**2. Basic Results**

**Definition 1** ([31])**.** *Given* 0 < *χ* < 1*. The CFD is given by,*

$${}^{C}D\_{a}^{\chi}g(s) = \frac{1}{\Gamma(1-\chi)}\frac{d}{ds}\int\_{a}^{s}(s-\omega)^{-\chi}(g(\omega)-g(a))d\omega. \tag{1}$$

**Definition 2** ([31])**.** *The MLF is defined by :*

$$E\_{\mathcal{X}}(s) = \sum\_{q=0}^{+\infty} \frac{s^q}{\Gamma(q\chi + 1)},$$

*with <sup>χ</sup>* <sup>&</sup>gt; <sup>0</sup>*, s* <sup>∈</sup> <sup>C</sup>*.*

**Lemma 1** ([21])**.** *We have for s* ≥ 0

$$\frac{s^{\chi}}{E\_{\chi}(\lambda s^{\chi})} \le \frac{\Gamma(\chi+1)}{\lambda} \gamma$$

*where* 0 < *χ* < 1 *and λ* > 0*.*

**Remark 1.** *The function d*(*t*) = *E<sup>χ</sup> <sup>b</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*χ satisfies CD<sup>χ</sup> <sup>a</sup> <sup>d</sup>*(*t*) = *bd*(*t*), *where b* <sup>∈</sup> <sup>R</sup>∗*.* **Definition 3.** *A mapping β* : *B* × *B* → [0, ∞] *is called a generalized metric on a nonempty set B if:*


**Theorem 1.** *Let* (*B*, *β*) *be a generalized complete metric space. Suppose that K* : *B* → *B is contractive with <sup>k</sup>* <sup>&</sup>lt; <sup>1</sup>*. If there is an integer <sup>k</sup>*<sup>0</sup> <sup>≥</sup> <sup>0</sup>*, such that <sup>β</sup>*(*Kk*0+1*b*0, *<sup>K</sup>k*<sup>0</sup> *<sup>b</sup>*0) <sup>&</sup>lt; <sup>∞</sup> *for some b*<sup>0</sup> ∈ *B, so:*


We consider the following system:

$$\begin{aligned} \, ^C D\_0^{\lambda\_2} x(t) - \, ^C D\_0^{\lambda\_1} x(t - \xi(t)) &= \, ^B B\_0 x(t) + B\_1 x(t - \xi(t)) \\ &+ B\_2 v(t) + F(t, x(t), x(t - \xi(t)), v(t)), t \ge 0, \end{aligned} \tag{2}$$

with the initial condition *x*(*s*) = *ζ*(*s*) for −*ς* ≤ *s* ≤ 0, with 0 < *λ*<sup>1</sup> ≤ *λ*<sup>2</sup> < 1, *ς*(*t*) is continuous, 0 <sup>≤</sup> *<sup>ς</sup>*(*t*) <sup>≤</sup> *<sup>ς</sup>*, *<sup>υ</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>p</sup>* is the disturbance, *<sup>ζ</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup> [−*ς*, 0], <sup>R</sup>*<sup>q</sup>* , *<sup>C</sup>* <sup>∈</sup> <sup>R</sup>*q*×*q*, *<sup>B</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*q*×*<sup>q</sup> <sup>B</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>*q*×*q*, *<sup>B</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>*q*×*p*.

The function *F* is continuous and satisfies:

$$\|\|F(\tau, \sigma\_1, \sigma\_2, \sigma\_3) - F(\tau, \psi\_1, \psi\_2, \psi\_3)\|\| \le f(\tau) \left( \|\sigma\_1 - \psi\_1\| + \|\sigma\_2 - \psi\_2\| + \|\sigma\_3 - \psi\_3\| \right), \tag{3}$$

and *<sup>F</sup>*(*τ*, 0, 0, 0) = 0, for all (*τ*, *<sup>σ</sup>*1, *<sup>σ</sup>*2, *<sup>σ</sup>*3, *<sup>ψ</sup>*1, *<sup>ψ</sup>*2, *<sup>ψ</sup>*3) <sup>∈</sup> <sup>R</sup><sup>+</sup> <sup>×</sup> <sup>R</sup>*<sup>q</sup>* <sup>×</sup> <sup>R</sup>*<sup>q</sup>* <sup>×</sup> <sup>R</sup>*<sup>p</sup>* <sup>×</sup> <sup>R</sup>*<sup>q</sup>* <sup>×</sup> <sup>R</sup>*<sup>q</sup>* <sup>×</sup> <sup>R</sup>*<sup>p</sup>* where *f* is a continuous function.

The function *υ* is continuous and satisfies:

$$\exists \varrho > 0 : \quad \upsilon^T(t)\upsilon(t) \le \varrho^2. \tag{4}$$

**Definition 4.** *The FOS (2) possesses FTS w.r.t.* {*γ*1, *γ*2, , *T*}*, γ*<sup>1</sup> < *γ*<sup>2</sup> *if*

*ζ* ≤ *γ*1,

*implies:*

*x*(*t*) ≤ *γ*2, ∀*t* ∈ [0, *T*], *for all υ satisfying (4), where ζ* = sup *τ*∈[−*ς*,0] *ζ*(*τ*)*.*
