*3.2. Existence Result Via Banach Contraction Principle*

Now, we prove the uniqueness of problem (5) by means of Banach contraction principle. Therefore, the following hypotheses are needed.

(*A*3) There exist constants *K*, *L* > 0 such that

$$|f(t, \boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}) - f(t, \bar{\boldsymbol{u}}, \bar{\boldsymbol{v}}, \bar{\boldsymbol{w}})| \le K(|\boldsymbol{u} - \bar{\boldsymbol{u}}| + |\boldsymbol{v} - \bar{\boldsymbol{v}}|) + L|\boldsymbol{w} - \bar{\boldsymbol{w}}|$$

for any *<sup>u</sup>*, *<sup>v</sup>*, *<sup>w</sup>*, *<sup>u</sup>*¯, *<sup>v</sup>*¯, *<sup>w</sup>*¯ <sup>∈</sup> <sup>R</sup> and *<sup>t</sup>* ∈ J .

(*A*4) Suppose that

$$\left(\frac{2K}{1-L}\right)\Omega < 1,$$

where

$$\Omega = \frac{|\delta|\Gamma(\rho+q)}{\Gamma(q)\Gamma(\rho+r)}\mathcal{B}(q,\rho+r)\sum\_{i=1}^{m}b\_{i}(\phi(\xi\_{i})-\phi(0))^{\rho+r+q-1} + \frac{\mathcal{B}(q,r)}{\Gamma(r)}(\phi(T)-\phi(0))^{r}.\tag{36}$$

**Theorem 4.** *Let* 0 < *r* < 1*,* 0 ≤ *p* ≤ 1 *and q* = *r* + *p* − *rp. Suppose that the hypotheses* (*A*1)*,* (*A*3) *and* (*A*4) *are satisfied. Then, problem* (5) *has a unique solution in the space* <sup>C</sup>*r*,*<sup>p</sup>* <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>].

**Proof.** Define the operator *<sup>F</sup>* : <sup>C</sup>1−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] → C1−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] by

$$\begin{split} (Fz)(t) &= \frac{\delta \Gamma(\rho+q)}{\Gamma(q)\Gamma(\rho+r)} (\phi(t) - \phi(0))^{q-1} \sum\_{i=1}^{m} b\_i \int\_{0^+}^{\mathbb{T}\_i} \phi'(s) (\phi(\xi\_i) - \phi(s))^{\rho+r-1} T\_z(s) ds \\ &+ \frac{1}{\Gamma(r)} \int\_{0^+}^{t} \phi'(s) (\phi(t) - \phi(s))^{r-1} T\_z(s) ds, \end{split} \tag{37}$$

then, clearly the operator *<sup>F</sup>* is well-defined. Let *<sup>z</sup>*1, *<sup>z</sup>*<sup>2</sup> ∈ C*r*,*<sup>p</sup>* <sup>1</sup>−*q*;*φ*[<sup>J</sup> , <sup>R</sup>] and *<sup>t</sup>* ∈ J , then, we have

$$\begin{split} & \left| \langle (Fz\_1)(t) - (Fz\_2)(t) \rangle (\phi(t) - \phi(0))^{1-q} \right| \\ & \leq \frac{|\delta| \Gamma(\rho + q)}{\Gamma(q) \Gamma(\rho + r)} \sum\_{i=1}^{m} b\_i \int\_{0^+}^{\overline{z}\_i} \phi'(s) (\phi(\zeta\_i) - \phi(s))^{\rho + r - 1} |T\_{z\_1}(s) - T\_{z\_2}(s)| ds \\ & \quad + \frac{1}{\Gamma(r)} (\phi(t) - \phi(0))^{1-q} \int\_{0^+}^{t} \phi'(s) (\phi(t) - \phi(s))^{r - 1} |T\_{z\_1}(s) - T\_{z\_2}(s)| ds \end{split} \tag{38}$$

and

$$\begin{split} |T\_{z\_1}(t) - T\_{z\_2}(t)| &= |f(t, z\_1(t), z\_1(\gamma t)), T\_{z\_1}(t) - f(t, z\_2(t), z\_2(\gamma t), T\_{z\_2}(t))| \\ &\le K(|z\_1(t) - z\_2(t)| + |z\_1(\gamma t) - z\_2(\gamma t)|) + L|(T\_{z\_1})(t) - (T\_{z\_2})(t)| \\ &\le \left(\frac{2K}{1 - L}\right)|z\_1(t) - z\_2(t)|. \end{split} \tag{39}$$

Thus, by substituting Equation (39) in Equation (38), we obtain

$$\begin{split} & \left| \langle (Fz\_1)(t) - (Fz\_2)(t) \rangle (\phi(t) - \phi(0))^{1-q} \right| \\ & \leq \frac{|\delta| \Gamma(\rho+q)}{\Gamma(q) \Gamma(\rho+r)} \sum\_{i=1}^{m} b\_i \left( \frac{2K}{(1-L)} \int\_{0+}^{\tilde{z}\_i} \phi'(s) (\phi(\xi\_i) - \phi(s))^{\rho+r-1} ds \right) \| z\_1(t) - z\_2(t) \|\_{\mathcal{C}1-q\phi} \\ & \quad + \frac{1}{\Gamma(r)} (\phi(t) - \phi(0))^{1-q} \frac{2K}{(1-L)} \left( \int\_{0+}^{t} \phi'(s) (\phi(t) - \phi(s))^{r-1} ds \right) \| z\_1(t) - z\_2(t) \|\_{\mathcal{C}1-q\phi} \\ & \leq \frac{2K}{(1-L)} \left( \frac{|\delta| \Gamma(\rho+q)}{\Gamma(q) \Gamma(\rho+r)} \mathcal{B}(q, \rho+r) \sum\_{i=1}^{m} b\_i (\phi(\tilde{z}\_i) - \phi(0))^{\rho+r+q-1} \\ & \quad + \frac{\mathcal{B}(q,r)}{\Gamma(r)} (\phi(T) - \phi(0))^{r} \right) \| z\_1(t) - z\_2(t) \|\_{\mathcal{C}1-q\phi} . \end{split} \tag{40}$$

Also,

$$\begin{split} \| \| (Fz\_1) - (Fz\_2) \| \|\_{\mathcal{L}\_{1-q\phi}} &\leq \frac{2K}{(1-L)} \left( \frac{|\delta| \Gamma(\rho+q)}{\Gamma(q) \Gamma(\rho+r)} \mathcal{B}(q,\rho+r) \sum\_{i=1}^{m} b\_i (\phi(\xi\_i) - \phi(0))^{\rho+r+q-1} \\ &+ \frac{\mathcal{B}(q,r)}{\Gamma(r)} (\phi(T) - \phi(0))^r \right) \| z\_1(t) - z\_2(t) \|\_{\mathcal{L}\_{1-q\phi}}. \end{split} \tag{41}$$

It follows from hypotheses (*A*4) that *F* is a contraction map. Therefore, by Banach contraction principle, we can conclude that problem (5) has a unique solution.
