*3.1. The Case λ*<sup>1</sup> < *λ*<sup>2</sup>

From Theorem 1 in [23], we have the solution of the FOS (2) is the solution of the following system

$$\begin{split} \mathbf{x}(t) &= \; \mathbb{V}(0) - \mathbb{C}\mathbb{I}(-\boldsymbol{\zeta}(0)) \frac{t^{\lambda\_2-\lambda\_1}}{\Gamma(\lambda\_2-\lambda\_1+1)} + \frac{1}{\Gamma(\lambda\_2-\lambda\_1)} \int\_0^t (t-s)^{\lambda\_2-\lambda\_1-1} \mathbb{C}\mathbf{x}(s-\boldsymbol{\zeta}(s)) ds \\ &+ \quad \frac{1}{\Gamma(\lambda\_2)} \int\_0^t (t-s)^{\lambda\_2-1} \Big[ B\_0 \mathbf{x}(s) + B\_1 \mathbf{x}(s-\boldsymbol{\zeta}(s)) \\ &+ \quad B\_2 \boldsymbol{\nu}(s) + F(s, \mathbf{x}(s), \mathbf{x}(s-\boldsymbol{\zeta}(s)), \boldsymbol{\nu}(s)) \Big] ds, \; 0 \le t \le T\_{\text{\textquotedblleft}} \end{split}$$

*x*(*t*) = *ζ*(*t*), −*ς* ≤ *t* ≤ 0.

**Theorem 2.** *The FOS (2) is FTS w.r.t.* {*γ*1, *γ*2, , *T*}*, γ*<sup>1</sup> < *γ*<sup>2</sup> *if there exist η*1, *η*<sup>2</sup> > 0*, such that*

$$G(\gamma\_1, \varrho) \le \gamma\_2. \tag{5}$$

*where*

$$\begin{aligned} G(\gamma\_1, \varrho) &= \begin{pmatrix} \delta + c\_1 E\_{\lambda\_2 - \lambda\_1} \left( (c + \eta\_1) T^{\lambda\_2 - \lambda\_1} \right) E\_{\lambda\_2} \left( (b\_0 + b\_1 + \eta\_2) T^{\lambda\_2} \right) \end{pmatrix} \gamma\_1 \\ &+ \ c\_2 E\_{\lambda\_2 - \lambda\_1} \left( (c + \eta\_1) T^{\lambda\_2 - \lambda\_1} \right) E\_{\lambda\_2} \left( (b\_0 + b\_1 + \eta\_2) T^{\lambda\_2} \right) \varrho, \end{aligned} \tag{6}$$

$$\begin{split} \delta &= 1 + c \frac{\tau^{\lambda\_2 - \lambda\_1}}{\Gamma(\lambda\_2 - \lambda\_1 + 1)}, c\_1 = \frac{1}{(1 - \eta)} \left( \frac{c \delta M\_1}{\Gamma(\lambda\_2 - \lambda\_1 + 1)} + \frac{b\_0 \delta M\_2}{\Gamma(\lambda\_2 + 1)} + \frac{b\_1 \delta M\_2}{\Gamma(\lambda\_2 + 1)} \right), \\ c\_2 &= \frac{b\_2 M\_2}{(1 - \eta) \Gamma(\lambda\_2 + 1)}, M\_1 = \sup\_{\tau \in [0, T]} \left( \frac{\tau^{\lambda\_2 - \lambda\_1}}{E\_{\lambda\_2 - \lambda\_1} \left( (c + \eta\_1) \tau^{\lambda\_2 - \lambda\_1} \right)} \right), \\ M\_2 &= \sup\_{\tau \in [0, T]} \left( \frac{\tau^{\lambda\_2}}{E\_{\lambda\_2} \left( (b\_0 + b\_1 + \eta\_2) \tau^{\lambda\_2} \right)} \right) \text{and } \eta = \left( \frac{c}{c + \eta\_1} + \frac{b\_0 + b\_1}{b\_0 + b\_1 + \eta\_2} \right). \end{split}$$

**Proof.** Let *<sup>ζ</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> [−*ς*, 0], <sup>R</sup>*<sup>q</sup>* , such that *ζ* ≤ *γ*1. Let F = *C* [−*ς*, *<sup>T</sup>*], <sup>R</sup>*<sup>q</sup>* and consider the metric *β* on F by

$$\beta(y\_1, y\_2) = \inf \left\{ r \in [0, \infty] : \|y\_1(t) - y\_2(t)\| \le r\emptyset(t), \forall t \in [-\emptyset, T] \right\},$$

where *<sup>g</sup>* is given by *<sup>g</sup>*(*τ*) = *<sup>E</sup>λ*2−*λ*<sup>1</sup> (*c* + *η*1)*τλ*2−*λ*<sup>1</sup> *Eλ*<sup>2</sup> (*b*<sup>0</sup> + *b*<sup>1</sup> + *η*2)*τλ*<sup>2</sup> for *τ* ∈ [0, *T*] and *g*(*τ*) = 1, for *τ* ∈ [−*ς*, 0].

We consider the operator: D : F→F, such that

$$\begin{aligned} (\mathcal{D}X)(w) &= \, \_\zeta \zeta(0) - \mathcal{C}\zeta(-\mathfrak{c}(0)) \frac{w^{\lambda\_2 - \lambda\_1}}{\Gamma(\lambda\_2 - \lambda\_1 + 1)} \\ &+ \frac{1}{\Gamma(\lambda\_2 - \lambda\_1)} \int\_0^w (w - s)^{\lambda\_2 - \lambda\_1 - 1} \mathcal{C}X(s - \mathfrak{c}(s)) \, ds \\ &+ \frac{1}{\Gamma(\lambda\_2)} \int\_0^w (w - s)^{\lambda\_2 - 1} \Big[ B\_0 X(s) + B\_1 X(s - \mathfrak{c}(s)) \\ &+ B\_2 v(s) + F(s, X(s), X(s - \mathfrak{c}(s)), v(s)) \Big] \, ds, \end{aligned} \tag{7}$$

for *w* ∈ [0, *T*] and (D*X*)(*w*) = *ζ*(*w*), for *w* ∈ [−*ς*, 0].

Note that, D is well defined, (F, *β*) is a generalized complete metric space, *β*(D*X*0, *X*0) < ∞, and {*X*<sup>1</sup> ∈ F : *β*(*X*0, *X*1) < ∞} = F, ∀*X*<sup>0</sup> ∈ F.

Let *X*1, *X*<sup>2</sup> ∈ F, for *w* ∈ [−*ς*, 0], we get (D*X*1)(*w*) − (D*X*2)(*w*) = 0. For *w* ∈ [0, *T*], we have

 (D*X*1)(*w*) <sup>−</sup> (D*X*2)(*w*) ≤ *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−*λ*1−<sup>1</sup> <sup>Γ</sup>(*λ*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*1) *<sup>c</sup>X*1(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) <sup>−</sup> *<sup>X</sup>*2(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*))*dr* + *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> Γ(*λ*2) *<sup>f</sup>*(*r*) + *B*0 *X*1(*r*) − *X*2(*r*) + *f*(*r*) + *B*1 *X*1(*r* − *ς*(*r*)) − *X*2(*r* − *ς*(*r*)) *dr* ≤ *c w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−*λ*1−<sup>1</sup> <sup>Γ</sup>(*λ*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*1) *X*1(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) <sup>−</sup> *<sup>X</sup>*2(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*))*dr* +*b*<sup>0</sup> *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> *X*1(*r*) <sup>−</sup> *<sup>X</sup>*2(*r*) <sup>Γ</sup>(*λ*2) *dr* +*b*<sup>1</sup> *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> *X*1(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) <sup>−</sup> *<sup>X</sup>*2(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) <sup>Γ</sup>(*λ*2) *dr*. (8)

Then,

 (D*X*1)(*w*) <sup>−</sup> (D*X*2)(*w*) ≤ *c w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−*λ*1−<sup>1</sup> Γ(*λ*<sup>2</sup> − *λ*1) *X*1(*r* − *ς*(*r*)) − *X*2(*r* − *ς*(*r*)) *<sup>g</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) *<sup>g</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*))*dr* + *b*0 Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> *X*1(*r*) <sup>−</sup> *<sup>X</sup>*2(*r*) *<sup>g</sup>*(*r*) *<sup>g</sup>*(*r*)*dr* + *b*1 Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> *X*1(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) <sup>−</sup> *<sup>X</sup>*2(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) *<sup>g</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*)) *<sup>g</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*))*dr* ≤ *cβ*(*X*1, *X*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−*λ*1−<sup>1</sup> <sup>Γ</sup>(*λ*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*1) *<sup>g</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*))*dr* + *b*0*β*(*X*1, *X*2) Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> *<sup>g</sup>*(*r*)*dr* + *b*1*β*(*X*1, *X*2) Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> *<sup>g</sup>*(*<sup>r</sup>* <sup>−</sup> *<sup>ς</sup>*(*r*))*dr*.

Therefore,

$$\begin{split} \left\|(\mathcal{D}X\_{1})(w)-(\mathcal{D}X\_{2})(w)\right\| &\leq \ &c\beta(X\_{1},X\_{2})\int\_{0}^{w}\frac{(w-\tau)^{\lambda\_{2}-\lambda\_{1}-1}}{\Gamma(\lambda\_{2}-\lambda\_{1})}g(\tau)d\tau\\ &+\ &\frac{(b\_{0}+b\_{1})\beta(X\_{1},X\_{2})}{\Gamma(\lambda\_{2})}\int\_{0}^{w}(w-\tau)^{\lambda\_{2}-1}g(\tau)d\tau\\ &\leq &c\beta(X\_{1},X\_{2})E\_{\lambda\_{2}}\left((b\_{0}+b\_{1}+\eta\_{2})w^{\lambda\_{2}}\right)\\ &\times\ &\int\_{0}^{w}\frac{(w-\tau)^{\lambda\_{2}-\lambda\_{1}-1}}{\Gamma(\lambda\_{2}-\lambda\_{1})}E\_{\lambda\_{2}-\lambda\_{1}}\left((c+\eta\_{1})\tau^{\lambda\_{2}-\lambda\_{1}}\right)d\tau\\ &+\ &(b\_{0}+b\_{1})\beta(X\_{1},X\_{2})E\_{\lambda\_{2}-\lambda\_{1}}\left((c+\eta\_{1})w^{\lambda\_{2}-\lambda\_{1}}\right)\\ &\times\ &\int\_{0}^{w}\frac{(w-\tau)^{\lambda\_{2}-1}}{\Gamma(\lambda\_{2})}E\_{\lambda\_{2}}\left((b\_{0}+b\_{1}+\eta\_{2})\tau^{\lambda\_{2}}\right)d\tau. \end{split}$$

Using Remark 1, we get

$$\begin{split} \left\|(\mathcal{D}X\_1)(w) - (\mathcal{D}X\_2)(w)\right\| &\leq \ &\frac{c}{c+\eta\_1}\beta(X\_1, X\_2)\varrho(w) + \frac{b\_0}{b\_0+b\_1+\eta\_2}\beta(X\_1, X\_2)\varrho(w) \\ &+ \ &\frac{b\_1}{b\_0+b\_1+\eta\_2}\beta(X\_1, X\_2)\varrho(w) \\ &\leq &(\frac{c}{c+\eta\_1} + \frac{b\_0+b\_1}{b\_0+b\_1+\eta\_2})\beta(X\_1, X\_2)\varrho(w). \end{split} \tag{9}$$

Then,

$$\frac{\left\| \left( \mathcal{D}X\_1 \right)(w) - \left( \mathcal{D}X\_2 \right)(w) \right\|}{g(w)} \le \left( \frac{c}{c + \eta\_1} + \frac{b\_0 + b\_1}{b\_0 + b\_1 + \eta\_2} \right) \beta(X\_1, X\_2).$$

Thus,

$$
\beta(\mathcal{D}X\_1, \mathcal{D}X\_2) \le (\frac{c}{c+\eta\_1} + \frac{b\_0 + b\_1}{b\_0 + b\_1 + \eta\_2}) \beta(X\_1, X\_2).
$$

Therefore, D is contractive.

Let *x*<sup>0</sup> be the function given by *x*0(*τ*) = *ζ*(*τ*), for *τ* ∈ [−*ς*, 0] and *x*0(*τ*) = *ζ*(0) − *Cζ* − *ς*(0) *<sup>τ</sup>λ*2−*λ*<sup>1</sup> <sup>Γ</sup>(*λ*2−*λ*1+1) for *<sup>τ</sup>* <sup>∈</sup> [0, *<sup>T</sup>*].

Then, we have

$$\|\varkappa\_0(\pi)\| \le \left(\|\zeta\| + c\|\zeta\|\right) \frac{T^{\lambda\_2 - \lambda\_1}}{\Gamma(\lambda\_2 - \lambda\_1 + 1)}\lambda\_1$$

for all *τ* ∈ [−*ς*, *T*]. For *τ* ∈ [−*ς*, 0], we get (D*x*0)(*τ*) − *x*0(*τ*) = 0. For *w* ∈ [0, *T*], we have

 (D*x*0)(*w*) <sup>−</sup> *<sup>x</sup>*0(*w*) ≤ *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>s</sup>*)*λ*2−*λ*1−<sup>1</sup> <sup>Γ</sup>(*λ*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*1) *<sup>c</sup>x*<sup>0</sup> *s* − *ς*(*s*) *ds* + 1 Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>s</sup>*)*λ*2−<sup>1</sup> *b*0*x*0(*s*) + *b*1*x*<sup>0</sup> *s* − *ς*(*s*) + *b*<sup>2</sup> *ds* ≤ *c ζ* <sup>+</sup> *<sup>c</sup>ζ <sup>T</sup>λ*2−*λ*<sup>1</sup> Γ(*λ*<sup>2</sup> − *λ*<sup>1</sup> + 1) *<sup>w</sup>λ*2−*λ*<sup>1</sup> Γ(*λ*<sup>2</sup> − *λ*<sup>1</sup> + 1) + *b*0 *ζ* <sup>+</sup> *<sup>c</sup>ζ <sup>T</sup>λ*2−*λ*<sup>1</sup> Γ(*λ*<sup>2</sup> − *λ*<sup>1</sup> + 1) + *b*<sup>1</sup> *ζ* <sup>+</sup> *<sup>c</sup>ζ <sup>T</sup>λ*2−*λ*<sup>1</sup> Γ(*λ*<sup>2</sup> − *λ*<sup>1</sup> + 1) + *b*<sup>2</sup> *wλ*<sup>2</sup> Γ(*λ*<sup>2</sup> + 1) <sup>≤</sup> *<sup>c</sup>ζ<sup>δ</sup> <sup>w</sup>λ*2−*λ*<sup>1</sup> Γ(*λ*<sup>2</sup> − *λ*<sup>1</sup> + 1) + *b*0*ζδ* + *b*1*ζδ* + *b*<sup>2</sup> *<sup>w</sup>λ*<sup>2</sup> Γ(*λ*<sup>2</sup> + 1) . (10)

Then

$$\begin{aligned} \left\| \frac{\left\| (\mathcal{D}\mathbf{x}\_{0})(w) - \mathbf{x}\_{0}(w) \right\|}{\operatorname\*{g}(w)} \right\| & \leq \frac{c \|\zeta\| \|\delta M\_{1}}{\Gamma(\lambda\_{2} - \lambda\_{1} + 1)} \\ & + \quad \left( b\_{0} \|\zeta\| \delta + b\_{1} \|\zeta\| \delta + b\_{2} \varrho \right) \frac{M\_{2}}{\Gamma(\lambda\_{2} + 1)}, \end{aligned} \tag{11}$$

for all *w* ∈ [0, *T*].

Therefore,

$$\begin{array}{rcl}\beta(\mathcal{D}\mathbf{x}\_{0\prime}\mathbf{x}\_{0}) & \leq & \frac{c\|\boldsymbol{\xi}\|\delta\mathbf{M}\_{1}}{\Gamma(\lambda\_{2}-\lambda\_{1}+1)}\\ &+ & \left(b\mathbf{o}\|\boldsymbol{\xi}\|\delta+b\_{1}\|\boldsymbol{\xi}\|\delta+b\_{2}\boldsymbol{\varrho}\right)\frac{\mathbf{M}\_{2}}{\Gamma(\lambda\_{2}+1)}.\end{array} \tag{12}$$

It follows from Theorem 1 that there is a unique solution *x* of (2) with initial conditions of *ζ*, such that

$$\begin{split} \beta(x\_0, x) &\leq \ \frac{1}{1 - \eta} \Big[ \frac{c \|\zeta\| \|\delta M\_1}{\Gamma(\lambda\_2 - \lambda\_1 + 1)} \\ &+ \ \left( b\_0 \|\|\zeta\| \delta + b\_1 \|\|\zeta\| \delta + b\_2 \varrho \right) \frac{M\_2}{\Gamma(\lambda\_2 + 1)} \Big] \\ &\leq \ c\_1 \gamma\_1 + c\_2 \varrho. \end{split} \tag{13}$$

Therefore,

$$\|\mathbf{x}\_0(t) - \mathbf{x}(t)\| \le \left(c\_1\gamma\_1 + c\_2\varrho\right) E\_{\lambda\_2-\lambda\_1}((c+\eta\_1)T^{\lambda\_2-\lambda\_1}) E\_{\lambda\_2}((b\_0+b\_1+\eta\_2)T^{\lambda\_2}),$$

for every *t* ∈ [0, *T*].

Then,

$$\begin{split} \|\mathbf{x}(t)\| &\leq \quad \|\mathbf{x}\_{0}(t)\| + \|\mathbf{x}(t) - \mathbf{x}\_{0}(t)\| \\ &\leq \quad \left(\delta + c\_{1}E\_{\lambda\_{2}-\lambda\_{1}}\left((c+\eta\_{1})T^{\lambda\_{2}-\lambda\_{1}}\right)E\_{\lambda\_{2}}\left((b\_{0}+b\_{1}+\eta\_{2})T^{\lambda\_{2}}\right)\right)\gamma\_{1} \\ &+ \quad c\_{2}E\_{\lambda\_{2}-\lambda\_{1}}\left((c+\eta\_{1})T^{\lambda\_{2}-\lambda\_{1}}\right)E\_{\lambda\_{2}}\left((b\_{0}+b\_{1}+\eta\_{2})T^{\lambda\_{2}}\right)\varrho\_{\prime} \end{split} \tag{14}$$

for every *t* ∈ [0, *T*].

Thus, *x*(*t*) ≤ *γ*2, for all *t* ∈ [0, *T*], if (5) is satisfied.

**Remark 2.** *Using Lemma 1, we get*

$$c\_1 \le \frac{1}{(1-\eta)} \left( \frac{c\delta}{c+\eta\_1} + \frac{b\_0\delta}{b\_0+b\_1+\eta\_2} + \frac{b\_1\delta}{b\_0+b\_1+\eta\_2} \right).$$

*and*

$$c\_2 \le \frac{1}{(1-\eta)} \frac{b\_2}{b\_0 + b\_1 + \eta\_2}.$$

*Let*

$$\vec{\varepsilon}\_{1} = \frac{1}{(1-\eta)} \left( \frac{c\delta}{c+\eta\_{1}} + \frac{b\_{0}\delta}{b\_{0}+b\_{1}+\eta\_{2}} + \frac{b\_{1}\delta}{b\_{0}+b\_{1}+\eta\_{2}} \right) \vec{\varepsilon}\_{1}$$

*and*

$$\tilde{c}\_2 = \frac{1}{(1-\eta)} \frac{b\_2}{b\_0 + b\_1 + \eta\_2}$$

*Therefore, the condition (5) can be relaxed by:*

$$\mathcal{G}(\gamma\_1, \varrho) \le \gamma\_2. \tag{15}$$

.

*where*

$$\begin{split} G(\gamma\_1, \varrho) &= \quad \left(\delta + \overline{\varepsilon}\_1 E\_{\lambda\_2 - \lambda\_1} \left( (c + \eta\_1) T^{\lambda\_2 - \lambda\_1} \right) E\_{\lambda\_2} \left( (b\_0 + b\_1 + \eta\_2) T^{\lambda\_2} \right) \right) \gamma\_1 \\ &+ \quad \overline{\varepsilon}\_2 E\_{\lambda\_2 - \lambda\_1} \left( (c + \eta\_1) T^{\lambda\_2 - \lambda\_1} \right) E\_{\lambda\_2} \left( (b\_0 + b\_1 + \eta\_2) T^{\lambda\_2} \right) \varrho. \end{split} \tag{16}$$

*3.2. The Case λ*<sup>1</sup> = *λ*<sup>2</sup> The solution of the FOS (2) is the solution of

$$\begin{aligned} \mathbf{x}(t) &= \; \zeta(0) + \mathcal{C} \Big( \mathbf{x}(t - \zeta(t)) - \zeta(-\zeta(0)) \Big) + \frac{1}{\Gamma(\lambda\_2)} \int\_0^t (t - s)^{\lambda\_2 - 1} \Big[ B\_0 \mathbf{x}(s) + B\_1 \mathbf{x}(s - \zeta(s)) \Big] ds \\ &+ \quad B\_2 \mathbf{v}(s) + F(s, \mathbf{x}(s), \mathbf{x}(s - \zeta(s)), \mathbf{v}(s)) \Big] ds, \; 0 \le t \le T, \\\\ \mathbf{x}(t) &= \zeta(t), -\zeta \le t \le 0. \end{aligned}$$

**Theorem 3.** *The FOS (2) is FTS w.r.t.* {*γ*1, *γ*2, , *T*}*, γ*<sup>1</sup> < *γ*<sup>2</sup> *if there exist θ* > 0*, such that*

*η* < 1,

*b*<sup>0</sup> + *b*<sup>1</sup> + *θ*

*and*

$$K(\gamma\_1, \varrho) \le \gamma\_2. \tag{17}$$

$$h\_0 + \underbrace{b\_0 + b\_1}\_{\text{max}}$$

*where*

$$K(\gamma\_1, \varrho) \quad = \left(1 + c\_1 E\_{\lambda\_2} \left( (b\_0 + b\_1 + \theta) T^{\lambda\_2} \right) \right) \gamma\_1$$

$$+ \quad c\_2 E\_{\lambda\_2} \left( (b\_0 + b\_1 + \theta) T^{\lambda\_2} \right) \big| \varrho \,. \tag{18}$$

 ,

$$\begin{aligned} c\_1 &= \frac{1}{(1-\eta)} \left( 2c + \frac{b\_0 M}{\Gamma(\lambda\_2 + 1)} + \frac{b\_1 M}{\Gamma(\lambda\_2 + 1)} \right), c\_2 = \frac{b\_2 M}{(1-\eta)\Gamma(\lambda\_2 + 1)} \text{ and } c\_1 \\ M &= \sup\_{\tau \in [0, T]} \left( \frac{\tau^{\lambda\_2}}{E\_{\lambda\_2} \left( (b\_0 + b\_1 + \theta) \tau^{\lambda\_2} \right)} \right). \end{aligned}$$

*η* = *c* +

**Proof.** Let *<sup>ζ</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> [−*ς*, 0], <sup>R</sup>*<sup>q</sup>* , such that *ζ* ≤ *γ*1. Let F = *C* [−*ς*, *<sup>T</sup>*], <sup>R</sup>*<sup>q</sup>* and consider the metric *β* on F by

$$\beta(y\_1, y\_2) = \inf \left\{ r \in [0, \infty] : \frac{||y\_1(l) - y\_2(l)||}{g(l)} \le r, \forall l \in [-\emptyset, T] \right\},$$

where *g* is given by *g*(*l*) = 1, for *l* ∈ [−*ς*, 0] and *g*(*l*) = *Eλ*<sup>2</sup> (*b*<sup>0</sup> + *b*<sup>1</sup> + *θ*)*l λ*2 for *l* ∈ [0, *T*]. We consider the operator: D : F→F, such that

$$\begin{aligned} (\mathcal{D}X)(w) &= \, \_\zeta \zeta(0) + \mathcal{C} \Big( X(w - \zeta(w)) - \zeta(-\zeta(0)) \Big) \\ &+ \, \frac{1}{\Gamma(\lambda\_2)} \int\_0^w (w - s)^{\lambda\_2 - 1} \Big[ B\_0 X(s) + B\_1 X(s - \zeta(s)) \\ &+ \, \quad B\_2 v(s) + F(s, X(s), X(s - \zeta(s)), v(s)) \Big] ds, \tag{19} \end{aligned} \tag{10}$$

for *w* ∈ [0, *T*] and (D*X*)(*w*) = *ζ*(*w*), for *w* ∈ [−*ς*, 0]. Note that, D is well defined, (F, *β*) is a generalized complete metric space, *β*(D*X*0, *X*0) < ∞, and {*X*<sup>1</sup> ∈ F : *β*(*X*0, *X*1) < ∞} = F, ∀*X*<sup>0</sup> ∈ F. Let *X*1, *X*<sup>2</sup> ∈ F, for *w* ∈ [−*ς*, 0], we get (D*X*1)(*w*) − (D*X*2)(*w*) = 0.

For *w* ∈ [0, *T*], we have

 (D*X*1)(*w*) <sup>−</sup> (D*X*2)(*w*) ≤ *cX*1(*w* − *ς*(*w*)) − *X*2(*w* − *ς*(*w*)) + *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>r</sup>*)*λ*2−<sup>1</sup> Γ(*λ*2) *<sup>f</sup>*(*r*) + *B*0 *X*1(*r*) − *X*2(*r*) + *f*(*r*) + *B*1 *X*1(*r* − *ς*(*r*)) − *X*2(*r* − *ς*(*r*)) *dr* ≤ *c X*1(*w* − *ς*(*w*)) − *X*2(*w* − *ς*(*w*)) *g w* − *ς*(*w*) *<sup>g</sup> w* − *ς*(*w*) +*b*<sup>0</sup> *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>u</sup>*)*λ*2−<sup>1</sup> Γ(*λ*2) *X*1(*u*) − *X*2(*u*) *<sup>g</sup>*(*u*) *<sup>g</sup>*(*u*)*du* +*b*<sup>1</sup> *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>u</sup>*)*λ*2−<sup>1</sup> Γ(*λ*2) *X*1(*u* − *ς*(*u*)) − *X*2(*u* − *ς*(*u*)) *g u* − *ς*(*u*) *<sup>g</sup> u* − *ς*(*u*) *du* ≤ *cβ*(*X*1, *X*2)*g w* − *ς*(*w*) + *b*0*β*(*X*1, *X*2) Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>u</sup>*)*λ*2−<sup>1</sup> *<sup>g</sup>*(*u*)*du* + *b*1*β*(*X*1, *X*2) Γ(*λ*2) *w* 0 (*<sup>w</sup>* <sup>−</sup> *<sup>u</sup>*)*λ*2−<sup>1</sup> *<sup>g</sup>*(*u*)*du*. (20)

Using Remark 1, we get

$$\begin{split} \left\|(\mathcal{D}X\_1)(w) - (\mathcal{D}X\_2)(w)\right\| &\leq \left\|c\beta(X\_1, X\_2)\varrho(w) + \frac{b\_0}{b\_0 + b\_1 + \theta}\beta(X\_1, X\_2)\varrho(w)\right\| \\ &+ \frac{b\_1}{b\_0 + b\_1 + \theta}\beta(X\_1, X\_2)\varrho(w) \\ &\leq \left(c + \frac{b\_0 + b\_1}{b\_0 + b\_1 + \theta}\right)\beta(X\_1, X\_2)\varrho(w). \end{split} \tag{21}$$

Then,

$$\left\| \frac{\left\| (\mathcal{D}X\_1)(w) - (\mathcal{D}X\_2)(w) \right\|}{g(w)} \le (c + \frac{b\_0 + b\_1}{b\_0 + b\_1 + \theta}) \beta(X\_1, X\_2)\_{\theta} \right\|$$

Thus,

$$
\beta(\mathcal{D}X\_1, \mathcal{D}X\_2) \le \left(c + \frac{b\_0 + b\_1}{b\_0 + b\_1 + \theta}\right) \beta(X\_1, X\_2).
$$

Therefore, D is contractive.

Let *x*<sup>0</sup> be the function given by *x*0(*τ*) = *ζ*(*τ*), for *τ* ∈ [−*ς*, 0] and *x*0(*τ*) = *ζ*(0) for *τ* ∈ [0, *T*].

Then, we have

$$\|\mathbf{x}\_0(\pi)\| \le \|\mathbf{\tilde{y}}\|\_{\mathsf{H}}$$

for all *t* ∈ [−*ς*, *T*]. For *τ* ∈ [−*ς*, 0], we get (D*x*0)(*τ*) − *x*0(*τ*) = 0. For *w* ∈ [0, *T*], we have

$$\begin{split} \left\|(\mathcal{D}\mathbf{x}\_{0})(\boldsymbol{w}) - \mathbf{x}\_{0}(\boldsymbol{w})\right\| &\quad \leq \ 2c\|\boldsymbol{\zeta}\| \\ &\quad + \ \frac{1}{\Gamma(\lambda\_{2})} \int\_{0}^{\boldsymbol{w}} (\boldsymbol{w} - \boldsymbol{s})^{\lambda\_{2}-1} \left[b\_{0} \|\mathbf{x}\_{0}(\boldsymbol{s})\| + b\_{1} \|\mathbf{x}\_{0}(\boldsymbol{s} - \boldsymbol{\zeta}(\boldsymbol{s}))\| + b\_{2}\boldsymbol{\varrho}\right] ds \\ &\leq \ 2c\|\boldsymbol{\zeta}\| + \frac{\boldsymbol{w}^{\lambda\_{2}}}{\Gamma(\lambda\_{2}+1)} \Big(b\_{0} \|\boldsymbol{\zeta}\| + b\_{1} \|\boldsymbol{\zeta}\| + b\_{2}\boldsymbol{\varrho}\Big). \end{split} \tag{22}$$

Then,

$$\begin{array}{rcl} \left\|(\mathcal{D}\mathbf{x}\_{0})(w) - \mathbf{x}\_{0}(w)\right\| & & \\ \hline \\ & & \mathcal{G}(w) \\ & & & \\ & & & \left(b\_{0}\|\zeta\| + b\_{1}\|\zeta\| + b\_{2}\varrho\right) \frac{\mathcal{M}}{\Gamma(\lambda\_{2} + 1)} \end{array} \tag{23}$$

for all *w* ∈ [0, *T*]. Therefore,

$$\begin{aligned} \left(\mathcal{C}(\mathcal{D}x\_0, x\_0)\right) &\leq \ 2c||\mathcal{G}||\\ &+ \quad \left(b\_0||\mathcal{G}|| + b\_1||\mathcal{G}|| + b\_2\varrho\right) \frac{M}{\Gamma(\lambda\_2 + 1)}.\end{aligned} \tag{24}$$

Theorem 1 implies that (2) has a unique solution *x* with initial conditions of *ζ*, such that

$$\begin{split} \beta(x\_0, x) &\leq \ \frac{1}{1-\eta} \left[ 2c \|\zeta\| \right] \\ &+ \ \left( b\_0 \|\zeta\| + b\_1 \|\zeta\| + b\_2 \varrho \right) \frac{\mathcal{M}}{\Gamma(\lambda\_2 + 1)} \Big] \\ &\leq \ c\_1 \gamma\_1 + c\_2 \varrho. \end{split} \tag{25}$$

Therefore,

$$\|\|\mathbf{x}\_0(t) - \mathbf{x}(t)\|\| \le \left(c\_1\gamma\_1 + c\_2\varrho\right)E\_{\lambda\_2}\left((b\mathbf{0} + b\_1 + \theta)T^{\lambda\_2}\right),$$

for all *t* ∈ [0, *T*]. Then,

$$\begin{split} \|\mathbf{x}(t)\| &\leq \quad \|(\mathbf{x} - \mathbf{x}\_{0})(t)\| + \|\mathbf{x}\_{0}(t)\| \\ &\leq \quad \left(1 + c\_{1}E\_{\lambda\_{2}}\left((b\_{0} + b\_{1} + \theta)T^{\lambda\_{2}}\right)\right)\gamma\_{1} \\ &+ \quad c\_{2}E\_{\lambda\_{2}}\left((b\_{0} + b\_{1} + \theta)T^{\lambda\_{2}}\right)\varrho. \end{split} \tag{26}$$

Thus, *x*(*t*) ≤ *γ*2, for all *t* ∈ [0, *T*], if (17) is satisfied.

**Remark 3.** *Using Lemma 1, we get*

$$c\_1 \le \frac{1}{(1-\eta)} \left( 2c + \frac{b\_0}{b\_0 + b\_1 + \theta} + \frac{b\_1}{\theta + b\_1 + b\_0} \right)$$

*and*

$$c\_2 \le \frac{1}{(1-\eta)} \frac{b\_2}{b\_0 + b\_1 + \theta}.$$

*Let us consider*

$$\mathcal{E}\_1 = \frac{1}{(1-\eta)} \left( 2c + \frac{b\_0}{\theta + b\_1 + b\_0} + \frac{b\_1}{b\_0 + b\_1 + \theta} \right)$$

*and*

$$\vec{\varepsilon}\_2 = \frac{1}{(1-\eta)} \frac{b\_2}{\theta + b\_1 + b\_0}$$

*Therefore, the condition (17) can be relaxed by:*

$$\mathcal{K}(\gamma\_1, \varrho) \le \gamma\_{2'} \tag{27}$$

.

*where*

$$\begin{aligned} \mathcal{R}(\gamma\_1, \varrho) &= \left(1 + \mathfrak{E}\_1 E\_{\lambda\_2} \left( (b\_0 + b\_1 + \theta) T^{\lambda\_2} \right) \right) \gamma\_1 \\ &+ \ \mathfrak{E}\_2 E\_{\lambda\_2} \left( (b\_0 + b\_1 + \theta) T^{\lambda\_2} \right) \varrho. \end{aligned} \tag{28}$$

**Remark 4.** *In the Theorem 3, c* < 1 *it is a necessary condition.*

**Remark 5.** *In the case when C* = 0*, we get the results in [21].*
