0$ ,  $\kappa > 0$ .} $$

*provided the right side is pointwise determined on* [*d*, +∞)*,* Γ(·) *is the gamma function.*

**Definition 2** ([19])**.** *The left-sided R-L fractional derivative of order κ* > 0, *m* − 1 ≤ *κ* < *m*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>*, for a function* <sup>G</sup> : [*d*, <sup>+</sup>∞) <sup>→</sup> <sup>R</sup> *is presented by*

$${}^{L}D\_{d+}^{\\\kappa} \mathcal{G}(\mathfrak{z}) = \frac{1}{\Gamma(m-\kappa)} \frac{d^m}{d\mathfrak{z}^m} \int\_d^{\mathfrak{z}} \frac{\mathcal{G}(w)}{(\mathfrak{z}-w)^{\kappa+1-m}} dw, \mathfrak{z} > d,$$

*where* Γ(·) *is the gamma function.*

**Definition 3** ([19])**.** *The left-sided Caputo derivative of the type of order κ* > 0, *m* − 1 ≤ *κ* < *<sup>m</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup> *for a function* <sup>G</sup> : [*d*, <sup>+</sup>∞) <sup>→</sup> <sup>R</sup>*, is defined as*

$${}^{C}D\_{d^{+}}^{\\\kappa} \mathcal{G}(\mathfrak{z}) = \frac{1}{\Gamma(m-\kappa)} \int\_{d}^{\mathfrak{z}} \frac{\mathcal{G}^{m}(w)}{(\mathfrak{z}-w)^{\kappa+1-m}} dw = I\_{d+}^{m-\kappa} \mathcal{G}^{m}(\mathfrak{z}), \mathfrak{z} > d\_{\kappa}$$

*where* Γ(·) *is the gamma function.*

**Definition 4** ([22])**.** *The left-sided HFD of order* 0 < *κ* < 1 *and type ε* ∈ [0, 1]*, of function* <sup>G</sup> : [*d*, <sup>+</sup>∞) <sup>→</sup> <sup>R</sup>*, is defined as*

$$D\_{d^{+}}^{\kappa,\varepsilon} \mathcal{G}(\mathfrak{z}) = [I\_{d^{+}}^{(1-\kappa)\varepsilon} D(I\_{d^{+}}^{(1-\kappa)(1-\varepsilon)} \mathcal{G})](\mathfrak{z}).$$

**Remark 1** ([22])**.** *1. If ε* = 0, 0 < *κ* < 1*, and d* = 0*, then the HFD corresponds to the classical R-L fractional derivative:*

$$D\_{0^{+}}^{\\\kappa,0} \mathcal{G}(\mathfrak{z}) = \frac{d}{d\mathfrak{z}} I\_{0+}^{1-\kappa} \mathcal{G}(\mathfrak{z}) = {}^{L}D\_{0^{+}}^{\kappa} \mathcal{G}(\mathfrak{z}).$$

*2. If ε* = 1, 0 < *κ* < 1*, and d* = 0*, then the HFD corresponds to the classical Caputo fractional derivative:*

$$D\_{0^{+}}^{\\\kappa,1} \mathcal{G}(\mathfrak{z}) = I\_{0^{+}}^{1-\kappa} \frac{d}{d\mathfrak{z}} \mathcal{G}(\mathfrak{z}) = {}^{\mathbb{C}}D\_{0^{+}}^{\kappa} \mathcal{G}(\mathfrak{z}).$$

**Definition 5** ([38])**.** *For* 0 < *ξ* < 1, 0 < *ω* < *<sup>π</sup>* <sup>2</sup> *,* <sup>Θ</sup>−*<sup>ξ</sup> <sup>ω</sup> is the family of closed linear operators, the sector S<sup>ω</sup>* <sup>=</sup> {*<sup>v</sup>* <sup>∈</sup> <sup>C</sup>\{0} *with* <sup>|</sup>*arg v*| ≤ *<sup>ω</sup>*}*, and* <sup>A</sup> : *<sup>D</sup>*(A) <sup>⊂</sup> *<sup>Y</sup>* <sup>→</sup> *Y, which satisfy*

*(i) σ*(A) ⊆ *S<sup>ω</sup> ;*

*(ii) For any ω* < *δ* < *π* ∃ Λ*<sup>δ</sup> is a constant, such that,*

$$\left\|(vI - \mathsf{A})^{-1}\right\| \leq \Lambda\_{\delta} |v|^{-\xi}$$

*then* <sup>A</sup> <sup>∈</sup> <sup>Θ</sup>−*<sup>ξ</sup> <sup>ω</sup> is called an almost sectorial operator on Y.*

**Lemma 1** ([38])**.** *Let* 0 < *ξ* < 1 *and* 0 < *ω* < *<sup>π</sup>* <sup>2</sup> , <sup>A</sup> <sup>∈</sup> <sup>Θ</sup>−*<sup>ξ</sup> <sup>ω</sup>* (*Y*)*. Then*


$$\mathsf{A}^{\theta}T(\mathfrak{z})\mathfrak{y} = \frac{1}{2\pi i} \int\_{\Gamma\_{\gamma}} z^{\theta} e^{-\mathfrak{z}z} \mathsf{R}(z;\mathsf{A})\mathfrak{y}dz,\text{ for all }\mathsf{y} \in \mathsf{Y}\_{\mathsf{A}}$$

*and, hence,* ∃ *is a constant* Λ = Λ (*β*, *θ*) > 0*, such that*

$$\left\|\mathsf{A}^{\theta}T(\mathfrak{z})\right\|\_{B(Y)} \leq \Lambda' \mathfrak{z}^{-\beta - \operatorname{Re}(\theta) - 1}, \operatorname{for all } \mathfrak{z} > 0;$$


*Consider the operator families* ' S*κ*(z) ( <sup>z</sup>∈*<sup>S</sup> <sup>π</sup>* <sup>2</sup> <sup>−</sup>*<sup>ω</sup>* , ' Q*κ*(z) ( <sup>z</sup>∈*<sup>S</sup> <sup>π</sup>* <sup>2</sup> <sup>−</sup>*<sup>ω</sup> is defined as follows:*

$$\begin{aligned} \mathcal{S}\_{\mathbb{K}}(\mathfrak{z}) &= \int\_0^\infty W\_{\mathbb{K}}(\nu) T(\mathfrak{z}^{\kappa}\nu) d\nu, \\ \mathcal{Q}\_{\mathbb{K}}(\mathfrak{z}) &= \int\_0^\infty \kappa \nu W\_{\mathbb{K}}(\nu) T(\mathfrak{z}^{\kappa}\nu) d\nu, \end{aligned}$$

*where Wκ*(*β*) *is the Wright-type function:*

$$\mathcal{W}\_{\mathbb{K}}(\beta) = \sum\_{n \in \mathbb{N}} \frac{(-\beta)^{n-1}}{\Gamma(1 - \kappa n)(n-1)!}, \qquad \beta \in \mathbb{C}.\tag{3}$$

*Let* −1 < *ι* < ∞, *p* > 0*, the succeeding properties are satisfied.*


**Theorem 1** ([19])**.** S*κ*(z) *and* Q*κ*(z) *are continuous in the uniform operator topology, for* z > 0*, for every c* > 0*, the continuity is uniform on* [*c*, ∞)*.*

**Definition 6** ([16])**.** *A multi-valued map* G *is called u.s.c. on Y if for each* y<sup>0</sup> ∈ *Y the set* G(y0) *is a non-empty, closed subset of Y, and if for each open set* U *of Y containing* G(y0)*, there exists an open neighborhood* V *of* y0*, such that* G(V) ⊆ U*.*

**Definition 7** ([16])**.** G *is said to be completely continuous if* G(*C*) *is relatively compact for each bounded subset C of Y. If a multi-valued map* G *is completely continuous with non-empty compact values, then* G *is upper semi-continuous if and only if* G *has a closed graph i.e.,* y*<sup>m</sup>* → y0*,* z*<sup>m</sup>* → z0*,* z*<sup>m</sup>* ∈ G(y*m*) *imply* z<sup>0</sup> ∈ G(y0)*.*

**Definition 8** ([16])**.** *A multi-valued mapping* <sup>G</sup> : *<sup>Y</sup>* <sup>→</sup> <sup>2</sup>*<sup>Y</sup> is said to be condensing, if for any bounded subset D* ⊂ *Y with β*(*D*) = 0*, we have β*(*F*(*D*)) < *β*(*D*)*, where β*(·) *denotes the Kuratowski measure of non-compactness, defined as follows:*

> *β*(*D*) = inf ' *d* > 0 : *D covered by a finite number of balls of radius d* ( .

**Lemma 2.** *System* (1)*–*(2) *is equivalent to an integral inclusion given by*

$$\begin{split} \mathbf{y}(\mathbf{y}) \in & \frac{\mathbf{y}\_{0} - \mathcal{N}(\mathbf{0}, \mathbf{y}(0))}{\Gamma(\boldsymbol{\varepsilon}(1-\boldsymbol{\kappa}) + \boldsymbol{\kappa})} \mathbf{y}^{(1-\boldsymbol{\kappa})(\boldsymbol{\varepsilon}-1)} + \mathcal{N}(\mathbf{y}, \mathbf{y}(\mathbf{y})) + \frac{1}{\Gamma(\boldsymbol{\kappa})} \int\_{0}^{\mathbf{y}} (\mathbf{y} - \boldsymbol{w})^{\boldsymbol{\kappa}-1} \mathbf{A} \mathcal{N}(\boldsymbol{w}, \mathbf{y}(\mathbf{w})) dw \\ & + \frac{1}{\Gamma(\boldsymbol{\kappa})} \int\_{0}^{\mathbf{y}} (\mathbf{y} - \boldsymbol{w})^{\boldsymbol{\kappa}-1} \left[ \mathbf{A} \mathbf{y}(\boldsymbol{w}) + \mathcal{G} \left( \boldsymbol{w}, \mathbf{y}(\mathbf{w}), \int\_{0}^{\mathbf{w}} e(\mathbf{w}, \mathbf{s}, \mathbf{y}(\mathbf{s})) ds \right) dw \right]. \end{split}$$

**Definition 9.** *By a mild solution of the Cauchy problem* (1)*–*(2)*, the function* y(z) ∈ *C*(J ,*Y*)*satisfies*

$$\begin{split} \mathbf{y}(\mathbf{y}) &= \mathcal{S}\_{\mathbf{x},\mathbf{\mathcal{E}}}(\mathbf{y}) \left[ \mathbf{y}\_{0} - \mathcal{N}(\mathbf{0}, \mathbf{y}(0)) \right] + \mathcal{N}(\mathbf{y}, \mathbf{y}(\mathbf{y})) + \int\_{0}^{\mathfrak{z}} \mathcal{K}\_{\mathbf{x}}(\mathfrak{z} - w) \mathsf{A} \mathcal{N}(w, \mathbf{y}(w)) dw \\ &+ \int\_{0}^{\mathfrak{z}} \mathcal{K}\_{\mathbf{x}}(\mathfrak{z} - w) \mathcal{G} \left( w, \mathbf{y}(w), \int\_{0}^{w} e(w, \mathbf{s}, \mathbf{y}(\mathbf{s})) ds \right) dw, \quad \mathfrak{z} \in \mathcal{J}\_{\mathbf{z}} \end{split}$$

*where* S*κ*,*ε*(z) = *I ε*(1−*κ*) <sup>0</sup> <sup>K</sup>*κ*(z), <sup>K</sup>*κ*(z) = <sup>z</sup>*κ*−1Q*κ*(z). **Lemma 3** ([32])**.** *For any fixed ν* > 0, Q*κ*(*ν*), K*κ*(*ν*) *and* S*κ*,*ε*(*ν*) *are linear operators, and for any y* ∈ *Y*,

$$\left\lVert \left\lVert \mathcal{Q}\_{\mathbf{x}}(\mathbf{y}) \right\rVert \right\rVert \leq L' \mathfrak{z}^{-\kappa + \kappa \mathfrak{z}^{\mathsf{z}}} \left\lVert \left\lVert \mathcal{K}\_{\mathbf{x}}(\mathbf{y}) \mathbf{y} \right\rVert \right\rVert \leq L' \mathfrak{z}^{-1 + \kappa \mathfrak{z}^{\mathsf{z}}} \left\lVert \left\lVert \mathbf{y} \right\rVert \right\rVert \cdot \left\lVert \left\lVert \mathcal{S}\_{\mathbf{x}, \mathbf{c}}(\mathbf{y}) \mathbf{y} \right\rVert \right\rVert \leq L'' \mathfrak{z}^{-1 + \varepsilon - \kappa \varepsilon + \kappa \mathfrak{z}^{\mathsf{z}}} \left\lVert \mathbf{y} \right\rVert\_{\mathbf{x}}$$

*where*

$$L' = \Lambda\_0 \frac{\Gamma(\mathfrak{f})}{\Gamma(\kappa \mathfrak{f})}, \\ L'' = \Lambda\_0 \frac{\Gamma(\mathfrak{f})}{\Gamma(\varepsilon(1-\kappa) + \kappa \mathfrak{f})}.$$

**Lemma 4** ([32])**.** *Let* ' *T*(z) ( z><sup>0</sup> *be equicontinuous, then* ' Q*κ*(z) ( z>0, ' K*κ*(z) ( z>0, *and* ' S*κ*,*ε*(z) ( z><sup>0</sup> *are strongly continuous, i.e., for any* <sup>y</sup> <sup>∈</sup> *Y and* <sup>z</sup><sup>2</sup> <sup>&</sup>gt; <sup>z</sup><sup>1</sup> <sup>&</sup>gt; 0,

$$\begin{cases} \left| \mathcal{Q}\_{\mathbf{x}} (\mathfrak{z}\_{2}) \mathbf{y} - \mathcal{Q}\_{\mathbf{x}} (\mathfrak{z}\_{1}) \mathbf{y} \right| \to 0, \left| \left| \mathcal{K}\_{\mathbf{x}} (\mathfrak{z}\_{2}) \mathbf{y} - \mathcal{K}\_{\mathbf{x}} (\mathfrak{z}\_{1}) \mathbf{y} \right| \right| \to 0 \\\left| \left| \mathcal{S}\_{\mathbf{x}, \varepsilon} (\mathfrak{z}\_{2}) \mathbf{y} - \mathcal{S}\_{\mathbf{x}, \varepsilon} (\mathfrak{z}\_{1}) \mathbf{y} \right| \right| \to 0, \text{ as } \mathfrak{z}\_{2} \to \mathfrak{z}\_{1}. \end{cases}$$

**Proposition 1** ([39])**.** *Let κ* ∈ (0, 1), *μ* ∈ (0, 1] *and for all* y ∈ *D*(A)*, there exists a* Λ*<sup>μ</sup>* > 0*, such that*

$$\left\| \mathbf{A}^{\mu} \mathcal{Q}\_{\mathbf{x}}(\mathbf{y}) \mathbf{y} \right\| \leq \frac{\kappa \Lambda\_{\mu} \Gamma(2-\mu)}{\mathfrak{z}^{\kappa \mu} \Gamma(1+\kappa(1-\mu))} \left\| \mathbf{y} \right\|\_{\mathsf{Y}} \, \mathbf{0} < \mathsf{y} < d.$$

**Lemma 5** ([40])**.** *Let* J *be a compact real interval and* P*bd*,*cv*,*cl*(*Y*) *be the set of all non-empty, bounded, convex, and closed subsets of Y. Let* <sup>G</sup> *be the <sup>L</sup>*1*-Carathéodory multi-valued map, measurable to* z *for each* y ∈ *Y, u.s.c. to* y *for each* z ∈ *C*(J ,*Y*)*, the set*

$$S\_{\mathcal{G}, \mathfrak{Y}} = \left\{ \mathbf{g} \in L^1(\mathcal{J}, Y) : \mathbf{g}(\mathfrak{z}) \in \mathcal{G} \left( \mathfrak{z}, \mathbf{y}(\mathfrak{z}), \int\_0^w e(w, \mathbf{s}, \mathbf{y}(\mathfrak{s})) ds \right) , \mathfrak{z} \in \mathcal{J} \right\},\tag{4}$$

*is non-empty. Let* <sup>Υ</sup>*be the linear continuous function from L*1(<sup>J</sup> ,*Y*) *to , then*

$$\mathbf{Y} \circ \mathbf{S}\_{\mathcal{G}} : \mathbb{C} \to \mathcal{P}\_{bd, \text{cv}, \mathcal{cl}}(\mathbb{C}), \quad \mathbf{y} \to (\mathbf{Y} \circ \mathbf{S}\_{\mathcal{G}})(\mathbf{y}) = \mathbf{Y}(\mathbf{S}\_{\mathcal{G}, \mathbf{y}}), \tag{5}$$

*is a closed graph operator in* × *.*

**Lemma 6** (Martelli's fixed point theorem [17])**.** *Let Y be a Banach space and F* : *Y* → P*bd*,*cv*,*cl*(*Y*) *be an upper semi-continuous and condensing map. If the set*

$$\mathcal{M} = \{ \mathbf{y} \in \mathcal{Y} : \lambda \mathbf{y} \in F(\mathbf{y}) \text{ for some } \lambda > 1 \}$$

*is bounded, then F has a fixed point.*

#### **3. Existence**

We need the succeeding hypotheses:


$$S\_{\mathcal{G}, \mathbf{y}} = \left\{ \mathbf{g} \in L^1(\mathcal{J}, \mathbf{y}) : \mathbf{g}(\mathfrak{z}) \in \mathcal{G} \left( \mathfrak{z}, \mathbf{y}(\mathfrak{z}), \int\_0^w e(w, \mathbf{s}, \mathbf{y}(\mathbf{s})) ds \right), \mathfrak{z} \in \mathcal{J} \right\},$$

is non-empty.

(b) For z ∈ J , G(z, ·, ·) : *Y* × *Y* → *Y*, *e*(z,*s*, ·) : *Y* → *Y* are continuous functions and for each y ∈ , G ·, y, % *<sup>e</sup>*) : J →I and *<sup>e</sup>*(·, ·, <sup>y</sup>) : I×J → *<sup>Y</sup>* are strongly measurable.

(c) There exists a function *φ*(z) ∈ *C*(J , R+) satisfying

$$\begin{split} \lim\_{\mathfrak{z}\to 0^{+}} \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\mathfrak{z}} I\_{0^{+}}^{\kappa\mathfrak{z}} \mathfrak{e}(\mathfrak{z}) &= 0 \\ \left\| \mathcal{G}(\mathfrak{z}, \mathfrak{z}\_{1}, \mathfrak{z}\_{2}) \right\| &= \sup \left\{ \left\| \mathcal{G} \right\| : \mathcal{G}(\mathfrak{z}) \in \mathcal{G} \left( \mathfrak{z}, \mathfrak{y}(\mathfrak{z}), \int\_{0}^{\mathfrak{z}} e(\mathfrak{z}, \mathfrak{s}, \mathfrak{y}(\mathfrak{s})) d\mathfrak{s} \right) \right\} \\ &\leq \mathfrak{e}(\mathfrak{z}) \Phi(\left\| \mathfrak{z}\_{1} \right\| + \left\| \mathfrak{z}\_{2} \right\|) . \end{split}$$

for a.e. <sup>z</sup> ∈ J and <sup>z</sup>1,z<sup>2</sup> <sup>∈</sup> *<sup>Y</sup>*, where <sup>Φ</sup> : <sup>R</sup><sup>+</sup> <sup>→</sup> (0, <sup>∞</sup>) is a continuous, additive, and non-decreasing function, satisfying Φ(*γ*1(z)(y)) ≤ *γ*1(z)Φ(y), where *γ* ∈ *C*(J , R+).

(d) There exists *ψ* ∈ *C*(J , R+), such that

$$\left\| \left| \int\_{0}^{\mathfrak{z}} e(\mathfrak{z}\_{\mathsf{z}} \mathsf{s}\_{\mathsf{z}} \mathsf{y}(s)) \right| \right\| \leq \psi(\mathfrak{z}) \|\mathsf{y}\| \text{ for each } \mathfrak{z} \in \mathcal{J}\_{\mathsf{z}} \text{ y } \in \mathsf{Y}. \text{ } $$

(*H*3) For any z ∈ J , multi-valued map N : J × *Y* → *Y* is a continuous function and there exists *<sup>μ</sup>* <sup>∈</sup> (0, 1), such that N ∈ *<sup>D</sup>*(A*μ*) and all <sup>y</sup> <sup>∈</sup> *<sup>Y</sup>*, <sup>z</sup> ∈ J , <sup>A</sup>*μ*<sup>N</sup> (z, ·) satisfy the following:

$$\left| \left| \mathbb{A}^{\mu} \mathcal{N} (\mathfrak{z} \operatorname{y} (\mathfrak{z})) \right| \right| \leq M\_{\mathcal{S}} \left( 1 + \mathfrak{z}^{1 - \varepsilon + \kappa \varepsilon - \kappa \tilde{\mathfrak{z}}} ||\mathfrak{y} (\mathfrak{z})|| \right) \text{ and } \left| \left| \mathbb{A}^{-\mu} \right| \right| \leq M\_{\mathcal{O}} \left( \mathfrak{z} , \mathfrak{y} \right) \in \mathcal{J} \times \mathcal{Y}.$$

(*H*4) N is completely continuous, and for any bounded set *D* ⊂ , the set {z → N (z, y(z)), y ∈ *D*} is equicontinuous in *Y*.

**Theorem 2.** *Assume that* (*H*1) − (*H*4) *hold. Then the HF system* (1)*–*(2) *has a mild solution on* J *, provided*

$$L' \int\_0^{\mathfrak{z}} (\mathfrak{z} - w)^{\kappa\_{\mathfrak{z}}^{\mathfrak{z}} - 1} \phi(\mathfrak{z}) \left( 1 + \psi(\mathfrak{z}) \right) dw < \int\_{M\_1^\*}^{\infty} \frac{du}{\Phi(u)} \lambda$$

*where*

$$M\_1^\* = d^{1 - \varepsilon + \kappa \varepsilon - \kappa \xi} \left[ L^{\prime \prime} d^{-1 + \varepsilon - \kappa \varepsilon + \kappa \xi} \left( \mathbf{y}\_0 - M\_0 M\_\mathcal{g} \right) + M\_0 M\_\mathcal{g} (1 + P) \right]^2$$

*and* <sup>y</sup><sup>0</sup> <sup>∈</sup> *<sup>D</sup>*(A*<sup>θ</sup>* ) *with <sup>θ</sup>* <sup>&</sup>gt; <sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*.

**Proof.** We define the multi-valued operator Ψ : X→P(X ) by

$$\begin{split} \Psi(\mathbf{y}(\mathbf{j})) &= \left\{ z \in \mathcal{X} : z(\mathbf{j}) = \mathbf{y}^{1-\varepsilon+\kappa x-\kappa \mathbf{\mathcal{I}}} \Big[ \mathcal{S}\_{\mathbf{x},\mathbf{\mathcal{E}}}(\mathbf{y}) \left[ \mathbf{y}\_{0} - \mathcal{N}(\mathbf{0}, \mathbf{y}(0)) \right] + \mathcal{N}(\mathbf{y}, \mathbf{y}(\mathbf{j})) \Big] \\ &+ \int\_{0}^{\mathbf{\mathcal{I}}} (\mathfrak{z} - w)^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathfrak{z} - w) \mathsf{A} \mathcal{N}(w, \mathbf{y}(w)) dw \\ &+ \int\_{0}^{\mathbf{\mathcal{I}}} (\mathfrak{z} - w)^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathfrak{z} - w) \mathcal{G} \Big( w, \mathbf{y}(w), \int\_{0}^{w} e(w, \mathbf{s}, \mathbf{y}(\mathbf{s})) ds \Big) \right\} dw, \mathfrak{z} \in (0, d] \Big\}. \end{split}$$

To show that the fixed point of Ψ exists. **Step:1** Convexity of Ψ(y) ∀ y ∈ *BP*(J ).

Let *<sup>z</sup>*1, *<sup>z</sup>*<sup>2</sup> ∈ {Ψy(z)} and *<sup>h</sup>*1, *<sup>h</sup>*<sup>2</sup> ∈ *<sup>S</sup>*G,<sup>y</sup> such that <sup>z</sup> ∈ J . We know

$$\begin{split} z\_{i} &= \mathbf{j}^{1-\varepsilon+\kappa\varepsilon-\kappa\_{5}^{\mathbb{Z}}} \Big[ \mathcal{G}\_{\mathbf{x},\boldsymbol{\varepsilon}}(\mathbf{y}) \Big[ \mathbf{y}\_{0} - \mathcal{N}(\mathbf{0},\mathbf{y}(0)) \Big] + \mathcal{N}(\mathbf{y},\mathbf{y}(\boldsymbol{\varepsilon})) \\ &+ \int\_{0}^{\boldsymbol{\mathfrak{z}}} (\boldsymbol{\mathfrak{y}} - \boldsymbol{w})^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathbf{y} - \boldsymbol{w}) \mathsf{A} \mathcal{N}(\mathbf{w},\mathbf{y}(\boldsymbol{w})) dw + \int\_{0}^{\boldsymbol{\mathfrak{z}}} (\boldsymbol{\mathfrak{y}} - \boldsymbol{w})^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathbf{y} - \boldsymbol{w}) h\_{i}(\boldsymbol{w}) dw \Big], \quad i = 1, 2, \end{split}$$

Let 0 ≤ *λ* ≤ 1; then for each of z ∈ J , we have

$$\begin{split} \lambda z\_1 + (1 - \lambda) z\_2(\mathfrak{z}) &= \mathfrak{z}^{1 - \varepsilon + \kappa \varepsilon - \kappa \mathbb{E}} \Big( \mathcal{S}\_{\mathbb{K}\mathscr{E}}(\mathfrak{z}) \left[ \mathbf{y}\_0 - \mathcal{N}(\mathbf{0}, \mathbf{y}(0)) \right] + \mathcal{N} \Big( \mathfrak{z}, \mathfrak{y}(\mathfrak{z}) \Big) \\ &+ \int\_0^\mathfrak{z} \left( \mathfrak{z} - w \right)^{\kappa - 1} \mathcal{Q}\_{\mathbb{K}}(\mathfrak{z} - w) \mathcal{A} \big( w, \mathfrak{y}(w) \big) dw \Big) \\ &+ \mathfrak{z}^{1 - \varepsilon + \kappa \varepsilon - \kappa \mathbb{E}} \int\_0^\mathfrak{z} \big( \mathfrak{z} - w \big)^{\kappa - 1} \mathcal{Q}\_{\mathbb{K}}(\mathfrak{z} - w) \Big[ \lambda h\_1(w) + (1 - \lambda) h\_2(w) \Big] dw. \end{split}$$

We know that N has a convex value, then *<sup>S</sup>*G,<sup>y</sup> is convex. So, *<sup>λ</sup>h*<sup>1</sup> + (<sup>1</sup> − *<sup>λ</sup>*)*h*<sup>2</sup> ∈ *<sup>S</sup>*G,y. Therefore,

$$
\lambda z\_1 + (1 - \lambda) z\_2 \in \Psi \mathbf{y}(\mathfrak{z})\_\prime
$$

hence Ψ is convex.

**Step 2:** Boundness of Ψ on *BP*(J ). Consider, ∀ y ∈ *BP*(J ), we have

 *z*(z) <sup>≤</sup> sup <sup>z</sup> 1−*ε*+*κε*−*κξ* S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + N z, y(z) + z 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*κ*(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*κ*(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* <sup>≤</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ* sup S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + <sup>N</sup> z, y(z) <sup>+</sup> sup <sup>z</sup> 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−<sup>1</sup> <sup>A</sup>1−*μ*Q*κ*(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*) <sup>A</sup>*μ*<sup>N</sup> *w*, y(*w*) *dw* <sup>+</sup> sup <sup>z</sup> 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−<sup>1</sup> Q*κ*(z − *w*) G *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*w*) *ds dw* <sup>≤</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ*\* *Ld*−1+*ε*−*κε*+*κξ* y<sup>0</sup> − *M*0*Mg* + *M*0*Mg*(1 + *P*) + <sup>+</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ*\* Λ1−*<sup>μ</sup> dκμ*Γ(1 + *μ*) *μ*Γ(1 + *κμ*) *Mg*(1 + *P*) + *L <sup>φ</sup>*(z)Φ(y)[<sup>1</sup> <sup>+</sup> *<sup>ψ</sup>*(z)] *<sup>d</sup>κξ κξ* + ≤ M<sup>∗</sup> <sup>1</sup> <sup>+</sup> *<sup>d</sup>ε*(1−*κ*)−*κξ*−<sup>1</sup> \* Λ1−*<sup>μ</sup> dκμ*Γ(1 + *μ*) *μ*Γ(1 + *κμ*) *Mg*(1 + *P*) + *L <sup>φ</sup>*(z)Φ(y)[<sup>1</sup> <sup>+</sup> *<sup>ψ</sup>*(z)] *<sup>d</sup>κξ κξ* + .

From Lemma 2 and hypotheses (*H*3), we have the boundness of the operators. Hence, it is bounded.

**Step 3:** Next, we show that the *z*(z) bounded maps are set to the equicontinuous set of *BP*(J ).

Consider 0 < <sup>z</sup><sup>1</sup> < <sup>z</sup><sup>2</sup> ≤ *<sup>d</sup>* and ∃G∈ *<sup>S</sup>*G,y, we have

 

*z*(z2) <sup>−</sup> *<sup>z</sup>*(z1) ≤ z 1−*ε*+*κε*−*κξ* 2 \* S*κ*,*ε*(z2) 0 y<sup>0</sup> − N (0, y(0))1 + N z2, y(z2) + z<sup>2</sup> 0 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z<sup>2</sup> 0 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw*<sup>+</sup> − z 1−*ε*+*κε*−*κξ* 1 \* S*κ*,*ε*(z1) 0 y<sup>0</sup> − N (0, y(0))1 + N z1, y(z1) + z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw*+ ≤ 0 z 1−*ε*+*κε*−*κξ* <sup>2</sup> S*κ*,*ε*(z2) − z 1−*ε*+*κε*−*κξ* <sup>1</sup> S*κ*,*ε*(z1) 10y<sup>0</sup> − N (0, y(0))1 + z 1−*ε*+*κε*−*κξ* <sup>2</sup> N (z2, y(z2)) − z 1−*ε*+*κε*−*κξ* <sup>1</sup> N (z1, y(z1)) + z 1−*ε*+*κε*−*κξ* 2 z<sup>1</sup> 0 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z 1−*ε*+*κε*−*κξ* 2 z<sup>2</sup> z1 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* − z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z 1−*ε*+*κε*−*κξ* 2 z<sup>1</sup> 0 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* + z 1−*ε*+*κε*−*κξ* 2 z<sup>2</sup> z1 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* − z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* ≤ 0 z 1−*ε*+*κε*−*κξ* <sup>2</sup> S*κ*,*ε*(z2) − z 1−*ε*+*κε*−*κξ* <sup>1</sup> S*κ*,*ε*(z1) 10y<sup>0</sup> − N (0, y(0))1 + z 1−*ε*+*κε*−*κξ* <sup>2</sup> N (z2, y(z2)) − z 1−*ε*+*κε*−*κξ* <sup>1</sup> N (z1, y(z1)) + z 1−*ε*+*κε*−*κξ* 2 z<sup>2</sup> z1 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>N</sup> *w*, y(*w*) *dw* + z 1−*ε*+*κε*−*κξ* 2 z<sup>1</sup> 0 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* − z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* − z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z 1−*ε*+*κε*−*κξ* 2 z<sup>2</sup> z1 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* + z 1−*ε*+*κε*−*κξ* 2 z<sup>1</sup> 0 (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* − z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* + z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>2</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* − z 1−*ε*+*κε*−*κξ* 1 z<sup>1</sup> 0 (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw* = 8 ∑ *i*=1 *Ii*.

Since S*κ*,*ε*(z)(y<sup>0</sup> − *M*0*Mg*) is strong-continuous, we have

$$I\_1 \text{ tends to } 0 \text{ as } \mathfrak{z}\_2 \to \mathfrak{z}\_1.$$

The equicontinuity of N ensures that

$$I\_2 \text{ tends to } 0, \text{ as } \mathfrak{z}\_2 \to \mathfrak{z}\_1.$$

$$\begin{aligned} I\_3 &= \left\| \mathfrak{z}\_2^{1-\varepsilon+\kappa\varepsilon-\kappa\mathfrak{z}} \int\_{\mathfrak{z}\_1}^{\mathfrak{z}\_2} (\mathfrak{z}\_2 - w)^{\kappa-1} \mathcal{Q}\_{\mathbb{X}}(\mathfrak{z}\_2 - w) \mathsf{A} \mathcal{N} \left( w, \mathfrak{y}(w) \right) dw \right\| \\ &\le \mathfrak{z}\_2^{1-\varepsilon+\kappa\varepsilon-\kappa\mathfrak{z}} \Lambda\_{1-\mu} M\_{\mathfrak{z}} (1+P) \frac{\Gamma(1+\mu)}{\mu \Gamma(1+\kappa\mu)} (\mathfrak{z}\_2 - \mathfrak{z}\_1)^{\kappa\mu} \end{aligned}$$

*Then*, *I*<sup>3</sup> *tends* 0 *as* z<sup>2</sup> → z1.

$$\begin{split} I\_{4} &= \left\| \mathbf{j}\_{2}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathbf{x}}} \int\_{0}^{\mathbf{j}\_{1}} (\mathbf{j}\_{2}-w)^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathbf{j}\_{2}-w) \mathsf{A} \mathcal{N} \begin{pmatrix} w, \mathbf{y}(w) \end{pmatrix} dw \\ &- \mathbf{j}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathbf{x}}} \int\_{0}^{\mathbf{j}\_{1}} (\mathbf{j}\_{1}-w)^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathbf{j}\_{2}-w) \mathsf{A} \mathcal{N} \begin{pmatrix} w, \mathbf{y}(w) \end{pmatrix} dw \right\| \\ &\leq \kappa \Lambda\_{1-\mu} M\_{\mathcal{S}} (1+P) \frac{\Gamma(1+\mu)}{\mu \Gamma(1+\kappa\mu)} \\ &\times \left\| \int\_{0}^{\mathbf{j}\_{1}} \left( \mathbf{j}\_{2}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathbf{x}}} (\mathbf{j}\_{2}-w)^{\kappa-1} - \mathbf{j}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathbf{x}}} (\mathbf{j}\_{1}-w)^{\kappa-1} \right) (\mathbf{j}\_{2}-w)^{\kappa(\mu-1)} dw \right\|. \end{split}$$
  $We$   $We$   $\text{ $\mu$  denotes  $\S\_{L}$  and  $0$  as  $\tilde{\chi}\_{2} \to \chi\_{1}$ . Also,  $\mu$ $ 

*We have*, *I*<sup>4</sup> *tends* 0 *as* z<sup>2</sup> → z1. *Also*,

$$\begin{split} I\_{5} = \left\| \mathfrak{z}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\mathbb{5}} \int\_{0}^{\mathfrak{z}\_{1}} \left( (\mathfrak{z}\_{1}-w)^{\kappa-1} Q\_{\kappa}(\mathfrak{z}\_{2}-w) \mathsf{A} \mathcal{N} \begin{pmatrix} w, \mathbf{y}(w) \end{pmatrix} \right) \right. \\ \left. - (\mathfrak{z}\_{1}-w)^{\kappa-1} Q\_{\kappa}(\mathfrak{z}\_{1}-w) \mathsf{A} \mathcal{N} \begin{pmatrix} w, \mathbf{y}(w) \end{pmatrix} \right) dw \right\|\, \\ \leq \left. M\_{0}' M\_{\mathfrak{z}} (1+P) \mathfrak{z}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\mathbb{5}} \int\_{0}^{\mathfrak{z}\_{1}} \left( \mathfrak{z}\_{1}-w \right)^{\kappa-1} \left| \left[ \mathcal{Q}\_{\kappa}(\mathfrak{z}\_{2}-w) - \mathcal{Q}\_{\kappa}(\mathfrak{z}\_{1}-w) \right] \right| \, . \end{split}$$

*By Theorem* 1 *and strong continuity o f* Q*κ*(z), *I*<sup>5</sup> *tends to* 0, *as* z<sup>2</sup> → z1.

$$\begin{split} I\_{6} &= \left\| \mathfrak{z}\_{2}^{1-\varepsilon+\kappa\varepsilon-\kappa\mathbb{5}} \int\_{\mathfrak{z}\_{1}}^{\mathfrak{z}\_{2}} (\mathfrak{z}\_{2}-w)^{\kappa-1} \mathcal{Q}\_{\kappa}(\mathfrak{z}\_{2}-w) \mathcal{G}\left(w,\mathfrak{y}(w),\int\_{0}^{w} e\{w,s,\mathfrak{y}(s)\} ds\right) dw \right\| \right\| \\ &\leq L' \left| \mathfrak{z}\_{2}^{1-\varepsilon+\kappa\varepsilon-\kappa\mathbb{5}} \int\_{\mathfrak{z}\_{1}}^{\mathfrak{z}\_{2}} (\mathfrak{z}\_{2}-w)^{\kappa\mathbb{5}} - \mathfrak{z} (w) \Phi(\mathfrak{y}) \left[1+\mathfrak{y}(\mathfrak{z})\right] dw \right| \\ &\leq L' \int\_{0}^{\mathfrak{z}\_{1}} \left[ \mathfrak{z}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\mathbb{5}} (\mathfrak{z}\_{1}-w)^{\kappa\mathbb{5}} - \mathfrak{z}\_{2}^{(1+\kappa\mathbb{5})(1-\kappa)} (\mathfrak{z}\_{2}-w)^{\kappa\mathbb{5}} - 1 \right] \\ &\quad \times \mathfrak{g}(w) \Phi(\mathfrak{y}) \left[1+\mathfrak{y}(\mathfrak{z})\right] dw. \end{split}$$

Then *I*<sup>6</sup> *tends to* 0 as z<sup>2</sup> → z<sup>1</sup> by using (*H*2) and the Lebesgue-dominated convergent theorem.

$$\begin{split} I\_{7} &= \left\| \mathfrak{z}\_{2}^{1-\varepsilon+\kappa\varepsilon-\kappa\_{5}^{\mathbb{C}}} \int\_{0}^{\mathfrak{z}\_{1}} (\mathfrak{z}\_{2}-w)^{\kappa-1} \mathcal{Q}\_{\kappa}(\mathfrak{z}\_{2}-w) \mathcal{G}\left(w,\mathbf{y}(w),\int\_{0}^{w} e(w,s,\mathbf{y}(s))ds\right) dw \\ & - \mathfrak{z}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\_{5}^{\mathbb{C}}} \int\_{0}^{\mathfrak{z}\_{1}} (\mathfrak{z}\_{1}-w)^{\kappa-1} \mathcal{Q}\_{\kappa}(\mathfrak{z}\_{2}-w) \mathcal{G}\left(w,\mathbf{y}(w),\int\_{0}^{w} e(w,s,\mathbf{y}(s))ds\right) dw \right\|\, \\ & \leq L' \int\_{0}^{\mathfrak{z}\_{1}} (\mathfrak{z}\_{2}-w)^{-\kappa+\kappa\varepsilon} \Big| \mathfrak{z}\_{2}^{1-\varepsilon+\kappa\varepsilon-\kappa\varepsilon} (\mathfrak{z}\_{2}-w)^{\kappa-1} - \mathfrak{z}\_{1}^{1-\varepsilon+\kappa\varepsilon-\kappa\varepsilon}(\mathfrak{z}\_{1}-w)^{\kappa-1} \Big| \\ & \quad \times \oint (w) \Phi(\mathfrak{y}) [1+\psi(\mathfrak{z})] dw, \end{split}$$

and % <sup>z</sup><sup>1</sup> <sup>0</sup> 2z (1+*κξ*)(1−*κ*) <sup>1</sup> (z<sup>1</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1*φ*(*w*)Φ(y)[<sup>1</sup> <sup>+</sup> *<sup>ψ</sup>*(z)]*dw* exists (*<sup>w</sup>* <sup>∈</sup> (0,z1]), then from Lebesgue's dominated convergence theorem, we obtain

$$\int\_{0}^{\mathfrak{J}\_{1}} (\mathfrak{z}\_{2} - w)^{-\mathbf{x} + \mathfrak{x}\_{\mathfrak{z}}^{\mathfrak{z}}} \left| \mathfrak{z}\_{2}^{1 - \varepsilon + \mathfrak{x}\varepsilon - \mathfrak{x}\_{\mathfrak{z}}^{\mathfrak{z}}} (\mathfrak{z}\_{2} - w)^{\mathfrak{x} - 1} - \mathfrak{z}\_{1}^{1 - \varepsilon + \mathfrak{x}\varepsilon - \mathfrak{x}\_{\mathfrak{z}}^{\mathfrak{z}}} (\mathfrak{z}\_{1} - w)^{\mathfrak{x} - 1} \right| \phi(w) \Phi(\mathfrak{y}) \left[ 1 + \psi(\mathfrak{y}) \right] dw$$
 
$$\to 0 \text{ as } \mathfrak{z} \to \mathfrak{z}\_{1\prime}$$

so we conclude limz2→z<sup>1</sup> *<sup>I</sup>*<sup>7</sup> <sup>=</sup> 0. For any > 0, we have

$$\begin{split} I\_{\mathbb{S}} &= \left\| \int\_{0}^{\mathbb{S}\mathbb{1}} \mathbf{1}\_{1}^{1-\varepsilon+\kappa\mathbf{x}-\kappa\mathbf{\tilde{x}}} \Big[ \mathcal{Q}\_{\mathbf{x}}(\mathbf{y}\_{2}-w) - \mathcal{Q}\_{\mathbf{x}}(\mathbf{y}\_{1}-w) \Big] (\mathbf{y}\_{1}-w)^{\kappa-1} \mathcal{G} \Big( w, \mathbf{y}(w), \int\_{0}^{w} e(w, s, \mathbf{y}(s)) ds \Big) dw \Big] \right\| \\ &\leq \mathbf{j}\_{1}^{1-\varepsilon+\kappa\mathbf{x}-\kappa\mathbf{\tilde{x}}+\kappa(1+\xi)} \int\_{0}^{\mathbb{S}\mathbf{1}} (\mathbf{y}\_{1}-w)^{\kappa\mathbf{\tilde{x}}-\mathbb{1}} \boldsymbol{\Phi}(w) \boldsymbol{\Phi}(\mathbf{y}) \Big[ 1+\boldsymbol{\Psi}(\mathbf{y}) \Big] dw \\ &\quad \times \sup\_{w \in \left[ \mathbb{Q}\_{\mathbf{x}}(\mathbf{y}\_{2}-w) - \mathcal{Q}\_{\mathbf{x}}(\mathbf{y}\_{1}-w) \right]} \| \\ &\quad + 2L' \int\_{\mathbb{S}\mathbf{1}-\varepsilon}^{\mathbb{S}\mathbf{1}} \boldsymbol{\vartheta}\_{1}^{1-\varepsilon+\kappa\mathbf{x}+\kappa\mathbf{\tilde{x}}} (\mathbf{y}\_{1}-w)^{\kappa\mathbf{\tilde{x}}-\mathbb{1}} \boldsymbol{\Phi}(w) \boldsymbol{\Phi}(\mathbf{y}) \Big[ 1+\boldsymbol{\Psi}(\mathbf{y}) \Big] dw. \end{split}$$

From Theorem (1) and limz2→z<sup>1</sup> *<sup>I</sup>*<sup>6</sup> <sup>=</sup> 0, we have *<sup>I</sup>*<sup>8</sup> <sup>→</sup> 0 independently of <sup>y</sup> <sup>∈</sup> *BP*(<sup>J</sup> ) as z<sup>2</sup> → z1, → 0. Hence, *z*(z2) − *z*(z1) → 0 independently of y ∈ *BP*(J ) as z<sup>2</sup> → z1. Therefore, {Ψy(z) : y ∈ *BP*(J )} is equicontinuous on J .

**Step 4:** Show the relative compact of *V*(z) = ' *z*(z) : *z* ∈ Ψ(*BP*(J ))( for z ∈ J .

$$\text{Let } 0 < \mathfrak{a} < \mathfrak{z}\_{\nu} \text{ and there is a positive value } q \text{, assume an operator } z'(\mathfrak{z}) \text{ on } B\_P(\mathcal{J}) \text{ by}$$

*z <sup>α</sup>*,*q*(z) = z <sup>1</sup>−*ε*+*κε*−*κξ*\* S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + N z, y(z) + z−*<sup>α</sup>* 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z−*<sup>α</sup>* 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*<sup>κ</sup>* (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw*<sup>+</sup> = z <sup>1</sup>−*ε*+*κε*−*κξ*\* S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + N z, y(z) + z−*<sup>α</sup>* 0 <sup>∞</sup> *q κθM<sup>κ</sup>* (*θ*)(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1*T*((<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*<sup>κ</sup> <sup>θ</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z−*<sup>α</sup>* 0 <sup>∞</sup> *q κθM<sup>κ</sup>* (*θ*)(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1*T*((<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*<sup>κ</sup> <sup>θ</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dθdw*<sup>+</sup> = z <sup>1</sup>−*ε*+*κε*−*κξ*\* S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + N z, y(z) + + *κ*z <sup>1</sup>−*ε*+*κε*−*κξT*(*α<sup>κ</sup> q*) z−*<sup>q</sup>* 0 <sup>∞</sup> *q <sup>θ</sup>M<sup>κ</sup>* (*θ*)(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−<sup>1</sup> <sup>×</sup> *<sup>T</sup>*((<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*<sup>κ</sup> <sup>θ</sup>* <sup>−</sup> *<sup>α</sup><sup>κ</sup> <sup>q</sup>*) 0 AN *w*, y(*w*) + G *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *dθdw*.

From the compactness of *<sup>T</sup>*(*ακq*), we note that *<sup>V</sup>α*,*<sup>ξ</sup>* (z) = {(*z <sup>α</sup>*,*q*(z))y(z) : y ∈ *BP*(J )} is pre-compact in Y. ∀ y ∈ *BP*(J ), we have

 *z*(z) <sup>−</sup> *<sup>z</sup> <sup>α</sup>*,*q*(z) ≤ *κ*z <sup>1</sup>−*ε*+*κε*−*κξ* <sup>z</sup> 0 *<sup>q</sup>* 0 *<sup>θ</sup>M<sup>κ</sup>* (*θ*)(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1*T*((<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*<sup>κ</sup> <sup>θ</sup>*) \* AN *w*, y(*w*) + G *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds*+*dθdw* + *κ*z <sup>1</sup>−*ε*+*κε*−*κξ* <sup>z</sup> z−*<sup>α</sup>* <sup>∞</sup> *q* (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1*θM<sup>κ</sup>* (*θ*)*T*((<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*<sup>κ</sup> <sup>θ</sup>*) \* AN *w*, y(*w*) + G *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds*+*dθdw* ≤ *κ*Λ0z <sup>1</sup>−*ε*+*κε*−*κξ* <sup>z</sup> 0 *<sup>q</sup>* 0 *<sup>θ</sup>M<sup>κ</sup>* (*θ*)(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−*<sup>κ</sup> <sup>θ</sup>ξ*−<sup>1</sup> × 0 *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ(y)[1 + *ψ*(z)]1 *dθdw* + z z−*<sup>α</sup>* <sup>∞</sup> *q* (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1*θM<sup>κ</sup>* (*θ*)(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−*<sup>κ</sup> <sup>θ</sup>ξ*−1<sup>0</sup> *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ(y)[1 + *ψ*(z)]1 *dw* ≤ *κ*Λ0z <sup>1</sup>−*ε*+*κε*−*κξ* <sup>z</sup> 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1<sup>0</sup> *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ(y)[1 + *ψ*(z)]1 *dw <sup>q</sup>* 0 *θ<sup>ξ</sup> M<sup>κ</sup>* (*θ*)*dθ* + z z−*<sup>α</sup>* (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1<sup>0</sup> *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ(y)[1 + *ψ*(z)]1 *dw* <sup>∞</sup> 0 *θ<sup>ξ</sup> M<sup>κ</sup>* (*θ*)*dθ* ≤ *κ*Λ0z <sup>1</sup>−*ε*+*κε*−*κξ* <sup>z</sup> 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1<sup>0</sup> *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ(y)[1 + *ψ*(z)]1 *dw <sup>q</sup>* 0 *θ<sup>ξ</sup> M<sup>κ</sup>* (*θ*)*dθ* + Γ(1 − *ξ*) Γ(1 − *κξ*) z z−*<sup>α</sup>* (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1<sup>0</sup> *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ(y)[1 + *ψ*(z)]1 *dw* → 0 as *α tends to* 0, *q tends to* 0.

So, *Vα*,*q*(z) = ' *zα*,*q*(z) : z ∈ *BP*(J ) ( are arbitrary closed to *V*(z) = ' *z*(z) : z ∈ *BP*(I) ( . Therefore, {*z*(z) : z ∈ *BP*(J )} is relatively compact by the Arzela–Ascoli theorem. Thus, the continuity of *z*(z) and relative compactness of {*z*(z) : z ∈ *BP*(J )} imply that *z*(z) is a completely continuous operator.

**Step 5:** Ψ has a closed graph.

Take y*<sup>n</sup>* → y<sup>∗</sup> as *n* → ∞, *zn*(z) ∈ Ψ(y*n*) and *zn* → *z*<sup>∗</sup> as *n* → ∞, we have to show that *<sup>z</sup>*<sup>∗</sup> ∈ <sup>Ψ</sup>(y∗). Since *zn* ∈ <sup>Ψ</sup>(y*n*) then ∃ a function G*<sup>n</sup>* ∈ *<sup>S</sup>*G,y*<sup>n</sup>* , such that

$$\begin{split} z\_{n}(\mathfrak{z}) &= \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\frac{\mathfrak{z}}{\mathfrak{z}}} \Big[ \mathcal{S}\_{\kappa\varepsilon}(\mathfrak{z}) \Big[ \mathfrak{y}\_{0} - \mathcal{N}(\mathfrak{0},\mathfrak{y}(0)) \Big] + \mathcal{N}(\mathfrak{z},\mathfrak{y}\_{n}(\mathfrak{z})) \\ &+ \int\_{0}^{\mathfrak{z}} (\mathfrak{z}-w)^{\kappa-1} \mathcal{Q}\_{\kappa}(\mathfrak{z}-w) \mathsf{A} \mathcal{N}\big(w,\mathfrak{y}\_{n}(w)\big) dw + \int\_{0}^{\mathfrak{z}} (\mathfrak{z}-w)^{\kappa-1} \mathcal{Q}\_{\kappa}(\mathfrak{z}-w) \mathcal{G}\_{n}(w) dw \Big]. \end{split}$$

We need to show that ∃ G∗ ∈ *S*G,y<sup>∗</sup> , such that

$$\begin{split} z\_{\ast}(\mathfrak{z}) &= \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\frac{\mathfrak{z}}{\mathfrak{z}}} \Big[ \mathcal{S}\_{\mathsf{x},\mathsf{c}}(\mathfrak{z}) \left[ \mathfrak{y}\_{0} - \mathcal{N}(\mathsf{0},\mathsf{y}(\mathsf{0})) \right] + \mathcal{N}(\mathfrak{z},\mathsf{y}\_{\ast}(\mathfrak{z})) \\ &+ \int\_{0}^{\mathfrak{z}} (\mathfrak{z}-w)^{\kappa-1} \mathcal{Q}\_{\mathsf{x}}(\mathfrak{z}-w) \mathsf{A} \mathcal{N}\left(w,\mathsf{y}\_{\ast}(w)\right) dw + \int\_{0}^{\mathfrak{z}} (\mathfrak{z}-w)^{\kappa-1} \mathcal{Q}\_{\mathsf{x}}(\mathfrak{z}-w) \mathcal{G}\_{\mathsf{x}}(w) dw \Big]. \end{split}$$

Clearly,

$$\begin{split} \left\| \left[ z\_{n}(\mathfrak{z}) - \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\xi} \left( \mathcal{S}\_{\mathbf{x},\varepsilon}(\mathfrak{z}) \left[ \mathfrak{y}\_{0} + \mathcal{N}(\mathfrak{0},\mathfrak{y}(0)) \right] - \mathcal{N} \left( \mathfrak{y}, \mathfrak{y}\_{n}(\mathfrak{z}) \right) \right) \right. \\ \left. \left. - \int\_{0}^{\mathfrak{z}} (\mathfrak{z} - w)^{\mathfrak{z}-1} Q\_{\mathfrak{x}} (\mathfrak{z} - w) \mathsf{A} \mathcal{N} \left( w, \mathfrak{y}\_{n}(w) \right) dw \right) \right] \\ - \left[ z\_{\ast}(\mathfrak{z}) - \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\xi} \left( \mathcal{S}\_{\mathbf{x},\varepsilon}(\mathfrak{z}) \left[ \mathfrak{y}\_{0} - \mathcal{N} \left( \mathfrak{0}, \mathfrak{y}(0) \right) \right] - \mathcal{N} \left( \mathfrak{y}, \mathfrak{y}\_{\ast}(\mathfrak{z}) \right) \right. \\ \left. - \int\_{0}^{\mathfrak{z}} (\mathfrak{z} - w)^{\mathfrak{z}-1} Q\_{\mathfrak{x}} (\mathfrak{z} - w) \mathsf{A} \mathcal{N} \left( w, \mathfrak{y}\_{\ast}(w) \right) dw \right) \right] \right] \right] \to 0 \text{ as } n \to \infty. \end{split}$$

Next, we define the operator Υ : *L* (J ,*Y*) → X ,

$$\mathcal{Y}(\mathbf{y})(\mathfrak{z}) = \int\_0^\mathfrak{z} (\mathfrak{z} - w)^{\kappa - 1} \mathcal{Q}\_\mathbf{x}(\mathfrak{z} - w) \mathcal{G}\left(w, \mathbf{y}(w), \int\_0^w e\left(w, \mathbf{s}, \mathbf{y}(w)\right) ds\right) dw.$$

We have (by (5)) that Υ ◦ *S*G,<sup>y</sup> is a closed graph operator. So, by referring to y*psilon*, we know

$$\begin{aligned} \left[ z\_n(\mathfrak{z}) - \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathbb{S}}} \left( \mathcal{S}\_{\kappa,\varepsilon}(\mathfrak{z}) \left[ \mathbf{y}\_0 + \mathcal{N} \left( \mathbf{0}, \mathbf{y}(0) \right) \right] - \mathcal{N} \left( \mathbf{y}, \mathbf{y}\_n(\mathfrak{z}) \right) \right) \\ - \int\_0^\mathfrak{\mathbb{S}} \left( \mathfrak{z} - w \right)^{\mathfrak{s}-1} \mathcal{Q}\_\kappa (\mathfrak{z} - w) \mathsf{A} \mathcal{N} \left( w, \mathbf{y}\_n(w) \right) dw \right) \right] \in \mathsf{Y}(S\_{\mathcal{G}\mathcal{X}n})\_{\leq 1} \end{aligned}$$

since G*<sup>n</sup>* → G∗, we follow from (5) that

$$\begin{aligned} \left[ z\_\* \left( \mathfrak{z} \right) - \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathfrak{z}}} \left( \mathcal{S}\_{\mathbf{x},\varepsilon} (\mathfrak{z}) \left[ \mathfrak{y}\_0 - \mathcal{N} \left( \mathbf{0}, \mathfrak{y}(\mathbf{0}) \right) \right] - \mathcal{N} \left( \mathfrak{y}, \mathfrak{y}\_\* (\mathfrak{z}) \right) \right) \\ - \int\_0^\mathfrak{z} \left( \mathfrak{z} - w \right) \mathfrak{z}^{-1} \mathcal{Q}\_\mathbf{x} \left( \mathfrak{z} - w \right) \mathsf{A} \mathcal{N} \left( w, \mathfrak{y}\_\* (w) \right) dw \right) \right] \in \mathrm{Y}(S\_{\mathcal{G},\mu\_\*}). \end{aligned}$$

Therefore, Ψ is a closed graph. **Step:6** Set Λ is bounded.

$$\Lambda = \{ \mathbf{y} \in \partial B\_P(\mathcal{J}) : \lambda \mathbf{y} = \mathbf{Y}(\mathbf{y}) \text{ for some } \lambda > 1 \}.$$

Let <sup>y</sup> ∈ <sup>Λ</sup>. Then *<sup>λ</sup><sup>w</sup>* ∈ <sup>Ψ</sup>(y) for some *<sup>λ</sup>* > 1. Thus, there exists G ∈ *<sup>S</sup>*G,<sup>y</sup> in ways that for each z ∈ [0, *d*] and <sup>A</sup>1−*<sup>μ</sup>* ≤ *M* <sup>0</sup>, we have

$$\begin{split} \mathbf{y}(\mathfrak{z}) &= \lambda^{-1} \mathfrak{z}^{1-\varepsilon+\kappa\varepsilon-\kappa\tilde{\mathbb{I}}} \Big[ \mathcal{S}\_{\mathbf{x},\mathfrak{c}}(\mathfrak{z}) \Big[ \mathbf{y}\_{0} - \mathcal{N}(\mathbf{0},\mathbf{y}(0)) \Big] + \mathcal{N}(\mathfrak{z},\mathfrak{y}(\mathfrak{z})) \\ &+ \int\_{0}^{\mathfrak{z}} (\mathfrak{z}-w)^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathfrak{z}-w) \mathsf{A} \mathcal{N}(w,\mathfrak{y}(w)) dw \\ &+ \int\_{0}^{\mathfrak{z}} (\mathfrak{z}-w)^{\kappa-1} \mathcal{Q}\_{\mathbf{x}}(\mathfrak{z}-w) \mathcal{G} \Big( w, \mathfrak{y}(w), \int\_{0}^{w} e(w, s, \mathfrak{y}(s)) ds \Big) \Big] dw. \end{split}$$

By assumptions (*H*2) − (*H*4), we have

 y(z) = *λ*−1z <sup>1</sup>−*ε*+*κε*−*κξ*\* S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + N z, y(z) + z 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*κ*(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)A<sup>N</sup> *w*, y(*w*) *dw* + z 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−1Q*κ*(<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)<sup>G</sup> *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds*+*dw* <sup>≤</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ*\* sup S*κ*,*ε*(z) 0 y<sup>0</sup> − N (0, y(0))1 + N z, y(z) <sup>+</sup> sup <sup>z</sup> 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κ*−<sup>1</sup> Q*κ*(z − *w*) AN *w*, y(*w*) + G *w*, y(*w*), *<sup>w</sup>* 0 *e w*,*s*, y(*s*) *ds dw*<sup>+</sup> <sup>≤</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ*\* *Ld*−1+*ε*−*κε*+*κξ* y<sup>0</sup> − *M*0*Mg* + *M*0*Mg*(1 + *P*) + + *d*1−*ε*+*κε*−*κξ L* z 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1<sup>0</sup> *M* <sup>0</sup>*Mg*(1 + *P*) + *φ*(*w*)Φ( y(*w*) )(1 + *ψ*(*w*))1 *dw* ≤ *M*<sup>∗</sup> <sup>1</sup> + *L M*∗ <sup>2</sup> <sup>+</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ <sup>L</sup>* z 0 (<sup>z</sup> <sup>−</sup> *<sup>w</sup>*)*κξ*−1*φ*(*w*)Φ( y(*w*) )(1 + *ψ*(*w*))*dw*, <sup>1</sup> <sup>=</sup> *<sup>d</sup>*1−*ε*+*κε*−*κξ*\* +

$$\begin{aligned} & \text{where } M\_1^\kappa = d^{1-\varepsilon+\kappa\varepsilon-\kappa\frac{\pi}{5}} \left[ L^\prime d^{-1+\varepsilon-\kappa\varepsilon+\kappa\frac{\pi}{5}} (\mathbf{y}\_0 - M\_0 M\_\mathcal{S}) + M\_0 M\_\mathcal{S} (1+P) \right] \\ & \text{and } M\_2^\kappa = d^{1-\varepsilon(1+\kappa\frac{\pi}{5})} \frac{M\_0^\prime M\_\mathcal{S} (1+P)}{\kappa \mathfrak{F}}. \end{aligned}$$

Consider the RHS of the above inequality as *γ*(z). Then, we have

$$\begin{aligned} \gamma(0) &= M\_1^\*, \; ||\mathfrak{y}(\mathfrak{z})|| \le \gamma(\mathfrak{z}), \; \mathfrak{z} \in [0, d], \\\gamma'(\mathfrak{z}) &= d^{1 - \varepsilon + \kappa \varepsilon - \kappa \mathfrak{z}} L'(w - \mathfrak{z})^{\kappa \mathfrak{z} - 1} \phi(\mathfrak{z}) \Phi(\|\mathfrak{y}(\mathfrak{z})\|) \left(1 + \psi(\mathfrak{z})\right). \end{aligned}$$

By the non-decreasing character of Φ, we obtain

$$\gamma'(\mathfrak{z}) = d^{1-\varepsilon+\kappa\varepsilon-\kappa\mathfrak{z}} L'(w-\mathfrak{z})^{\kappa\mathfrak{z}-1} \phi(\mathfrak{z}) \Phi(\gamma(\mathfrak{z})) \left(1+\psi(\mathfrak{z})\right).$$

Then the above inequality implies (for each z ∈ J ) that

$$\int\_{\gamma(0)}^{\gamma(\mathfrak{z})} \frac{d\mathfrak{u}}{\Phi(\mathfrak{u})} \leq L' \int\_0^{\mathfrak{z}} (\mathfrak{z} - w)^{\kappa \mathfrak{z} - 1} \phi(\mathfrak{z}) \left(1 + \psi(\mathfrak{z})\right) dw \\ < \int\_{M\_1^\*}^{\infty} \frac{d\mathfrak{u}}{\Phi(\mathfrak{u})}.$$

This inequality implies that there exists a constant L, such that *γ*(z) ≤ L, z ∈ J , and, hence, y(z) ≤ L. From this we notice that set Λ is bounded. Therefore, by [17], Martelli's fixed point theorem Ψ has a fixed point, which is the mild solution of the system (1)–(2).

#### **4. Example**

As an idea of how our findings may be used, think about the following Hilfer fractional neutral integro-differential inclusion,

$$\begin{aligned} D\_{0^{+}}^{\frac{4}{\beta},\varepsilon} \left[ \Delta(\mathfrak{z},v) - \overline{\mathcal{N}}(\mathfrak{z},\Delta(\mathfrak{z},v)) \right] & \in \frac{\partial^{2}}{\partial \mathfrak{z}^{2}} \Delta(\mathfrak{z},v) + \mathcal{G}(\mathfrak{z},\Delta(\mathfrak{z},v),(E\Delta)(\mathfrak{z},v)) \mathfrak{z} \in (0,d], v \in [0,\pi], \\ \Delta(\mathfrak{z},0) = \Delta(\mathfrak{z},\pi) &= 0 \mathfrak{z} \in [0,d], \\ I^{(1-\frac{4}{\beta})(1-\varepsilon)} \mathfrak{y}(w,0) &= \mathfrak{y}\_{0}(v), v \in [0,\pi], \end{aligned} \tag{6}$$

where *<sup>D</sup>*<sup>4</sup> 7 ,*ε* <sup>0</sup><sup>+</sup> is the *HFD* of order <sup>4</sup> <sup>7</sup> , type *<sup>ε</sup>*, *<sup>I</sup>*(1<sup>−</sup> <sup>4</sup> <sup>7</sup> )(1−*ε*) is the Riemann–Liouville integral of order <sup>3</sup> <sup>7</sup> (<sup>1</sup> <sup>−</sup> *<sup>ε</sup>*), <sup>G</sup>¯ z, Δ(z, *v*),(*E*Δ)(z, *v*) ,(*E*Δ)(z, *<sup>v</sup>*), *and* <sup>N</sup>¯ (z, <sup>Δ</sup>(z, *<sup>v</sup>*)) are the required functions.

To write the system (6) in the abstract form of (1)–(2), we chose the space *Y* = *L*2[0, *π*]. Define an almost sectorial operator <sup>A</sup> by <sup>A</sup><sup>Δ</sup> <sup>=</sup> <sup>Δ</sup>zz with the domain

$$D(\mathsf{A}) = \left\{ \Delta \in \mathcal{Y} : \frac{\partial \Delta}{\partial \mathfrak{J}}, \frac{\partial^2 \Delta}{\partial \mathfrak{J}^2} \in \mathcal{Y} : \Delta(\mathfrak{J}, 0) = \Delta(\mathfrak{J}, \pi) = 0 \right\}.$$

Then A produces a compact semigroup that is analytic and self-adjoint, T(z)z ≥ 0. Additionally, the discrete spectrum of <sup>A</sup> contains eigenvalues of *<sup>k</sup>*2, *<sup>k</sup>* <sup>∈</sup> <sup>N</sup> and orthogonal eigenvectors *<sup>ζ</sup>k*(*z*) = <sup>6</sup> <sup>2</sup> *<sup>π</sup>* sin(*kz*), then

$$\mathbb{A}z = \sum\_{k=0}^{\infty} k^2 \langle z, \zeta\_k \rangle \zeta\_k.$$

Moreover, we have each *<sup>v</sup>* <sup>∈</sup> *<sup>Y</sup>*, <sup>T</sup>(z)*<sup>v</sup>* <sup>=</sup> <sup>∑</sup><sup>∞</sup> *<sup>k</sup>*=<sup>1</sup> *<sup>ζ</sup>*−*k*2z*v*, *<sup>ζ</sup>kζk*. In particular, <sup>T</sup>(·) is uniformly stable semigroup and T(z) ≤ *M*, which satisfies (*H*1).

<sup>y</sup>(z)(*v*) = <sup>Δ</sup>(z, *<sup>v</sup>*), <sup>z</sup> ∈ J = [0, *<sup>d</sup>*], *<sup>v</sup>* <sup>∈</sup> [0, *<sup>π</sup>*]. Take <sup>y</sup> <sup>∈</sup> *<sup>Y</sup>* <sup>=</sup> *<sup>L</sup>*2[0, *<sup>π</sup>*], *<sup>v</sup>* <sup>∈</sup> [0, *<sup>π</sup>*], we consider the multi-valued mapping G : J × *Y* × *Y* → *Y*,

$$\begin{aligned} \mathcal{G}\left(\mathfrak{z}\cdot\mathfrak{z}(\mathfrak{z}),(\operatorname{Ey})(\mathfrak{z})\right) &= \mathcal{G}\left(\mathfrak{z}\cdot\Delta(\mathfrak{z}\cdot\mathfrak{v}),(\operatorname{E\Delta})(\mathfrak{z}\cdot\mathfrak{v})\right) \\ &= \frac{e^{-\mathfrak{z}}}{1+e^{-\mathfrak{z}}}\sin\left(w(\mathfrak{z}\cdot\mathfrak{v})+\int\_{0}^{\mathfrak{z}}\cos(\mathfrak{z}s)\Delta(s,\mathfrak{v})ds\right), \end{aligned}$$

where

$$(E\mathbf{y})(\mathfrak{z})(v) = \int\_0^\mathfrak{z} e(\mathfrak{z}, \mathfrak{s}, \Delta(\mathfrak{s}, v)) d\mathfrak{s} = \int\_0^\mathfrak{z} \cos(\mathfrak{z}s) \Delta(\mathfrak{s}, v) d\mathfrak{s}.$$

Since, mapping G is measurable, upper semi-continuous, and strongly measurable,

$$
\overline{\mathcal{G}}\left(\mathfrak{z}\,\Delta(\mathfrak{z}\,v),(E\Delta)(\mathfrak{z}\,v)\right) \le \mathcal{M}\_1^\*.
$$

So G is satisfied (*H*2). Additionally, N : J × *Y* → *Y* must have completely continuous mapping, which is defined as N (z, *u*(z)) = N (z, Δ(z, *v*)), satisfying the necessary hypotheses. Therefore, the required mapping satisfied all hypotheses. As a result, the nonlocal Cauchy problem (1)–(2) may be used to rephrase the fractional system (6). It is clear that the boundary of G z, Δ(z, *u*),(*E*Δ)(z, *u*) is uniform. The problem has a mild solution on J , according to Theorem 2 .

#### **5. Conclusions**

In this study, Martelli's fixed point theorem was used to examine the possibility of a mild solution for an abstract Hilfer fractional differential system via almost sectorial operators. Adequate criteria were applied to the present findings and were satisfied. The controllability of the Hilfer fractional neutral derivative (via almost sectorial operators) will be investigated in the future using a fixed point technique.

**Author Contributions:** Conceptualisation, C.B.S.V.B. and R.U.; methodology, C.B.S.V.B.; validation, C.B.S.V.B. and R.U.; formal analysis, C.B.S.V.B.; investigation, R.U.; resources, C.B.S.V.B.; writing original draft preparation, C.B.S.V.B.; writing review and editing, R.U.; visualisation, R.U.; supervision, R.U.; project administration, R.U. All authors have read and agreed to the published version of the manuscript.

**Funding:** There are no funders to report for this submission.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

**Acknowledgments:** The authors are grateful to the reviewers of this article who provided insightful comments and advice that allowed us to revise and improve the content of the paper. The first author would like to thank the management of VIT University for providing a teaching cum research assistant fellowship.

**Conflicts of Interest:** The authors have no conflict of interest to declare.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

HFD Hilfer fractional derivative

HF Hilfer fractional

#### **References**


## *Article* **Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary**

**Areej Bin Sultan †, Mohamed Jleli † and Bessem Samet \*,†**

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia; 437203645@student.ksu.edu.sa (A.B.S.); jleli@ksu.edu.sa (M.J.)

**\*** Correspondence: bsamet@ksu.edu.sa

† These authors contributed equally to this work.

**Abstract:** We first consider the damped wave inequality *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>t*<sup>2</sup> <sup>−</sup> *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>t</sup>* <sup>≥</sup> *<sup>x</sup>σ*|*u*<sup>|</sup> *<sup>p</sup>*, *t* > 0, *<sup>x</sup>* <sup>∈</sup> (0,L), where *<sup>L</sup>* > 0, *<sup>σ</sup>* <sup>∈</sup> <sup>R</sup>, and *<sup>p</sup>* <sup>&</sup>gt; 1, under the Dirichlet boundary conditions (*u*(*t*, 0), *u*(*t*, *L*))= (*f*(*t*), *g*(*t*)), *t* > 0. We establish sufficient conditions depending on *σ*, *p*, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: *g* ≡ 0 and *g*(*t*) = *t <sup>γ</sup>*, *<sup>γ</sup>* <sup>&</sup>gt; <sup>−</sup>1. Next, we extend our study to the time-fractional analogue of the above problem, namely, the timefractional damped wave inequality *<sup>∂</sup>α<sup>u</sup> <sup>∂</sup>t<sup>α</sup>* <sup>−</sup> *<sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>∂</sup>β<sup>u</sup> <sup>∂</sup>t<sup>β</sup>* <sup>≥</sup> *<sup>x</sup>σ*|*u*<sup>|</sup> *<sup>p</sup>*, *<sup>t</sup>* <sup>&</sup>gt; 0, *<sup>x</sup>* <sup>∈</sup> (0, *<sup>L</sup>*), where *<sup>α</sup>* <sup>∈</sup> (1, 2), *<sup>β</sup>* <sup>∈</sup> (0, 1), and *<sup>∂</sup><sup>τ</sup> <sup>∂</sup>t<sup>τ</sup>* is the time-Caputo fractional derivative of order *τ*, *τ* ∈ {*α*, *β*}. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm.

**Keywords:** time-fractional damped wave inequalities; bounded domain; singularity; nonexistence

**MSC:** 35B44; 35B33; 26A33

#### **1. Introduction**

In this paper, we first consider the damped wave inequality

$$\begin{cases} \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial t} \ge \mathbf{x}^\sigma |u|^p, \quad t > 0, \, \mathbf{x} \in (0, L), \\\\ (u(t, 0), u(t, L)) = (f(t), g(t)), \quad t > 0, \\\\ \left( u(0, x), \frac{\partial u}{\partial t}(0, x) \right) = (u\_0(x), u\_1(x)), \quad \mathbf{x} \in (0, L), \end{cases} \tag{1}$$

where *<sup>L</sup>* <sup>&</sup>gt; 0, *<sup>σ</sup>* <sup>∈</sup> <sup>R</sup>, and *<sup>p</sup>* <sup>&</sup>gt; 1. It is supposed that *<sup>u</sup>*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*]), *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞)), and *g*(*t*) = *Cgt <sup>γ</sup>*, where *Cg* <sup>≥</sup> 0 and *<sup>γ</sup>* <sup>&</sup>gt; <sup>−</sup>1, are constants. Namely, we establish sufficient conditions depending on the initial values, the boundary conditions, *p*, and *σ*, under which (1) admits no global weak solution, in a sense that will be specified later.

Next, we study the time-fractional analogue of (1), namely the time-fractional damped wave inequality

**Citation:** Bin Sultan, A.; Jleli, M.; Samet, B. Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary. *Fractal Fract.* **2021**, *5*, 258. https://doi.org/10.3390/ fractalfract5040258

Academic Editor: Rodica Luca

Received: 1 November 2021 Accepted: 29 November 2021 Published: 6 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

$$\begin{cases} \begin{aligned} \frac{\partial^{\alpha}u}{\partial t^{\alpha}} - \frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{\beta}u}{\partial t^{\beta}} \ge \mathbf{x}^{\sigma}|u|^{p}, \quad t>0, \mathbf{x}\in(0,L), \\\\ (u(t,0), u(t,L)) = (f(t), g(t)), \quad t>0, \\\\ \left(u(0,\mathbf{x}), \frac{\partial u}{\partial t}(0,\mathbf{x})\right) = (u\_{0}(\mathbf{x}), u\_{1}(\mathbf{x})), \quad \mathbf{x}\in(0,L), \end{aligned} \end{cases} \tag{2}$$

where *<sup>α</sup>* <sup>∈</sup> (1, 2), *<sup>β</sup>* <sup>∈</sup> (0, 1), and *<sup>∂</sup><sup>τ</sup> <sup>∂</sup>t<sup>τ</sup>* , *τ* ∈ {*α*, *β*}, is the time-Caputo fractional derivative of order *τ*.

The investigation of the question of blow-up of solutions to initial boundary value problems for semilinear wave equations started in the 1970s. For example, Tsutsumi [1] considered the nonlinear damped wave equation

$$\frac{\partial^2 u}{\partial t^2} - \Delta u + b \frac{\partial u}{\partial t} = F(u)\_{\prime\prime}$$

under homogeneous Dirichlet boundary conditions, where *b* ≥ 0 and

$$F(s)s - 2(2\kappa + 1)\int\_0^s F(\tau) \,d\tau \ge d\_0|s|^{\rho+2}, \quad s \in \mathbb{R}\_+$$

for some *κ* > 0 and *ρ* > 0. By means of the energy method, the author established sufficient conditions for the blow-up of solutions. In [2], using a concavity argument, Levine established sufficient conditions for the blow-up of solutions to an abstract Cauchy problem in a Hilbert space, of the form

$$P\frac{\partial^2 \mu}{\partial t^2} + Au + Q\frac{\partial \mu}{\partial t} = F(\mu)\_\nu$$

where *P* and *A* are positive symmetric operators and *F* is a nonlinear operator satisfying certain conditions. Later, the concavity method was used and developed by many authors in order to study more general problems. For further blow-up results for nonlinear wave equations, obtained by means of the energy/concavity method, see e.g., [3–11] and the references therein.

Fractional operators arise in various applications, such as chemistry, biology, continuum mechanics, anomalous diffusion, and materials science, see for instance [12–16]. Consequently, many mathematicians dealt with the study of fractional differential equations in both theoretical and numerical aspects, see e.g., [17–21].

In [22], Kirane and Tatar considered the time-fractional damped wave equation

$$\begin{cases} \frac{\partial^2 u}{\partial t^2} - \Delta u + \frac{\partial^{1+a} u}{\partial t^{1+a}} = a|u|^{p-1}u, \quad t > 0, \text{x} \in \Omega, \\\\ u(t, \mathbf{x}) = 0, \quad t > 0, \text{x} \in \partial\Omega, \\\\ \left( u(0, \mathbf{x}), \frac{\partial u}{\partial t}(0, \mathbf{x}) \right) = (u\_0(\mathbf{x}), u\_1(\mathbf{x})), \quad \mathbf{x} \in \Omega, \end{cases} \tag{3}$$

where *<sup>p</sup>* <sup>&</sup>gt; 1, *<sup>α</sup>* <sup>∈</sup> (−1, 1), and <sup>Ω</sup> is a bounded domain of <sup>R</sup>*N*. Using some arguments based on Fourier transforms and the Hardy–Littlewood inequality, it was shown that the energy grows exponentially for sufficiently large initial data.

By combining an argument due to Georgiev and Todorova [23] with the techniques used in [22], Tatar [24] proved that the solutions to (3) blow up in finite-time for sufficiently large initial data.

In all the above cited references, the blow-up results were obtained for sufficiently large initial data. In this paper, we use a different approach than those used in the above

mentioned references. Namely, our approach is based on the test function method introduced by Mitidieri and Pohozaev [25]. Taking into consideration the boundedness of the domain as well as the boundary conditions, adequate test functions are used to obtain sufficient conditions for the nonexistence of global weak solutions to problems (1) and (2). Notice that our results hold without assuming that the initial values are large with respect to a certain norm.

Let us mention also that recently, methods for the numerical diagnostics of the solution's blow-up have been actively developing (see e.g., [26–28]), which make it possible to refine the theoretical estimates.

The rest of the paper is organized as follows: In Section 2, we provide some preliminaries on fractional calculus, and some useful lemmas. We state our main results in Section 3. The proofs are presented in Section 4.

#### **2. Preliminaries on Fractional Calculus**

For the reader's convenience, we recall below some notions from fractional calculus, see e.g., [17,20].

Let *<sup>T</sup>* <sup>&</sup>gt; 0 be fixed. Given *<sup>ρ</sup>* <sup>&</sup>gt; 0 and *<sup>v</sup>* <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>T</sup>*]), the left-sided and right-sided Riemann–Liouville fractional integrals of order *ρ* of *v*, are defined, respectively, by

$$(l\_0^\rho v)(t) = \frac{1}{\Gamma(\rho)} \int\_0^t (t - s)^{\rho - 1} v(s) \, ds \quad \text{and} \quad (l\_T^\rho v)(t) = \frac{1}{\Gamma(\rho)} \int\_t^T (s - t)^{\rho - 1} v(s) \, ds,$$

for almost everywhere *t* ∈ [0, *T*], where Γ denotes the Gamma function. It can be easily seen that, if *v* ∈ *C*([0, *T*]), then

$$\lim\_{t \to 0^+} (I\_0^\rho v)(t) = \lim\_{t \to T^-} (I\_T^\rho v)(t) = 0.$$

In this case, we may consider *I ρ* <sup>0</sup> *v* and *I ρ <sup>T</sup>v* as continuous functions in [0, *T*], by taking

$$(I\_0^\rho v)(0) = (I\_T^\rho v)(T) = 0.$$

Given a positive integer *<sup>n</sup>*, *<sup>τ</sup>* <sup>∈</sup> (*<sup>n</sup>* <sup>−</sup> 1, *<sup>n</sup>*), and *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>n*([0, *<sup>T</sup>*]), the (left-sided) Caputo fractional derivative of order *τ* of *v*, is defined by

$$\frac{d^\tau v}{dt^\tau}(t) = \left(I\_0^{n-\tau} \frac{d^n v}{dt^n}\right)(t) = \frac{1}{\Gamma(n-\tau)} \int\_0^t (t-s)^{n-\tau-1} \frac{d^n v}{dt^n}(s) \, ds \,\prime$$

for all *t* ∈ [0, *L*].

We have the following integration by parts rule.

**Lemma 1** (see the Corollary in [17], p. 67)**.** *Let <sup>ρ</sup>* <sup>&</sup>gt; <sup>0</sup>*, <sup>q</sup>*,*<sup>r</sup>* <sup>≥</sup> <sup>1</sup>*, and* <sup>1</sup> *<sup>q</sup>* <sup>+</sup> <sup>1</sup> *<sup>r</sup>* ≤ 1 + *ρ (q* = 1*, <sup>r</sup>* <sup>=</sup> <sup>1</sup>*, in the case* <sup>1</sup> *<sup>q</sup>* <sup>+</sup> <sup>1</sup> *<sup>r</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>ρ</sup>). If* (*v*, *<sup>w</sup>*) <sup>∈</sup> *<sup>L</sup>q*([0, *<sup>T</sup>*]) <sup>×</sup> *<sup>L</sup>r*([0, *<sup>T</sup>*])*, then*

$$\int\_0^T (I\_0^\rho v)(t) w(t) \, dt = \int\_0^T v(t) (I\_T^\rho w)(t) \, dt.$$

**Lemma 2.** *For sufficiently large λ, let*

$$\eta(t) = T^{-\lambda}(T - t)^{\lambda}, \quad 0 \le t \le T. \tag{4}$$

*Let ρ* ∈ (0, 1)*. Then*

(*I*

$$(\prescript{\rho}{}{\eta})(t) \quad = \begin{array}{c} \Gamma(\lambda + 1) \\ \overline{\Gamma(\rho + \lambda + 1)} \end{array} T^{-\lambda}(T - t)^{\rho + \lambda} \Big| \tag{5}$$

$$(I\_T^\rho \eta)'(t) = -\frac{\Gamma(\lambda + 1)}{\Gamma(\rho + \lambda)} T^{-\lambda} (T - t)^{\rho + \lambda - 1},\tag{6}$$

$$(l\_T^\rho \eta)''(t) \quad = \begin{array}{c} \Gamma(\lambda + 1) \\ \hline \Gamma(\rho + \lambda - 1) \end{array} T^{-\lambda} (T - t)^{\rho + \lambda - 2}. \tag{7}$$

**Proof.** We have

$$\begin{aligned} \left(I\_T^\rho \eta\right)(t) &= \frac{1}{\Gamma(\rho)} \int\_t^T (s-t)^{\rho-1} \eta(s) \, ds \\ &= \frac{T^{-\lambda}}{\Gamma(\rho)} \int\_t^T (s-t)^{\rho-1} (T-s)^\lambda \, ds \\ &= \frac{T^{-\lambda}}{\Gamma(\rho)} \int\_t^T (s-t)^{\rho-1} ((T-t)-(s-t))^\lambda \, ds \\ &= \frac{T^{-\lambda}(T-t)^\lambda}{\Gamma(\rho)} \int\_t^T (s-t)^{\rho-1} \left(1-\frac{s-t}{T-t}\right)^\lambda \, ds. \end{aligned}$$

Using the change of variable *z* = *<sup>s</sup>*−*<sup>t</sup> <sup>T</sup>*−*<sup>t</sup>* , we obtain

$$\begin{aligned} (d\_T^\rho \eta)(t) &= \frac{T^{-\lambda}(T-t)^{\lambda+\rho}}{\Gamma(\rho)} \int\_0^1 z^{\rho-1} (1-z)^\lambda \, dz \\ &= \frac{T^{-\lambda}(T-t)^{\lambda+\rho}}{\Gamma(\rho)} B(\rho, \lambda+1) \end{aligned}$$

where *B* denotes the Beta function. Using the property (see e.g., [20])

$$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}, \quad a, b > 0,$$

we obtain

$$\begin{aligned} (I\_T^\rho \eta)(t) &= \frac{T^{-\lambda}(T-t)^{\lambda+\rho}}{\Gamma(\rho)} \frac{\Gamma(\rho)\Gamma(\lambda+1)}{\Gamma(\rho+\lambda+1)} \\ &= \frac{\Gamma(\lambda+1)}{\Gamma(\rho+\lambda+1)} T^{-\lambda}(T-t)^{\rho+\lambda} \end{aligned}$$

which proves (5).

Next, calculating the derivative of *I ρ <sup>T</sup>η*, we obtain

$$(I\_T^\rho \eta)'(t) = -\frac{(\rho + \lambda)\Gamma(\lambda + 1)}{\Gamma(\rho + \lambda + 1)} T^{-\lambda}(T - t)^{\rho + \lambda - 1}.$$

On the other hand, by the property (see e.g., [20])

$$
\Gamma(a+1) = a\Gamma(a), \quad a > 0,\tag{8}
$$

we obtain

$$
\Gamma(\rho + \lambda + 1) = (\rho + \lambda)\Gamma(\rho + \lambda).
$$

Hence, we deduce that

$$(I\_T^\rho \eta)'(t) = -\frac{\Gamma(\lambda + 1)}{\Gamma(\rho + \lambda)} T^{-\lambda} (T - t)^{\rho + \lambda - 1} \lambda$$

which proves (6).

Differentiating (*I ρ <sup>T</sup>η*) and using (8), we obtain

$$\begin{aligned} \left(I\_T^{\rho}\eta\right)''(t) &= \frac{(\rho+\lambda-1)\Gamma(\lambda+1)}{\Gamma(\rho+\lambda)}T^{-\lambda}(T-t)^{\rho+\lambda-2} \\ &= \frac{(\rho+\lambda-1)\Gamma(\lambda+1)}{(\rho+\lambda-1)\Gamma(\rho+\lambda-1)}T^{-\lambda}(T-t)^{\rho+\lambda-2} \\ &= \frac{\Gamma(\lambda+1)}{\Gamma(\rho+\lambda-1)}T^{-\lambda}(T-t)^{\rho+\lambda-2} \end{aligned}$$

which proves (7).

The following inequality will be useful later.

**Lemma 3** (Young's Inequality with Epsilon, see [29], p. 36)**.** *Let ε* > 0 *and p* > 1*. Then, for all a*, *b* ≥ 0*, there holds*

$$ab \le \varepsilon a^p + \mathcal{C}\_{\varepsilon, p} b^{\frac{p}{p-1}} \rho$$

*where Cε*,*<sup>p</sup>* = (*<sup>p</sup>* <sup>−</sup> <sup>1</sup>)*p*−1(*εp*) −1 *p*−1 *.*

**Remark 1.** *For a function <sup>u</sup>* : (0, <sup>∞</sup>) <sup>×</sup> (0, *<sup>L</sup>*) <sup>→</sup> <sup>R</sup>*, the notation <sup>∂</sup>α<sup>u</sup> <sup>∂</sup>t<sup>α</sup> used in* (2)*, where* 1 < *α* < 2*, means the following:*

$$\frac{\partial^{\alpha}u}{\partial t^{\alpha}}(t,x) = \left(I\_0^{2-\alpha}\frac{\partial^2 u}{\partial t^2}(\cdot,x)\right)(t), \quad t>0, \, 0 < x < L\_{\lambda}$$

*i.e.,*

$$\frac{\partial^{\alpha}u}{\partial t^{\alpha}}(t,\mathfrak{x}) = \frac{1}{\Gamma(2-\alpha)} \int\_{a}^{t} (t-s)^{1-\alpha} \frac{\partial^{2}u}{\partial t^{2}}(s,\mathfrak{x}) \, ds.$$

*Similarly, the notation <sup>∂</sup>β<sup>u</sup> <sup>∂</sup>t<sup>β</sup> used in* (2)*, where* <sup>0</sup> < *<sup>β</sup>* < <sup>1</sup>*, means the following:*

$$\frac{\partial^{\beta}u}{\partial t^{\beta}}(t,\boldsymbol{x}) = \left(I\_0^{1-\beta}\frac{\partial u}{\partial t}(\cdot,\boldsymbol{x})\right)(t), \quad t>0, \, 0 < \boldsymbol{x} < L\_{\star}$$

*i.e.,*

$$\frac{\partial^{\beta}u}{\partial t^{\beta}}(t,\varkappa) = \frac{1}{\Gamma(1-\beta)}\int\_{a}^{t}(t-s)^{-\beta}\frac{\partial u}{\partial t}(s,\varkappa)\,ds.$$

#### **3. Statement of the Main Results**

We first consider problem (1). Let

$$Q = [0, \infty) \times [0, L].$$

We introduce the test function space

$$\Phi = \left\{ \varphi \in \mathbb{C}^2(Q) \, : \, \varphi \ge 0, \, \varphi(\cdot, 0) = \varphi(\cdot, L) \equiv 0, \, \varphi(t, \cdot) \equiv 0 \text{ for sufficiently large } t \right\}.$$

**Definition 1.** *Let <sup>u</sup>*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*]) *and <sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞))*. We say that u is a global weak solution to* (1)*, if*

(i) *<sup>x</sup>σ*|*u*<sup>|</sup> *<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*(*Q*)*, u* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*(*Q*)*;* (ii) *for every ϕ* ∈ Φ*,*

$$\begin{split} & \int\_{Q} \mathbf{x}^{\sigma} |u|^{p} \boldsymbol{\varrho} \, d\mathbf{x} \, dt + \int\_{0}^{\infty} \Big( f(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, 0) - \mathbf{g}(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, L) \Big) \, dt \\ & \quad + \int\_{0}^{L} \Big( u\_{1}(\mathbf{x}) \boldsymbol{\varrho}(0, \mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial \boldsymbol{\varrho}}{\partial t}(0, \mathbf{x}) + u\_{0}(\mathbf{x}) \boldsymbol{\varrho}(0, \mathbf{x}) \Big) \, d\mathbf{x} \\ & \quad \leq - \int\_{Q} \boldsymbol{u} \frac{\partial^{2} \boldsymbol{\varrho}}{\partial \mathbf{x}^{2}} \, d\mathbf{x} \, dt + \int\_{Q} \boldsymbol{u} \frac{\partial^{2} \boldsymbol{\varrho}}{\partial t^{2}} \, d\mathbf{x} \, dt - \int\_{Q} \boldsymbol{u} \frac{\partial \boldsymbol{\varrho}}{\partial t} \, d\mathbf{x} \, dt. \end{split} \tag{9}$$

**Remark 2.** *The weak formulation* (9) *is obtained by multiplying the differential inequality in* (1) *by ϕ, integrating over Q, and using the initial conditions in* (1)*. So, clearly, any global solution to* (1) *is a global weak solution to* (1) *in the sense of Definition 1.*

We first consider the case *g* ≡ 0.

**Theorem 1.** *Let u*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*])*, f* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞))*, and g* ≡ 0*. Suppose that*

$$\int\_{0}^{L} (u\_0(\mathbf{x}) + u\_1(\mathbf{x}))(L - \mathbf{x}) \, d\mathbf{x} > 0. \tag{10}$$

*If*

$$
\sigma < -(p+1),
\tag{11}
$$

*then* (1) *admits no global weak solution.*

**Remark 3.** *Comparing with the existing results in the literature, in Theorem 1, it is not required that the initial data are sufficiently large with respect to a certain norm. The same remark holds for the next theorems.*

**Example 1.** *Consider problem* (1) *with*

$$f(t) = \frac{1}{\sqrt{t}}, \ t > 0, \quad \text{g} \equiv 0, \quad u\_0(\mathbf{x}) = -(L - \mathbf{x}), \quad u\_1(\mathbf{x}) = 2(L - \mathbf{x}), \quad \sigma = -4, \quad p = 2.1$$

*Then, all the assumptions of Theorem 1 are satisfied. Consequently, we deduce that* (1) *admits no global weak solution.*

Next, we consider the case when

$$\mathbf{g}(t) = \mathbb{C}\_{\mathcal{S}} t^{\gamma}, \quad \gamma > -1, \quad t > 0,\tag{12}$$

where *Cg* > 0 is a constant.

**Theorem 2.** *Let <sup>u</sup>*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*])*, <sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞))*, and g be the function defined by* (12)*. If one of the following conditions is satisfied:*


*then* (1) *admits no global weak solution.*

**Example 2.** *Consider problem* (1) *with*

$$f(t) = \frac{e^t}{\sqrt{t}}, \ t > 0, \quad u\_0(\mathbf{x}) = \mathbf{x}, \quad u\_1(\mathbf{x}) = \mathbf{x}^2, \quad g(t) = \sqrt{t}, \ t > 0, \quad \sigma = -2, \quad p = 2.1$$

*Then, by the statement (ii) of Theorem 2, we deduce that* (1) *admits no global weak solution.*

Consider now problem (2). For all *T* > 0, let

$$Q\_T = [0, T] \times [0, L].$$

We introduce the test function space

$$\Phi\_T = \left\{ \varphi \in \mathbb{C}^2(Q\_T) \, : \, \varphi \ge 0, \, \varphi(\cdot, 0) = \varrho(\cdot, L) \equiv 0, \, \frac{\partial(I\_T^{2-a}\varphi)}{\partial t}(T, \cdot) \equiv 0 \right\}.$$

**Definition 2.** *Let <sup>u</sup>*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*]) *and <sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞))*. We say that u is a global weak solution to* (2)*, if*

(i) *<sup>x</sup>σ*|*u*<sup>|</sup> *<sup>p</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*(*Q*)*, u* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*(*Q*)*;*

(ii) *for all T* > 0 *and ϕ* ∈ Φ*T,*

$$\begin{split} & \int\_{Q\_T} \mathbf{x}^T |u|^p \boldsymbol{\varrho} \, d\mathbf{x} \, dt + \int\_0^T \left( f(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, 0) - \boldsymbol{\varrho}(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, L) \right) dt \\ & \quad + \int\_0^L \left( u\_1(\mathbf{x}) (I\_T^{2-a} \boldsymbol{\varrho})(0, \mathbf{x}) - u\_0(\mathbf{x}) \frac{\partial (I\_T^{2-a} \boldsymbol{\varrho})}{\partial t}(0, \mathbf{x}) + u\_0(\mathbf{x}) (I\_T^{1-\beta} \boldsymbol{\varrho})(0, \mathbf{x}) \right) d\mathbf{x} \\ & \leq - \int\_{Q\_T} u \frac{\partial^2 \boldsymbol{\varrho}}{\partial \mathbf{x}^2} \, d\mathbf{x} \, dt + \int\_{Q\_T} u \frac{\partial^2 (I\_T^{2-a} \boldsymbol{\varrho})}{\partial t^2} \, d\mathbf{x} \, dt - \int\_{Q\_T} u \frac{\partial (I\_T^{1-\beta} \boldsymbol{\varrho})}{\partial t} \, d\mathbf{x} \, dt. \end{split} \tag{13}$$

**Remark 4.** *The weak formulation* (13) *is obtained by multiplying the differential inequality in* (2) *by ϕ, integrating over QT, using the initial conditions in* (2)*, and using the fractional integration by parts rule provided by Lemma 1. So, clearly, any global solution to* (2) *is a global weak solution to* (2) *in the sense of Definition 2.*

As for problem (1), we first consider the case *g* ≡ 0.

**Theorem 3.** *Let u*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*])*, f* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞))*, and g* ≡ 0*. If*

$$
\sigma < -(p+1)\_\prime
$$

*and one of the following conditions is satisfied:*

$$\alpha < \beta + 1, \quad \int\_0^L u\_1(\mathbf{x}) (L - \mathbf{x}) \, d\mathbf{x} > 0; \tag{14}$$

$$\mathfrak{a} = \mathfrak{F} + 1, \quad \int\_0^L (u\_0(\mathfrak{x}) + u\_1(\mathfrak{x}))(L - \mathfrak{x}) \, d\mathfrak{x} > 0; \tag{15}$$

$$
\alpha > \beta + 1, \quad \int\_0^L u\_0(\mathbf{x}) (L - \mathbf{x}) \, d\mathbf{x} > 0,\tag{16}
$$

*then* (2) *admits no global weak solution.*

**Example 3.** *Consider problem* (2) *with*

$$f(t) = \frac{1}{\sqrt{t}}, \ t > 0, \quad u\_0 \equiv 0, \quad u\_1(x) = 2(L - x), \quad a = \frac{3}{2}, \quad \beta = \frac{3}{4}, \quad \sigma = -4, \quad p = 2.1$$

*Since* (14) *is satisfied and σ* < −(*p* + 1)*, by Theorem 3, we deduce that* (2) *admits no global weak solution.*

Next, we consider the inhomogeneous case, where the function *g* is given by (12).

**Theorem 4.** *Let u*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*])*, f* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞))*, and g be the function defined by* (12)*. If*

$$
\alpha > \max\{1 - \gamma, 1\}, \quad \beta > \max\{-\gamma, 0\}, \tag{17}
$$

*and one of the following conditions is satisfied:*

(i) *σ* < −(*p* + 1)*;* (ii) *σ* ≥ −(*p* + 1)*, γ* > 0*,*

*then* (2) *admits no global weak solution.*

**Example 4.** *Consider problem* (2) *with*

$$f(t) = \frac{1}{\sqrt{t}}, \ t > 0, \quad u\_0(\mathbf{x}) = -\mathbf{x}, \quad u\_1(\mathbf{x}) = \mathbf{x}^2, \quad g(t) = t^{\frac{2}{3}}, \ t > 0, \quad \mathbf{a} = \frac{3}{2}, \quad \beta = \frac{1}{2}\sqrt{t}$$

*and*

$$
\sigma = -\mathbf{3}, \quad p = \mathbf{3}.
$$

*Then* (17) *is satisfied, σ* ≥ −(*p* + 1)*, and γ* > 0*. Then, by Theorem 4, we deduce that* (2) *admits no global weak solution.*

#### **4. Proof of the Main Results**

Throughout this section, any positive constant independent on *T* and *R*, is denoted by *C*. Namely, in the proofs, we use several asymptotic estimates as *T* → ∞ and *R* → ∞; therefore, the value of any positive constant independent of *T* and *R* has no influence in our analysis.

#### *4.1. Proof of Theorem 1*

**Proof.** Suppose that *u* is a global weak solution to (1). Then, by (9), for every *ϕ* ∈ Φ, there holds

$$\begin{split} &\int\_{Q} \mathbf{x}^{\sigma} |u|^{p} \varrho \, d\mathbf{x} \, dt + \int\_{0}^{\infty} \left( f(t) \frac{\partial \varrho}{\partial \mathbf{x}}(t, 0) - g(t) \frac{\partial \varrho}{\partial \mathbf{x}}(t, L) \right) dt \\ &+ \int\_{0}^{L} \left( u\_{1}(\mathbf{x}) \varrho(0, \mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial \varrho}{\partial t}(0, \mathbf{x}) + u\_{0}(\mathbf{x}) \varrho(0, \mathbf{x}) \right) d\mathbf{x} \\ &\leq \int\_{Q} |u| \left| \frac{\partial^{2} \varrho}{\partial \mathbf{x}^{2}} \right| d\mathbf{x} \, dt + \int\_{Q} |u| \left| \frac{\partial^{2} \varrho}{\partial t^{2}} \right| d\mathbf{x} \, dt + \int\_{Q} |u| \left| \frac{\partial \varrho}{\partial t} \right| d\mathbf{x} \, dt. \end{split} \tag{18}$$

On the other hand, using Lemma 3 with *ε* = <sup>1</sup> <sup>3</sup> and adequate choices of *a* and *b*, we obtain

$$\int\_{Q} |u| \left| \frac{\partial^2 \varrho}{\partial \mathbf{x}^2} \right| d\mathbf{x} \, dt \quad \leq \quad \frac{1}{3} \int\_{Q} \mathbf{x}^r |u|^p \varrho \, d\mathbf{x} \, dt + \mathbb{C} \int\_{Q} \mathbf{x}^{\frac{-r}{p-1}} \varrho^{\frac{-1}{p-1}} \left| \frac{\partial^2 \varrho}{\partial \mathbf{x}^2} \right|\_{p}^{\frac{p}{p-1}} d\mathbf{x} \, dt,\tag{19}$$

$$\int\_{Q} |u| \left| \frac{\partial^2 \varrho}{\partial t^2} \right| \, d\mathbf{x} \, dt \quad \leq \quad \frac{1}{3} \int\_{Q} \mathbf{x}^{\sigma} |u|^p \, \varrho \, d\mathbf{x} \, dt + \mathbb{C} \int\_{Q} \mathbf{x}^{\frac{-\sigma}{p-1}} \, \varrho^{\frac{-1}{p-1}} \left| \frac{\partial^2 \varrho}{\partial t^2} \right|^{\frac{p-1}{p-1}} d\mathbf{x} \, dt,\tag{20}$$

$$\int\_{Q} |u| \left| \frac{\partial \boldsymbol{\varrho}}{\partial t} \right| d\boldsymbol{x} \, dt \quad \leq \quad \frac{1}{3} \int\_{Q} \mathbf{x}^{\sigma} |u|^{p} \boldsymbol{\varrho} \, d\boldsymbol{x} \, dt + \mathbb{C} \int\_{Q} \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left| \frac{\partial \boldsymbol{\varrho}}{\partial t} \right|^{\frac{p}{p-1}} d\boldsymbol{x} \, dt. \tag{21}$$

Using (18)–(21), we obtain

$$\begin{split} &\int\_{0}^{\infty} \left( f(t) \frac{\partial \varphi}{\partial \mathbf{x}}(t, 0) - g(t) \frac{\partial \varphi}{\partial \mathbf{x}}(t, L) \right) dt \\ &+ \int\_{0}^{L} \left( u\_{1}(\mathbf{x}) \varphi(0, \mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial \varphi}{\partial t}(0, \mathbf{x}) + u\_{0}(\mathbf{x}) \varphi(0, \mathbf{x}) \right) d\mathbf{x} \\ &\leq C \sum\_{j=1}^{3} I\_{j}(\boldsymbol{\varphi})\_{\mathbf{y}} \end{split} \tag{22}$$

where

$$I\_1(\boldsymbol{\varrho}) \quad = \int\_Q \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left| \frac{\partial^2 \boldsymbol{\varrho}}{\partial \boldsymbol{\varkappa}^2} \right|^{\frac{p}{p-1}} \boldsymbol{\varkappa}$$

$$I\_2(\boldsymbol{\varrho}) \quad = \int\_Q \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left| \frac{\partial^2 \boldsymbol{\varrho}}{\partial t^2} \right|^{\frac{p}{p-1}} \boldsymbol{\varkappa}$$

$$I\_3(\boldsymbol{\varrho}) \quad = \quad \int\_Q \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left| \frac{\partial \boldsymbol{\varrho}}{\partial t} \right|^{\frac{p}{p-1}} \boldsymbol{\varkappa}$$

Consider now two cut-off functions *<sup>ξ</sup>*, *<sup>μ</sup>* <sup>∈</sup> <sup>C</sup>∞([0, <sup>∞</sup>)) satisfying the following properties:

$$0 \le \xi, \mu \le 1, \quad \xi(s) = \begin{cases} 1 & \text{if } \quad 0 \le s \le \frac{1}{2} \\\ 0 & \text{if } \quad s \ge 1 \end{cases}, \quad \mu(s) = \begin{cases} \ 0 & \text{if } \quad 0 \le s \le \frac{1}{2} \\\ 1 & \text{if } \quad s \ge 1 \end{cases}.$$

For sufficiently large and *R*, let

$$
\varphi\_1(t) = \mathfrak{J}^\ell(R^{-\theta}t), \quad \varphi\_2(\mathbf{x}) = (L-\mathbf{x})\mu^\ell(\mathbf{Rx}), \quad t \ge 0, \mathbf{x} \in [0, L], \tag{23}
$$

where *θ* > 0 is a constant that will be determined later. Consider the function

$$
\varphi(t, \mathbf{x}) = \varphi\_1(t)\varphi\_2(\mathbf{x}), \quad t \ge 0, \mathbf{x} \in [0, L]. \tag{24}
$$

By the properties of the cut-off functions *ξ* and *μ*, it can be easily seen that the function *ϕ* defined by (24), belongs to Φ. Thus, the estimate (22) holds for this function.

Now, let us estimate the terms *Ij*(*ϕ*), *j* = 1, 2, 3. For *j* = 1, by (24), we obtain

$$I\_1(\varphi) = \left(\int\_0^\infty \varphi\_1(t) \, dt\right) \left(\int\_0^L \mathbf{x}^{\frac{-\sigma}{p-1}} \, \varphi\_2^{\frac{-1}{p-1}}(\mathbf{x}) |\varphi\_2''(\mathbf{x})|^{\frac{p}{p-1}} \, d\mathbf{x}\right) := I\_1^{(1)}(\varphi\_1) I\_1^{(2)}(\varphi\_2). \tag{25}$$

On the other hand, by the definitions of the function *ϕ*<sup>1</sup> and the cut-off function *ξ*, there holds

$$\begin{split} I\_1^{(1)}(\varphi\_1) &= \quad \int\_0^\infty \tilde{\xi}^\ell \left( \mathcal{R}^{-\theta} t \right) dt \\ &= \quad \int\_0^{\mathcal{R}^\theta} \tilde{\xi}^\ell \left( \mathcal{R}^{-\theta} t \right) dt \\ &\leq \quad \mathcal{R}^\theta. \end{split} \tag{26}$$

By the definitions of the function *ϕ*<sup>2</sup> and the cut-off function *μ*, we obtain

$$\begin{array}{rcl}\eta\_2''(\mathbf{x}) &=& \ell \mathbf{R}^2 \boldsymbol{\mu}^{\ell-2}(\mathbf{R}\mathbf{x}) \times \\ & \left[ (\mathbf{L} - \mathbf{x}) \left( (\ell - 1) \boldsymbol{\mu}^{\prime 2}(\mathbf{R}\mathbf{x}) + \boldsymbol{\mu}(\mathbf{R}\mathbf{x}) \boldsymbol{\mu}^{\prime \prime}(\mathbf{R}\mathbf{x}) \right) - 2 \mathbf{R}^{-1} \boldsymbol{\mu}(\mathbf{R}\mathbf{x}) \boldsymbol{\mu}^{\prime}(\mathbf{R}\mathbf{x}) \right] \boldsymbol{\chi}\_{\left[ \frac{1}{2} \mathbf{R}^{-1}, \boldsymbol{R}^{-1} \right]}(\mathbf{x}) & \end{array}$$

which yields

$$|q\_{2}^{\prime\prime}(\mathbf{x})| \le \mathsf{CR}^{2} \mu^{\ell-2}(\mathbf{R}\mathbf{x}) \chi\_{\left[\frac{1}{2}\mathbf{R}^{-1}, \mathbf{R}^{-1}\right]}(\mathbf{x})\_{\prime}$$

where *<sup>χ</sup>*[ <sup>1</sup> <sup>2</sup> *<sup>R</sup>*−1,*R*−<sup>1</sup>] is the indicator function of the interval 1 <sup>2</sup>*R*−1, *<sup>R</sup>*−<sup>1</sup> . Then, there holds

$$\begin{split} \mathbb{E}\_{1}^{(2)}(\varphi\_{2}) &\leq \quad \mathsf{CR}^{\frac{2p}{p-1}} \int\_{\frac{1}{2}\mathbb{R}^{-1}}^{\mathbb{R}^{-1}} \mathsf{x}^{\frac{-\sigma}{p-1}} (L-\mathsf{x})^{\frac{-1}{p-1}} \mu^{\ell-\frac{2p}{p-1}} (\mathsf{Rx}) \, d\mathsf{x} \\ &\leq \quad \mathsf{CR}^{\frac{2p}{p-1}} \int\_{\frac{1}{2}\mathbb{R}^{-1}}^{\mathbb{R}^{-1}} \mathsf{x}^{\frac{-\sigma}{p-1}} \, d\mathsf{x} \\ &\leq \quad \mathsf{CR}^{\frac{\sigma}{p-1} + \frac{2p}{p-1} - 1} . \end{split}$$

Thus, it follows from (25)–(27) that

*I*

$$I\_1(\varphi) \le \mathcal{CR}^{\theta + \frac{p+1+\sigma}{p-1}}.\tag{28}$$

For *j* = 2, *Ij*(*ϕ*) can be written as

$$I\_2(\varphi) = \left( \int\_0^\infty \varphi\_1^{\frac{-1}{p-1}}(t) |\varphi\_1'(t)|^{\frac{p}{p-1}} dt \right) \left( \int\_0^L \mathbf{x}^{\frac{-\sigma}{p-1}} \varphi\_2(\mathbf{x}) \, d\mathbf{x} \right) := I\_2^{(1)}(\varphi\_1) I\_2^{(2)}(\varphi\_2). \tag{29}$$

By the definitions of the function *ϕ*<sup>1</sup> and the cut-off function *ξ*, we obtain

$$\varphi\_1''(t) = \ell R^{-2\theta} \tilde{\xi}^{\ell-2}(R^{-\theta}t) \left[ (\ell - 1) \tilde{\xi}^{\ell 2}(R^{-\theta}t) + \tilde{\xi}^{\ell - 1}(R^{-\theta}t) \tilde{\xi}^{\prime \prime}(R^{-\theta}t) \right] \chi\_{\left[ \frac{1}{2}R^{\theta}, R^{\theta} \right]}(t) \,\iota$$

which yields

$$|\varphi\_1^{\prime\prime}(t)| \le \mathsf{CR}^{-2\theta} \tilde{\xi}^{\ell-2}(\mathsf{R}^{-\theta}t) \chi\_{\left[\frac{1}{2}\mathsf{R}^{\theta},\mathsf{R}^{\theta}\right]}(t).$$

Thus, there holds

$$\begin{array}{rcl}I\_2^{(1)}(\varphi\_1) & \leq & \mathcal{CR}^{\frac{-2\theta p}{p-1}} \int\_{\frac{1}{2}R^{\theta}}^{R^{\theta}} \mathfrak{f}^{\ell-\frac{2p}{p-1}}(R^{-\theta}t) \, dt\\ & \leq & \mathcal{CR}^{\theta\left(1-\frac{2p}{p-1}\right)}.\end{array} \tag{30}$$

Moreover, we have

$$\begin{aligned} \left(\,\_2^{(2)}(\varphi\_2)\right) &= \quad \int\_0^L \ge \,\_{p-1}^{\frac{-\sigma}{p-1}} \varphi\_2(\mathbf{x}) \, d\mathbf{x} \\ &= \quad \int\_{\frac{1}{2}R^{-1}}^L \ge \,\_{p-1}^{\frac{-\sigma}{p-1}} (L-\mathbf{x})\mu^\ell(R\mathbf{x}) \, d\mathbf{x} \\ &\le \quad \, \,\_C \int\_{\frac{1}{2}R^{-1}}^L \ge \,\_{p-1}^{\frac{-\sigma}{p-1}} \, d\mathbf{x}. \end{aligned}$$

On the other hand, by (11), we have *σ* < *p* − 1, thus we deduce that

*I*

$$
\sigma\_2^{(2)}(\varphi\_2) \le \mathcal{C}.\tag{31}
$$

Combining (29)–(31), there holds

*I*

$$I\_2(\varphi) \le \mathcal{CR}^{\theta \left(1 - \frac{2p}{p-1}\right)}.\tag{32}$$

Now, let us estimate *I*3(*ϕ*). This term can be written as

$$I\_3(\boldsymbol{\varrho}) = \left( \int\_0^\infty \boldsymbol{\varrho}\_1^{\frac{-1}{p-1}}(t) |\boldsymbol{\varrho}\_1'(t)|^{\frac{p}{p-1}} \, dt \right) \left( \int\_0^L \boldsymbol{\chi}^{\frac{-r}{p-1}} \boldsymbol{\varrho}\_2(\boldsymbol{\chi}) \, d\boldsymbol{x} \right) := I\_3^{(1)}(\boldsymbol{\varrho}\_1) I\_3^{(2)}(\boldsymbol{\varrho}\_2). \tag{33}$$

A similar calculation as above yields

$$I\_3^{(1)}(\\\varphi\_1) \le \mathcal{CR}^{\theta \left(1 - \frac{p}{p-1}\right)}.\tag{34}$$

Observe that *I* (2) <sup>3</sup> (*ϕ*2) = *I* (2) <sup>2</sup> (*ϕ*2). Thus, by (31), (33), and (34), we obtain

$$I\_{\mathbb{B}}(\boldsymbol{\varrho}) \le \mathsf{CR}^{\theta \left(1 - \frac{p}{p-1}\right)}.\tag{35}$$

Next, combining (28), (32), and (35), we obtain

$$\sum\_{j=1}^{3} I\_j(\varphi) \le \mathcal{C} \left( R^{\theta + \frac{p+1+\upsilon}{p-1}} + R^{\theta \left( 1 - \frac{p}{p-1} \right)} \right). \tag{36}$$

Let *θ* be such that

that is,

$$
\theta + \frac{p+1+\sigma}{p-1} = \theta \left( 1 - \frac{p}{p-1} \right),
$$

$$
\theta = \frac{-(p+1) - \sigma}{p}.
$$

Notice that by (11), we have *θ* > 0. Then, (36) reduces to

$$\sum\_{j=1}^{3} I\_j(\boldsymbol{\varphi}) \le \mathcal{CR}^{\theta \left(1 - \frac{p}{p-1}\right)}.\tag{37}$$

Next, let us estimate the terms from the right side of (22). Observe that by the definition of the function *ϕ*, and the properties of the cut-off function *μ*, we have

$$\frac{\partial \varphi}{\partial x}(t,0) = 0, \quad t > 0.$$

Moreover, since *g* ≡ 0, there holds

$$\int\_{0}^{\infty} \left( f(t) \frac{\partial \varrho}{\partial \mathfrak{x}}(t, 0) - g(t) \frac{\partial \varrho}{\partial \mathfrak{x}}(t, L) \right) dt = 0. \tag{38}$$

By the properties of the cut-off function *ξ*, we have

$$
\varphi(0, \mathfrak{x}) = \varphi\_2(\mathfrak{x}), \quad \frac{\partial \varphi}{\partial t}(0, \mathfrak{x}) = 0, \quad \mathfrak{x} \in (0, L).
$$

Thus, we obtain

$$\begin{aligned} &\int\_0^L \left( u\_1(\mathbf{x}) \varrho(0, \mathbf{x}) - u\_0(\mathbf{x}) \frac{\partial \varrho}{\partial t}(0, \mathbf{x}) + u\_0(\mathbf{x}) \varrho(0, \mathbf{x}) \right) d\mathbf{x} \\ &= \int\_0^L (u\_0(\mathbf{x}) + u\_1(\mathbf{x})) \varrho(0, \mathbf{x}) \, d\mathbf{x} \\ &= \int\_0^L (u\_0(\mathbf{x}) + u\_1(\mathbf{x})) \varrho\_2(\mathbf{x}) \, d\mathbf{x} \\ &= \int\_0^L (u\_0(\mathbf{x}) + u\_1(\mathbf{x})) (L - \mathbf{x}) \mu^\ell(\mathbf{R} \mathbf{x}) \, d\mathbf{x}. \end{aligned}$$

Then, taking into consideration that *<sup>u</sup>*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*]), by the dominated convergence theorem, we obtain

$$\begin{split} &\lim\_{R\to\infty} \int\_{0}^{L} \left( \mu\_{1}(\mathbf{x})\,\varrho(0,\mathbf{x}) - \mu\_{0}(\mathbf{x})\frac{\partial\varrho}{\partial t}(0,\mathbf{x}) + \mu\_{0}(\mathbf{x})\,\varrho(0,\mathbf{x}) \right) d\mathbf{x} \\ &= \int\_{0}^{L} \left( \mu\_{0}(\mathbf{x}) + \mu\_{1}(\mathbf{x}) \right) (L-\mathbf{x}) \,d\mathbf{x}. \end{split} \tag{39}$$

Hence, by (10), for sufficiently large *R*, there holds

$$\int\_{0}^{L} \left( \mu\_{1}(\mathbf{x}) \varrho(0, \mathbf{x}) - \mu\_{0}(\mathbf{x}) \frac{\partial \varrho}{\partial \mathbf{l}}(0, \mathbf{x}) + \mu\_{0}(\mathbf{x}) \varrho(0, \mathbf{x}) \right) d\mathbf{x} \geq \frac{1}{2} \int\_{0}^{L} (\mu\_{0}(\mathbf{x}) + \mu\_{1}(\mathbf{x}))(L - \mathbf{x}) \, d\mathbf{x}. \tag{40}$$

Next, combining (22), (37), (38), and (40), we obtain

$$\frac{1}{2} \int\_0^L (\mu\_0(\mathfrak{x}) + \mu\_1(\mathfrak{x}))(L - \mathfrak{x}) \, d\mathfrak{x} \le \mathcal{CR}^{\theta \left(1 - \frac{p}{p-1}\right)} \, . $$

Passing to the limit as *R* → ∞ in the above inequality, we obtain

$$\frac{1}{2} \int\_{0}^{L} (u\_0(\mathfrak{x}) + u\_1(\mathfrak{x}))(L - \mathfrak{x}) \, d\mathfrak{x} \le 0,$$

which contradicts (10). Consequently, (1) admits no global weak solution. The proof is completed.

#### *4.2. Proof of Theorem 2*

**Proof.** As was performed previously, suppose that *u* is a global weak solution to (1). From the proof of Theorem 1, for sufficiently large *R*, there holds

$$\begin{split} & -\int\_{0}^{\infty} g(t) \frac{\partial \rho}{\partial \mathbf{x}}(t, L) \, dt \\ & + \int\_{0}^{L} \left( u\_{1}(\mathbf{x}) \rho(0, \mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial \rho}{\partial t}(0, \mathbf{x}) + u\_{0}(\mathbf{x}) \rho(0, \mathbf{x}) \right) \, d\mathbf{x} \\ & \leq C \left( R^{\theta + \frac{p+1+\sigma}{p-1}} + R^{\theta \left( 1 - \frac{p}{p-1} \right)} \int\_{\frac{1}{2} R^{-1}}^{L} \mathbf{x}^{\frac{-\sigma}{p-1}} \, d\mathbf{x} \right), \end{split} \tag{41}$$

where *θ* > 0 and *ϕ* is the function defined by (24). On the other hand, by the definition of the function *ϕ*, for sufficiently large *R*, there holds

$$\frac{\partial \rho}{\partial x}(t, L) = -\rho\_1(t), \quad t > 0,$$

which yields

$$\begin{aligned} -\int\_0^\infty g(t) \frac{\partial \varrho}{\partial x}(t, L) \, dt &= \int\_0^\infty g(t) \varrho\_1(t) \, dt \\ &= \quad \, \, \mathrm{C} \int\_0^\infty t^\gamma \tilde{\xi}^\ell(\mathrm{R}^{-\theta} t) \, dt \\ &\ge \quad \, \, \mathrm{C} \int\_0^{\frac{1}{2}R^\theta} t^\gamma \, dt \\ &= \quad \, \, \mathrm{C} R^{\theta(\gamma+1)} .\end{aligned}$$

Then, by (41), we deduce that

$$\begin{split} &\mathbb{C} + R^{-\theta(\gamma+1)} \int\_{0}^{L} \Big( u\_{1}(\mathbf{x}) \boldsymbol{\varrho}(0, \mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial \boldsymbol{\varrho}}{\partial t}(0, \mathbf{x}) + u\_{0}(\mathbf{x}) \boldsymbol{\varrho}(0, \mathbf{x}) \Big) d\mathbf{x} \\ &\leq \mathbb{C} \Big( R^{-\theta\gamma + \frac{p+1+\sigma}{p-1}} + R^{-\theta\left(\gamma + \frac{p}{p-1}\right)} \int\_{\frac{1}{2}R^{-1}}^{L} \boldsymbol{\chi}^{\frac{-\sigma}{p-1}} d\mathbf{x} \Big). \end{split} \tag{42}$$

Let *σ* < −(*p* + 1). In this case, (42) reduces to

$$\begin{split} &C + R^{-\theta(\gamma+1)} \int\_{0}^{L} \left( u\_{1}(\mathbf{x}) \boldsymbol{\varrho}(0, \mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial \boldsymbol{\varrho}}{\partial t}(0, \mathbf{x}) + u\_{0}(\mathbf{x}) \boldsymbol{\varrho}(0, \mathbf{x}) \right) d\mathbf{x} \\ &\leq C \left( R^{-\theta\gamma + \frac{p+1+\sigma}{p-1}} + R^{-\theta\left(\gamma + \frac{p}{p-1}\right)} \right). \end{split} \tag{43}$$

Taking *θ* > 0 so that

$$
\theta \gamma > \frac{p+1+\sigma}{p-1},
\tag{44}
$$

passing to the limit as *R* → ∞ in (43), and using (39), we obtain a contradiction with *C* > 0. This proves part (i) of Theorem 2.

Let *σ* ≥ −(*p* + 1) and *γ* > 0.

If −(*p* + 1) ≤ *σ* < *p* − 1, then (43) holds. Since *γ* > 0, there exists *θ* > 0 such that (44) holds. Thus, passing to the limit as *R* → ∞ in (43), we obtain a contradiction. If *σ* = *p* − 1, then (42) yields

$$\begin{aligned} &\mathcal{C} + R^{-\theta(\gamma+1)} \int\_0^L \left( u\_1(\mathbf{x}) \varphi(0, \mathbf{x}) - u\_0(\mathbf{x}) \frac{\partial \varphi}{\partial t}(0, \mathbf{x}) + u\_0(\mathbf{x}) \varphi(0, \mathbf{x}) \right) d\mathbf{x} \\ &\leq \mathcal{C} \left( R^{-\theta\gamma + \frac{p+1+\sigma}{p-1}} + R^{-\theta\left(\gamma + \frac{p}{p-1}\right)} \ln R \right). \end{aligned}$$

As in the previous case, since *γ* > 0, there exists *θ* > 0 such that (44) holds. Thus, passing to the limit as *R* → ∞ in the above inequality, we obtain a contradiction. If *σ* > *p* − 1, then (42) yields

$$\begin{aligned} &\mathbb{C} + R^{-\theta(\gamma+1)} \int\_0^L \left( u\_1(\mathbf{x}) \varphi(0, \mathbf{x}) - u\_0(\mathbf{x}) \frac{\partial \varphi}{\partial t}(0, \mathbf{x}) + u\_0(\mathbf{x}) \varphi(0, \mathbf{x}) \right) d\mathbf{x} \\ &\leq \mathcal{C} \left( R^{-\theta\gamma + \frac{p+1+\sigma}{p-1}} + R^{-\theta\left(\gamma + \frac{p}{p-1}\right) + \frac{\sigma}{p-1} - 1} \right). \end{aligned}$$

Taking *θ* such that (44) is satisfied, and passing to the limit as *R* → ∞ in the above inequality, a contradiction follows. Thus, part (ii) of Theorem 2 is proved.

#### *4.3. Proof of Theorem 3*

**Proof.** Suppose that *u* is a global weak solution to (2). Then, by (13), for every *T* > 0 and *ϕ* ∈ Φ*T*, there holds

$$\begin{split} &\int\_{Q\_T} \mathbf{x}^T |u|^p \, q \, \mathrm{d}x \, dt + \int\_0^T \left( f(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, 0) - \boldsymbol{\varrho}(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, L) \right) dt \\ &+ \int\_0^L \left( u\_1(\mathbf{x}) (I\_T^{2-n} \boldsymbol{\varrho})(0, \mathbf{x}) - u\_0(\mathbf{x}) \frac{\partial (I\_T^{2-n} \boldsymbol{\varrho})}{\partial t}(0, \mathbf{x}) + u\_0(\mathbf{x}) (I\_T^{1-\tilde{\varrho}} \boldsymbol{\varrho})(0, \mathbf{x}) \right) d\mathbf{x} \\ &\leq \int\_{Q\_T} |u| \left| \frac{\partial^2 \boldsymbol{\varrho}}{\partial \mathbf{x}^2} \right| \, d\mathbf{x} \, dt + \int\_{Q\_T} |u| \left| \frac{\partial^2 (I\_T^{2-n} \boldsymbol{\varrho})}{\partial t^2} \right| \, d\mathbf{x} \, dt + \int\_{Q\_T} |u| \left| \frac{\partial (I\_T^{1-\tilde{\varrho}} \boldsymbol{\varrho})}{\partial t} \right| \, d\mathbf{x} \, dt. \end{split} \tag{45}$$

On the other hand, using Lemma 3 with *ε* = <sup>1</sup> <sup>3</sup> and adequate choices of *a* and *b*, we obtain

$$\begin{split} &\int\_{Q\_T} |u| \left| \frac{\partial^2 \varrho}{\partial \mathbf{x}^2} \right| \, d\mathbf{x} \, dt \\ &\leq \frac{1}{3} \int\_{Q\_T} \mathbf{x}^\sigma |u|^p \, \varrho \, d\mathbf{x} \, dt + \mathbb{C} \int\_{Q\_T} \mathbf{x}^{\frac{-\sigma}{p-1}} \varrho^{\frac{-1}{p-1}} \left| \frac{\partial^2 \varrho}{\partial \mathbf{x}^2} \right|^{\frac{p}{p-1}} \, d\mathbf{x} \, dt, \\ &\int\_{Q\_T} |u| \left| \frac{\partial^2 (I\_T^{2-a} \varrho)}{\partial t^2} \right| \, d\mathbf{x} \, dt \\ &\leq \frac{1}{3} \int\_{Q\_T} \mathbf{x}^\sigma |u|^p \varrho \, d\mathbf{x} \, dt + \mathbb{C} \int\_{Q\_T} \mathbf{x}^{\frac{-\sigma}{p-1}} \varrho^{\frac{-1}{p-1}} \left| \frac{\partial^2 (I\_T^{2-a} \varrho)}{\partial t^2} \right|^{\frac{p}{p-1}} \, d\mathbf{x} \, dt, \end{split} \tag{47}$$

and

$$\begin{split} & \int\_{Q\_T} |u| \left| \frac{\partial(I\_T^{1-\beta}\varphi)}{\partial t} \right| \, d\mathbf{x} \, dt \\ & \leq \frac{1}{3} \int\_{Q\_T} \mathbf{x}^{\sigma} |u|^p \boldsymbol{\varrho} \, d\mathbf{x} \, dt + \mathbb{C} \int\_{Q\_T} \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left| \frac{\partial(I\_T^{1-\beta}\boldsymbol{\varrho})}{\partial t} \right|^{\frac{p}{p-1}} \, d\mathbf{x} \, dt. \end{split} \tag{48}$$

Using (45)–(48), we obtain

$$\begin{split} &\int\_{0}^{T} \left( f(t) \frac{\partial \boldsymbol{\varrho}}{\partial \boldsymbol{x}}(t, 0) - \boldsymbol{g}(t) \frac{\partial \boldsymbol{\varrho}}{\partial \boldsymbol{x}}(t, L) \right) dt \\ &+ \int\_{0}^{L} \left( \boldsymbol{u}\_{1}(\boldsymbol{x}) (\boldsymbol{I}\_{T}^{2-a} \boldsymbol{\varrho})(0, \boldsymbol{x}) - \boldsymbol{u}\_{0}(\boldsymbol{x}) \frac{\partial (\boldsymbol{I}\_{T}^{2-a} \boldsymbol{\varrho})}{\partial t}(0, \boldsymbol{x}) + \boldsymbol{u}\_{0}(\boldsymbol{x}) (\boldsymbol{I}\_{T}^{1-\tilde{\rho}} \boldsymbol{\varrho})(0, \boldsymbol{x}) \right) d\boldsymbol{x} \\ &\leq \sum\_{j=1}^{3} \boldsymbol{J}\_{\tilde{l}}(\boldsymbol{\varrho})\_{\prime} \end{split} \tag{49}$$

where

$$\begin{array}{rcl}f\_1(\boldsymbol{\varrho})&=&\int\_{Q\_T} \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left|\frac{\partial^2 \boldsymbol{\varrho}}{\partial \boldsymbol{x}^2}\right|^{\frac{p}{p-1}} d\boldsymbol{x} \, dt, \\\\ f\_2(\boldsymbol{\varrho})&=&\int\_{Q\_T} \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left|\frac{\partial^2 (I\_T^{2-\alpha} \boldsymbol{\varrho})}{\partial t^2}\right|^{\frac{p}{p-1}} d\boldsymbol{x} \, dt, \\\\ f\_3(\boldsymbol{\varrho})&=&\int\_{Q\_T} \mathbf{x}^{\frac{-\sigma}{p-1}} \boldsymbol{\varrho}^{\frac{-1}{p-1}} \left|\frac{\partial (I\_T^{1-\beta} \boldsymbol{\varrho})}{\partial t}\right|^{\frac{p}{p-1}} d\boldsymbol{x} \, dt. \end{array}$$

For sufficiently large *T*, *λ*, , and *R*, let

$$
\varphi(t, \mathbf{x}) = \eta(t)\varphi\_2(\mathbf{x}), \quad t \ge 0, \ x \in [0, L], \tag{50}
$$

where *η* is the function defined by (4), and *ϕ*<sup>2</sup> is the function given by (23). Using Lemma 2 and the properties of the cut-off function *μ*, it can be easily seen that the function *ϕ* defined by (50), belongs to Φ*T*. Thus, (49) holds for this function.

Let us estimate the terms *Jj*(*ϕ*), *j* = 1, 2, 3. For *j* = 1, by (50), we have

$$J\_1(\varphi) = \left(\int\_0^T \eta(t) \, dt\right) \left(\int\_0^L \mathbf{x}^{\frac{-\sigma}{p-1}} \varphi\_2^{\frac{-1}{p-1}}(\mathbf{x}) |\varphi\_2''(\mathbf{x})|^{\frac{p}{p-1}} \, d\mathbf{x}\right). \tag{51}$$

An elementary calculation shows that

$$\int\_0^T \eta(t) \, dt = \frac{T}{\lambda + 1}.\tag{52}$$

Hence, using (27), (51), and (52), we obtain

$$J\_1(\varphi) \le CTR^{\frac{\sigma + 2p}{p-1} - 1}.\tag{53}$$

For *j* = 2, we have

$$J\_2(\varphi) = \left( \int\_0^T \eta^{\frac{-1}{p-1}}(t) |(I\_T^{2-a}\eta)^{\prime\prime}(t)|^{\frac{p}{p-1}} \, dt \right) \left( \int\_0^L \mathbf{x}^{\frac{-\sigma}{p-1}} \varphi\_2(\mathbf{x}) \, d\mathbf{x} \right). \tag{54}$$

Moreover, by Lemma 2, we obtain

$$|\eta^{\frac{-1}{p-1}}(t)|(I\_T^{2-\alpha}\eta)^{\prime\prime}(t)|^{\frac{p}{p-1}} = \left[\frac{\Gamma(\lambda+1)}{\Gamma(1-\alpha+\lambda)}\right]^{\frac{p}{p-1}}T^{-\lambda}(T-t)^{\lambda-\frac{ap}{p-1}}.$$

Integrating over (0, *T*), there holds

$$\int\_0^T \eta^{\frac{-1}{p-1}}(t) |(I\_T^{2-a}\eta)''(t)|^{\frac{p}{p-1}} dt = \mathcal{C}T^{\frac{-ap}{p-1}+1}.\tag{55}$$

Next, taking into consideration that *σ* < −(*p* + 1) (so *σ* < *p* − 1), it follows from (31), (54), and (55) that

$$J\_2(\varphi) \le C T^{1 - \frac{ap}{p-1}}.\tag{56}$$

Proceeding as above, we obtain

$$J\_3(\varphi) \le C T^{1 - \frac{\beta p}{p-1}}.\tag{57}$$

Hence, by (53), (56), and (57), we obtain

$$\sum\_{j=1}^{3} f\_j(\varphi) \le \mathcal{C} \left( T R^{\frac{\sigma + 2p}{p-1} - 1} + T^{1 - \frac{\beta p}{p-1}} \right). \tag{58}$$

Consider now the terms from the right side of (49). By (50) and the properties of the cut-off function *μ*, since *g* ≡ 0, there holds

$$\int\_{0}^{T} \left( f(t) \frac{\partial \varrho}{\partial \mathfrak{x}}(t, 0) - \mathfrak{g}(t) \frac{\partial \varrho}{\partial \mathfrak{x}}(t, L) \right) dt = 0. \tag{59}$$

On the other hand, using (50) and Lemma 2, for all *x* ∈ [0, *L*], we obtain

$$\begin{array}{rcl}(I\_{T}^{2-\alpha}\varrho)(0,\mathsf{x})&=&\frac{\Gamma(\lambda+1)}{\Gamma(3-\alpha+\lambda)}T^{2-\alpha}\varrho\_{2}(\mathsf{x})&:=&\mathsf{C}\_{1}T^{2-\alpha}\varrho\_{2}(\mathsf{x}),\\\frac{\partial(I\_{T}^{2-\alpha}\varrho)}{\partial t}(0,\mathsf{x})&=&-\frac{\Gamma(\lambda+1)}{\Gamma(2-\alpha+\lambda)}T^{1-\alpha}\varrho\_{2}(\mathsf{x})&:=&-\mathsf{C}\_{2}T^{1-\alpha}\varrho\_{2}(\mathsf{x}),\\(I\_{T}^{1-\beta}\varrho)(0,\mathsf{x})&=&\frac{\Gamma(\lambda+1)}{\Gamma(2-\beta+\lambda)}T^{1-\beta}\varrho\_{2}(\mathsf{x})&:=&\mathsf{C}\_{3}T^{1-\beta}\varrho\_{2}(\mathsf{x}).\end{array}$$

Consequently, we obtain

$$\begin{split} &\int\_{0}^{L} \left( u\_{1}(\mathbf{x}) (I\_{T}^{2-\mathfrak{a}}\boldsymbol{\varrho})(0,\mathbf{x}) - u\_{0}(\mathbf{x}) \frac{\partial (I\_{T}^{2-\mathfrak{a}}\boldsymbol{\varrho})}{\partial t}(0,\mathbf{x}) + u\_{0}(\mathbf{x}) (I\_{T}^{1-\mathfrak{beta}}\boldsymbol{\varrho})(0,\mathbf{x}) \right) d\mathbf{x} \\ &= \int\_{0}^{L} \left( \mathbb{C}\_{1} T^{2-\mathfrak{a}} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} T^{1-\mathfrak{a}} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} T^{1-\mathfrak{beta}} u\_{0}(\mathbf{x}) \right) \boldsymbol{\varrho}\_{2}(\mathbf{x}) \, d\mathbf{x} \\ &= \int\_{0}^{L} \left( \mathbb{C}\_{1} T^{2-\mathfrak{a}} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} T^{1-\mathfrak{a}} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} T^{1-\mathfrak{beta}} u\_{0}(\mathbf{x}) \right) (L-\mathbf{x}) \boldsymbol{\mu}^{\ell}(\mathbf{Rx}) \, d\mathbf{x}. \end{split} \tag{60}$$

Thus, combining (49), (58)–(60), we obtain

$$\begin{aligned} &\int\_0^L \left( \mathbb{C}\_1 T^{2-a} u\_1(\mathbf{x}) + \mathbb{C}\_2 T^{1-a} u\_0(\mathbf{x}) + \mathbb{C}\_3 T^{1-\beta} u\_0(\mathbf{x}) \right) (L-\mathbf{x}) \mu^\ell(\mathbf{R} \mathbf{x}) \, d\mathbf{x} \\ &\le \mathbb{C} \left( T R^{\frac{\sigma+2p}{p-1}-1} + T^{1-\frac{\beta p}{p-1}} \right). \end{aligned}$$

Next, taking *T* = *Rθ*, where *θ* > 0 is a constant that will be determined later, the above inequality reduces to

$$\begin{split} &\int\_{0}^{L} \left( \mathbb{C}\_{1} \mathbb{R}^{\theta(2-a)} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} \mathbb{R}^{\theta(1-a)} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} \mathbb{R}^{\theta(1-\theta)} u\_{0}(\mathbf{x}) \right) (L-\mathbf{x}) \mu^{\ell}(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \\ &\leq \mathbb{C} \left( \mathbb{R}^{\theta + \frac{\sigma + 2p}{p-1} - 1} + \mathbb{R}^{\theta \left(1 - \frac{\beta p}{p-1}\right)} \right). \end{split} \tag{61}$$

Suppose that (14) holds. In this case, we obtain

$$\begin{split} &\lim\_{R\to\infty}R^{-\theta(2-a)}\int\_{0}^{L}\Big(\mathsf{C}\_{1}R^{\theta(2-a)}u\_{1}(\mathbf{x})+\mathsf{C}\_{2}R^{\theta(1-a)}u\_{0}(\mathbf{x})+\mathsf{C}\_{3}R^{\theta(1-\tilde{\theta})}u\_{0}(\mathbf{x})\Big)(L-\mathbf{x})\mu^{\ell}(R\mathbf{x})\,d\mathbf{x} \\ &=\mathsf{C}\_{1}\int\_{0}^{L}u\_{1}(\mathbf{x})(L-\mathbf{x})\,d\mathbf{x} \\ &>0. \end{split}$$

Hence, for sufficiently large *R*,

$$\int\_{0}^{L} \left( \mathbb{C}\_{1} R^{\theta(2-a)} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} R^{\theta(1-a)} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} R^{\theta(1-\beta)} u\_{0}(\mathbf{x}) \right) (L-\mathbf{x}) \mu^{\ell}(\mathbf{Rx}) \, d\mathbf{x} \geq \mathbb{C} R^{\theta(2-a)}.\tag{62}$$

Combining (61) with (62), we obtain

$$C \le R^{\theta(a-1) + \frac{\sigma + 2p}{p-1} - 1} + R^{\theta\left(a - \frac{\beta p}{p-1} - 1\right)}.\tag{63}$$

Observe that, since *α* < *β* + 1, we have

$$
\alpha - \frac{\beta p}{p - 1} - 1 < 0.
$$

Hence, taking into consideration that *σ* < −(*p* + 1), picking *θ* > 0 so that

$$
\theta < \frac{-(p+1) - \sigma}{(p-1)(\alpha -1)}\gamma
$$

and passing to the limit as *R* → ∞ in (63), we obtain a contradiction with *C* > 0. Suppose that (15) holds. Then,

$$(I\_T^{2-\alpha}\varphi)(0,\mathfrak{x}) = (I\_T^{1-\beta}\varphi)(0,\mathfrak{x}).$$

Thus, (61) reduces to

$$\begin{split} &\int\_{0}^{L} \left( \mathbb{C}\_{1} R^{\theta(2-\mathfrak{a})} (u\_{0}(\mathfrak{x}) + u\_{1}(\mathfrak{x})) + \mathbb{C}\_{2} R^{\theta(1-\mathfrak{a})} u\_{0}(\mathfrak{x}) \right) (L-\mathfrak{x}) \mu^{\ell}(\mathbb{R}\mathfrak{x}) \, d\mathfrak{x} \\ &\leq \mathbb{C} \left( R^{\theta + \frac{\sigma + 2p}{p-1} - 1} + R^{\theta \left( 1 - \frac{\beta p}{p-1} \right)} \right) . \end{split} \tag{64}$$

Moreover, we have

$$\begin{aligned} &\lim\_{R\to\infty}R^{-\theta(2-\alpha)}\int\_{0}^{L}\Big(\mathbb{C}\_{1}R^{\theta(2-\alpha)}(u\_{0}(\mathbf{x})+u\_{1}(\mathbf{x}))+\mathbb{C}\_{2}R^{\theta(1-\alpha)}u\_{0}(\mathbf{x})\Big)(L-\mathbf{x})\mu^{\ell}(\mathbf{R}\mathbf{x})\,d\mathbf{x} \\ &=\mathbb{C}\_{1}\int\_{0}^{L}(u\_{0}(\mathbf{x})+u\_{1}(\mathbf{x}))(L-\mathbf{x})\,d\mathbf{x} \\ &>0,\end{aligned}$$

which yields

$$\int\_0^L \left( \mathbb{C}\_1 \mathbb{R}^{\theta(2-a)} (u\_0(\mathbf{x}) + u\_1(\mathbf{x})) + \mathbb{C}\_2 \mathbb{R}^{\theta(1-a)} u\_0(\mathbf{x}) \right) (L-\mathbf{x}) \mu^\ell(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \ge \mathbb{C} \mathcal{R}^{\theta(2-a)} \, d\mathbf{x}$$

for sufficiently large *R*. Hence, using (64), and following the same argument as above, a contradiction follows.

Finally, suppose that (16) holds. In this case, we obtain

$$\begin{aligned} &\lim\_{R\to\infty}R^{-\theta(1-\theta)}\int\_{0}^{L}\Big(\mathsf{C}\_{1}R^{\theta(2-a)}u\_{1}(\mathbf{x})+\mathsf{C}\_{2}R^{\theta(1-a)}u\_{0}(\mathbf{x})+\mathsf{C}\_{3}R^{\theta(1-\tilde{\theta})}u\_{0}(\mathbf{x})\Big)(L-\mathbf{x})\mu^{\ell}(R\mathbf{x})\,d\mathbf{x} \\ &=\mathsf{C}\_{3}\int\_{0}^{L}u\_{0}(\mathbf{x})(L-\mathbf{x})\,d\mathbf{x} \\ &>0.\end{aligned}$$

Hence, for sufficiently large *R*,

$$\int\_{0}^{L} \left( \mathbb{C}\_{1} R^{\theta(2-a)} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} R^{\theta(1-a)} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} R^{\theta(1-\beta)} u\_{0}(\mathbf{x}) \right) (L-\mathbf{x}) \mu^{\ell}(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \geq \mathbb{C} R^{\theta(1-\beta)}.\tag{65}$$

Combining (61) with (65), we obtain

$$C \le R^{\theta\beta + \frac{\sigma + 2p}{p-1} - 1} + R^{\frac{-\theta\beta}{p-1}}.\tag{66}$$

Taking *θ* > 0 such that

$$
\theta < \frac{-\sigma - (p+1)}{\beta(p-1)}\nu
$$

and passing to the limit as *R* → ∞ in (66), a contradiction follows. This completes the proof of Theorem 3.

#### *4.4. Proof of Theorem 4*

**Proof.** Suppose that *u* is a global weak solution to (2). From the proof of Theorem 3, for sufficiently large *T* and *R*, there holds

$$\begin{split} & -\int\_{0}^{T} \mathbf{g}(t) \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}}(t, \mathcal{L}) \, dt \\ & + \int\_{0}^{L} \Big( \mathsf{C}\_{1} T^{2-\alpha} u\_{1}(\mathbf{x}) + \mathsf{C}\_{2} T^{1-\alpha} u\_{0}(\mathbf{x}) + \mathsf{C}\_{3} T^{1-\beta} u\_{0}(\mathbf{x}) \Big) (\mathsf{L} - \mathsf{x}) \mu^{\ell}(\mathcal{R} \mathbf{x}) \, d\mathbf{x} \\ & \leq \mathsf{C} \Big( T \mathcal{R}^{\frac{\sigma + 2p}{p-1} - 1} + T^{1-\frac{\beta p}{p-1}} \int\_{\frac{1}{2} \mathcal{R}^{-1}}^{L} \mathbf{x}^{\frac{-\sigma}{p-1}} \, d\mathbf{x} \Big) , \end{split} \tag{67}$$

where *ϕ* is the function defined by (50). On the other hand, by (50) and the properties of the cut-off function *μ*, we have

$$\begin{aligned} -\int\_0^T g(t) \frac{\partial \rho}{\partial \mathbf{x}}(t, L) \, dt &= \int\_0^T g(t) \eta(t) \, dt \\ &= \quad \, \, T^{-\lambda} \int\_0^T t^\gamma (T - t)^\lambda \, dt \\ &= \quad \, \, \, B(\gamma + 1, \lambda + 1) T^{\gamma + 1} \\ &:= \quad \, \, \, C T^{\gamma + 1} \end{aligned}$$

where *B* denotes the Beta function. Thus, by (67), we obtain

$$\begin{aligned} &\mathcal{C} + \int\_0^L \left( \mathcal{C}\_1 T^{1-a-\gamma} u\_1(\mathbf{x}) + \mathcal{C}\_2 T^{-\gamma-a} u\_0(\mathbf{x}) + \mathcal{C}\_3 T^{-\beta-\gamma} u\_0(\mathbf{x}) \right) (L-\mathbf{x}) \mu^\ell(\mathcal{R}\mathbf{x}) \, d\mathbf{x} \\ &\leq \mathcal{C} \left( T^{-\gamma} \mathcal{R}^{\frac{\sigma+2p}{p-1}-1} + T^{-\frac{\beta p}{p-1}-\gamma} \int\_{\frac{1}{2}R^{-1}}^L \mathbf{x}^{\frac{-\sigma}{p-1}} \, d\mathbf{x} \right) . \end{aligned}$$

Taking *T* = *Rθ*, where *θ* > 0 is a constant that will be determined later, the above inequality reduces to

$$\begin{split} &C + \int\_{0}^{L} \left( \mathbb{C}\_{1} R^{\theta(1-\mathfrak{a}-\gamma)} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} R^{-\theta(\gamma+\mathfrak{a})} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} R^{-\theta(\beta+\gamma)} u\_{0}(\mathbf{x}) \right) (L-\mathbf{x}) \mu^{\ell}(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \\ &\leq \mathcal{C} \Big( R^{-\theta\gamma + \frac{\sigma+2\rho}{p-1}-1} + R^{-\theta\left(\frac{\theta p}{p-1} + \gamma\right)} \int\_{\frac{1}{2}R^{-1}}^{L} \mathbf{x}^{\frac{-\sigma}{p-1}} \, d\mathbf{x} \Big). \end{split} \tag{68}$$

Let *σ* < −(*p* + 1). In this case, for sufficiently large *R*, there holds

$$\int\_{\frac{1}{2}R^{-1}}^{L} \varkappa^{\frac{-\sigma}{p-1}} \,d\varkappa \le C.$$

Hence, (68) yields

$$\begin{split} &C + \int\_{0}^{L} \Big( \mathsf{C}\_{1} R^{\theta(1-\alpha-\gamma)} u\_{1}(\mathbf{x}) + \mathsf{C}\_{2} R^{-\theta(\gamma+a)} u\_{0}(\mathbf{x}) + \mathsf{C}\_{3} R^{-\theta(\beta+\gamma)} u\_{0}(\mathbf{x}) \Big) (L-\mathbf{x}) \mu^{\ell}(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \\ &\leq C \Big( R^{-\theta\gamma + \frac{\sigma+2\rho}{p-1} - 1} + R^{-\theta\left(\frac{\beta p}{p-1} + \gamma\right)} \Big). \end{split} \tag{69}$$

Since by (17), *β* + *γ* > 0, there holds

$$\frac{\beta p}{p-1} + \gamma > 0.$$

Thus, taking *θ* > 0 so that

$$
\theta \gamma > \frac{\sigma + p + 1}{p - 1},
\tag{70}
$$

using (17), and passing to the limit as *R* → ∞ in (69), we obtain a contradiction with *C* > 0. This proves part (i) of Theorem 4.

Let *σ* ≥ −(*p* + 1) and *γ* > 0.

If −(*p* + 1) ≤ *σ* < *p* − 1, then (69) holds. Since *γ* > 0, there exists *θ* > 0 satisfying (70). Thus, passing to the limit as *R* → ∞ in (69), a contradiction follows. If *σ* = *p* − 1, then (68) yields

$$\begin{split} &\mathbb{C} + \int\_{0}^{L} \left( \mathbb{C}\_{1} R^{\theta(1-\alpha-\gamma)} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} R^{-\theta(\gamma+a)} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} R^{-\theta(\beta+\gamma)} u\_{0}(\mathbf{x}) \right) (L-\mathbf{x}) \mu^{\ell}(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \\ &\leq \mathbb{C} \Big{(} R^{-\theta\gamma + \frac{\delta+2\rho}{p-1}-1} + R^{-\theta\left(\frac{\delta p}{p-1} + \gamma\right)} \ln R \Big{)}. \end{split} \tag{71}$$

As in the previous case, since *γ* > 0, there exists *θ* > 0 satisfying (70). Thus, passing to the limit as *R* → ∞ in (71), a contradiction follows. If *σ* > *p* − 1, then (68) yields

$$\begin{split} &\mathbb{C} + \int\_{0}^{L} \Big( \mathbb{C}\_{1} R^{\theta(1-\alpha-\gamma)} u\_{1}(\mathbf{x}) + \mathbb{C}\_{2} R^{-\theta(\gamma+\alpha)} u\_{0}(\mathbf{x}) + \mathbb{C}\_{3} R^{-\theta(\beta+\gamma)} u\_{0}(\mathbf{x}) \Big) (L-\mathbf{x}) \mu^{\ell}(\mathbf{R}\mathbf{x}) \, d\mathbf{x} \\ &\leq \mathbb{C} \Big( R^{-\theta\gamma + \frac{\sigma+2p}{p-1}-1} + R^{-\theta\left(\frac{\beta p}{p-1} + \gamma\right) + \frac{\sigma}{p-1}-1} \Big). \end{split} \tag{72}$$

So, taking *θ* > 0 satisfying (70) and

$$
\theta \left( \frac{\beta p}{p - 1} + \gamma \right) > \frac{\sigma}{p - 1} - 1,
$$

and passing to the limit as *R* → ∞ in (72), a contradiction follows. This proves part (ii) of Theorem 4.

#### **5. Conclusions**

Using the test function method, sufficient conditions for the nonexistence of global weak solutions to problems (1) and (2) are obtained. For each problem, an adequate choice of a test function is made, taking into consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm.

In this paper, we treated only the one dimensional case. It will be interesting to study problems (1) and (2) in a bounded domain <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup>*<sup>N</sup>* under different types of boundary conditions, such as Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions.

**Author Contributions:** Investigation, A.B.S.; Supervision M.J. and B.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** The second author is supported by the Researchers Supporting Project number (RSP-2021/57), King Saud University, Riyadh, Saudi Arabia.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Nadia Allouch 1, John R. Graef 2,\* and Samira Hamani <sup>1</sup>**

Mostaganem 27000, Algeria; nadia.allouch.etu@univ-mosta.dz (N.A.); hamani\_samira@yahoo.fr (S.H.)

<sup>2</sup> Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

**\*** Correspondence: john-graef@utc.edu; Tel.: +1-423-425-4545

**Abstract:** The authors investigate the existence of solutions to a class of boundary value problems for fractional *q*-difference equations in a Banach space that involves a *q*-derivative of the Caputo type and nonlinear integral boundary conditions. Their result is based on Mönch's fixed point theorem and the technique of measures of noncompactness. This approach has proved to be an interesting and useful approach to studying such problems. Some basic concepts from the fractional *q*-calculus are introduced, including *q*-derivatives and *q*-integrals. An example of the main result is included as well as some suggestions for future research.

**Keywords:** boundary value problems; fractional *q*-difference equations; Caputo fractional *q*-difference derivative; measure of noncompactness; Mönch's fixed point theorem

**MSC:** 26A33; 34A37

**Citation:** Allouch, N.; Graef, J.R.; Hamani, S. Boundary Value Problem for Fractional *q*-Difference Equations with Integral Conditions in Banach Spaces. *Fractal Fract.* **2022**, *6*, 237. https://doi.org/10.3390/ fractalfract6050237

Academic Editor: Rodica Luca

Received: 8 March 2022 Accepted: 23 April 2022 Published: 25 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Fractional differential equations play an essential role when attempting to model phenomena in a number of areas and have recently been studied by researchers in engineering, physics, chemistry, biology, economics, and control theory. For additional details see, for example, the monographs of Hilfer [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4], Samko et al. [5], and Tarasov [6] as well as the references they contain. The existence of solutions to fractional boundary value problems is currently a very active area of research as can be seen, for example, from the recent papers of Ahmad et al. [7], Agarwal et al. [8], Benchohra et al. [9], Benhamida et al. [10], Hamini et al. [11], and Zahed et al. [12].

Considerable attention has been given to the problem of existence of solutions to boundary value problems for fractional differential equations in Banach spaces, and we refer the reader to the recent contributions in [13–15].

The *q*-difference calculus, or quantum calculus, was first introduced by Jackson in 1910 [16,17]. The basic definitions and properties of the *q*-difference calculus can be found in [18,19]. Later, Al-Salam [20] and Agarwal [21] proposed the study of the fractional *q*-difference calculus. Fractional *q*-difference calculus by itself and nonlinear fractional *q*-difference boundary value problems have appeared as the object of study for a number of researchers. Recent developments on the fractional *q*-difference calculus and boundary value problems for such can be found in [7,22–25] and the references therein.

In this paper, we study the existence of solutions to the boundary value problem (BVP for short) for fractional *q*-difference equations with nonlinear integral conditions

$$(\prescript{C}{}{D}\_q^a y)(t) = f(t, y(t)), \text{ for a.e.} \ t \in J = [0, T], \quad 1 < a \le 2,\tag{1}$$

$$y(0) - y'(0) = \int\_0^T \mathbf{g}(s, y(s)) ds,\tag{2}$$

$$y(T) + y'(T) = \int\_0^T h(s, y(s)) ds,\tag{3}$$

where *<sup>T</sup>* <sup>&</sup>gt; 0, *<sup>q</sup>* <sup>∈</sup> (0, 1), *CD<sup>α</sup> <sup>q</sup>* is the Caputo fractional *q*-difference derivative of order 1 < *α* ≤ 2, and *f* , *g*, *h* : *J* × *E* → *E* are given functions and *g* and *h* are continuous.

In our investigation of the existence of solutions to the problem above, we utilize the method associated with the technique of measures of noncompactness and Mönch's fixed point theorem. This approach turns out to be very useful in proving the existence of solutions for several different types of equations. The method of using measures of noncompactness was mainly initiated in the monograph of Banas and Goebel [26], and subsequently developed and used in many papers; see, for example, Banas et al. [27], Guo et al. [28], Akhmerov et al. [29], Mönch [30], Mönch and Von Harten [31], and Szufla [32].

This paper is structured as follows. In Section 2, we introduce some preliminary concepts including basic definitions and properties from fractional q-calculus and some properties of the Kuratowski measure of noncompactness. In Section 3, the existence of solutions to problem (1)–(3) is proved by using Mönch's fixed point theorem. Section 4 contains an example to illustrate our main results. The final section contains some concluding remarks and suggestions for future research.

#### **2. Materials and Methods**

We begin by introducing definitions, notations, and some preliminary facts that are used in the remainder of this paper.

Let *J* = [0, *T*], *T* > 0, and consider the Banach space *C*(*J*, *E*) of continuous functions from *J* into *E* with the norm

$$||y||\_{\infty} = \sup\{|y(t)|\ :\ t \in J\}.$$

We let *<sup>C</sup>*2(*J*, *<sup>E</sup>*) be the space of differentiable functions *<sup>y</sup>* : *<sup>J</sup>* <sup>→</sup> *<sup>E</sup>*, whose first and second derivatives are continuous, and let *L*1(*J*, *E*) be the Banach space of measurable functions *y* : *J* → *E* that are Bochner integrable with the norm

$$\|y\|\_{L^1} = \int\_I |y(t)|dt.$$

Let *<sup>L</sup>*∞(*J*, *<sup>E</sup>*) be the Banach space of bounded measurable functions *<sup>y</sup>* : *<sup>J</sup>* <sup>→</sup> *<sup>E</sup>* equipped with the norm

$$\|y\|\_{L^{\infty}} = \inf\{c > 0 \; : \; \|y(t)\| \le c \; \; a.e \; t \in J\}.$$

We now recall some definitions and properties from the fractional q-calculus [18,19]. For *<sup>a</sup>* <sup>∈</sup> <sup>R</sup> and 0 <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; 1, we set

$$[a]\_q = \frac{1 - q^a}{1 - q}.$$

The q-analogue of the power (*<sup>a</sup>* <sup>−</sup> *<sup>b</sup>*)(*n*) is given by

$$(a-b)^{(0)} = 1,\ (a-b)^{(n)} = \prod\_{k=0}^{n-1} (a - bq^k),\ a, b \in \mathbb{R},\ n \in \mathbb{N}.$$

In general,

$$(a-b)^{(\alpha)} = a^{\alpha} \prod\_{k=0}^{\infty} \left( \frac{a - bq^k}{a - bq^{k+\alpha}} \right), \ a, b, \alpha \in \mathbb{R}.$$

Note that if *b* = 0, then *a*(*α*) = *aα*.

**Definition 1** ([19])**.** *The q-gamma function is defined by*

$$\Gamma\_q(\alpha) = \frac{(1-q)^{(\alpha-1)}}{(1-q)^{\alpha-1}}, \ \alpha \in \mathbb{R} - \{0, -1, -2, \dots\}.$$

We wish to point out that the q-gamma function satisfies the relation Γ*q*(*α* + 1) = [*α*]*q*Γ*q*(*α*).

**Definition 2** ([19])**.** *The q-derivative of order <sup>n</sup>* <sup>∈</sup> <sup>N</sup> *of a function <sup>f</sup>* : *<sup>J</sup>* <sup>→</sup> <sup>R</sup> *is defined by* (*D*<sup>0</sup> *<sup>q</sup> f*)(*t*) = *f*(*t*)*,*

$$(D\_q f)(t) = (D\_q^1 f)(t) = \frac{f(t) - f(qt)}{(1 - q)t}, \; t \neq 0, \; (D\_q f)(0) = \lim\_{t \to 0} (D\_q f)(t).$$

*and*

$$(D\_q^n f)(t) = (D\_q^1 D\_q^{n-1} f)(t), \; t \in \mathcal{J}, \; n \in \{1, 2, \ldots\}.$$

Now set *Jt* <sup>=</sup> {*tq<sup>n</sup>* : *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>}∪{0}.

**Definition 3** ([19])**.** *The q-integral of a function f* : *Jt* <sup>→</sup> <sup>R</sup> *is defined by*

$$(I\_{\emptyset}f)(t) = \int\_0^t f(s)d\_{\emptyset}s = \sum\_{n=0}^\infty t(1-q)q^n f(tq^n)s$$

*provided that the series converges.*

We note that (*Dq Iq f*)(*t*) = *f*(*t*), while if *f* is continuous at 0, then

$$(I\_q D\_q f)(t) = f(t) - f(0).$$

**Definition 4** ([21])**.** *The Riemann–Liouville fractional q-integral of order <sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *of a function <sup>f</sup>* : *<sup>J</sup>* <sup>→</sup> <sup>R</sup> *is defined by* (*I*<sup>0</sup> *<sup>q</sup> f*)(*t*) = *f*(*t*)*, and*

$$(I\_q^\alpha f)(t) = \int\_0^t \frac{(t - qs)^{(\alpha - 1)}}{\Gamma\_q(\alpha)} f(s) d\_q s, \ t \in J.$$

Note that for *α* = 1, we have (*I*<sup>1</sup> *<sup>q</sup> f*)(*t*)=(*Iq f*)(*t*).

**Lemma 1** ([33])**.** *For <sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *and <sup>β</sup>* <sup>∈</sup> (−1, <sup>+</sup>∞)*, we have*

$$(I\_q^\alpha(t-a)^{(\beta)})(t) = \frac{\Gamma\_q(\beta+1)}{\Gamma\_q(\alpha+\beta+1)}(t-a)^{(\alpha+\beta)},\ 0 < a < t < T.$$

*In particular,*

$$(I\_q^\alpha 1)(t) = \frac{1}{\Gamma\_q(\alpha + 1)} t^{(\alpha)}.1$$

In what follows, we let [*α*] denote the integer part of *α*.

**Definition 5** ([34])**.** *The Riemann–Liouville fractional q-derivative of order <sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *of a function <sup>f</sup>* : *<sup>J</sup>* <sup>→</sup> <sup>R</sup> *is defined by* (*D*<sup>0</sup> *<sup>q</sup> f*)(*t*) = *f*(*t*)*, and*

$$(D\_q^\alpha f)(t) = (D\_q^{[\alpha]} I\_q^{[\alpha]-\alpha} f)(t), \ t \in J.$$

**Definition 6** ([34])**.** *The Caputo fractional q-derivative of order <sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *of a function <sup>f</sup>* : *<sup>J</sup>* <sup>→</sup> <sup>R</sup> *is defined by* (*D*<sup>0</sup> *<sup>q</sup> f*)(*t*) = *f*(*t*)*, and*

$$(^{\mathbb{C}}D\_q^{\alpha}f)(t) = (I\_q^{[\alpha]-\alpha}D\_q^{[\alpha]}f)(t), \ t \in J.$$

**Lemma 2** ([34])**.** *Let <sup>α</sup>, <sup>β</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *and let f be a function defined on J. Then:*

*(1)* (*I<sup>α</sup> q I β <sup>q</sup> f*)(*t*)=(*I α*+*β <sup>q</sup> f*)(*t*)*; (2)* (*D<sup>α</sup> q Iα <sup>q</sup> f*)(*t*) = *f*(*t*)*.*

**Lemma 3** ([34])**.** *Let <sup>α</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> *and let f be a function defined on J. Then:*

$$(I\_q^{\alpha} \, ^\mathbb{C}D\_q^{\alpha} f)(t) = f(t) - \sum\_{k=0}^{[\alpha]-1} \frac{t^k}{\Gamma\_q(k+1)} (D\_q^k f)(0).$$

*In particular, if α* ∈ (0, 1)*, then*

$$(I\_q^\alpha \, ^C D\_q^\alpha f)(t) = f(t) - f(0).$$

Next, we recall the definition of the Kuratowski measure of noncompactness and summarize some of the main properties of this measure.

**Definition 7** ([26])**.** *Let E be a Banach space and let* Ω*<sup>E</sup> be the family of bounded subsets of E. The Kuratowski measure of noncompactness is the map μ* : Ω*<sup>E</sup>* → [0, ∞) *defined by*

$$\mu(B) = \inf \{ \varepsilon > 0 : B \subset \cup\_{i=1}^m B\_i \text{ and } \operatorname{diam}(B\_i) \le \varepsilon \}, \text{ where } B \in \Omega\_{\mathbb{E}}.$$

**Property 1** ([26])**.** *The Kuratowski measure of noncompactness satisfies:*


*Here B and conB denote the closure and the convex hull of the bounded set B, respectively.*

**Definition 8.** *The map f* : *J* × *E* → *E is Carathéodory if*


For a given set *V* of functions *v* : *J* → *E*, let

*V*(*t*) = {*v*(*t*) : *v* ∈ *V*}, *t* ∈ *J*, *V*(*J*) = {*v*(*t*) : *v* ∈ *V*, *t* ∈ *J*}.

We next recall Mönch's fixed point theorem.

**Theorem 1** ([30,35])**.** *Let D be a bounded, closed, and convex subset of a Banach space E such that* 0 ∈ *D, and let N be a continuous mapping of D into itself. If the implication*

*V* = *conN*(*V*) *or V* = *N*(*V*) ∪ {0} *implies μ*(*V*) = 0,

*holds for every subset V of D, then N has a fixed point.*

The next lemma is a useful result.

**Lemma 4** ([28])**.** *If V* ⊂ *C*(*J*, *E*) *is a bounded and equicontinuous set, then*

$$\begin{aligned} \text{1.} \quad & \text{The function } t \to \mu(V(t)) \text{ is continuous on } \mathcal{J}.\\ \text{2.} \quad & \mu\left(\left\{\int\_{\mathcal{J}} y(t)dt \; : \; y \in V\right\}\right) \le \int\_{\mathcal{J}} \mu(V(t))dt. \end{aligned}$$

#### **3. Results**

We now define what is meant by a solution of the problem (1)–(3).

**Definition 9.** *A function <sup>y</sup>* <sup>∈</sup> *<sup>C</sup>*2(*J*, *<sup>E</sup>*) *is said to be a solution of the problem (1)–(3) if <sup>y</sup> satisfies the equation* (*CD<sup>α</sup> <sup>q</sup> y*)(*t*) = *f*(*t*, *y*(*t*)) *on J, and satisfies the boundary conditions (2) and (3).*

In order to prove the existence of solutions to the problem (1)–(3), we need the following lemma.

**Lemma 5.** *Let σ, ρ*1*, ρ*<sup>2</sup> : *J* → *E be continuous functions. The solution of the boundary value problem*

$$(\prescript{C}{}{D}\_q^a y)(t) = \sigma(t), \; t \in J = [0, T], \quad 1 < a \le 2,\tag{4}$$

$$y(0) - y'(0) = \int\_0^T \rho\_1(s)ds,\tag{5}$$

$$y(T) + y'(T) = \int\_0^T \rho\_2(s)ds,\tag{6}$$

*is given by*

$$y(t) = K(t) + \int\_0^T H(t, s)\sigma(s)d\_q s,\tag{7}$$

*where*

$$K(t) = \frac{(1+T-t)}{(2+T)} \int\_0^T \rho\_1(s)ds + \frac{(1+t)}{(2+T)} \int\_0^T \rho\_2(s)ds,\tag{8}$$

*and*

$$H(t,s) = \begin{cases} \frac{(t-qs)^{(a-1)}}{\Gamma\_q(a)} - \frac{(1+t)(T-qs)^{(a-1)}}{(2+T)\Gamma\_q(a)} - \frac{(1+t)(T-qs)^{(a-2)}}{(2+T)\Gamma\_q(a-1)}, & 0 \le s < t, \\\\ -\frac{(1+t)(T-qs)^{(a-1)}}{(2+T)\Gamma\_q(a)} - \frac{(1+t)(T-qs)^{(a-2)}}{(2+T)\Gamma\_q(a-1)}, & t \le s \le T. \end{cases} \tag{9}$$

**Proof.** Applying the Riemann–Liouville fractional *q*-integral of order *α* to both sides of Equation (4), and by using Lemma 3, we have

$$y(t) = \int\_0^t \frac{(t - qs)^{(\alpha - 1)}}{\Gamma\_q(\alpha)} \sigma(s) d\_q s + c\_0 + c\_1 t. \tag{10}$$

Using the boundary conditions (5) and (6), we obtain

$$c\_0 - c\_1 = \int\_0^T \rho\_1(s)ds,\tag{11}$$

and

 $c\_0 + (1+T)c\_1 + 
\int\_0^T \frac{(t-qs)^{(a-1)}}{\Gamma\_q(a)} \sigma(s) d\_q s$ 
$$+ 
\int\_0^T \frac{(t-qs)^{(a-2)}}{\Gamma\_q(a-1)} \sigma(s) d\_q s = 
\int\_0^T \rho\_2(s) ds. \tag{12}$$

Equations (11) and (12) give

$$c\_1 = \frac{1}{(2+T)} \left( \int\_0^T \rho\_2(s) ds - \int\_0^T \rho\_1(s) ds - \int\_0^T \frac{(t-qs)^{(a-1)}}{\Gamma\_q(a)} \sigma(s) d\_q s \right. \\ \left. \left. \begin{array}{c} (t \text{-} qs)^{(a-1)} \\ \Gamma\_q(a) \end{array} \right| ds \right) \tag{13}$$
 
$$- \int\_0^T \frac{(t-qs)^{(a-2)}}{\Gamma\_q(a-1)} \sigma(s) d\_q s \Big), \tag{13}$$

and

$$c\_0 = \frac{(1+T)}{(2+T)} \int\_0^T \rho\_1(s)ds + \frac{1}{(2+T)} \left( \int\_0^T \rho\_2(s)ds - \int\_0^T \frac{(t-qs)^{(a-1)}}{\Gamma\_q(a)} \sigma(s)d\_qs \right)$$

$$- \int\_0^T \frac{(t-qs)^{(a-2)}}{\Gamma\_q(a-1)} \sigma(s)d\_qs \Big). \tag{14}$$

From (10), (13), and (14) and using the fact that % *<sup>T</sup>* <sup>0</sup> <sup>=</sup> % *<sup>t</sup>* <sup>0</sup> <sup>+</sup> % *<sup>T</sup> <sup>t</sup>* , we have

$$y(t) = K(t) + \int\_0^T H(t, s)\sigma(s)d\_q s\_\prime$$

where

$$K(t) = \frac{(1+T-t)}{(2+T)} \int\_0^T \rho\_1(s)ds + \frac{(1+t)}{(2+T)} \int\_0^T \rho\_2(s)ds\,\rho$$

and

$$H(t,s) = \begin{cases} \frac{(t-qs)^{(a-1)}}{\Gamma\_q(a)} - \frac{(1+t)(T-qs)^{(a-1)}}{(2+T)\Gamma\_q(a)} - \frac{(1+t)(T-qs)^{(a-2)}}{(2+T)\Gamma\_q(a-1)}, & 0 \le s < t, \\\\ -\frac{(1+t)(T-qs)^{(a-1)}}{(2+T)\Gamma\_q(a)} - \frac{(1+t)(T-qs)^{(a-2)}}{(2+T)\Gamma\_q(a-1)}, & t \le s \le T, \end{cases}$$

which is what we wanted to show.

We now prove an existence result for the problem (1)–(3) by applying Mönch's fixed point theorem (Theorem 1 above).

Let

$$H^\* = \sup\_{(t,s)\in I\times J} |H(t,s)|.$$

**Theorem 2.** *Assume that the following conditions hold.*

*(P1) The functions f , g, h* : *J* × *E* → *E satisfy Carathéodory conditions.*

*(P2) There exists pf , pg, ph* <sup>∈</sup> *<sup>L</sup>*∞(*J*, <sup>R</sup>+) *such that*

 *f*(*t*, *y*) ≤ *pf*(*t*) *y* , *f or a*.*e*. *t* ∈ *J and all y* ∈ *E*,


*(P3) For almost all t* ∈ *J and each bounded set B* ⊂ *E, we have*

$$
\mu(f(t, B)) \le p\_f(t)\mu(B), \text{ for a.e.} \ t \in \mathcal{J},
$$

$$
\mu(\mathcal{g}(t, B)) \le p\_{\mathcal{S}}(t)\mu(B), \text{ for a.e.} \ t \in \mathcal{J},
$$

$$
\mu(h(t, B)) \le p\_h(t)\mu(B), \text{ for a.e.} \ t \in \mathcal{J}.
$$

*Then, the BVP (1)–(3) has at least one solution in C*2(*J*, *E*)*, provided*

$$\frac{T(1+T)}{\left(2+T\right)}\left(||p\_{\mathcal{S}}||\_{L^{\infty}} + ||p\_{h}||\_{L^{\infty}}\right) + H^{\*}T||p\_{f}||\_{L^{\infty}} < 1.\tag{15}$$

**Proof.** In order to transform problem (1)–(3) into a fixed point type problem, consider the operator

$$N: \mathcal{C}^2(\mathcal{J}, \mathcal{E}) \longrightarrow \mathcal{C}^2(\mathcal{J}, \mathcal{E})$$

defined by

$$(Ny)(t) \quad = \quad K(t) + \int\_0^T H(t,s)f(s,y(s))d\_\eta s,\tag{16}$$

where

$$K(t) = \frac{(1+T-t)}{(2+T)} \int\_0^T g(s, y(s))ds + \frac{(1+t)}{(2+T)} \int\_0^T h(s, y(s))ds,$$

and H(t,s) is given by (9). It is easy to see that the fixed points of *N* are solutions of (1)–(3). Let *R* > 0 and consider

$$D\_R = \{ y \in \mathbb{C}^2(f, E) \; : \; ||y||\_{\infty} \le R \}. \tag{17}$$

Clearly, *DR* is a closed, bounded, and convex subset of *C*2(*J*, *E*). We show that *N* satisfies the hypotheses of Mönch's fixed point theorem. We give the proof in three steps.

**Step 1:** *<sup>N</sup> is continuous.* Let {*yn*}*n*∈<sup>N</sup> be a sequence with *yn* <sup>→</sup> *<sup>y</sup>* in *<sup>C</sup>*2(*J*, *<sup>E</sup>*). For each *t* ∈ *J*, we have

$$\begin{aligned} |(Ny\_n)(t) - (Ny)(t)| &\leq \quad \frac{(1+T-t)}{(2+T)} \int\_0^T |g(s, y\_n(s)) - g(s, y(s))| ds \\ &+ \frac{(1+t)}{(2+T)} \int\_0^T |h(s, y\_n(s)) - h(s, y(s))| ds \\ &+ \int\_0^T |H(t, s)| |f(s, y\_n(s)) - f(s, y(s))| d\_\theta s. \end{aligned}$$

Hence,

$$\begin{aligned} \|N(y\_n) - N(y)\| &\le \frac{T(1+T)}{(2+T)} \|g(s, y\_n(s)) - g(s, y(s))\| \\ &+ \frac{T(1+T)}{(2+T)} \|h(s, y\_n(s)) - h(s, y(s))\| \\ &+ H^\*T \|f(s, y\_n(s)) - f(s, y(s))\|. \end{aligned}$$

Let *ρ* > 0 be such that

 *yn* <sup>∞</sup> ≤ *ρ*, *y* <sup>∞</sup> ≤ *ρ*.

By (P2), we have

$$||f(\mathbf{s}, y\_n(\mathbf{s})) - f(\mathbf{s}, y(\mathbf{s}))|| \le 2\rho p\_f(\mathbf{s}) := \sigma\_f(\mathbf{s})\_\prime$$

$$\begin{aligned} ||\varrho(s, y\_\eta(s)) - \varrho(s, y(s))|| &\le 2\rho p\_\mathcal{S}(s) := \sigma\_\mathcal{S}(s), \\ ||h(s, y\_\eta(s)) - h(s, y(s))|| &\le 2\rho p\_\mathcal{h}(s) := \sigma\_\mathcal{h}(s). \end{aligned}$$

and *<sup>σ</sup>f*(*s*), *<sup>σ</sup>g*(*s*), *<sup>σ</sup>h*(*s*) <sup>∈</sup> *<sup>L</sup>*1(*J*, <sup>R</sup>+). Since the functions *<sup>f</sup>* , *<sup>g</sup>*, and *<sup>h</sup>* satisfy Carathéodory conditions, the Lebesgue-dominated convergence theorem implies that

$$||N(y\_n) - N(y)||\_\infty \to 0 \text{ as } n \to \infty.$$

Consequently, *N* is continuous on *C*2(*J*, *E*).

**Step 2:** *N maps DR into itself.* Now, for any *y* ∈ *DR*, (P2) and (15) imply that for each *t* ∈ *J*,

$$\begin{split} \|(Ny)(t)\| &\leq \frac{(1+T-t)}{(2+T)} \int\_{0}^{T} \|g(s,y(s))\| \|ds + \frac{(1+t)}{(2+T)} \int\_{0}^{T} \|h(s,y(s))\| \|ds\\ &\quad + \int\_{0}^{T} |H(t,s)| \| \|f(s,y(s))\| \|d\_{q}s,\\ \leq & \frac{(1+T-t)}{(2+T)} \int\_{0}^{T} p\_{\mathcal{S}}(s) \|y\| ds + \frac{(1+t)}{(2+T)} \int\_{0}^{T} p\_{h}(s) \|y\| ds\\ &\quad + \int\_{0}^{T} |H(t,s)| p\_{f}(s) \|y\| \|d\_{q}s,\\ \leq & R\left(\frac{T(1+T)}{(2+T)} \|p\_{\mathcal{S}}\|\_{L^{\infty}} + \frac{T(1+T)}{(2+T)} \|p\_{h}\|\_{L^{\infty}} + H^{\*}T \|p\_{f}\|\_{L^{\infty}}\right),\\ \leq R. \end{split}$$

**Step 3:** *N*(*DR*) *is bounded and equicontinuous.* In view of Step 2, it is clear that *N*(*DR*) is bounded. To show the equicontinuity of *N*(*DR*), let *t*1, *t*<sup>2</sup> ∈ *J*, *t*<sup>1</sup> < *t*2, and *y* ∈ *DR*. Then,

$$\begin{split} \|(Ny)(t\_2) - (Ny)(t\_1)\| &= \left\| \frac{(t\_1 - t\_2)}{(2 + T)} \int\_0^T g(s, y(s)) ds + \frac{(t\_2 - t\_1)}{(2 + T)} \int\_0^T h(s, y(s)) ds \right\| \\ &\quad + \int\_0^T (H(t\_2, s) - H(t\_1, s)) f(s, y(s)) d\_q s \Big\|\_{\ell} \\ &\leq \frac{(t\_1 - t\_2)}{(2 + T)} \int\_0^T \|g(s, y(s))\| ds + \frac{(t\_2 - t\_1)}{(2 + T)} \int\_0^T \|h(s, y(s))\| \, ds \\ &\quad + \int\_0^T |H(t\_2, s) - H(t\_1, s)| \|f(s, y(s))\| \|d\_q s. \end{split}$$

By (P2), we have

$$\begin{split} \|(Ny)(t\_2) - (Ny)(t\_1)\| &\leq \frac{(t\_1 - t\_2)}{(2 + T)} \int\_0^T p\_\mathcal{S}(s) \|y\| ds + \frac{(t\_2 - t\_1)}{(2 + T)} \int\_0^T p\_h(s) \|y\| \, ds \\ &\quad + \int\_0^T |H(t\_2, s) - H(t\_1, s)| p\_f(s) \|y\| \|d\_q s\_\prime \\ &\leq RT \frac{(t\_1 - t\_2)}{(2 + T)} \|p\_\mathcal{S}\|\_{L^\infty} + RT \frac{(t\_2 - t\_1)}{(2 + T)} \|p\_h\|\_{L^\infty} \\ &\quad + R \|p\_f\|\_{L^\infty} \int\_0^T |H(t\_2, s) - H(t\_1, s)| \, d\_q s. \end{split}$$

As *t*<sup>1</sup> → *t*2, the right-hand side of the above inequality tends to zero, which shows the equicontinuity of *N*(*DR*).

Now, let *V* ⊂ *DR* be such that *V* ⊂ *con*(*N*(*V*) ∪ {0}). Since *V* is bounded and equicontinuous, the function *v* → *v*(*t*) = *μ*(*V*(*t*)) is continuous on *J*. Moreover, (P3), Lemma 4, and properties of the measure *μ* imply that for each *t* ∈ *J*,

$$\begin{split} v(t) &\leq \mu(N(V)(t) \cup \{0\}), \\ &\leq \mu(N(V)(t)), \\ &\leq \frac{(1+T-t)}{(2+T)} \int\_{0}^{T} p\_{\mathcal{S}}(s)\mu(V(s))ds + \frac{(1+t)}{(2+T)} \int\_{0}^{T} p\_{\mathcal{h}}(s)\mu(V(s))ds \\ &\quad + \int\_{0}^{T} |H(t,s)|p\_{f}(s)\mu(V(s))d\_{q}s, \\ &\leq ||v||\_{\infty} \left[\frac{T(1+T)}{(2+T)} \left(||p\_{\mathcal{S}}||\_{L^{\infty}} + ||p\_{h}||\_{L^{\infty}}\right) + H^{\*}T||p\_{f}||\_{L^{\infty}}\right]. \end{split}$$

This means that

$$||v||\_{\infty} \left(1 - \left[\frac{T(1+T)}{(2+T)} \left(||p\_{\mathcal{S}}||\_{L^{\infty}} + ||p\_h||\_{L^{\infty}}\right) + H^\*T||p\_f||\_{L^{\infty}}\right]\right) \le 0.$$

From (15), we see that *v* <sup>∞</sup> = 0, so *v*(*t*) = 0 for *t* ∈ *J*, and hence, *V*(*t*) is relatively compact in *E*. The Ascoli–Arzelà theorem yields that *V* is relatively compact in *DR*. Applying Theorem 1, we see that *N* has a fixed point that in turn is a solution of (1)–(3).

#### **4. Example**

Let

$$E = l^1 = \{ (y\_1, y\_2, \dots, y\_n, \dots) \; : \; \sum\_{n=1}^{\infty} y\_n < \infty \} . $$

be our Banach space with the norm

$$\|y\|\_{E} = \sum\_{n=1}^{\infty} |y\_n|.$$

Consider the boundary value problem for fractional <sup>1</sup> <sup>4</sup> -difference equations given by

$$(\prescript{C}{}{D}\_{\frac{1}{4}}^{\frac{3}{2}}y)(t) = \frac{1}{(e^t + 5)} y\_n(t), \text{ for a.e.} \ t \in \mathcal{J} = [0, 1], \ 1 < a \le 2,\tag{18}$$

$$y(0) - y'(0) = \int\_0^1 \frac{s^3 - 1}{9} y\_n(s) ds \,\tag{19}$$

$$y(1) + y'(1) = \int\_0^1 \frac{s^3 + 1}{6} y\_n(s) ds. \tag{20}$$

Here, *α* = <sup>3</sup> <sup>2</sup> , *<sup>q</sup>* <sup>=</sup> <sup>1</sup> <sup>4</sup> , *T* = 1, and

$$f\_n(t, y) = \frac{1}{e^t + 5} y\_{n\prime} \ (t, y) \in \mathcal{J} \times \mathcal{E}\_{\prime}$$

$$g\_n(t, y) = \frac{t^3 - 1}{9} y\_{n\prime} \ (t, y) \in \mathcal{J} \times \mathcal{E}\_{\prime}$$

and

$$h\_n(t, y) = \frac{t^3 + 1}{6} y\_{n\nu} \ (t, y) \in \mathfrak{J} \times \mathbb{Z}\_{\nu}$$

where

$$y = (y\_{1\prime}y\_{2\prime}\ldots y\_{m\prime}\ldots \ldots)\_{\prime}$$

$$f = (f\_{1\prime}f\_{2\prime}\ldots f\_{m\prime}\ldots \ldots)\_{\prime}$$

$$\mathbf{g} = (\mathbf{g}\_{1\prime}\mathbf{g}\_{2\prime}\dots\mathbf{g}\_{n\prime}\dots)\_{\prime}$$

and

$$h = (h\_1, h\_2, \dots, h\_{n\nu} \dots).$$

Clearly, conditions (P1) and (P2) hold with

$$p\_f(t) = \frac{1}{e^t + 5'} \quad p\_\mathcal{S}(t) = \frac{t^3}{9'} \quad p\_h(t) = \frac{t^3}{6}.$$

From (9), we have

$$H^\* = \sup\_{(t,s)\in I\times J} |H(t,s)| = \frac{5}{3\Gamma\_{\frac{1}{4}}(\frac{3}{2})} + \frac{2}{3\Gamma\_{\frac{1}{4}}(\frac{1}{2})}.$$

To see that condition (15) is satisfied with *T* = 1, notice that

$$\begin{aligned} \frac{T(1+T)}{(2+T)} \left( \|p\_{\mathcal{S}}\|\_{L^{\infty}} + \|p\_h\|\_{L^{\infty}} \right) + H^\* T \|p\_f\|\_{L^{\infty}} \\ &= \frac{2}{3} \left( \frac{1}{9} + \frac{1}{6} \right) + \left( \frac{5}{3\Gamma\_{\frac{1}{4}}(\frac{3}{2})} + \frac{2}{3\Gamma\_{\frac{1}{4}}(\frac{1}{2})} \right) \frac{1}{6} \simeq 0.5564 < 1.5 \end{aligned}$$

Then, by Theorem 2, the problem (18)–(20) has a solution on [0, 1].

#### **5. Discussion**

In this work, we proved the existence of solutions to a fractional *q*-difference equation with nonlinear integral type boundary conditions in Banach spaces using a method involving the Kuratowski measure of noncompactness and Mönch's fixed point theorem. An example was presented to illustrate the effectiveness of the results.

An interesting direction for future research of course would be to consider fractional *q*-difference equations of order 0 < *α* ≤ 1 and orders greater than the 1 < *α* ≤ 2 considered here. Another direction would be to consider Riemann–Stieltjes integral-type boundary conditions. Adding impulsive effects to the problem would expand the ares of possible applications as well.

**Author Contributions:** Conceptualization, N.A., J.R.G. and S.H.; methodology, N.A., J.R.G. and S.H.; formal analysis, N.A., J.R.G. and S.H.; investigation, N.A., J.R.G. and S.H.; writing—original draft preparation, N.A. and S.H.; writing—review and editing, J.R.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Solvability Criterion for Fractional** *q***-Integro-Difference System with Riemann-Stieltjes Integrals Conditions**

**Changlong Yu 1,2,\* , Si Wang <sup>2</sup> , Jufang Wang <sup>2</sup> and Jing Li 1,\***

	- **\*** Correspondence: yuchanglong@emails.bjut.edu.cn (C.Y.); leejing@bjut.edu.cn (J.L.)

**Abstract:** Due to the great application potential of fractional *q*-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional *q*-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional *q*-integro-difference system with the nonlocal boundary conditions involving diverse fractional *q*-derivatives and Riemann-Stieltjes *q*-integrals. We acquire the existence results of solutions for the systems by applying Schauder fixed point theorem, Krasnoselskii's fixed point theorem, Schaefer's fixed point theorem and nonlinear alternative for single-valued maps, and a uniqueness result is obtained through the Banach contraction mapping principle. Finally, we give some examples to illustrate the main results.

**Keywords:** *q*-calculus; fractional *q*-integro-difference system; solvability; Riemann-Stieltjes *q*-integrals; fixed point theorems

#### **1. Introduction**

In the early twentieth century, Jackson [1] proposed a new mathematical direction of *q*-calculus, and it plays an indispensable role in the fields of nuclear, conformal quantum mechanics and dynamics. In the 1960s, Agarwal [2] and Al-Salam [3] put forward a novel concept of fractional *q*-calculus, its relevant application and development can be seen in the literature [4–6]. Compared with classical *q*-calculus, fractional *q*-calculus can more accurately describe some phenomena in nature, and many practical problems can be abstracted into fractional *q*-difference equations or a system of fractional *q*-difference equations by mathematical modeling. In recent years, abundant theoretical achievements have been made in the research of boundary value problems (BVPs) for fractional *q*difference equations, according to the literature [7–16] and the references therein.

Riemann-Stieltjes integral is a generalization of Riemann integral. As well as we known, the classical Riemann-Stieltjes integral can be widely applied in several areas of analysis, such as probability theory, stochastic processes, physics, econometrics, biometrics and informetrics and so on. BVPs with Riemann-Stieltjes integral boundary condition (BC) have been considered as both multi-point and integral type BCs are treated in a single framework. In recent years, some interesting results about the existence of solutions for nonlinear fractional differential equations with the Riemann-Stieltjes integral BC have been researched, see [17,18] and the references therein.

Nowadays, the system of nonlinear fractional differential equations has important applications in engineering, economy and other fields. This is mainly because the effect of using fractional calculus to solve problems is more practical and efficient than that of classical calculus. Over the years, the BVPs for a system of fractional differential equations have developed rapidly, and numerous mature conclusions have been obtained, which can be referred to the literature [19–25].

**Citation:** Yu, C.; Wang, S.; Wang, J.; Li, J. Solvability Criterion for Fractional *q*-Integro-Difference System with Riemann-Stieltjes Integrals Conditions. *Fractal Fract.* **2022**, *6*, 554. https://doi.org/ 10.3390/fractalfract6100554

Academic Editor: Rodica Luca

Received: 23 August 2022 Accepted: 27 September 2022 Published: 29 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In [24], Tudorache, A. and Luca, R. applied the Guo-Krasnoselskii fixed point theorem to study the existence of solutions for a system of fractional differential equations with *p*-Laplacian operators

$$\begin{cases} \begin{aligned} &D\_{0^{+}}^{\mathfrak{a}\_{1}}(\boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_{1}}(\boldsymbol{D}\_{0\_{+}}^{\mathcal{S}\_{1}}\boldsymbol{x}(t))) + \lambda f(t, \boldsymbol{x}(t), \boldsymbol{y}(t)) = \boldsymbol{0}, \qquad t \in (0, 1), \\ &D\_{0^{+}}^{\mathfrak{a}\_{2}}(\boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_{2}}(\boldsymbol{D}\_{0\_{+}}^{\mathcal{S}\_{2}}\boldsymbol{y}(t))) + \boldsymbol{\mu}g(t, \boldsymbol{x}(t), \boldsymbol{y}(t)) = \boldsymbol{0}, \qquad t \in (0, 1). \end{aligned} \end{cases}$$

with the nonlocal BCs

$$\begin{cases} \varkappa^{(j)}(0) = 0, \; j = 0, \dots, n - 2; \; D\_{0^{+}}^{\beta\_{1}}\varkappa(0) = 0, \\\ D\_{0^{+}}^{\gamma\_{0}}\varkappa(1) = \sum\_{i=1}^{p} \int\_{0}^{1} D\_{0^{+}}^{\gamma\_{i}}\varkappa(t) dH\_{i}t, \\\ \varkappa^{(j)}(0) = 0, \; j = 0, \dots, m - 2; \; D\_{0^{+}}^{\beta\_{2}}\varkappa(0) = 0, \\\ D\_{0^{+}}^{\delta\_{0}}\varkappa(1) = \sum\_{i=1}^{q} \int\_{0}^{1} D\_{0^{+}}^{\delta\_{i}}\varkappa(t) dK\_{i}t. \end{cases}$$

In [25], Luca, R. considered the existence of solutions of the nonlinear system of fractional differential equations by using a variety of fixed point theorems

$$\begin{cases} D\_{0^{+}}^{a}\mathbf{x}(t) + f(t, \mathbf{x}(t), \mathbf{y}(t), \operatorname{I}\_{0^{+}}^{\theta\_{1}}\mathbf{x}(t), \operatorname{I}\_{0^{+}}^{\sigma\_{1}}\mathbf{y}(t)) = \mathbf{0}, & t \in (0, 1), \\\ D\_{0^{+}}^{\beta}\mathbf{y}(t) + \operatorname{g}(t, \mathbf{x}(t), \operatorname{y}(t), \operatorname{I}\_{0^{+}}^{\theta\_{2}}\mathbf{x}(t), \operatorname{I}\_{0^{+}}^{\sigma\_{2}}\mathbf{y}(t)) = \mathbf{0}, & t \in (0, 1), \end{cases}$$

with the nonlocal BCs

$$\begin{cases} \begin{array}{c} \mathfrak{x}(0) = \mathfrak{x}'(0) = \dots = \mathfrak{x}^{(n-2)}(0) = 0, \ D\_{0^{+}}^{\gamma\_{0}}\mathfrak{x}(1) = \sum\_{i=1}^{p} \int\_{0}^{1} D\_{0^{+}}^{\gamma\_{i}}\mathfrak{y}(t) dH\_{i}(t), \\\ y(0) = y'(0) = \dots = y^{(m-2)}(0) = 0, \ D\_{0^{+}}^{\delta\_{0}}\mathfrak{y}(1) = \sum\_{i=1}^{q} \int\_{0}^{1} D\_{0^{+}}^{\delta\_{i}}\mathfrak{x}(t) dK\_{i}(t). \end{array} \end{cases}$$

Despite quite a number of contributions dealing with the solvability for the system of classical fractional difference equations. However, as the generalization of the above system, limited work has been done in the nonlinear system of fractional *q*-difference equations. In particular, there is little research on the existence and uniqueness of solutions for the system of fractional *q*-difference equations with Riemann-Stieltjes integral BC. To fill this gap, we investigate the system of nonlinear fractional *q*-difference equations

$$\begin{cases} \left(D\_q^a u\right)(t) + P(t, u(t), v(t), I\_q^{\omega\_1} u(t), I\_q^{\delta\_1} v(t)) = 0, \\\left(D\_q^\delta v\right)(t) + Q(t, u(t), v(t), I\_q^{\omega\_2} u(t), I\_q^{\delta\_2} v(t)) = 0, \end{cases} \tag{1}$$

with the nonlocal BCs

$$\begin{cases} \boldsymbol{u}(0) = D\_{\boldsymbol{q}} \boldsymbol{u}(0) = \dots = D\_{\boldsymbol{q}}^{\boldsymbol{n}-2} \boldsymbol{u}(0) = 0, \ D\_{\boldsymbol{q}}^{\boldsymbol{\zeta}\_{0}} \boldsymbol{u}(1) = \int\_{0}^{1} D\_{\boldsymbol{q}}^{\boldsymbol{\zeta}} \boldsymbol{v}(t) d\_{\boldsymbol{q}} \boldsymbol{H}(t), \\\ v(0) = D\_{\boldsymbol{q}} v(0) = \dots = D\_{\boldsymbol{q}}^{\boldsymbol{m}-2} v(0) = 0, \ D\_{\boldsymbol{q}}^{\boldsymbol{\zeta}\_{0}} v(1) = \int\_{0}^{1} D\_{\boldsymbol{q}}^{\boldsymbol{\zeta}} \boldsymbol{u}(t) d\_{\boldsymbol{q}} \boldsymbol{K}(t), \end{cases} \tag{2}$$

where *<sup>t</sup>* <sup>∈</sup> (0, 1), 0 <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; 1, *<sup>α</sup>* <sup>∈</sup> (*<sup>n</sup>* <sup>−</sup> 1, *<sup>n</sup>*], *<sup>β</sup>* <sup>∈</sup> (*<sup>z</sup>* <sup>−</sup> 1, *<sup>z</sup>*], *<sup>n</sup>*, *<sup>z</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>n</sup>* <sup>≥</sup> 2 and *<sup>z</sup>* <sup>≥</sup> 2, *<sup>ω</sup>*1, *<sup>ω</sup>*2, *<sup>δ</sup>*1, *<sup>δ</sup>*<sup>2</sup> <sup>&</sup>gt; 0, 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>&</sup>lt; *<sup>β</sup>* <sup>−</sup>1, 0 <sup>≤</sup> *<sup>ξ</sup>* <sup>&</sup>lt; *<sup>α</sup>* <sup>−</sup>1, *<sup>ζ</sup>*<sup>0</sup> <sup>∈</sup> [0, *<sup>α</sup>* <sup>−</sup>1), *<sup>ξ</sup>*<sup>0</sup> <sup>∈</sup> [0, *<sup>β</sup>* <sup>−</sup>1), *<sup>D</sup><sup>i</sup> <sup>q</sup>* denotes the Riemann-Liouville *q*-derivative of order *i* (*i* = *α*, *β*, *ζ*0, *ζ*, *ξ*0, *ξ*), *I <sup>q</sup>* is the Riemann-Liouville *q*-integral of order ( = *ω*1, *ω*2, *δ*1, *δ*2), *P* and *Q* are nonlinear functions. The BCs include Riemann-Stieltjes integrals, where *H*(*t*), *K*(*t*) are the bounded variation functions. In the case where *H*(*t*) = *K*(*t*) = *t* , the Riemann–Stieltjes integrals in (2) reduce to the classical *q*-integral.

The present paper is bulit up as follows. The second part offers the necessary definitions, lemmas and theorems needed in the following. The third part obtains the important conclusions by applying various fixed point theorems, including nine theorems or corollaries. In the final part, four examples are provided to verify our main results.

#### **2. Preliminaries**

In this section, we present some definitions, lemmas and theorems.

**Definition 1** ([11])**.** *Let β* ≥ 0 *and f be a function defined on* [0, 1]*. The fractional q-integral of the Riemann-Liouville type is*

$$(I\_q^\beta f)(s) = \frac{1}{\Gamma\_q(\beta)} \int\_0^s (s - qt)^{(\beta - 1)} f(t) d\_q t, \ \beta > 0, \ s \in [0, 1].$$

*Obviously,* (*I β <sup>q</sup> f*)(*s*)=(*Iq f*)(*s*)*, when β* = 1*.*

**Definition 2** ([11])**.** *The fractional q-derivative of the Riemann-Liouville type of order β* ≥ 0 *is defined by* (*D*<sup>0</sup> *<sup>q</sup> f*)(*s*) = *f*(*s*) *and*

$$(D\_q^\beta f)(s) = (D\_q^l I\_q^{l-\beta} f)(s), \quad \beta > 0, \quad s \in [0, 1]\_\prime$$

*where l is the smallest integer greater than or equal to β.*

**Lemma 1** ([11])**.** *Let α*, *β* ≥ 0 *and f be a function defined on* [0, 1]*. Then, the following formulas hold:*

*1.* (*I β <sup>q</sup> I<sup>α</sup> <sup>q</sup> f*)(*x*)=(*I α*+*β <sup>q</sup> f*)(*x*), *2.* (*D<sup>α</sup> q Iα <sup>q</sup> f*)(*x*) = *f*(*x*).

**Lemma 2** ([11])**.** *Let α* > 0 *and p be a positive integer. Then, the following equality holds:*

$$(I\_q^\alpha D\_q^p f)(\mathbf{x}) = (D\_q^p I\_q^\alpha f)(\mathbf{x}) - \sum\_{k=0}^{p-1} \frac{\mathbf{x}^{\alpha-p+k}}{\Gamma\_q(\alpha+k-p+1)} (D\_q^k f)(0).$$

**Lemma 3.** *If x* ∈ *C*[0, 1]*, then for κ* > 0*, we get*

$$|I\_q^\kappa \mathfrak{x}(t)| \le \frac{\|\|\mathfrak{x}\|\|}{\Gamma\_q(\kappa)}\lambda$$

*where x* <sup>=</sup> sup*t*∈[0,1] <sup>|</sup>*x*(*t*)|*.*

**Proof.** According to Definition 1, this lemma clearly holds.

**Definition 3** ([15])**.** *The function <sup>f</sup>* : *<sup>I</sup>* <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>→</sup> <sup>R</sup> *is called an S-Carathe*´*odory function if and only if*


**Theorem 1** ([26])**.** *(Schauder fixed point theorem) Let D be a bounded closed convex set in E (D does not necessarily have an interior point), and A* : *D* → *D is completely continuous, then A must have a fixed point in D.*

**Theorem 2** ([12])**.** *(Krasnoselskii's fixed point theorem) Let K be a closed convex and nonempty subset of a Banach space X. Let T*, *S be the operators such that*


*Then, there exists z* ∈ *K such that z* = *Tz* + *Sz*.

**Theorem 3** ([16])**.** *(Schaefer's fixed point theorem) Let T be a continuous and compact mapping of a Banach space X into itself, such that the set E* = {*x*|*x* ∈ *X* : *x* = *λTx*, 0 ≤ *λ* ≤ 1} *is bounded. Then T has a fixed point.*

**Theorem 4** ([15])**.** *(Nonlinear alternative for single-valued maps) Let E be a Banach space, let C be a closed and convex subset of E, and let U be an open subset of C and* 0 ∈ *U. Suppose that F* : *U* → *C is a continuous, compact (that is, F*(*U*) *is a relatively compact subset of C) map. Then either*


Throughout this paper, we adopt the following assumptions:

(H1) The functions *<sup>P</sup>*, *<sup>Q</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup>4, <sup>R</sup>) and for *xi*, *yi* <sup>∈</sup> <sup>R</sup>, there exist *Li*(*t*), *li*(*t*) ∈ *C*([0, 1], [0, +∞)), *i* = 1, 2, 3, 4, such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - P(t, \mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3, \mathbf{y}\_4) \right| &\leq \sum\_{i=1}^4 L\_i(t) |\mathbf{x}\_i - \mathbf{y}\_i|, \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - Q(t, \mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3, \mathbf{y}\_4) \right| &\leq \sum\_{i=1}^4 l\_i(t) |\mathbf{x}\_i - \mathbf{y}\_i|. \end{aligned}$$

(H <sup>1</sup>) The functions *<sup>P</sup>*, *<sup>Q</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup>4, <sup>R</sup>) and for *xi*, *yi* <sup>∈</sup> <sup>R</sup>, there exist real constants *Li*, *li* > 0, *i* = 1, 2, 3, 4, such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - P(t, \mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3, \mathbf{y}\_4) \right| &\leq \sum\_{i=1}^4 L\_i |\mathbf{x}\_i - \mathbf{y}\_i| . \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - Q(t, \mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3, \mathbf{y}\_4) \right| &\leq \sum\_{i=1}^4 l\_i |\mathbf{x}\_i - \mathbf{y}\_i|. \end{aligned}$$

(H <sup>1</sup> ) The functions *<sup>P</sup>*, *<sup>Q</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup>4, <sup>R</sup>) and for *xi*, *yi* <sup>∈</sup> <sup>R</sup>, there exist real functions *<sup>ρ</sup>i*(*t*), *i*(*t*) <sup>∈</sup> *<sup>C</sup>*([0, 1], <sup>R</sup>+), *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - P(t, \mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3, \mathbf{y}\_4) \right| &\leq \sum\_{i=1}^4 \rho\_i(t) |\mathbf{x}\_i - \mathbf{y}\_i|, \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - Q(t, \mathbf{y}\_1, \mathbf{y}\_2, \mathbf{y}\_3, \mathbf{y}\_4) \right| &\leq \sum\_{i=1}^4 \varrho\_i(t) |\mathbf{x}\_i - \mathbf{y}\_i|. \end{aligned}$$

(H2) The functions *<sup>P</sup>*, *<sup>Q</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup>4, <sup>R</sup>), and for *xi* <sup>∈</sup> <sup>R</sup>, there exist functions *ci*(*t*), *di*(*t*) <sup>∈</sup> *<sup>C</sup>*([0, 1], <sup>R</sup>+), and *hi*, *mi* <sup>∈</sup> (0, 1), *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, such that

$$\begin{aligned} |P(t, \mathbf{x}\_1, \mathbf{x}\_{2\prime}, \mathbf{x}\_{3\prime}, \mathbf{x}\_4)| &\leq c\_0(t) + \sum\_{i=1}^4 c\_i(t) |\mathbf{x}\_i|^{h\_i} \\ |Q(t, \mathbf{x}\_1, \mathbf{x}\_{2\prime}, \mathbf{x}\_{3\prime}, \mathbf{x}\_4)| &\leq d\_0(t) + \sum\_{i=1}^4 d\_i(t) |\mathbf{x}\_i|^{m\_i} \cdot \end{aligned}$$

(H3) The functions *<sup>P</sup>*, *<sup>Q</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup>4, <sup>R</sup>), and for *xi* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, there exist functions *<sup>σ</sup>*1(*t*), *<sup>σ</sup>*2(*t*) <sup>∈</sup> *<sup>C</sup>*([0, 1], <sup>R</sup>+) such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sigma\_1(t), \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sigma\_2(t). \end{aligned}$$

(H4) The functions *<sup>P</sup>*, *<sup>Q</sup>* : [0, 1] <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>→</sup> <sup>R</sup> and for a.e. *<sup>t</sup>* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup>, there exist *<sup>r</sup>*1(*t*),*r*2(*t*), *Li*(*t*), *li*(*t*) <sup>∈</sup> *<sup>C</sup>*([0, 1], <sup>R</sup>+), *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sum\_{i=1}^4 L\_i(t) |\mathbf{x}\_i| + r\_1(t), \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sum\_{i=1}^4 l\_i(t) |\mathbf{x}\_i| + r\_2(t). \end{aligned}$$

(H <sup>4</sup>) The functions *<sup>P</sup>*, *<sup>Q</sup>* : [0, 1] <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>→</sup> <sup>R</sup> and for a.e. *<sup>t</sup>* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup>, there exist non-negative real numbers *Li*, *li* (*i* = 1, 2, 3, 4), and *r*1, *r*2, where at least one of *r*<sup>1</sup> and *r*<sup>2</sup> is positive, such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sum\_{i=1}^4 L\_i |\mathbf{x}\_i| + r\_1, \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sum\_{i=1}^4 l\_i |\mathbf{x}\_i| + r\_2. \end{aligned}$$

(H5) The functions *<sup>P</sup>*, *<sup>Q</sup>* : [0, 1] <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>→</sup> <sup>R</sup> and for a.e. *<sup>t</sup>* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup>, there exist functions *pi*(*t*), *qi*(*t*) <sup>∈</sup> *<sup>C</sup>*([0, 1], <sup>R</sup>+), where *pi*(*t*), *qi*(*t*) have at least one non-zero function, and there exist nondecreasing functions *<sup>ϕ</sup>i*, *<sup>η</sup><sup>i</sup>* <sup>∈</sup> *<sup>C</sup>*([0, <sup>∞</sup>), <sup>R</sup>+), *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, such that

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sum\_{i=1}^4 p\_i(t) q\_i(|\mathbf{x}\_i|) + p\_0(t), \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \sum\_{i=1}^4 q\_i(t) \eta\_i(|\mathbf{x}\_i|) + q\_0(t). \end{aligned}$$

For convenience, we denote

*C*<sup>1</sup> =1 + 1 Γ*q*(*ω*1) , *C*<sup>2</sup> = 1 + 1 Γ*q*(*δ*1) , *C*<sup>3</sup> = max{*C*1, *C*2}, *C*<sup>4</sup> =1 + 1 Γ*q*(*ω*2) , *C*<sup>5</sup> = 1 + 1 Γ*q*(*δ*2) , *C*<sup>6</sup> = max{*C*4, *C*5}, *<sup>C</sup>*<sup>7</sup> <sup>=</sup> <sup>1</sup> <sup>Γ</sup>*q*(*α*) <sup>+</sup> Γ*q*(*β*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + Γ*q*(*β*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) · <sup>1</sup> 0 *dqK*(*s*) , *<sup>C</sup>*<sup>8</sup> <sup>=</sup> <sup>1</sup> <sup>Γ</sup>*q*(*β*) <sup>+</sup> Γ*q*(*α*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + Γ*q*(*α*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *s <sup>α</sup>*−*ξ*−1*dqK*(*s*) · <sup>1</sup> 0 *dqH*(*s*) , *<sup>C</sup>*<sup>9</sup> <sup>=</sup> <sup>Γ</sup>*q*(*α*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*α* − *ξ*) <sup>1</sup> 0 *s <sup>α</sup>*−*ξ*−1*dqK*(*s*) + Γ*q*(*α*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*α* − *ξ*) <sup>1</sup> 0 *dqK*(*s*) , *<sup>C</sup>*<sup>10</sup> <sup>=</sup> <sup>Γ</sup>*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) , *<sup>C</sup>*<sup>11</sup> <sup>=</sup>*C*<sup>7</sup> <sup>−</sup> <sup>1</sup> Γ*q*(*α*) , *<sup>C</sup>*<sup>12</sup> <sup>=</sup> *<sup>C</sup>*<sup>8</sup> <sup>−</sup> <sup>1</sup> Γ*q*(*β*) . <sup>Ω</sup><sup>1</sup> <sup>=</sup> <sup>Γ</sup>*q*(*β*) Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *<sup>s</sup>β*−*ζ*−1*dqH*(*s*), <sup>Ω</sup><sup>2</sup> <sup>=</sup> <sup>Γ</sup>*q*(*α*) Γ*q*(*α* − *ξ*) <sup>1</sup> 0 *s <sup>α</sup>*−*ξ*−1*dqK*(*s*), <sup>Ω</sup> <sup>=</sup> <sup>Γ</sup>*q*(*α*)Γ*q*(*β*) <sup>Γ</sup>*q*(*<sup>α</sup>* <sup>−</sup> *<sup>ζ</sup>*0)Γ*q*(*<sup>β</sup>* <sup>−</sup> *<sup>ξ</sup>*0) <sup>−</sup> <sup>Ω</sup>1Ω2. (3)

#### **3. Criterion of Uniqueness and Existence**

In this section, we show some existence and uniqueness results for the Systems (1)–(2).

**Lemma 4.** *Let <sup>h</sup>*, *<sup>k</sup>* <sup>∈</sup> *<sup>C</sup>*(0, 1) <sup>∩</sup> *<sup>L</sup>*1(0, 1) *and* <sup>Ω</sup> <sup>=</sup> <sup>0</sup>*, then the system of fractional q-difference equations*

$$\begin{cases} \, \, \_D^a u(t) + h(t) = 0, \, \quad t \in (0, 1), \\\, \_D^\beta v(t) + k(t) = 0, \, \quad t \in (0, 1), \end{cases} \tag{4}$$

*with the coupled BCs (2) has a unique solution* (*u*(*t*), *v*(*t*))*, namely*

*<sup>u</sup>*(*t*) = <sup>−</sup> <sup>1</sup> Γ*q*(*α*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*α*−1) *h*(*s*)*dqs* + *t α*−1 Ω \* Ω<sup>1</sup> Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) *k*(*s*)*dqs* <sup>−</sup> <sup>Ω</sup><sup>1</sup> Γ*q*(*α* − *ξ*) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) *h*(*τ*)*dqτ dqK*(*s*) + Γ*q*(*β*) Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) *h*(*s*)*dqs* <sup>−</sup> <sup>Γ</sup>*q*(*β*) Γ*q*(*β* − *ζ*)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) *k*(*τ*)*dqτ dqH*(*s*) + , *<sup>v</sup>*(*t*) = <sup>−</sup> <sup>1</sup> Γ*q*(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*β*−1) *k*(*s*)*dqs* + *tβ*−<sup>1</sup> Ω \* Ω<sup>2</sup> Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) *h*(*s*)*dqs* <sup>−</sup> <sup>Ω</sup><sup>2</sup> Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) *k*(*τ*)*dqτ dqH*(*s*) + Γ*q*(*α*) Γ*q*(*β* − *ξ*0)Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) *k*(*s*)*dqs* <sup>−</sup> <sup>Γ</sup>*q*(*α*) Γ*q*(*α* − *ξ*)Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) *h*(*τ*)*dqτ dqK*(*s*) + , *t* ∈ [0, 1].

**Proof.** The proof is similar to the Lemma 2.1 in [24].

Let *U* = *C*[0, 1] and *V* = *U* × *U* be the Banach spaces with the norms *u* <sup>=</sup> sup*t*∈[0,1] <sup>|</sup>*u*(*t*)<sup>|</sup> and (*u*, *v*) *V*= *u* + *v* , respectively. Nowdays, we introduce the operator T : *V* → *V*, where T (*x*, *y*)=(T1(*x*, *y*), T2(*x*, *y*)) for (*x*, *y*) ∈ *V*, and T1, T<sup>2</sup> : *V* → *U* are defined by

<sup>T</sup>1(*u*, *<sup>v</sup>*)(*t*) = <sup>−</sup> <sup>1</sup> Γ*q*(*α*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*α*−1) *Fuv*(*s*)*dqs* + Ω1*t α*−1 ΩΓ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) · *Guv*(*s*)*dqs* <sup>−</sup> <sup>Ω</sup>1*<sup>t</sup> α*−1 ΩΓ*q*(*α* − *ξ*) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) *Fuv*(*τ*)*dqτ dqK*(*s*) + Γ*q*(*β*)*t α*−1 ΩΓ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) *Fuv*(*s*)*dqs* <sup>−</sup> <sup>Γ</sup>*q*(*β*)*<sup>t</sup> α*−1 ΩΓ*q*(*β* − *ζ*)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) *Guv*(*τ*)*dqτ dqH*(*s*),

and

<sup>T</sup>2(*u*, *<sup>v</sup>*)(*t*) = <sup>−</sup> <sup>1</sup> Γ*q*(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*β*−1) *Guv*(*s*)*dqs* + Ω2*tβ*−<sup>1</sup> ΩΓ*q*(*α* − *ζ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) · *Fuv*(*s*)*dqs* <sup>−</sup> <sup>Ω</sup>2*tβ*−<sup>1</sup> ΩΓ*q*(*β* − *ζ*) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) *Guv*(*τ*)*dqτ* + *dqH*(*s*) + Γ*q*(*α*)*tβ*−<sup>1</sup> ΩΓ*q*(*β* − *ξ*0)Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) *Guv*(*s*)*dqs* <sup>−</sup> <sup>Γ</sup>*q*(*α*)*tβ*−<sup>1</sup> ΩΓ*q*(*α* − *ξ*)Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) *Fuv*(*τ*)*dqτ dqK*(*s*),

for *t* ∈ [0, 1] and (*u*, *v*) ∈ *V*, where

$$F\_{\mathsf{u}\upsilon}(\mathsf{s}) = P(\mathsf{s},\mathsf{u}(\mathsf{s}),\upsilon(\mathsf{s}), I\_{\mathsf{q}}^{\omega\_1}\mathsf{u}(\mathsf{s}), I\_{\mathsf{q}}^{\delta\_1}\upsilon(\mathsf{s})),\ G\_{\mathsf{u}\upsilon}(\mathsf{s}) = Q(\mathsf{s},\mathsf{u}(\mathsf{s}),\upsilon(\mathsf{s}), I\_{\mathsf{q}}^{\omega\_2}\mathsf{u}(\mathsf{s}), I\_{\mathsf{q}}^{\delta\_2}\upsilon(\mathsf{s})).$$

According to Lemma 4, it is easy to see that (*u*(*t*), *v*(*t*)) is a solution of the Systems (1)–(2) if and only if (*u*(*t*), *v*(*t*)) is a fixed point of operator T .

At first, we prove the existence and uniqueness theorem of the Systems (1)–(2) by Banach contraction mapping principle.

**Theorem 5.** *Suppose that* (H1) *holds. If* Ω = 0*, and*

$$
\Lambda = \Lambda\_1 \mathcal{C}\_3 (\mathcal{C}\_7 + \mathcal{C}\_9) + \Lambda\_2 \mathcal{C}\_6 (\mathcal{C}\_8 + \mathcal{C}\_{10}) < 1,
$$

*where* <sup>Λ</sup><sup>1</sup> = max*t*∈[0,1] ' ∑4 *<sup>i</sup>*=<sup>1</sup> *Li*(*t*) ( , <sup>Λ</sup><sup>2</sup> = max*t*∈[0,1] ' ∑4 *<sup>i</sup>*=<sup>1</sup> *li*(*t*) ( *. Then the Systems (1)–(2) has a unique solution.*

**Proof.** Let *r* > 0 such that

$$r = \frac{\mathbb{C}\_0(\mathbb{C}\_7 + \mathbb{C}\_9) + \mathbb{C}\_0(\mathbb{C}\_8 + \mathbb{C}\_{10})}{1 - \Lambda\_1 \mathbb{C}\_3(\mathbb{C}\_7 + \mathbb{C}\_9) - \Lambda\_2 \mathbb{C}\_6(\mathbb{C}\_8 + \mathbb{C}\_{10})},$$

where *<sup>C</sup>*<sup>0</sup> <sup>=</sup> sup*t*∈[0,1] <sup>|</sup>*P*(*t*, 0, 0, 0, 0)|, *<sup>C</sup>*<sup>0</sup> <sup>=</sup> sup*t*∈[0,1] <sup>|</sup>*Q*(*t*, 0, 0, 0, 0)|.

We divide two steps to prove the theorem.

(i) Our first task is to show that T maps bounded sets into bounded sets in *V*.

Let *Br* = ' (*u*, *v*) ∈ *V*, (*u*, *v*) *<sup>V</sup>*≤ *r* ( be a bounded set in *V* and (*u*, *v*) ∈ *Br*. Then we show that T (*Br*) ⊂ *Br*. By (H1) and Lemma 3, we get

$$\begin{split} \left| F\_{\text{uv}}(t) \right| &\leq \left| P(t, u(t), v(t), I\_{q}^{\omega\_{1}} u(t), I\_{q}^{\delta\_{1}} v(t)) - P(t, 0, 0, 0, 0) \right| + \left| P(t, 0, 0, 0, 0) \right| \\ &\leq \left[ L\_{1}(t) \left| u(t) \right| + L\_{2}(t) \left| v(t) \right| + L\_{3}(t) \left| I\_{q}^{\omega\_{1}} u(t) \right| + L\_{4}(t) \left| I\_{q}^{\delta\_{1}} v(t) \right| \right] + \mathcal{C}\_{0} \\ &\leq \Lambda\_{1} \left[ \left| \left| u \right| \right| + \left| \left| v \right| \right| + \frac{\left\| \left| u \right| \right|}{\Gamma\_{q} (\omega\_{1})} + \frac{\left\| \left| v \right| \right|}{\Gamma\_{q} (\delta\_{1})} \right] + \mathcal{C}\_{0} \\ &= \Lambda\_{1} (\mathcal{C}\_{1} \parallel u \parallel + \mathcal{C}\_{2} \parallel v \parallel) + \mathcal{C}\_{0} \\ &\leq \Lambda\_{1} \mathcal{C}\_{3} \parallel (u, v) \parallel\_{V} + \mathcal{C}\_{0} \leq \Lambda\_{1} \mathcal{C}\_{3} r + \mathcal{C}\_{0}, \end{split}$$

similarly,

$$\left| \mathcal{G}\_{\mu\upsilon}(t) \right| \leq \Lambda\_2 \mathcal{C}\_6 r + \tilde{\mathcal{C}}\_0.$$

According to the expression of operators T<sup>1</sup> and T2, we obtain


thus, we have

$$\|\|\left|\mathcal{T}\_{1}(\mu, v)\right\|\| \le \mathcal{C}\_{7}(\Lambda\_{1}\mathcal{C}\_{3}r + \mathcal{C}\_{0}) + \mathcal{C}\_{10}(\Lambda\_{2}\mathcal{C}\_{6}r + \bar{\mathcal{C}}\_{0}),\tag{5}$$

in like wise,

$$\|\|\ \mathcal{T}\_2(u,v) \parallel\| \le \mathcal{C}\_9(\Lambda\_1 \mathcal{C}\_3 r + \mathcal{C}\_0) + \mathcal{C}\_8(\Lambda\_2 \mathcal{C}\_6 r + \bar{\mathcal{C}}\_0). \tag{6}$$

Using (5) and (6), we obtain that for ∀(*u*, *v*) ∈ *Br*,

$$\begin{aligned} \parallel \mathcal{T}(\boldsymbol{u}, \boldsymbol{v}) \parallel\_{V} &= \parallel \,\,\mathcal{T}\_{1}(\boldsymbol{u}, \boldsymbol{v}) \parallel + \parallel \,\,\mathcal{T}\_{2}(\boldsymbol{u}, \boldsymbol{v}) \parallel \\ &\leq (\Lambda\_{1} \mathcal{C}\_{3} \boldsymbol{r} + \mathcal{C}\_{0})(\mathcal{C}\_{7} + \mathcal{C}\_{9}) + (\Lambda\_{2} \mathcal{C}\_{6} \boldsymbol{r} + \tilde{\mathcal{C}}\_{0})(\mathcal{C}\_{8} + \mathcal{C}\_{10}) = \boldsymbol{r}, \end{aligned}$$

that is T (*Br*) ⊂ *Br*.

(ii) The next step is to prove that operator T is a contraction. For (*ui*, *vi*) ∈ *Br*(*i* = 1, 2), *t* ∈ [0, 1], we get


Since

$$\begin{aligned} \left| F\_{\boldsymbol{u}\_{1}\boldsymbol{v}\_{1}}(\boldsymbol{s}) - F\_{\boldsymbol{u}\_{2}\boldsymbol{v}\_{2}}(\boldsymbol{s}) \right| &\leq \left[ L\_{1}(\boldsymbol{s}) \big| \boldsymbol{u}\_{1}(\boldsymbol{s}) - \boldsymbol{u}\_{2}(\boldsymbol{s}) \big| + L\_{2}(\boldsymbol{s}) \big| \boldsymbol{v}\_{1}(\boldsymbol{s}) - \boldsymbol{v}\_{2}(\boldsymbol{s}) \big| \right. \\ &\left. + L\_{3}(\boldsymbol{s}) \big| \big| I\_{q}^{\omega\_{1}}\boldsymbol{u}\_{1}(\boldsymbol{s}) - I\_{q}^{\omega\_{1}}\boldsymbol{u}\_{2}(\boldsymbol{s}) \big| + L\_{4}(\boldsymbol{s}) \big| I\_{q}^{\delta\_{1}}\boldsymbol{v}\_{1}(\boldsymbol{s}) - I\_{q}^{\delta\_{1}}\boldsymbol{v}\_{2}(\boldsymbol{s}) \big| \right] \\ &\leq \Lambda\_{1}(\mathbb{C}\_{1} \parallel \boldsymbol{u}\_{1} - \boldsymbol{u}\_{2} \parallel + \mathbb{C}\_{2} \parallel \boldsymbol{v}\_{1} - \boldsymbol{v}\_{2} \parallel) \\ &\leq \Lambda\_{1}\mathbb{C}\_{3} \parallel (\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - (\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \parallel\_{V\_{\boldsymbol{V}}} \end{aligned}$$

and

$$\left| \left| G\_{\mathcal{U}\_1 \mathcal{U}\_1}(\mathbf{s}) - G\_{\mathcal{U}\_2 \mathcal{U}\_2}(\mathbf{s}) \right| \right| \leq \Lambda\_2 \mathsf{C}\_6 \left| \left| \left( \mu\_1, v\_1 \right) - \left( \mu\_2, v\_2 \right) \right| \right| \_{V} \cdot \bar{\mu}$$

By (7), we have

 T1(*u*1, *<sup>v</sup>*1)(*t*) − T1(*u*2, *<sup>v</sup>*2)(*t*) ≤ 1 Γ*q*(*α*) <sup>1</sup> 0 *Fu*1*v*<sup>1</sup> (*s*) <sup>−</sup> *Fu*2*v*<sup>2</sup> (*s*) *dqs* <sup>+</sup> Γ*q*(*β*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 *Fu*1*v*<sup>1</sup> (*s*) <sup>−</sup> *Fu*2*v*<sup>2</sup> (*s*) *dqs* + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *Gu*1*v*<sup>1</sup> (*s*) <sup>−</sup> *Gu*2*v*<sup>2</sup> (*s*) *dqs* · <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 \* <sup>1</sup> 0 *Fu*1*v*<sup>1</sup> (*τ*) <sup>−</sup> *Fu*2*v*<sup>2</sup> (*τ*) *dq<sup>τ</sup>* + *dqK*(*s*) · <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) · <sup>1</sup> 0 \* <sup>1</sup> 0 *Gu*1*v*<sup>1</sup> (*τ*) <sup>−</sup> *Gu*2*v*<sup>2</sup> (*τ*) *dq*(*τ*) + *dqH*(*s*) ≤Λ1*C*<sup>3</sup> (*u*1, *v*1) − (*u*2, *v*2) *V* \* 1 <sup>Γ</sup>*q*(*α*) <sup>+</sup> Γ*q*(*β*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + Γ*q*(*β*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + + Λ2*C*<sup>6</sup> (*u*1, *v*1) − (*u*2, *v*2) *V* \* Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) + ,

hence, we deduce

$$\left\| \left| \begin{array}{c} \mathcal{T}\_{1}(\boldsymbol{u}\_{1},\boldsymbol{v}\_{1}) - \mathcal{T}\_{1}(\boldsymbol{u}\_{2},\boldsymbol{v}\_{2}) \; \middle| \; \left| \leq \left( \Lambda\_{1}\mathcal{C}\_{3}\mathcal{C}\_{7} + \Lambda\_{2}\mathcal{C}\_{6}\mathcal{C}\_{10} \right) \parallel \left( \boldsymbol{u}\_{1},\boldsymbol{v}\_{1} \right) - \left( \boldsymbol{u}\_{2},\boldsymbol{v}\_{2} \right) \right\| \; \left| \; \left| \; \boldsymbol{V} \end{array} \right. \tag{8}$$

For the same way, we can obtain

$$\left\| \left| \begin{array}{c} \mathcal{T}\_{2}(\boldsymbol{u}\_{1},\boldsymbol{v}\_{1}) - \mathcal{T}\_{2}(\boldsymbol{u}\_{2},\boldsymbol{v}\_{2}) \right\| \leq \left( \Lambda\_{1}\mathcal{C}\_{3}\mathcal{C}\_{9} + \Lambda\_{2}\mathcal{C}\_{6}\mathcal{C}\_{8} \right) \left\| \begin{array}{c} (\boldsymbol{u}\_{1},\boldsymbol{v}\_{1}) - (\boldsymbol{u}\_{2},\boldsymbol{v}\_{2}) \end{array} \right\| \boldsymbol{v} \right. . \tag{9}$$

From (8) and (9), we have

$$\begin{aligned} & \left\| \begin{array}{l} \mathcal{T}(\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - \mathcal{T}(\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \, \big| \, \big| \, \boldsymbol{V} \\ = & \left\| \begin{array}{l} \mathcal{T}\_{1}(\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - \mathcal{T}\_{1}(\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \, \big| \, \big| + \left\| \begin{array}{l} \mathcal{T}\_{2}(\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - \mathcal{T}\_{2}(\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \, \big| \, \boldsymbol{v} \\ \leq & \left[ \Lambda\_{1} \mathsf{C}\_{3}(\mathsf{C}\_{7} + \mathsf{C}\_{9}) + \Lambda\_{2} \mathsf{C}\_{6}(\mathsf{C}\_{8} + \mathsf{C}\_{10}) \right] \right\| \, \left( \boldsymbol{u}\_{1}, \boldsymbol{v}\_{1} \right) - \left( \boldsymbol{u}\_{2}, \boldsymbol{v}\_{2} \right) \, \big| \, \boldsymbol{v}\_{1} \\ = & \Lambda \, \left\| \, \left( \boldsymbol{u}\_{1}, \boldsymbol{v}\_{1} \right) - \left( \boldsymbol{u}\_{2}, \boldsymbol{v}\_{2} \right) \right\| \, \boldsymbol{v} \end{array} . \end{aligned} $$

Due to Λ < 1, it follows that T (*u*1, *v*1) − T (*u*2, *v*2) *V*< (*u*1, *v*1) − (*u*2, *v*2) *<sup>V</sup>*, so operator T is a contraction. Hence, we obtain that the Systems (1)–(2) has a unique solution (*u*, *v*) ∈ *Br* by using Banach contraction mapping principle. The proof is completed.

**Corollary 1.** *Suppose that* (H <sup>1</sup>) *holds. If* Ω = 0*, and*

$$
\Lambda^\* = \Lambda\_3 \mathbb{C}\_3 (\mathbb{C}\_7 + \mathbb{C}\_9) + \Lambda\_4 \mathbb{C}\_6 (\mathbb{C}\_8 + \mathbb{C}\_{10}) < 1,
$$

*where* Λ<sup>3</sup> = ∑<sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *Li*, <sup>Λ</sup><sup>4</sup> = <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *li. Then the Systems (1)–(2) has a unique solution.*

**Corollary 2.** *Suppose that* (H <sup>1</sup> ) *holds. If* Ω = 0*, and*

<sup>Λ</sup> <sup>=</sup> <sup>Λ</sup>5*C*3(*C*<sup>7</sup> <sup>+</sup> *<sup>C</sup>*9) + <sup>Λ</sup>6*C*6(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10) <sup>&</sup>lt; 1,

*where* <sup>Λ</sup><sup>5</sup> <sup>=</sup> sup*t*∈[0,1] {∑<sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *<sup>ρ</sup>i*(*t*)}, <sup>Λ</sup><sup>6</sup> <sup>=</sup> sup*t*∈[0,1] {∑<sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *i*(*t*)}*. Then the Systems (1)–(2) has a unique solution.*

Next, we apply several kinds of fixed point theorems to achieve the existence results of solutions for the Systems (1)–(2).

**Theorem 6.** *Suppose that* (H2) *and* Ω = 0 *hold. Then the System (1)–(2) has at least one solution.*

**Proof.** Let *BR* = {(*u*, *v*) ∈ *V*, (*u*, *v*) *<sup>V</sup>*≤ *R*}, and we denote

*<sup>R</sup>*<sup>1</sup> <sup>=</sup> max! *c*<sup>0</sup> + *c*<sup>1</sup> (N1)*h*1<sup>+</sup> *c*<sup>2</sup> (N2)*h*2<sup>+</sup> *c*<sup>3</sup> N<sup>1</sup> Γ*q*(*ω*1) *h*<sup>3</sup> + *c*<sup>4</sup> N<sup>2</sup> Γ*q*(*δ*1) *h*<sup>4</sup> " *C*7, ! *d*<sup>0</sup> + *d*<sup>1</sup> (N1)*m*1<sup>+</sup> *d*<sup>2</sup> (N2)*m*<sup>2</sup> + *d*<sup>3</sup> N<sup>1</sup> Γ*q*(*ω*2) *m*<sup>3</sup> + *d*<sup>4</sup> N<sup>2</sup> Γ*q*(*δ*2) *m*<sup>4</sup> " *<sup>C</sup>*10# , *<sup>R</sup>*<sup>2</sup> <sup>=</sup> max! *c*<sup>0</sup> + *c*<sup>1</sup> (N1)*h*1<sup>+</sup> *c*<sup>2</sup> (N2)*h*2<sup>+</sup> *c*<sup>3</sup> N<sup>1</sup> Γ*q*(*ω*1) *h*<sup>3</sup> + *c*<sup>4</sup> N<sup>2</sup> Γ*q*(*δ*1) *h*<sup>4</sup> " *C*9, ! *d*<sup>0</sup> + *d*<sup>1</sup> (N1)*m*1<sup>+</sup> *d*<sup>2</sup> (N2)*m*<sup>2</sup> + *d*<sup>3</sup> N<sup>1</sup> Γ*q*(*ω*2) *m*<sup>3</sup> + *d*<sup>4</sup> N<sup>2</sup> Γ*q*(*δ*2) *m*<sup>4</sup> " *C*8 # , *R* = 2 max' *R*1, *R*<sup>2</sup> ( .

where there exist <sup>N</sup>1, <sup>N</sup><sup>2</sup> <sup>∈</sup> <sup>R</sup> such that <sup>|</sup>*u*(*t*)|≤N1, <sup>|</sup>*v*(*t*)|≤N2.

Firstly, we show that T maps bounded sets into bounded sets in *V*. For (*u*, *v*) ∈ *BR*, we obtain

$$\begin{split} \|\|\mathcal{T}\_{1}(\boldsymbol{u},\boldsymbol{v})\|\| \leq & \Big[ \|\|\boldsymbol{c}\_{0}\|\| + \|\|\boldsymbol{c}\_{1}\|\|\left(\mathcal{N}\_{1}\right)^{h\_{1}} + \|\|\boldsymbol{c}\_{2}\|\|\left(\mathcal{N}\_{2}\right)^{h\_{2}} + \|\|\boldsymbol{c}\_{3}\|\|\left(\frac{\mathcal{N}\_{1}}{\Gamma\_{q}(\omega\_{1})}\right)^{h\_{3}} \\ & + \|\|\boldsymbol{c}\_{4}\|\|\left(\frac{\mathcal{N}\_{2}}{\Gamma\_{q}(\delta\_{1})}\right)^{h\_{4}} \Big] \mathbb{C}\_{7} + \Big[ \|\|\boldsymbol{d}\_{0}\|\| + \|\|\boldsymbol{d}\_{1}\|\|\left(\mathcal{N}\_{1}\right)^{m\_{1}} + \|\|\boldsymbol{d}\_{2}\|\|\left(\mathcal{N}\_{2}\right)^{m\_{2}} \Big] \\ & + \|\|\boldsymbol{d}\_{3}\|\|\left(\frac{\mathcal{N}\_{1}}{\Gamma\_{q}(\omega\_{2})}\right)^{m\_{3}} + \|\|\boldsymbol{d}\_{4}\|\|\left(\frac{\mathcal{N}\_{2}}{\Gamma\_{q}(\delta\_{2})}\right)^{m\_{4}} \Big] \mathbb{C}\_{10} \leq 2\mathcal{R}\_{1}, \end{split}$$

similarly, T2(*u*, *v*) ≤ 2*R*2, then

$$\parallel \mathcal{T}(\mathfrak{u}, \mathfrak{v}) \parallel\_{V} = \parallel \mathcal{T}\_{1}(\mathfrak{u}, \mathfrak{v}) \parallel + \parallel \mathcal{T}\_{2}(\mathfrak{u}, \mathfrak{v}) \parallel \leq \mathbb{R}, \ (\mathfrak{u}, \mathfrak{v}) \in B\_{\mathbb{R} \times \mathbb{R}}$$

as above, we obtain T (*BR*) ⊂ *BR*.

Secondly, we prove that T maps bounded sets into equicontinuous sets of *V*. Let N = max{N1, N2}, for simplicity of presentation, we denote that

$$\begin{split} \Psi\_{\mathcal{N}} &= \sup\_{t \in [0,1]} \left\{ |P(t,\boldsymbol{\mu},\boldsymbol{\upsilon},\boldsymbol{\chi},\boldsymbol{y})| \, |\, |\boldsymbol{u}| \leq \mathcal{N} \, \, |\, \boldsymbol{v}| \leq \mathcal{N} \, \, |\, \boldsymbol{x}| \leq \frac{\mathcal{N}}{\Gamma\_{q}(\omega\_{1})'} \, \, |\boldsymbol{y}| \leq \frac{\mathcal{N}}{\Gamma\_{q}(\delta\_{1})} \right\}, \\ \Theta\_{\mathcal{N}} &= \sup\_{t \in [0,1]} \left\{ |Q(t,\boldsymbol{\mu},\boldsymbol{\upsilon},\boldsymbol{\chi},\boldsymbol{y})| \, |\boldsymbol{u}| \leq \mathcal{N} \, \, |\, \boldsymbol{v}| \leq \mathcal{N} \, \, |\, \boldsymbol{x}| \leq \frac{\mathcal{N}}{\Gamma\_{q}(\omega\_{2})'} \, \, |\boldsymbol{y}| \leq \frac{\mathcal{N}}{\Gamma\_{q}(\delta\_{2})} \right\}. \end{split}$$

then for (*u*, *v*) ∈ *BR* and *t*1, *t*<sup>2</sup> ∈ [0, 1] with *t*<sup>1</sup> < *t*2, we have

$$\begin{split} & \left| \left| \left. T\_{1}(u,v)(t\_{2}) - \left. T\_{1}(u,v)(t\_{1}) \right| \right| \right| \\ & \leq \frac{\mathbb{1}\_{\mathcal{N}}}{\Gamma\_{q}(a+1)} (t\_{2}^{a} - t\_{1}^{a}) + \mathbb{1}\_{\mathcal{N}} (t\_{2}^{a-1} - t\_{1}^{a-1}) \left[ \frac{\Gamma\_{q}(\beta)}{|\Omega| \Gamma\_{q}(a-\xi)\Gamma\_{q}(\beta-\xi)} \right| \int\_{0}^{1} s^{\beta-\zeta-1} d\_{q}H(s) \right| \\ & \quad \cdot \left| \int\_{0}^{1} d\_{q}K(s) \right| + \frac{\Gamma\_{q}(\beta)}{|\Omega| \Gamma\_{q}(a-\xi\_{0})\Gamma\_{q}(\beta-\tilde{\xi}\_{0})} \Big| + \Theta\_{\mathcal{N}} (t\_{2}^{a-1} - t\_{1}^{a-1}) \left[ \frac{\Gamma\_{q}(\beta)}{|\Omega| \Gamma\_{q}(\beta-\tilde{\xi}\_{0})\Gamma\_{q}(\beta-\tilde{\xi})} \right] \\ & \quad \cdot \left| \int\_{0}^{1} s^{\beta-\tilde{\xi}-1} d\_{q}H(s) \right| + \frac{\Gamma\_{q}(\beta)}{|\Omega| \Gamma\_{q}(\beta-\tilde{\xi}\_{0})\Gamma\_{q}(\beta-\tilde{\xi})} \Big| \int\_{0}^{1} d\_{q}H(s) \right| \\ &= \frac{\mathbb{1}\_{\mathcal{N}}}{\Gamma\_{q}(a+1)} (t\_{2}^{a} - t\_{1}^{a}) + (\Psi\_{\mathcal{N}} \mathcal{C}\_{11} + \Theta\_{\mathcal{N}} \mathcal{C}\_{10}) (t\_{2}^{a-1} - t\_{1}^{a-1}). \end{split}$$

The same can be proved that

$$\left| \left| \mathcal{T}\_2(u,v)(t\_2) - \mathcal{T}\_2(u,v)(t\_1) \right| \leq \frac{\Theta\_N}{\Gamma\_q(\beta+1)}(t\_2^{\beta} - t\_1^{\beta}) + (\Psi\_N \mathbb{C}\_9 + \Theta\_N \mathbb{C}\_{12})(t\_2^{\beta-1} - t\_1^{\beta-1}). \right| $$

Hence, we conclude

$$|\mathcal{T}\_1(\boldsymbol{u}, \boldsymbol{v})(t\_2) - \mathcal{T}\_1(\boldsymbol{u}, \boldsymbol{v})(t\_1)| \to 0, \quad |\mathcal{T}\_2(\boldsymbol{u}, \boldsymbol{v})(t\_2) - \mathcal{T}\_2(\boldsymbol{u}, \boldsymbol{v})(t\_1)| \to 0,$$

as *t*<sup>2</sup> → *t*1, (*u*, *v*) ∈ *BR*. Thus, T (*BR*) is equicontinuous. According to the Arzela-Ascoli theorem, it follows that the set T (*BR*) is relatively compact. Therefore, T is compact on *BR*. By Theorem 1, we get that the System (1)–(2) has at least one solution. The proof is completed.

**Theorem 7.** *Suppose that* (H <sup>1</sup>) *and* (H3) *hold. If* Ω = 0*, and*

$$
\overline{\Lambda} = \Lambda\_3 \mathbb{C}\_3 \frac{1}{\Gamma\_q(\alpha)} + \Lambda\_4 \mathbb{C}\_6 \frac{1}{\Gamma\_q(\beta)} < 1.
$$

*Then the System (1)–(2) has at least one solution.*

**Proof.** Take *r*<sup>0</sup> > 0 such that

$$r\_0 \ge (\mathbb{C}\_7 + \mathbb{C}\_9) \parallel \sigma\_1 \parallel + (\mathbb{C}\_8 + \mathbb{C}\_{10}) \parallel \sigma\_2 \parallel \dots$$

Let *Br*<sup>0</sup> = {(*u*, *v*) ∈ *V*, (*u*, *v*) *<sup>V</sup>*≤ *r*0}, and let the operators be X = (X1, X2) : *Br*<sup>0</sup> → *V* and Y = (Y1, Y2) : *Br*<sup>0</sup> → *V*, where X1, X2, Y1, Y<sup>2</sup> : *Br*<sup>0</sup> → *U* are denoted by

<sup>X</sup>1(*u*, *<sup>v</sup>*)(*t*) = <sup>−</sup> <sup>1</sup> Γ*q*(*α*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*α*−1) *Fuv*(*s*)*dqs*, <sup>Y</sup>1(*u*, *<sup>v</sup>*)(*t*) = <sup>Ω</sup>1*<sup>t</sup> α*−1 ΩΓ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) *Guv*(*s*)*dqs* <sup>−</sup> <sup>Ω</sup>1*<sup>t</sup> α*−1 ΩΓ*q*(*α* − *ξ*) <sup>1</sup> 0 \* *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) *Fuv*(*τ*)*dqτ* + *dqK*(*s*) + Γ*q*(*β*)*t α*−1 ΩΓ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) *Fuv*(*s*)*dqs* <sup>−</sup> <sup>Γ</sup>*q*(*β*)*<sup>t</sup> α*−1 ΩΓ*q*(*β* − *ζ*)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 \* *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) *Guv*(*τ*)*dqτ* + *dqH*(*s*), <sup>X</sup>2(*u*, *<sup>v</sup>*)(*t*) = <sup>−</sup> <sup>1</sup> Γ*q*(*β*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*β*−1) *Guv*(*s*)*dqs*, <sup>Y</sup>2(*u*, *<sup>v</sup>*)(*t*) = <sup>Ω</sup>2*tβ*−<sup>1</sup> ΩΓ*q*(*α* − *ζ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) *Fuv*(*s*)*dqs* <sup>−</sup> <sup>Ω</sup>2*tβ*−<sup>1</sup> ΩΓ*q*(*β* − *ζ*) <sup>1</sup> 0 \* *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) *Guv*(*τ*)*dqτ* + *dqH*(*s*) + Γ*q*(*α*)*tβ*−<sup>1</sup> ΩΓ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) *Guv*(*s*)*dqs* <sup>−</sup> <sup>Γ</sup>*q*(*α*)*tβ*−<sup>1</sup> ΩΓ*q*(*α* − *ξ*)Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 \* *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) *Fuv*(*τ*)*dqτ* + *dqK*(*s*),

where *t* ∈ [0, 1], (*u*, *v*) ∈ *Br*<sup>0</sup> . Thus, T<sup>1</sup> = X<sup>1</sup> + Y1, T<sup>2</sup> = X<sup>2</sup> + Y<sup>2</sup> and T = X + Y. By (H3), we know that ∀(*u*1, *v*1),(*u*2, *v*2) ∈ *Br*<sup>0</sup> ,

 X (*u*1, *v*1) + Y(*u*2, *v*2) *V* ≤ X (*u*1, *v*1) *<sup>V</sup>* + Y(*u*2, *v*2) *V* = X1(*u*1, *v*1) + X2(*u*1, *v*1) + Y1(*u*2, *v*2) + Y2(*u*2, *v*2) ≤ 1 <sup>Γ</sup>*q*(*α*) *σ*<sup>1</sup> + 1 <sup>Γ</sup>*q*(*β*) *σ*<sup>2</sup> + *σ*<sup>1</sup> \* Γ*q*(*β*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqK*(*s*) · <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + + *σ*<sup>2</sup> \* Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) · <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) + + *σ*<sup>1</sup> · \* Γ*q*(*α*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*α* − *ξ*) <sup>1</sup> 0 *s <sup>α</sup>*−*ξ*−1*dqK*(*s*) + Γ*q*(*α*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*α* − *ζ*0) <sup>1</sup> 0 *dqK*(*s*) + + *σ*<sup>2</sup> \* Γ*q*(*α*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) + Γ*q*(*α*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + =(*C*<sup>7</sup> + *C*9) *σ*<sup>1</sup> +(*C*<sup>8</sup> + *C*10) *σ*<sup>2</sup> ≤ *r*0.

For ∀(*u*1, *v*1),(*u*2, *v*2) ∈ *Br*<sup>0</sup> , and Λ < 1, we have

$$\begin{split} & \quad \parallel \mathcal{X}(\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - \mathcal{X}(\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \parallel\_{V} \\ &= \parallel \mathcal{X}\_{1}(\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - \mathcal{X}\_{1}(\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \parallel + \parallel \mathcal{X}\_{2}(\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - \mathcal{X}\_{2}(\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \parallel \\ & \leq (\Lambda\_{3} \mathbb{C}\_{3} \frac{1}{\Gamma\_{q}(\boldsymbol{a})} + \Lambda\_{4} \mathbb{C}\_{6} \frac{1}{\Gamma\_{q}(\boldsymbol{\beta})}) (\parallel \boldsymbol{u}\_{1} - \boldsymbol{u}\_{2} \parallel + \parallel \boldsymbol{v}\_{1} - \boldsymbol{v}\_{2} \parallel) \\ &= \overline{\Lambda} \parallel (\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - (\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \parallel\_{V} \\ & < \parallel (\boldsymbol{u}\_{1}, \boldsymbol{v}\_{1}) - (\boldsymbol{u}\_{2}, \boldsymbol{v}\_{2}) \parallel\_{V} .\end{split}$$

Hence, the operator X is a contraction.

Owing to the continuity of *P* and *Q*, Y is continuous. Next, we need to verify that Y is a compact operator. Due to ∀(*u*, *v*) ∈ *Br*<sup>0</sup> ,

$$\parallel \mathcal{Y}(\mathfrak{u}, \mathfrak{v}) \parallel\_{V} = \parallel \mathcal{Y}\_{1}(\mathfrak{u}, \mathfrak{v}) \parallel + \parallel \mathcal{Y}\_{2}(\mathfrak{u}, \mathfrak{v}) \parallel \leq (\mathsf{C}\mathfrak{g} + \mathsf{C}\_{11}) \parallel \sigma\_{1} \parallel + (\mathsf{C}\_{10} + \mathsf{C}\_{12}) \parallel \sigma\_{2} \parallel\_{V}$$

we have derived that the functions from Y are uniformly bounded.

We can show the equicontinuous of the functions from Y(*Br*<sup>0</sup> ). We denote that

$$\begin{aligned} \Psi\_{r\_0} &= \sup\_{t \in [0,1]} \left\{ |P(t,\boldsymbol{\mu},\boldsymbol{\upsilon},\boldsymbol{\upsilon},\boldsymbol{y})| \, |\, |\boldsymbol{u}| \le r\_0 \, \, |\boldsymbol{v}| \le r\_0 \, \, |\, \boldsymbol{x}| \le \frac{r\_0}{\Gamma\_q(\omega\_1)'} \, \, |\boldsymbol{y}| \le \frac{r\_0}{\Gamma\_q(\delta\_1)} \right\}, \\ \Theta\_{r\_0} &= \sup\_{t \in [0,1]} \left\{ |Q(t,\boldsymbol{\mu},\boldsymbol{\upsilon},\boldsymbol{\upsilon},\boldsymbol{y})| \, |\, |\boldsymbol{u}| \le r\_0 \, \, |\boldsymbol{v}| \le r\_0 \, \, |\boldsymbol{x}| \le \frac{r\_0}{\Gamma\_q(\omega\_2)'} \, \, |\boldsymbol{y}| \le \frac{r\_0}{\Gamma\_q(\delta\_2)} \right\}. \end{aligned}$$

for (*u*, *v*) ∈ *Br*<sup>0</sup> and *t*1, *t*<sup>2</sup> ∈ [0, 1] with *t*<sup>1</sup> < *t*2. An argument similar to the one used in the proof of Theorem 6 shows that

$$|\mathcal{Y}\_1(u,v)(t\_2) - \mathcal{Y}\_1(u,v)(t\_1)| \to 0, \quad |\mathcal{Y}\_2(u,v)(t\_2) - \mathcal{Y}\_2(u,v)(t\_1)| \to 0,$$

as *t*<sup>2</sup> → *t*1, (*u*, *v*) ∈ *Br*<sup>0</sup> . Therefore, Y(*Br*<sup>0</sup> ) is equicontinuous. Then, we can see that Y(*Br*<sup>0</sup> ) is relatively compact. Hence, Y is compact on *Br*<sup>0</sup> . Using Theorem 2, we know that the System (1)–(2) has at least one solution. The proof is completed.

**Remark 1.** *Evidently, we prove that the operator* X *is a contraction, the operator* Y *is compact and continuous in Theorem 7. An alternative method of proof is to show that* X *is compact and continuous,* Y *is a contraction, that is Theorem 8.*

**Theorem 8.** *Suppose that* (H <sup>1</sup>) *and* (H3) *hold. If* Ω = 0*, and*

$$
\hat{\Lambda} = \Lambda\_3 \mathbb{C}\_3 (\mathbb{C}\_9 + \mathbb{C}\_{11}) + \Lambda\_4 \mathbb{C}\_6 (\mathbb{C}\_8 + \mathbb{C}\_{10}) < 1.
$$

*Then the Systems (1)–(2) has at least one solution.*

**Proof.** On the basis of Remark 1, this theorem can be proved by the same method as employed in Theorem 7.

**Theorem 9.** *Suppose that P*, *Q are S-Caratheodory functions and* ´ (H4) *hold. If* Ω = 0*, and*

$$\Xi = \max\{\mathcal{C}\_{13\prime}\mathcal{C}\_{14}\} < 1,$$

*where C*<sup>13</sup> = (*j*<sup>1</sup> + *<sup>j</sup>*<sup>3</sup> <sup>Γ</sup>*q*(*ω*1))(*C*<sup>7</sup> <sup>+</sup> *<sup>C</sup>*9)+(*k*<sup>1</sup> <sup>+</sup> *<sup>k</sup>*<sup>3</sup> <sup>Γ</sup>*q*(*ω*2))(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10), *<sup>C</sup>*<sup>14</sup> = (*j*<sup>2</sup> <sup>+</sup> *<sup>j</sup>*<sup>4</sup> <sup>Γ</sup>*q*(*δ*1))(*C*<sup>7</sup> <sup>+</sup> *C*9)+(*k*<sup>2</sup> + *<sup>k</sup>*<sup>4</sup> <sup>Γ</sup>1(*δ*2))(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10)*, and there exist <sup>A</sup>*1, *<sup>A</sup>*2, *ji*, *ki* <sup>&</sup>gt; <sup>0</sup> *such that* <sup>|</sup>*r*1(*t*)| ≤ *<sup>A</sup>*1, |*r*2(*t*)| ≤ *A*2*,* |*Li*(*t*)| ≤ *ji and* |*li*(*t*)| ≤ *ki* (*i* = 1, 2, 3, 4)*. Then, the System (1)–(2) has at least one solution.*

**Proof.** The main point of Theorem 9 is to prove T is completely continuous. Firstly, for the continuity of functions *P* and *Q*, we obtain that the operator T is continuous. Secondly, we show that T is compact.

Let the set Φ ⊂ *V* be bounded. Then, there exist integrable functions *M*1(*t*) and *<sup>M</sup>*2(*t*) <sup>∈</sup> *<sup>L</sup>*1([0, 1], <sup>R</sup>+) such that for <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, 1], (*u*, *<sup>v</sup>*) <sup>∈</sup> <sup>Φ</sup>, we have

$$\begin{aligned} \left| P(t, u(t), v(t), I\_q^{\omega\_1} u(t), I\_q^{\delta\_1} v(t)) \right| &\le M\_1(t), \\ \left| Q(t, u(t), v(t), I\_q^{\omega\_2} u(t), I\_q^{\delta\_2} v(t)) \right| &\le M\_2(t). \end{aligned}$$

According to the Theorem 5, we get

$$\begin{aligned} \left| F\_{\mathsf{u}\upsilon}(t) \right| &= \left| P(t, \mathsf{u}(t), \upsilon(t), I\_{q}^{\omega\_{1}}\mathsf{u}(t), I\_{q}^{\delta\_{1}}\upsilon(t)) \right| \leq \left| \left| M\_{1} \right| \right|\_{L^{1}}, \\ \left| G\_{\mathsf{u}\upsilon}(t) \right| &= \left| Q(t, \mathsf{u}(t), \upsilon(t), I\_{q}^{\omega\_{2}}\mathsf{u}(t), I\_{q}^{\delta\_{2}}\upsilon(t)) \right| \leq \left| \left| M\_{2} \right| \right|\_{L^{1}}. \end{aligned}$$

where *u <sup>L</sup>*<sup>1</sup><sup>=</sup> % <sup>1</sup> <sup>0</sup> |*u*(*t*)|*dqt*. Then

 T1(*u*, *v*) ≤ *M*<sup>1</sup> *L*1 \* 1 <sup>Γ</sup>*q*(*α*) <sup>+</sup> Γ*q*(*β*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + Γ*q*(*β*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) · <sup>1</sup> 0 *dqK*(*s*) + + *M*<sup>2</sup> *L*1 \* Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) + = *M*<sup>1</sup> *<sup>L</sup>*<sup>1</sup> *C*7+ *M*<sup>2</sup> *<sup>L</sup>*<sup>1</sup> *C*10,

in a similar manner, we have

$$\parallel \ T\_2(u, v) \parallel \leq \parallel M\_1 \parallel\_{L^1} \mathbf{C}\_{\Theta} + \parallel M\_2 \parallel\_{L^1} \mathbf{C}\_{\Theta} \cdot$$

so ∀(*u*, *v*) ∈ Φ,

 T (*u*, *v*) *V*= T1(*u*, *v*) + T2(*u*, *v*) ≤ *M*<sup>1</sup> *<sup>L</sup>*<sup>1</sup> (*C*<sup>7</sup> + *C*9)+ *M*<sup>2</sup> *<sup>L</sup>*<sup>1</sup> (*C*<sup>10</sup> + *C*8), therefore, T (Φ) is uniformly bounded.

Another step is to show that T (Φ) is equicontinuous. Proceeding as in the proof of Theorem 6, we obtain T1(*u*, *<sup>v</sup>*)(*t*2) − T1(*u*, *<sup>v</sup>*)(*t*1) <sup>→</sup> 0 and T2(*u*, *<sup>v</sup>*)(*t*2) − T2(*u*, *<sup>v</sup>*)(*t*1) → 0, as *t*<sup>2</sup> → *t*1, (*u*, *v*) ∈ Φ. Thus, T (Φ) is equicontinuous. At the same time, we can also obtain that T is completely continuous.

Finally, we illustrate that S = {(*u*, *v*) ∈ *V*, (*u*, *v*) = *λ*T (*u*, *v*), 0 ≤ *λ* ≤ 1} is bounded. Let (*u*, *v*) ∈ S, then ∀*t* ∈ [0, 1], we have *u*(*t*) = *λ*T1(*u*, *v*)(*t*), *v*(*t*) = *λ*T2(*u*, *v*)(*t*). For simplicity, we denote that

$$\begin{aligned} \hat{F}\_{\mathsf{u}\upsilon}(s) &= r\_1(s) + L\_1(s) \big| u(s) \big| + L\_2(s) \big| v(s) \big| + L\_3(s) \big| I\_q^{\omega\_1} u(s) \big| + L\_4(s) \big| I\_q^{\delta\_1} v(s) \big| , \\ \hat{G}\_{\mathsf{u}\upsilon}(s) &= r\_2(s) + l\_1(s) \big| u(s) \big| + l\_2(s) \big| v(s) \big| + l\_3(s) \big| I\_q^{\omega\_2} u(s) \big| + l\_4(s) \big| I\_q^{\delta\_2} v(s) \big| , \end{aligned}$$

so,

$$\begin{aligned} \hat{F}\_{\mathsf{u}\upsilon}(s) &\leq A\_1 + j\_1 \left| u(s) \right| + j\_2 \left| v(s) \right| + j\_3 \left| I\_{\eta}^{\omega\_1} u(s) \right| + j\_4 \left| I\_{\eta}^{\delta\_1} v(s) \right|, \\ \hat{G}\_{\mathsf{u}\upsilon}(s) &\leq A\_2 + k\_1 \left| u(s) \right| + k\_2 \left| v(s) \right| + k\_3 \left| I\_{\eta}^{\omega\_2} u(s) \right| + k\_4 \left| I\_{\eta}^{\delta\_2} v(s) \right|. \end{aligned}$$

then

 *u*(*t*) ≤ T1(*u*, *<sup>v</sup>*)(*t*) ≤ 1 Γ*q*(*α*) *<sup>t</sup>* 0 (*<sup>t</sup>* <sup>−</sup> *qs*)(*α*−1) *F*ˆ *uv*(*s*)*dqs* + Γ*q*(*β*)*t α*−1 |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*α*−*ζ*0−1) · *F*ˆ *uv*(*s*)*dqs* + Γ*q*(*β*)*t α*−1 |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *qs*)(*β*−*ξ*0−1) · *<sup>G</sup>*ˆ*uv*(*s*)*dqs* <sup>+</sup> Γ*q*(*β*)*t α*−1 |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) <sup>1</sup> 0 \* *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*α*−*ξ*−1) · *F*ˆ *uv*(*τ*)*dqτ* + *dqK*(*s*) + Γ*q*(*β*)*t α*−1 |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 \* *<sup>s</sup>* 0 (*<sup>s</sup>* <sup>−</sup> *<sup>q</sup>τ*)(*β*−*ζ*−1) · *<sup>G</sup>*ˆ*uv*(*τ*)*dq<sup>τ</sup>* + *dqH*(*s*) ,

hence,

 *u* <sup>≤</sup> <sup>1</sup> 0 *F*ˆ *uv*(*s*) *dqs* \* 1 <sup>Γ</sup>*q*(*α*) <sup>+</sup> Γ*q*(*β*) |Ω|Γ*q*(*α* − *ξ*)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) · <sup>1</sup> 0 *dqK*(*s*) + Γ*a*(*β*) |Ω|Γ*q*(*α* − *ζ*0)Γ*q*(*β* − *ξ*0) + + <sup>1</sup> 0 *G*ˆ*uv*(*s*) *dqs* · \* Γ*q*(*β*) <sup>|</sup>Ω|Γ*q*(*<sup>β</sup>* <sup>−</sup> *<sup>ξ</sup>*0)Γ*q*(*<sup>β</sup>* <sup>−</sup> *<sup>ζ</sup>*) · <sup>1</sup> 0 *sβ*−*ζ*−1*dqH*(*s*) + Γ*q*(*β*) |Ω|Γ*q*(*β* − *ξ*0)Γ*q*(*β* − *ζ*) <sup>1</sup> 0 *dqH*(*s*) + =(*A*<sup>1</sup> + *j*<sup>1</sup> *u* +*j*<sup>2</sup> *v* <sup>+</sup> *<sup>j</sup>*<sup>3</sup> <sup>Γ</sup>*q*(*ω*1) *u* <sup>+</sup> *<sup>j</sup>*<sup>4</sup> <sup>Γ</sup>*q*(*δ*1) *v* )*C*<sup>7</sup> + (*A*<sup>2</sup> + *k*<sup>1</sup> *u* +*k*<sup>2</sup> *v* + *k*3 <sup>Γ</sup>*q*(*ω*2) *u* + *k*4 <sup>Γ</sup>*q*(*δ*2) *v* )*C*10. (10)

Similarly,

$$\begin{aligned} \left| \left| \left| \left| \boldsymbol{\upsilon} \right| \right| \right| \leq & \left( A\_1 + j\_1 \left( \left| \left| \boldsymbol{u} \right| \right| \right) + j\_2 \left( \left| \left| \boldsymbol{\upsilon} \right| \right) \right) + \frac{j\_3}{\Gamma\_q(\omega\_1)} \parallel \left| \left| \boldsymbol{u} \right| \right| \right) + \frac{j\_4}{\Gamma\_q(\delta\_1)} \parallel \left| \left| \left| \boldsymbol{\upsilon} \right| \right| \right| \mathbf{C}\_9 \\ &+ \left( A\_2 + k\_1 \left( \left| \left| \left| \boldsymbol{u} \right| \right| \right| + k\_2 \left( \left| \left| \boldsymbol{\upsilon} \right| \right) \right) + \frac{k\_3}{\Gamma\_q(\omega\_2)} \parallel \left| \left| \left| \boldsymbol{u} \right| \right| \right| + \frac{k\_4}{\Gamma\_q(\delta\_2)} \parallel \left| \left| \left| \left| \boldsymbol{\upsilon} \right| \right| \right| \right) \mathbf{C}\_8 \end{aligned} \tag{11}$$

by means of (11) and (12), we have

$$\begin{aligned} \parallel \left( \mu\_\prime v \right) \parallel\_V &= \parallel \mu \parallel + \parallel v \parallel \\ \leq &A\_1 \left( \mathbf{C}\_7 + \mathbf{C}\_9 \right) + A\_2 \left( \mathbf{C}\_8 + \mathbf{C}\_{10} \right) + \mathbf{C}\_{13} \parallel \mu \parallel + \mathbf{C}\_{14} \parallel v \parallel \\ \leq &A\_1 \left( \mathbf{C}\_7 + \mathbf{C}\_9 \right) + A\_2 \left( \mathbf{C}\_8 + \mathbf{C}\_{10} \right) + \Xi \parallel \left( \mu\_\prime v \right) \parallel\_V \cdot \end{aligned}$$

Due to Ξ < 1, we get

$$\|\|\left(u\_{\prime}v\right)\|\|\_{V} \le \left[A\_{1}\left(\mathbb{C}\_{7} + \mathbb{C}\_{9}\right) + A\_{2}\left(\mathbb{C}\_{8} + \mathbb{C}\_{10}\right)\right] \left(1 - \Xi\right)^{-1}, \ (u\_{\prime}v) \in \mathcal{S}\_{\prime}$$

thus, S is bounded.

By Theorem 3, it is time to say that the Systems (1)–(2) has at least one solution. Hence, the statements in Theorem 9 are proved.

**Corollary 3.** *Suppose that P*, *Q are S-Caratheodory functions and* ´ (H <sup>4</sup>) *hold. If* Ω = 0*, and*

$$
\hat{\Xi} = \max \{ \mathcal{C}\_{15}, \mathcal{C}\_{16} \} < 1,
$$

*where C*<sup>15</sup> = *L*1(*C*<sup>7</sup> + *C*9) + *<sup>L</sup>*<sup>3</sup> <sup>Γ</sup>*q*(*ω*1)(*C*<sup>7</sup> <sup>+</sup> *<sup>C</sup>*9) + *<sup>l</sup>*1(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10) + *<sup>l</sup>*<sup>3</sup> <sup>Γ</sup>*q*(*ω*2)(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10), *C*<sup>16</sup> = *L*2(*C*<sup>7</sup> + *C*9) + *<sup>L</sup>*<sup>4</sup> <sup>Γ</sup>*q*(*δ*1)(*C*<sup>7</sup> <sup>+</sup> *<sup>C</sup>*9) + *<sup>l</sup>*2(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10) + *<sup>l</sup>*<sup>4</sup> <sup>Γ</sup>*q*(*δ*2)(*C*<sup>8</sup> <sup>+</sup> *<sup>C</sup>*10). *Then the Systems (1)–(2) has at least one solution.*

**Theorem 10.** *Suppose that P*, *Q are S-Carathe*´*odory functions and* (H5) *hold. If* Ω = 0 *and there exists* Π > 0 *such that*

\* *p*<sup>0</sup> + *p*<sup>1</sup> *ϕ*1(Π)+ *p*<sup>2</sup> *ϕ*2(Π)+ *p*<sup>3</sup> *ϕ*<sup>3</sup> Π Γ*q*(*ω*1) + *p*<sup>4</sup> *ϕ*<sup>4</sup> Π Γ*q*(*δ*1) + (*C*<sup>7</sup> <sup>+</sup> *<sup>C</sup>*9) + \* *q*<sup>0</sup> + *q*<sup>1</sup> *η*1(Π) + *q*<sup>2</sup> *η*2(Π)+ *q*<sup>3</sup> *η*<sup>3</sup> Π Γ*q*(*ω*2) + *q*<sup>4</sup> *η*<sup>4</sup> Π Γ*q*(*δ*2) + (*C*<sup>8</sup> + *C*10) < Π.

*Then the Systems (1)–(2) has at least one solution.*

**Proof.** Let *B*<sup>Π</sup> = {(*u*, *v*) ∈ *V*, (*u*, *v*) *<sup>V</sup>*≤ Π}. Firstly, we prove that T : *B*<sup>Π</sup> → *B*Π. For (*u*, *v*) ∈ *B*<sup>Π</sup> and *t* ∈ [0, 1], we have

$$\begin{split} \parallel \mathcal{T}\_{\mathrm{I}}(\mu, v) \parallel \leq & \mathrm{C}\_{7} \left[ \parallel \ p\_{0} \parallel + \parallel p\_{1} \parallel \varrho\_{1}(\mathrm{II}) + \parallel \ p\_{2} \parallel \varrho\_{2}(\mathrm{II}) + \parallel \ p\_{3} \parallel \varrho\_{3} \right(\frac{\mathrm{II}}{\Gamma\_{q}(\omega\_{1})}) \right. \\ &+ \parallel \ p\_{4} \parallel \varrho\_{4} \left( \frac{\mathrm{II}}{\Gamma\_{q}(\delta\_{1})} \right) \big] + \mathrm{C}\_{10} \left[ \parallel \ q\_{0} \parallel + \parallel q\_{1} \parallel \eta\_{1}(\mathrm{II}) + \parallel \ q\_{2} \parallel \eta\_{2}(\mathrm{II}) \right] \\ &+ \parallel \ q\_{3} \parallel \eta\_{3} \left( \frac{\mathrm{II}}{\Gamma\_{q}(\omega\_{2})} \right) + \parallel \ q\_{4} \parallel \eta\_{4} \left( \frac{\mathrm{II}}{\Gamma\_{q}(\delta\_{2})} \right) \bigg]. \end{split}$$

and

$$\begin{split} \|\|\mathcal{T}\_{\mathsf{T}}(\mu,v)\|\| \leq & \mathsf{C}\_{9} \Big[ \|\|p\_{0}\|\| + \|\|p\_{1}\|\|\,q\_{1}(\Pi) + \|\|p\_{2}\|\|\,q\_{2}(\Pi) + \|\|p\_{3}\|\|\,q\_{3}(\frac{\Pi}{\Gamma\_{q}(\omega\_{1})}) \\ &+ \|\|\,p\_{4}\|\|\,q\_{4}\Big(\frac{\Pi}{\Gamma\_{q}(\delta\_{1})}\Big) \Big] + \mathsf{C}\_{8} \Big[ \|\|q\_{0}\|\| + \|\|q\_{1}\|\|\,\eta\_{1}(\Pi) + \|\|q\_{2}\|\|\,\eta\_{2}(\Pi) \\ &+ \|\|\,q\_{3}\|\|\,\eta\_{3}\Big(\frac{\Pi}{\Gamma\_{q}(\omega\_{2})}\Big) + \|\|\,q\_{4}\|\|\,\eta\_{4}\Big(\frac{\Pi}{\Gamma\_{q}(\delta\_{2})}\Big) \Big]. \end{split}$$

For (*u*, *v*) ∈ *B*Π, we have

$$\begin{split} \|\|\mathcal{T}(u,v)\|\|\_{\mathcal{V}} &= \|\|\mathcal{T}\_{1}(u,v)\|\| + \|\|\mathcal{T}\_{2}(u,v)\|\| \\ &\leq (\mathsf{C}\_{7} + \mathsf{C}\_{9}) \left[ \|\|p\_{0}\|\| + \|\|p\_{1}\|\|\,\varrho\_{1}(\Pi) + \|\|p\_{2}\|\|\,\varrho\_{2}(\Pi) + \|\|p\_{3}\|\|\,\varrho\_{3}\Big(\frac{\Pi}{\varGamma\_{q}(\omega\_{1})}\Big) \right. \\ &\left. + \|\|\,p\_{4}\|\|\,\varrho\_{4}\Big(\frac{\Pi}{\varGamma\_{q}(\delta\_{1})}\Big) \right] + (\mathsf{C}\_{10} + \mathsf{C}\_{8}) \left[ \|\|q\_{0}\|\| + \|\|q\_{1}\|\|\,\eta\_{1}(\Pi) + \|\|q\_{2}\|\|\,\eta\_{2}(\Pi) \\ &\quad + \|\|\,q\_{3}\|\|\,\eta\_{3}\Big(\frac{\Pi}{\varGamma\_{q}(\omega\_{2})}\Big) + \|\|\,q\_{4}\|\|\,\eta\_{4}\Big(\frac{\Pi}{\varGamma\_{q}(\delta\_{2})}\Big) \right] < \Pi. \end{split}$$

Consequently, T (*B*Π) ⊂ *B*Π. At the same time, it is easy to see that T is completely continuous, which can be derived in the same way as employed in Theorem 6.

Furthermore, assume that there exists (*u*, *v*) ∈ *∂B*<sup>Π</sup> such that (*u*, *v*) = *λ*T (*u*, *v*) for *λ* ∈ (0, 1), it is simple to get (*u*, *v*) *V*≤ T (*u*, *v*) *<sup>V</sup>*< Π, this leads to a contradiction for (*u*, *v*) ∈ *∂B*Π. Therefore, by applying Theorem 4, we deduce that T has a fixed point (*u*, *v*) ∈ *B*Π, which is a solution of the Systems (1)–(2). The proof is completed.

#### **4. Application Examples**

In this section, for the system with the different nonlinearity terms, some examples are appreciated to illustrate our main results.

We consider the following system of fractional *q*-difference equations:

$$\begin{cases} \begin{array}{ll} \left(D\_{q}^{\frac{3}{2}}u\right)(t) + P(t,u(t),v(t),I\_{q}^{\frac{1}{4}}u(t),I\_{q}^{\frac{4}{3}}v(t)) = 0, & t \in (0,1), \\ \left(D\_{q}^{\frac{7}{2}}v\right)(t) + Q(t,u(t),v(t),I\_{q}^{\frac{3}{4}}u(t),I\_{q}^{\frac{5}{4}}v(t)) = 0, & t \in (0,1), \end{array} \end{cases} \tag{12}$$

with the nonlocal BCs

$$\begin{cases} \boldsymbol{u}(0) = 0, \quad \boldsymbol{D}\_{\boldsymbol{q}}^{\frac{1}{2}} \boldsymbol{u}(1) = \int\_{0}^{1} \boldsymbol{D}\_{\boldsymbol{q}}^{\frac{5}{2}} \boldsymbol{v}(t) d\_{\boldsymbol{q}}(-t^{2}),\\ \boldsymbol{v}(0) = \boldsymbol{D}\_{\boldsymbol{q}} \boldsymbol{v}(0) = 0, \quad \boldsymbol{D}\_{\boldsymbol{q}}^{5} \boldsymbol{v}(1) = \int\_{0}^{1} \boldsymbol{D}\_{\boldsymbol{q}}^{\frac{1}{2}} \boldsymbol{u}(t) d\_{\boldsymbol{q}} t, \end{cases} \tag{13}$$

where *α* = <sup>3</sup> <sup>2</sup> , *<sup>β</sup>* <sup>=</sup> <sup>5</sup> <sup>2</sup> , *<sup>ω</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>4</sup> , *<sup>δ</sup>*<sup>1</sup> <sup>=</sup> <sup>4</sup> <sup>3</sup> , *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> <sup>9</sup> <sup>4</sup> , *<sup>δ</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>3</sup> , *<sup>ζ</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>5</sup> , *<sup>ξ</sup>*<sup>0</sup> <sup>=</sup> <sup>7</sup> <sup>5</sup> , *<sup>ζ</sup>* <sup>=</sup> <sup>5</sup> <sup>4</sup> , *<sup>ξ</sup>* <sup>=</sup> <sup>1</sup> 6 , *q* = <sup>1</sup> <sup>2</sup> , *H*(*t*) = −*t* 2, *K*(*t*) = *t*.

After a simple caculation, we obtain Ω = 2.58954375 = 0, *C*<sup>1</sup> = 1.34100597, *C*<sup>2</sup> = 2.08201688, *C*<sup>3</sup> = *C*2, *C*<sup>4</sup> = 1.92455621, *C*<sup>5</sup> = 1.79251862, *C*<sup>6</sup> = *C*4, *C*<sup>7</sup> = 2.29230629, *C*<sup>8</sup> = 1.75022309, *C*<sup>9</sup> = 0.7590784, *C*<sup>10</sup> = 1.42695841, *C*<sup>11</sup> = 1.20641206, *C*<sup>12</sup> = 0.91031044.

**Example 1.** *Consider the nonlinear terms of the system*

$$\begin{aligned} P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) &= e^t + \frac{t}{36} \cos \mathbf{x}\_1 - \frac{t}{54} \sin \mathbf{x}\_2 + \frac{1}{63 + t} \arctan \mathbf{x}\_3 - \frac{\mathbf{x}\_4}{(t + 9)^2}, \\ Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) &= \frac{1}{\sqrt{5 + t^2}} - \frac{t}{48} \sin \mathbf{x}\_1 + \frac{t}{64} \cos \mathbf{x}\_2 - \frac{1}{36 + t} \arctan \mathbf{x}\_3 + \frac{\mathbf{x}\_4}{t^2 + 56}, \end{aligned}$$

*where t* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup> (*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4). *For xi*, *yi* <sup>∈</sup> <sup>R</sup> (*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4), *we obtain*

$$\begin{aligned} & \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - P(t, y\_1, y\_2, y\_3, y\_4) \right| \\ & \leq \frac{t}{36} |\mathbf{x}\_1 - y\_1| + \frac{t}{54} |\mathbf{x}\_2 - y\_2| + \frac{1}{t+63} |\mathbf{x}\_3 - y\_3| + \frac{1}{(t+9)^2} |\mathbf{x}\_4 - y\_4| \leq \Lambda\_1 \sum\_{i=1}^4 |\mathbf{x}\_i - y\_i| \\ & \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - Q(t, y\_1, y\_2, y\_3, y\_4) \right| \\ & \leq \frac{t}{48} |\mathbf{x}\_1 - y\_1| + \frac{t}{64} |\mathbf{x}\_2 - y\_2| + \frac{1}{t+36} |\mathbf{x}\_3 - y\_3| + \frac{1}{56 + t^2} |\mathbf{x}\_4 - y\_4| \leq \Lambda\_2 \sum\_{i=1}^4 |\mathbf{x}\_i - y\_i|. \end{aligned}$$

It is obvious that *L*1(*t*) = *<sup>t</sup>* <sup>36</sup> , *<sup>L</sup>*2(*t*) = *<sup>t</sup>* <sup>54</sup> , *<sup>L</sup>*3(*t*) = <sup>1</sup> <sup>63</sup>+*<sup>t</sup>* , *<sup>L</sup>*4(*t*) = <sup>1</sup> (*t*+9)<sup>2</sup> and *<sup>l</sup>*1(*t*) = *<sup>t</sup>* 48 , *l*2(*t*) = *<sup>t</sup>* <sup>64</sup> , *<sup>l</sup>*3(*t*) = <sup>1</sup> <sup>36</sup>+*<sup>t</sup>* , *<sup>l</sup>*4(*t*) = <sup>1</sup> *<sup>t</sup>*2+<sup>56</sup> . By a simple computation, we obtain Λ<sup>1</sup> = 0.07451499, Λ<sup>2</sup> = 0.08209325 and Λ = 0.97536897 < 1, respectively. By Theorem 5, the Systems (12)–(13) has a unique solution.

**Example 2.** *Consider the nonlinear terms of the system*

$$P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = (3t + 5)^2 + \frac{t}{30} |\mathbf{x}\_1|^{\frac{1}{3}} + \frac{t}{t^2 + 9} \arctan|\mathbf{x}\_2|^{\frac{1}{2}} + \frac{1}{5 + 8t} \sin|\mathbf{x}\_3|^{\frac{3}{4}}$$

$$-\frac{1}{(4t+9)^2}|\mathbf{x}\_4|^{\frac{1}{5}},$$

$$Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = \boldsymbol{\varepsilon}^t + \frac{1}{46(t+1)}|\mathbf{x}\_1|^{\frac{3}{5}} + \frac{t}{37}|\mathbf{x}\_2|^{\frac{1}{5}} + \frac{1}{t+29}\sin|\mathbf{x}\_3|^{\frac{2}{3}} - 8t\arctan|\mathbf{x}\_4|^{\frac{5}{6}}.$$

*where t* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup> (*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4). *It is clear that*

$$\begin{split} \left| P(t, \mathbf{x}\_{1}, \mathbf{x}\_{2}, \mathbf{x}\_{3}, \mathbf{x}\_{4}) \right| &\leq (3t+5)^{2} + \frac{t}{30} |\mathbf{x}\_{1}|^{\frac{1}{3}} + \frac{t}{t^{2}+9} |\mathbf{x}\_{2}|^{\frac{1}{2}} + \frac{1}{8t+5} |\mathbf{x}\_{3}|^{\frac{3}{4}} + \frac{1}{(4t+9)^{2}} |\mathbf{x}\_{4}|^{\frac{1}{5}},\\ \left| Q(t, \mathbf{x}\_{1}, \mathbf{x}\_{2}, \mathbf{x}\_{3}, \mathbf{x}\_{4}) \right| &\leq e^{t} + \frac{1}{46(t+1)} |\mathbf{x}\_{1}|^{\frac{3}{5}} + \frac{t}{37} |\mathbf{x}\_{2}|^{\frac{1}{5}} + \frac{1}{t+29} |\mathbf{x}\_{3}|^{\frac{2}{5}} + 8t |\mathbf{x}\_{4}|^{\frac{5}{5}}. \end{split}$$

Therefore, the assumption (H2) is satisfied with *<sup>c</sup>*0(*t*)=(3*<sup>t</sup>* <sup>+</sup> <sup>5</sup>)2, *<sup>c</sup>*1(*t*) = *<sup>t</sup>* 30 , *c*2(*t*) = *<sup>t</sup> <sup>t</sup>*2+<sup>9</sup> , *<sup>c</sup>*3(*t*) = <sup>1</sup> <sup>8</sup>*t*+<sup>5</sup> , *<sup>c</sup>*4(*t*) = <sup>1</sup> (4*t*+9)<sup>2</sup> , *<sup>d</sup>*0(*t*) = *<sup>e</sup><sup>t</sup>* , *d*1(*t*) = <sup>1</sup> <sup>46</sup>(*t*+1), *<sup>d</sup>*2(*t*) = *<sup>t</sup>* 37 , *d*3(*t*) = <sup>1</sup> *<sup>t</sup>*+<sup>29</sup> , and *d*4(*t*) = 8*t*. By Theorem 6, the Systems (12)–(13) has at least one solution.

**Example 3.** *Consider the nonlinear terms of the system*

$$P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = e^t + \frac{t}{40} \arctan \mathbf{x}\_1 - \frac{1}{(t+6)^2} \sin \mathbf{x}\_2 + \frac{1}{4(t+9)} \sin \mathbf{x}\_3 - \frac{t}{32} \cos \mathbf{x}\_4,$$

$$Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) = \frac{5t}{6+t^2} - \frac{3t}{56} \cos \mathbf{x}\_1 + \frac{1}{t+28} \sin \mathbf{x}\_2 - \frac{t}{72} \sin^2 \mathbf{x}\_3 + \frac{t}{18} \arctan \mathbf{x}\_4.$$

*where t* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup> (*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4). *For* <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, 1], *xi*, *yi* <sup>∈</sup> <sup>R</sup> (*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4), *We obtain*

$$\begin{split} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - P(t, y\_1, y\_2, y\_3, y\_4) \right| &\leq \frac{1}{40} |\mathbf{x}\_1 - y\_1| + \frac{1}{36} |\mathbf{x}\_2 - y\_2| + \frac{1}{36} |\mathbf{x}\_3 - y\_3| \\ &+ \frac{1}{32} |\mathbf{x}\_4 - y\_4|, \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) - Q(t, y\_1, y\_2, y\_3, y\_4) \right| &\leq \frac{3}{56} |\mathbf{x}\_1 - y\_1| + \frac{1}{28} |\mathbf{x}\_2 - y\_2| + \frac{1}{36} |\mathbf{x}\_3 - y\_3| \\ &+ \frac{1}{18} |\mathbf{x}\_4 - y\_4|. \end{split}$$

*and*

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq c^t + \frac{\pi t}{80} + \frac{1}{(t+6)^2} + \frac{1}{4(t+9)} + \frac{t}{32}, \\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \frac{5t}{6+t^2} + \frac{3t}{56} + \frac{1}{t+28} + \frac{t}{72} + \frac{\pi t}{36}, \end{aligned}$$

It is obvious that *L*<sup>1</sup> = <sup>1</sup> <sup>40</sup> , *<sup>L</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>36</sup> , *<sup>L</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>36</sup> , *<sup>L</sup>*<sup>4</sup> <sup>=</sup> <sup>1</sup> <sup>32</sup> , *<sup>l</sup>*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>56</sup> , *<sup>l</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>28</sup> , *<sup>l</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> 36 , *l*<sup>4</sup> = <sup>1</sup> <sup>18</sup> . By a simple computation, we have Λ<sup>3</sup> = 0.11180556, Λ<sup>4</sup> = 0.17261905, and Λ = 0.53180725 < 1, respectively. Therefore, the assumptions (H <sup>1</sup>), and (H3) are satisfied, by Theorem 7, the Systems (12)–(13) has at least one solution.

**Example 4.** *Consider the nonlinear terms of the system*

$$\begin{aligned} P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) &= \frac{t}{20} + \frac{6t}{35} \sin \mathbf{x}\_1 - \frac{1}{4t + 9\sqrt{6}} \sin 2\mathbf{x}\_2, \\ Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) &= \frac{2t}{3} + \frac{5t}{56} \sin \mathbf{x}\_1 - \frac{3}{2(7\sqrt{5} + t)} \sin 2\mathbf{x}\_2. \end{aligned}$$

*where t* <sup>∈</sup> [0, 1], *xi* <sup>∈</sup> <sup>R</sup> (*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4). *It is clear that*

$$\begin{aligned} \left| P(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \frac{1}{20} + \frac{6}{35} |\mathbf{x}\_1| + \frac{2}{9\sqrt{6}} |\mathbf{x}\_2|, \\\\ \left| Q(t, \mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3, \mathbf{x}\_4) \right| &\leq \frac{2}{3} + \frac{5}{56} |\mathbf{x}\_1| + \frac{3}{7\sqrt{5}} |\mathbf{x}\_2|. \end{aligned}$$

Hence, *L*<sup>1</sup> = <sup>6</sup> <sup>35</sup> , *<sup>L</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> 9 √6 , *L*<sup>3</sup> = *L*<sup>4</sup> = 0,*r*<sup>1</sup> = <sup>1</sup> <sup>20</sup> , *<sup>l</sup>*<sup>1</sup> <sup>=</sup> <sup>5</sup> <sup>56</sup> , *<sup>l</sup>*<sup>2</sup> <sup>=</sup> <sup>3</sup> 7 √5 , *l*<sup>3</sup> = *l*<sup>4</sup> = 0, *r*<sup>2</sup> = <sup>2</sup> <sup>3</sup> . By a simple computation, we obtain *C*<sup>15</sup> = 0.80677144, *C*<sup>16</sup> = 0.88577528, and Ξˆ = 0.88577528 < 1, respectively. By Corollary 3, the Systems (12)–(13) has at least one solution.

7

#### **5. Discussion**

The system of fractional *q*-difference equations plays an extremely crucial role in many fields, such as quantum mechanics, dynamical systems, black holes, mathematical physics equations and so on, see [2,3,5,6,27–30] and the references therein. In this article, we are concerned with the solvability of a system of fractional *q*-difference equations with Riemann-Stieltjes integrals conditions based on some classical fixed point theorems. We obtain the multiple existence and uniqueness conclusions for the Systems (1)–(2). As a matter of fact, in the limit *q* → 1−, the system studied in this paper reduces to the classical system of fractional differential equations. It follows that the results we have discussed are the generalization of the classical analysis, they can extend classical theory in order to expand the range of the possible applications. In the future, we will devote ourselves to finding new inspirations and outstanding methods to overcome the more complex practical problems associated with the system of fractional *q*-difference equations. Moreover, we will investigate numerical methods for this kind of system.

**Author Contributions:** Conceptualization, C.Y. and J.W.; methodology, C.Y. and S.W.; validation, C.Y., J.W. and J.L.; formal analysis, J.W.; resources, S.W.; data curation, C.Y.; writing—original draft preparation, S.W.; writing—review and editing, C.Y., J.W. and J.L.; supervision, C.Y. and J.L.; funding acquisition, C.Y., J.W. and J.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research project is supported by National Natural Science Foundation of China (12272011, 11772007), Beijing Natural Science Foundation (Z180005, 1172002), Natural Science Foundation of Hebei Province (A2015208114) and the Foundation of Hebei Education Department (QN2017063).

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank referees for their extraordinary comments, which help to enrich the content of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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