Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems

Abstract

Fractional multi-delay differential systems, as an important mathematical model, can effectively describe viscoelastic materials and non-local delay responses in ecosystem population dynamics. It has a wide and profound application background in interdisciplinary fields such as physics, biomedicine, and intelligent control. Through literature review and classification, it is evident that for the fractional multi-delay differential system, the existence and uniqueness of the solution and Ulam-Hyers stability (UHS), Ulam-Hyers-Rassias stability (UHRS) of the fractional multi-delay differential system are rarely studied by using the multi-delayed perturbation of two parameter Mittag-Leffler typematrix function. In this paper, we first establish the existence and uniqueness of the solution for the Riemann-Liouville fractional multi-delay differential system on finite intervals using the Banach and Schauder fixed point theorems. Second, we demonstrate the existence and uniqueness of the solution for the system on the unbounded intervals in the weighted function space. Furthermore, we investigate UHS and UHRS for the nonlinear fractional multi-delay differential system in unbounded intervals. Finally, numerical examples are provided to validate the key theoretical results.

1. Introduction

Fractional calculus, a mathematical theory generalizing differentiation and integration to arbitrary orders, extends classical integer-order calculus. Its nonlocality implies that the derivative or integral of a function at a point depends not only on the local value but also on the global behavior of the function. Consequently, fractional differential systems (FDSs) are particularly effective in modeling complex systems with memory and hereditary characteristics, such as viscoelastic materials and biological phenomena [1]. As a pivotal branch of modern differential theory, fractional calculus provides unique advantages in practical application modeling. These advantages have driven its widespread adoption across disciplines, including engineering, biomedicine, electrical engineering, and communication systems [2,3,4]. The inherent precision and flexibility of FDSs not only improve model accuracy but also offer novel perspectives for problem analysis and solution design.

As a fundamental characteristic of dynamical systems, time delays are universally present in both natural and engineering systems. These delays are primarily reflected in the significant temporal lag between the feedback of system output signals and the input signals. Delay effects are prevalent in processes such as energy transfer, material transport, and information transmission. The presence of delays can alter the original properties of a system; for example, they can make the system unstable and induce oscillations or chaotic behavior [5]. Therefore, delays have become critical factors in diverse fields, including control engineering, biological systems, economic management, and medicine [6,7,8].

Fractional delay differential systems (FDDSs) extend classical FDSs through the introduction of delay terms. This extension not only preserves the fractional derivatives but also accounts for the influence of historical states on the current system state. Such systems have been shown to be particularly effective in simulating memory relaxation behaviors in viscoelastic materials and nonlocal delayed responses in ecological population dynamics, offering unique advantages for analyzing processes with hereditary characteristics and propagation delays. Compared to ordinary differential systems (ODSs), FDDSs provide a more rigorous mathematical framework for characterizing real-world dynamical behaviors in biological systems and engineering applications [9,10,11]. In the framework of the qualitative theory of FDDSs, the main research include the explicit solution representation and the basic analytical properties, including the existence, uniqueness, stability and controllability of the solution [12,13,14,15].

In 2012, Medved and Pospíšil [16] studied the following multi-delay differential systems:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\nu }^{\prime }(t)=A\nu (t)+B_1\nu (t-h_1)+B_2\nu (t-h_2)+\cdots +B_N\nu (t-h_N)+f(t),t⩾0,\hfill \\ \nu (t)=\psi (t),-h⩽t⩽0,\hfill \end{array}\right.\end{array}\end{array}$$

(1)

where $A, B_m\in {\mathbb{R}}^{n\times n}$ are constant matrices that pairwise commute, $h_m>0$, $m=1, 2, \cdots ,N$, $\psi \in C^1([-h,0],{\mathbb{R}}^n)$, and $h:=max\{h_1,\cdots ,h_N\}$. In [16], the authors pioneered a multi-delay matrix exponential approach to derive an explicit solution for system (1). It is worth mentioning that the author overcomes the difficulty of constructing the fundamental solution matrix caused by multiple delays.

In 2017–2018, Li and Wang [17,18] constructed the Mittag-Leffler type matrix polynomial function with fractional delay and studied its related properties. Based on these properties, they subsequently provided the explicit solutions for the Caputo type linear homogeneous FDDS and the Riemann-Liouville type FDDS.

In 2022, Mahmudov [19] obtained the multi-delayed perturbation of two parameter Mittag-Leffler type matrix function by introducing the concept of multivariate determining matrix equation, which extends the classical Mittag-Leffler type matrix functions and delayed Mittag-Leffler type matrix functions. Then he presented an explicit analytical solution for the following system of order $l<\alpha \le l-1$:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }y)(t)=By(t)+\sum _{i=1}^d{B}_{i}y(t-{\vartheta }_i)+g(t),t\in (0,T],{\vartheta }_i>0,\hfill \\ & {(}^{RL}{D}_{0^{+}}^{\alpha -j}y)(t){|}_{t=0}=\phi (0)=q_j,j=1,2,\cdots ,l,\hfill \\ & y(t)=\phi (t),-\vartheta \le t<0,\hfill \end{array}\right.\end{array}\end{array}$$

(2)

where ${}^{RL}D_{0^{+}}^{\alpha }$ denotes the Riemann-Liouville fractional derivative, $l\in \mathbb{N}$, $T>0$, $B,B_1$, $\cdots ,B_d\in {\mathbb{R}}^{n\times n}$, $\vartheta :=max\{{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_d\}$, $\phi :[-\vartheta ,0]\to {\mathbb{R}}^n$ an arbitrary differentiable function that determines initial conditions and $g\in C((0,T],{\mathbb{R}}^n)$.

The stability of differential systems is a crucial research topic in mathematics, spanning multiple disciplines and playing a key role in practical applications. Ulam-type stability, with its theoretical depth and practical value, has attracted significant academic attention in recent years. In 1940, Ulam [20] first put forward a question about the stability of functional equations at a conference at the University of Wisconsin. The first answer to the question of Ulam was given by Hyers [21] in 1941 in Banach space. In 1978, Rassias [22] weakened the boundedness condition of Cauchy difference and obtained the UHRS, which marked a new breakthrough. Recently, Wang [23] applied the Henry-Gronwall inequality to establish four types of Ulam stability results for Caputo FDS over the infinite time intervals. In [24], Ali investigated the controllability and generalized UHS of variable coefficient linear Riemann-Liouville type FDS in Banach space. Existence and uniqueness of solutions of the initial value problem for nonlinear second order impulsive differential equations have been established in [25] and the authors considered UHRS of the system. For more details, refer to the relevant studies [26,27,28,29,30].

Motivated by [19,25], we study the existence, uniqueness, and Ulam-type stability of a fractional differential system with multi-delay:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }y)(t)=By(t)+\sum _{i=1}^d{B}_{i}y(t-{\vartheta }_i)+g(t,y(t)),t>0,{\vartheta }_i>0,\hfill \\ & {(}^{RL}{D}_{0^{+}}^{\alpha -1}y)(t){|}_{t=0}=\phi (0)=y_0,\hfill \\ & y(t)=\phi (t),-\vartheta \le t<0,\hfill \end{array}\right.\end{array}\end{array}$$

(3)

where $0<\alpha <1$, $\phi \in C([-\vartheta ,0],{\mathbb{R}}^n)$, $g\in C(J\times {\mathbb{R}}^n,{\mathbb{R}}^n)$ and $J:=(0,\infty )$.

By ([19], Theorem 4.2), the solution $y(t)$ of system (3) can be expressed as

$$y(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell ,$$

(4)

where $Y_{\alpha ,\alpha }(t)$ is defined as (5) when $\beta =\alpha$.

The organization of this paper is as follows. In Section 2, we introduce necessary notations and definitions and establish some vital lemmas. In Section 3, We present the existence and uniqueness of the solution for the fractional multi-delay differential system in the $C_{\gamma }$ space. In Section 4, the UHS and UHRS of the system (3) are discussed in the weighted space $C_{\omega }$. In Section 5, we provide some examples to illustrate the main theorems.

2. Preliminaries

Set $x\in {\mathbb{R}}^n$ and $A\in {\mathbb{R}}^{n\times n}$, we introduce vector norm $\parallel x\parallel ={\sum }_{i=1}^n\left|x_i\right|$ and matrix norm $\parallel A\parallel ={max}_{1\le j\le n}{\sum }_{i=1}^n\left|a_{ij}\right|$. Let $C([a,b],{\mathbb{R}}^n)$ be the Banach space of continuous function with $\parallel y\parallel ={max}_{a\le t\le b}\parallel y(t)\parallel$. For any $0<\gamma <1$, we denote $C_{\gamma }([a,b],{\mathbb{R}}^n)=\{y\in C((a,b],{\mathbb{R}}^n):{(·-a)}^{\gamma }y(·)\in C([a,b],{\mathbb{R}}^n)\}$. Then $C_{\gamma }([a,b],{\mathbb{R}}^n)$ is a Banach space with ${\parallel y\parallel }_{C_{\gamma }}={max}_{a\le t\le b}\parallel {(t-a)}^{\gamma }y(t)\parallel$. Let $C_{\omega }(J,{\mathbb{R}}^n)$ be the Banach space composed of all bounded functions on $C(J,{\mathbb{R}}^n)$, and define the norm ${\parallel ·\parallel }_{C_{\omega }}$ as ${\parallel y\parallel }_{C_{\omega }}:=sup\{e^{-\omega t}\parallel y(t)\parallel ,t\in J\}$, where $\omega \in \mathbb{R}$ is a given constant. Denote $\lambda =\parallel B\parallel +{\sum }_{j=1}^d\parallel B_j\parallel$.

Definition 1

(see [9]). For a function $y:[a,\infty )\to \mathbb{R}$, its Riemann-Liouville derivative of order $0<\alpha <1$ can be defined as

$$\begin{array}{c}\hfill {(}^{RL}{D}_{a^{+}}^{\alpha }y)(x)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dx}}{\int }_a^x{(x-t)}^{-\alpha }y(t)dt,x>a.\end{array}$$

Definition 2

(see [9]). For a function $y:[a,\infty )\to \mathbb{R}$, its fractional integral of order $0<\alpha <1$ can be defined as

$$\begin{array}{c}\hfill (I_{a^{+}}^{\alpha }y)(x)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^x{(x-t)}^{\alpha -1}y(t)dt,x>a.\end{array}$$

Definition 3

(see [19]). The coefficient matrices $Q_k(s_1,\cdots ,s_d)$, $k=1,2,\cdots$, satisfies the following multivariate determining matrix equation

$$\begin{array}{ccc}\hfill Q_{k+1}(s_1,\cdots ,s_d)& =& B{Q}_k(s_1,\cdots ,s_d)+\sum _{i=1}^d{B}_i{Q}_k(s_1,\cdots ,s_i-{\vartheta }_i,\cdots ,s_d),\hfill \\ \hfill Q_0(s_1,\cdots ,s_d)& =& Q_k(-{\vartheta }_1,\cdots ,s_d)=\cdots =Q_k(s_1,\cdots ,-{\vartheta }_d)=\Theta ,Q_1(0,\cdots ,0)=I,\hfill \\ \hfill k& =& 0,1,2,\cdots ,and{s}_i=0,{\vartheta }_i,2{\vartheta }_i,\cdots ,\hfill \end{array}$$

where I is an identity matrix, Θ is a zero matrix.

In [19], the author introduced a shift operator ${\mathcal{T}}^{\vartheta }(\vartheta \in \mathbb{R})$ which takes a function f to its translation:

$$\begin{array}{c}\hfill {\mathcal{T}}^{\vartheta }f(t)=e^{\vartheta {\displaystyle \frac{d}{dt}}}f:=f(t+\vartheta ).\end{array}$$

We now recall the multi-delayed perturbation of two parameter Mittag-Leffler type matrix function $Y_{\alpha ,\beta }(t)$ by using the shift operator ${\mathcal{T}}^{\vartheta }$.

Definition 4

(see [19]). Let $\alpha ,\beta >0$. The multi-delayed perturbation of two parameter Mittag-Leffler type matrix function $Y_{\alpha ,\beta }(·):\mathbb{R}\to {\mathbb{R}}^{n\times n}$ generated by $B,B_1,\cdots ,B_d$ is defined by

$$\begin{array}{c}\hfill Y_{\alpha ,\beta }(t)=\left\{\begin{array}{c}\Theta ,t\in [-\vartheta ,0),\hfill \\ {\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \sum _{\begin{array}{c}{i}_1+\cdots +i_d\le k\\ i_1,\cdots ,i_d\ge 0\end{array}}}{Q}_{k+1}(i_1{\vartheta }_1,i_2{\vartheta }_2,\cdots ,i_d{\vartheta }_d)e^{-(i_1{\vartheta }_1+\cdots +i_d{\vartheta }_d){\displaystyle \frac{d}{dt}}}{\displaystyle \frac{{(t)}_{+}^{k\alpha +\beta -1}}{\Gamma (k\alpha +\beta )}},t\in [0,\infty ),\hfill \end{array}\right.\end{array}$$

(5)

where ${(·)}_{+}:=max\{0,·\}$. It follows from that $Q_{k+1}(i_1{\vartheta }_1,i_2{\vartheta }_2,\cdots ,i_d{\vartheta }_d)=\Theta$, if $i_1+\cdots +i_d\ge k+1$, $i_1,\cdots ,i_d\ge 0$.

Definition 5

(see [9]). Let $\alpha ,\beta >0$, the Mittag-Leffler function of two parameters $E_{\alpha ,\beta }({·}^{\alpha })$ is defined by

$$E_{\alpha ,\beta }(z^{\alpha })=\sum _{k=0}^{\infty }{\displaystyle \frac{z^{k\alpha }}{\Gamma (k\alpha +\beta )}},z\in \mathbb{C}.$$

Lemma 1.

For any $t\in (0,T]$, $0<\alpha <1$, $\beta >0$ and $\alpha +\beta \ge 1$, one has

$$\parallel Y_{\alpha ,\beta }(t)\parallel \le t^{\beta -1}{E}_{\alpha ,\beta }(\lambda t^{\alpha }).$$

Proof. 

According to (5) and ([19], Remark 3.5), we have

$$\parallel Y_{\alpha ,\beta }(t)\parallel \le {\displaystyle \sum _{k=0}^{\infty }}\sum _{\begin{array}{c}{i}_0+i_1+\dots +i_d=k\\ i_0,i_1,\dots ,i_d\ge 0\end{array}}{\displaystyle \frac{k!}{i_0!i_1!\dots i_d!}}{\parallel B\parallel }^{i_0}{\displaystyle \prod _{j=1}^d}\parallel B_j^{i_j}\parallel {\displaystyle \frac{{\left(t-{\displaystyle \sum _{j=1}^d}{i}_j{\vartheta }_j\right)}_{+}^{k\alpha +\beta -1}}{\Gamma (k\alpha +\beta )}}.$$

For any $h_i>0$ and $i_j\ge 0$, we obtain

$$\begin{array}{ccc}\hfill \parallel Y_{\alpha ,\beta }(t)\parallel & \le & {\displaystyle \sum _{k=0}^{\infty }}\sum _{\begin{array}{c}{i}_0+i_1+\dots +i_d=k\\ i_0,i_1,\dots ,i_d\ge 0\end{array}}{\displaystyle \frac{k!}{i_0!i_1!\dots i_d!}}{\parallel B\parallel }^{i_0}{\displaystyle \prod _{j=1}^d}{\parallel B_j\parallel }^{i_j}{\displaystyle \frac{{(t)}_{+}^{k\alpha +\beta -1}}{\Gamma (k\alpha +\beta )}}\hfill \\ & \le & {\displaystyle \sum _{k=0}^{\infty }}(\parallel B\parallel +{\displaystyle \sum _{j=1}^d}\parallel B_j{\parallel )}^k{\displaystyle \frac{t^{k\alpha +\beta -1}}{\Gamma (k\alpha +\beta )}}=t^{\beta -1}{E}_{\alpha ,\beta }(\lambda t^{\alpha }).\hfill \end{array}$$

The proof is completed. □

Remark 1.

Let $\beta =1$, one has $\parallel Y_{\alpha ,1}(t)\parallel \le E_{\alpha ,1}(\lambda t^{\alpha })$. Let ${\displaystyle \frac{1}{2}}\le \alpha =\beta <1$, one has $\parallel Y_{\alpha ,\alpha }(t)\parallel \le t^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })$.

Lemma 2.

For any $t>0$ and ${\displaystyle \frac{1}{2}}\le \alpha <1$, one has

$$\begin{array}{ccc}\hfill {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel d\ell & \le & t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha }),\hfill \\ \hfill {\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}\parallel Y_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )\parallel d\ell & \le & \left\{\begin{array}{c}{t}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha })-{(t-{\vartheta }_i)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t-{\vartheta }_i)}^{\alpha }),t\ge {\vartheta }_i,\hfill \\ t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha }),t<{\vartheta }_i.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Let $0\le \ell \le t$ and $0\le t-\ell <t$, we have

$$\begin{array}{ccc}\hfill {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel d\ell & \le & {\int }_0^t\sum _{k=0}^{\infty }{\lambda }^k{\displaystyle \frac{{(t-\ell )}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}d\ell \le {\int }_0^t\sum _{k=0}^{\infty }{\lambda }^k{\displaystyle \frac{{(t-\ell )}^{k\alpha +\alpha -1}}{\Gamma (\alpha k+\alpha )}}d\ell \hfill \\ & \le & \sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k}{\Gamma (k\alpha +\alpha )}}{\int }_0^t{(t-\ell )}^{k\alpha +\alpha -1}d\ell \le \sum _{k=0}^{\infty }{\lambda }^k{\displaystyle \frac{t^{k\alpha +\alpha }}{\Gamma (k\alpha +\alpha +1)}}\hfill \\ & \le & t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha }).\hfill \end{array}$$

Let $t\ge {\vartheta }_i$, we have $min(t-{\vartheta }_i,0)=0$, which deduce $0\le t-{\vartheta }_i\le t-{\vartheta }_i-\ell \le t$. Thus

$$\begin{array}{ccc}\hfill {\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}\parallel Y_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )\parallel d\ell & =& {\int }_{-{\vartheta }_i}^0\parallel Y_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )\parallel d\ell \hfill \\ & \le & {\int }_{-{\vartheta }_i}^0\sum _{k=0}^{\infty }{\lambda }^k{\displaystyle \frac{{(t-{\vartheta }_i-\ell )}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}d\ell \hfill \\ & \le & \sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k}{\Gamma (k\alpha +\alpha )}}{\int }_{-{\vartheta }_i}^0{(t-{\vartheta }_i-\ell )}^{k\alpha +\alpha -1}d\ell \hfill \\ & =& \sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{t}^{k\alpha +\alpha }}{\Gamma (k\alpha +\alpha +1)}}-\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{(t-{\vartheta }_i)}^{k\alpha +\alpha }}{\Gamma (k\alpha +\alpha +1)}}\hfill \\ & =& t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha })-{(t-{\vartheta }_i)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t-{\vartheta }_i)}^{\alpha }).\hfill \end{array}$$

Similarly, when $t<{\vartheta }_i$, we gain

$${\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}\parallel Y_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )\parallel d\ell ={\int }_{-{\vartheta }_i}^{t-{\vartheta }_i}\parallel Y_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )\parallel d\ell \le t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha }).$$

The proof is completed. □

Lemma 3

(See [31]). For any $\alpha \in (0,1)$, the following propositions hold.

(i)

For all $z>0$, we have

$$E_{\alpha }(z)={\displaystyle \frac{1}{\alpha }}exp(z^{{\displaystyle \frac{1}{\alpha }}})+{\int }_0^{\infty }{K}_{\alpha }(r,z)dr,$$

where

$$K_{\alpha }(r,z)=-{\displaystyle \frac{sin(\pi \alpha )}{\pi \alpha }}{\displaystyle \frac{zexp(-r^{\frac{1}{\alpha }})}{r^2-2rzcos(\pi \alpha )+z^2}}.$$

(ii)

For all $z>0$, we have

$$E_{\alpha ,\alpha }(z)={\displaystyle \frac{1}{\alpha }}{z}^{{\displaystyle \frac{1}{\alpha }}-1}exp(z^{{\displaystyle \frac{1}{\alpha }}})+{\int }_0^{\infty }{K}_{\alpha ,\alpha }(r,z)dr.$$

(iii)

For all $z<0$, we have

$$E_{\alpha ,\alpha }(z)={\int }_0^{\infty }{K}_{\alpha ,\alpha }(r,z)dr,$$

where

$$K_{\alpha ,\alpha }(r,z)={\displaystyle \frac{sin(\pi \alpha )}{\pi \alpha }}{\displaystyle \frac{r^{\frac{1}{\alpha }}exp(-r^{\frac{1}{\alpha }})}{r^2-2rzcos(\pi \alpha )+z^2}}.$$

Proof. 

For the proof process, please refer to the reference [32]. □

Lemma 4.

Let $\lambda >0$, for any $\alpha \in (0,1)$ and $t>0$, we have

$$|t^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })-{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}exp({\lambda }^{{\displaystyle \frac{1}{\alpha }}}t)|\le {\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^2\pi sin(\pi \alpha )}}{t}^{-\alpha -1}.$$

Proof. 

According to Lemma 3, we have

$$t^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })={\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}+t^{\alpha -1}{\int }_0^{\infty }{\displaystyle \frac{sin(\pi \alpha )}{\pi \alpha }}{\displaystyle \frac{r^{\frac{1}{\alpha }}exp(-r^{\frac{1}{\alpha }})}{r^2-2r\lambda t^{\alpha }cos(\pi \alpha )+z^2}}dr.$$

Utilizing the following inequality

$$r^2-2r\lambda t^{\alpha }cos(\pi \alpha )+{\lambda }^2{t}^{2\alpha }\ge (1-{cos}^2(\pi \alpha )){\lambda }^2{t}^{2\alpha }={sin}^2(\pi \alpha ){\lambda }^2{t}^{2\alpha },$$

we have

$$\begin{array}{ccc}\hfill |t^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })-{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}|& =& t^{\alpha -1}{\int }_0^{\infty }\left|{\displaystyle \frac{sin(\pi \alpha )}{\pi \alpha }}{\displaystyle \frac{r^{\frac{1}{\alpha }}exp(-r^{\frac{1}{\alpha }})}{r^2-2r\lambda t^{\alpha }cos(\pi \alpha )+t^2}}\right|dr\hfill \\ & \le & {\displaystyle \frac{t^{\alpha -1}sin(\pi \alpha )}{\pi \alpha }}{\int }_0^{\infty }\left|{\displaystyle \frac{r^{\frac{1}{\alpha }}exp(-r^{\frac{1}{\alpha }})}{{sin}^2(\pi \alpha ){\lambda }^2{t}^{2\alpha }}}\right|dr\hfill \\ & \le & {\displaystyle \frac{t^{\alpha -1}sin(\pi \alpha )}{\pi \alpha {sin}^2(\pi \alpha ){\lambda }^2{t}^{2\alpha }}}{\int }_0^{\infty }{r}^{{\displaystyle \frac{1}{\alpha }}}exp(-r^{{\displaystyle \frac{1}{\alpha }}})dr\hfill \\ & =& {\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^2\pi sin(\pi \alpha )}}{t}^{-\alpha -1}.\hfill \end{array}$$

□

Lemma 5.

For any $t>0$ and ${\displaystyle \frac{1}{2}}\le \alpha <1$, we have

$${\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel d\ell \le {\displaystyle \frac{1}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}.$$

Proof. 

For $0\le \ell \le t$, we can deduce $0\le t-\ell <t$. By using Lemma 4, we obtain

$$\begin{array}{ccc}\hfill {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel d\ell & \le & {\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })d\ell \hfill \\ & \le & {\int }_0^t|{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })-{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}(t-\ell )}+{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}(t-\ell )}|d\ell \hfill \\ & \le & {\int }_0^t\left({\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^2\pi sin(\pi \alpha )}}{(t-\ell )}^{-\alpha -1}+{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}(t-\ell )}\right)d\ell \hfill \\ & \le & -t^{-\alpha }{\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^2\pi \alpha sin(\pi \alpha )}}+{\displaystyle \frac{1}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}-{\displaystyle \frac{1}{\alpha \lambda }}\hfill \\ & \le & {\displaystyle \frac{1}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}.\hfill \end{array}$$

□

3. Existence and Uniqueness Results

In this section, we mainly prove the existence and uniqueness of the solution of the system in $C_{\gamma }$ space. For every $t\in [-\vartheta ,T]$, we define

$$\Psi (t)=\left\{\begin{array}{c}{\displaystyle \sum _{i=1}^d}\parallel B_i\parallel t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha }),t<{\vartheta }_i,\hfill \\ {\displaystyle \sum _{i=1}^d}\parallel B_i\parallel (t^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda t^{\alpha })-{(t-{\vartheta }_i)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t-{\vartheta }_i)}^{\alpha })),t\ge {\vartheta }_i.\hfill \end{array}\right.$$

Consider the following assumptions:

$\left[H_1\right]$ There exists a constant $L>0$ such that

$$\parallel g(t,y)-g(t,z)\parallel \le L\parallel y-z\parallel ,t\in (0,T],y,z\in {\mathbb{R}}^n.$$

$\left[H_2\right]$ $\eta =L{T}^{\alpha }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\mathbb{B}[1-\gamma ,\alpha ]<1$.

Let ${\mathfrak{B}}_r=\{y\in C_{\gamma }((0,T],{\mathbb{R}}^n){:\parallel y\parallel }_{C_{\gamma }}\le r,r\ge {\displaystyle \frac{\mu }{1-\eta }}\}$, where $\mu =T^{\alpha +\gamma -1}{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\parallel y_0\parallel +T^{\gamma }\Psi (T)\parallel \phi \parallel +T^{\alpha +\gamma }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\parallel \tilde{g}\parallel$ and $\parallel \tilde{g}\parallel :=\underset{t>0}{sup}\parallel g(t,0)\parallel <\infty$. Now, we use Schauder fixed point theorem to establish an existence theorem.

Theorem 1.

If $\alpha +\gamma >1$, $\left[H_1\right]$ and $\left[H_2\right]$ are held, then (3) has at least one solution $y\in C_{\gamma }((0,],{\mathbb{R}}^n)$.

Proof. 

Define an operator $\mathcal{F}$ on ${\mathfrak{B}}_r$ as follows:

$$(\mathcal{F}y)(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell ,$$

(6)

Step 1.

We show that $\mathcal{F}({\mathfrak{B}}_r)\subset {\mathfrak{B}}_r$. For any $t\in (0,T]$ and $y\in {\mathfrak{B}}_r$, by Lemmas 1 and 2, we have

$$\begin{array}{ccc}& & \parallel t^{\gamma }(\mathcal{F}y)(t)\parallel \hfill \\ & \le & \parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t)y_0\parallel +\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )\parallel d\ell +{\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & \le & \parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t)\parallel \parallel y_0\parallel +\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )\parallel \parallel B_i\parallel \parallel \phi (\ell )\parallel d\ell \hfill \\ & & +{\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel \parallel g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & \le & t^{\alpha +\gamma -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\parallel y_0\parallel +t^{\gamma }\Psi (t)\parallel \phi \parallel +{\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel (\parallel g(\ell ,y(\ell ))-g(t,\mathbf{0})\parallel +\parallel g(\ell ,\mathbf{0})\parallel )d\ell \hfill \\ & \le & t^{\alpha +\gamma -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\parallel y_0\parallel +t^{\gamma }\Psi (t)\parallel \phi \parallel +{\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel (L\parallel y(\ell )\parallel +\parallel \tilde{g}\parallel )\hfill \\ & \le & t^{\alpha +\gamma -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\parallel y_0\parallel +t^{\gamma }\Psi (t)\parallel \phi \parallel +L{t}^{\gamma }{E}_{\alpha ,\alpha }(\lambda t^{\alpha }){\int }_0^t{(t-\ell )}^{\alpha -1}{\ell }^{-\gamma }\parallel {\ell }^{\gamma }y(\ell )\parallel d\ell \hfill \\ & & +t^{\alpha +\gamma }{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\parallel \tilde{g}\parallel \hfill \\ & \le & t^{\alpha +\gamma -1}{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\parallel y_0\parallel +t^{\gamma }\Psi (t)\parallel \phi \parallel +L{t}^{\alpha }{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\mathbb{B}[1-\gamma ,\alpha ]{\parallel y\parallel }_{C_{\gamma }}+t^{\alpha +\gamma }{E}_{\alpha ,\alpha }(\lambda t^{\alpha })\parallel \tilde{g}\parallel \hfill \\ & \le & T^{\alpha +\gamma -1}{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\parallel y_0\parallel +T^{\gamma }\Psi (T)\parallel \phi \parallel +T^{\alpha +\gamma }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\parallel \tilde{g}\parallel +L{T}^{\alpha }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\mathbb{B}[1-\gamma ,\alpha ]{\parallel y\parallel }_{C_{\gamma }}\hfill \\ & \le & \mu +\eta r\le r.\hfill \end{array}$$

Step 2.

We prove $\mathcal{F}$ is continuous.

Let $y_n$ be a Cauchy sequence with $y_n\to y$ in ${\mathfrak{B}}_r$. For each $t\in (0,T]$,

$$\begin{array}{ccc}\hfill \parallel t^{\gamma }((\mathcal{F}{y}_n)(t)-(\mathcal{F}y)(t))\parallel & \le & {\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel \parallel g(\ell ,y_n(\ell ))-g(t,y(\ell ))\parallel d\ell \hfill \\ & \le & {\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel L\parallel y_n(\ell )-y(\ell )\parallel d\ell \hfill \\ & \le & L{t}^{\gamma }{E}_{\alpha ,\alpha }(\lambda t^{\alpha }){\int }_0^t{(t-\ell )}^{\alpha -1}{\ell }^{-\gamma }\parallel {\ell }^{\gamma }(y_n(\ell )-y(\ell ))\parallel d\ell \hfill \\ & \le & L{t}^{\alpha }{E}_{\alpha ,\alpha }(\lambda t^{\alpha })B[1-\gamma ,\alpha ]{\parallel y_n-y\parallel }_{C_{\gamma }}\hfill \\ & \le & L{T}^{\alpha }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })B[1-\gamma ,\alpha ]{\parallel y_n-y\parallel }_{C_{\gamma }}.\hfill \end{array}$$

This implies that $\mathcal{F}$ is continuous.

Step 3.

We show $\mathcal{F}({\mathfrak{B}}_r)$ is equicontinuous.

Without loss of generality, let $0<t_1<t_2<T$. For any $y\in {\mathfrak{B}}_r$, we have

$$\begin{array}{ccc}& & \parallel t^{\gamma }((\mathcal{F}y)(t_2)-(\mathcal{F}{y}_1)(t))\parallel \hfill \\ & \le & t^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2)-Y_{\alpha ,\alpha }(t_1)\parallel \parallel y_0\parallel \hfill \\ & & +\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t_1-{\vartheta }_i,0)}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )-Y_{\alpha ,\alpha }(t_1-{\vartheta }_i-\ell )\parallel \parallel B_i\parallel \parallel \phi (\ell )\parallel d\ell \hfill \\ & & +\sum _{i=1}^d{\int }_{min(t_1-{\vartheta }_i,0)}^{min(t_2-{\vartheta }_i,0)}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )\parallel \parallel B_i\parallel \parallel \phi (\ell )\parallel d\ell \hfill \\ & & +{\int }_0^{t_1}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-\ell )-Y_{\alpha ,\alpha }(t_1-\ell )\parallel \parallel g(\ell ,y(\ell ))\parallel d\ell +{\int }_{t_1}^{t_2}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-\ell )\parallel \parallel g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & :=& {\mathcal{I}}_1+{\mathcal{I}}_2+{\mathcal{I}}_3+{\mathcal{I}}_4+{\mathcal{I}}_5.\hfill \end{array}$$

Let $t_2\to t_1$, we obtain

$$\begin{array}{ccc}\hfill Y_{\alpha ,\alpha }(t_2)& \to & Y_{\alpha ,\alpha }(t_1),\hfill \\ \hfill Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )& \to & Y_{\alpha ,\alpha }(t_1-{\vartheta }_i-\ell ),\hfill \\ \hfill Y_{\alpha ,\alpha }(t_2-\ell )& \to & Y_{\alpha ,\alpha }(t_1-\ell ),\hfill \end{array}$$

this yields that ${\mathcal{I}}_1\to 0,{\mathcal{I}}_2\to 0$ and ${\mathcal{I}}_4\to 0$ as $t_2\to t_1$. For ${\mathcal{I}}_5$, one can get

$$\begin{array}{ccc}\hfill {\mathcal{I}}_5& =& {\int }_{t_1}^{t_2}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-\ell )\parallel \parallel g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & \le & t^{\gamma }\parallel g\parallel {\int }_{t_1}^{t_2}\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{(t_2-\ell )}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}d\ell \hfill \\ & \le & t^{\gamma }{(t_2-t_1)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t_2-t_1)}^{\alpha })\parallel g\parallel \to 0as{t}_2\to t_1,\hfill \end{array}$$

For

$${\mathcal{I}}_3=\sum _{i=1}^d{\int }_{min(t_1-{\vartheta }_i,0)}^{min(t_2-{\vartheta }_i,0)}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )\parallel \parallel B_i\parallel \parallel \phi \parallel d\ell ,$$

we need to consider the following three cases:

(i)

When $T>t_2>t_1>{\vartheta }_i>0$, We have $min(t_2-{\vartheta }_i,0)=min(t_1-{\vartheta }_i,0)=0$. Thus ${\mathcal{J}}_3=0$.

(ii)

When $T>t_2>{\vartheta }_i>t_1>0$, We have $min(t_2-{\vartheta }_i,0)=0,min(t_1-{\vartheta }_i,0)=t_1-{\vartheta }_i$. So $t_1-{\vartheta }_i<\ell <0$, which deduce $0<t_2-{\vartheta }_i<t_2-{\vartheta }_i-\ell <t_2-t_1$. Thus,

$$\begin{array}{ccc}\hfill {\mathcal{I}}_3& =& \sum _{i=1}^d{\int }_{t_1-{\vartheta }_i}^0{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )\parallel \parallel B_i\parallel \parallel \phi \parallel d\ell \hfill \\ & \le & t^{\gamma }\parallel \phi \parallel \sum _{i=1}^d\parallel B_i\parallel {\int }_{t_1-{\vartheta }_i}^0\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{(t_2-{\vartheta }_i-\ell )}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}d\ell \hfill \\ & \le & t^{\gamma }\parallel \phi \parallel \sum _{i=1}^d\parallel B_i\parallel \left({(t_2-t_1)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t_2-t_1)}^{\alpha })-{(t_2-{\vartheta }_i)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t_2-{\vartheta }_i)}^{\alpha })\right)\hfill \\ & \le & t^{\gamma }\parallel \phi \parallel \sum _{i=1}^d\parallel B_i\parallel {(t_2-t_1)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t_2-t_1)}^{\alpha })\to 0as{t}_2\to t_1.\hfill \end{array}$$

(iii)

When $T>{\vartheta }_i>t_2>t_1>0$, we have $min(t_2-{\vartheta }_i,0)=t_2-{\vartheta }_i,min(t_1-{\vartheta }_i,0)=t_1-{\vartheta }_i$. So $t_1-{\vartheta }_i<\ell <t_2-{\vartheta }_i$, which deduce $0<t_2-{\vartheta }_i-\ell <t_2-t_1$. Thus,

$$\begin{array}{ccc}\hfill {\mathcal{I}}_3& =& \sum _{i=1}^d{\int }_{t_1-{\vartheta }_i}^{t_2-{\vartheta }_i}{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )\parallel \parallel B_i\parallel \parallel \phi \parallel d\ell \hfill \\ & =& t^{\gamma }\parallel \phi \parallel \sum _{i=1}^d\parallel B_i\parallel \underset{t_1-{\vartheta }_i}{\overset{t_2-{\vartheta }_i}{\int }}\parallel Y_{\alpha ,\alpha }(t_2-{\vartheta }_i-\ell )\parallel d\ell \hfill \\ & \le & t^{\gamma }\parallel \phi \parallel \sum _{i=1}^d\parallel B_i\parallel {\int }_{t_1-{\vartheta }_i}^{t_2-{\vartheta }_i}{(t_2-{\vartheta }_i-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t_2-{\vartheta }_i-\ell )}^{\alpha })d\ell \hfill \\ & \le & t^{\gamma }\parallel \phi \parallel \sum _{i=1}^d\parallel B_i\parallel {(t_2-t_1)}^{\alpha }{E}_{\alpha ,\alpha +1}(\lambda {(t_2-t_1)}^{\alpha })\to 0as{t}_2\to t_1.\hfill \end{array}$$

As a result, we immediately obtain that $\parallel t^{\gamma }((\mathcal{F}y)(t_2)-(\mathcal{F}{y}_1)(t))\parallel \to 0$ as $t_2\to t_1$. Thus, $\mathcal{F}({\mathfrak{B}}_r)$ is equicontinuous. From the Arzela-Ascoli theorem, $\mathcal{F}({\mathfrak{B}}_r)$ is relatively compact. Schauder fixed point theorem guarantees that $\mathcal{F}$ has a fixed point in ${\mathfrak{B}}_r$.

□

Next, we use the Banach fixed point theorem to establish an uniqueness result.

Theorem 2.

If $\alpha +\gamma >1$, $\left[H_1\right]$ and $\left[H_2\right]$ are satisfied, then (3) has a unique solution $y\in C_{\gamma }((0,T],{\mathbb{R}}^n)$.

Proof. 

Consider the operator $\mathcal{F}$ defined in (6) on $C_{\gamma }((0,T],{\mathbb{R}}^n)$. For $y,z\in C_{\gamma }((0,T],{\mathbb{R}}^n)$ and each $t\in (0,T]$, one has

$$\begin{array}{ccc}\hfill \parallel t^{\gamma }((\mathcal{F}y)(t)-(\mathcal{F}z)(t))\parallel & \le & {\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel \parallel g(\ell ,y(\ell ))-g(t,z(\ell ))\parallel d\ell \hfill \\ & \le & {\int }_0^t\parallel t^{\gamma }{Y}_{\alpha ,\alpha }(t-\ell )\parallel L\parallel y(\ell )-z(\ell )\parallel d\ell \hfill \\ & \le & L{t}^{\gamma }{E}_{\alpha ,\alpha }(\lambda t^{\alpha }){\int }_0^t{(t-\ell )}^{\alpha -1}{\ell }^{-\gamma }\parallel {\ell }^{\gamma }(y(\ell )-z(\ell ))\parallel d\ell \hfill \\ & \le & L{t}^{\alpha }{E}_{\alpha ,\alpha }(\lambda t^{\alpha })B[1-\gamma ,\alpha ]{\parallel y-z\parallel }_{C_{\gamma }}\hfill \\ & \le & L{T}^{\alpha }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })B[1-\gamma ,\alpha ]{\parallel y-z\parallel }_{C_{\gamma }}.\hfill \end{array}$$

From $\left[H_2\right]$, we know that $\begin{array}{c}\hfill {\parallel \mathcal{F}y-\mathcal{F}z\parallel }_{C_{\gamma }}\le {\eta \parallel y-z\parallel }_{C_{\gamma }}<{\parallel y-z\parallel }_{C_{\gamma }}\end{array}$. Applying the contraction mapping principle, the operator $\mathcal{F}$ has one and only one fixed point which is the solution of system (3). □

4. Ulam-Type Stability Analysis

In this part, we first prove the theorem of existence and uniqueness by the Banach fixed point theorem in $C_{\omega }$. Then, we establish the Ulam-type stability results of nonlinear fractional multi-delay differential system on the infinite time interval.

4.1. Uniqueness of the Solution for System (3)

Now, the Banach fixed point theorem is utilized to establish the theorem of existence and uniqueness. For the sake of convenience, the following assumptions are given:

$\left[H_3\right]$ Let $g\in C(J\times {\mathbb{R}}^n,{\mathbb{R}}^n)$, there exists a constant $L_g>0$ such that

$$\parallel g(t,y)-g(t,z)\parallel \le L_g{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\parallel y-z\parallel ,t>0,y,z\in {\mathbb{R}}^n.$$

$\left[H_4\right]$ $L_g<\alpha \lambda$.

Theorem 3.

If $\left[H_3\right]$ and $\left[H_4\right]$ are held, then (3) has a unique solution $y\in C_{\omega }(J,{\mathbb{R}}^n)$, where $\omega ={\lambda }^{{\displaystyle \frac{1}{\alpha }}}$.

Proof. 

For all $t>0$, the operator $\mathcal{T}:C_{\omega }(J,{\mathbb{R}}^n)\to C_{\omega }(J,{\mathbb{R}}^n)$ can be written as:

$$(\mathcal{T}y)(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell ,$$

(7)

Putting $\omega ={\lambda }^{{\displaystyle \frac{1}{\alpha }}}$. For $\forall y,y^{*}\in C_{\omega }(J,{\mathbb{R}}^n)$, $t\in J$, by Lemma 5, we have

$$\begin{array}{ccc}& & \parallel (\mathcal{T}{y}^{*})(t)-(\mathcal{T}y)(t)\parallel \hfill \\ & \le & {\int }_0^t{t}^{\gamma }\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel \parallel g(\ell ,y^{*}(\ell ))-g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & \le & {\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })L_g{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }\parallel y^{*}(\ell )-y(\ell ))\parallel d\ell \hfill \\ & \le & L_g{\int }_0^t|{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })-{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}+{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}|d\ell {\parallel y^{*}-y\parallel }_{C_{\omega }}\hfill \\ & \le & L_g{\int }_0^t\left({\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^2\pi \alpha sin(\pi \alpha )}}{t}^{-\alpha -1}+{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\right)d\ell {\parallel y^{*}-y\parallel }_{C_{\omega }}\hfill \\ & \le & L_g\left(-t^{-\alpha }{\displaystyle \frac{\Gamma (\alpha +1)}{{\lambda }^2\pi {\alpha }^{2}sin(\pi \alpha )}}+{\displaystyle \frac{1}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}-{\displaystyle \frac{1}{\alpha \lambda }}\right){\parallel y^{*}-y\parallel }_{C_{\omega }}\hfill \\ & \le & {\displaystyle \frac{L_g}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}{\parallel y^{*}-y\parallel }_{C_{\omega }}.\hfill \end{array}$$

Thus,

$$e^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\parallel (\mathcal{T}{y}^{*})(t)-(\mathcal{T}y)(t)\parallel \le {\displaystyle \frac{L_g}{\alpha \lambda }}{\parallel y^{*}-y\parallel }_{C_{\omega }},\forall t>0.$$

According to $\left[H_4\right]$, the mapping $\mathcal{T}:C_{\omega }(J,{\mathbb{R}}^n)\to C_{\omega }(J,{\mathbb{R}}^n)$ is a contraction mapping. Therefore, there exists a unique fixed point of $\mathcal{T}$ in $C_{\omega }(J,{\mathbb{R}}^n)$, which corresponds to the solution of system (3). □

4.2. Ulam-Hyers Stablity

Let $\epsilon >0$ and $\tilde{J}=[-\vartheta ,0]\cup J$. Consider (3) and the following inequalities:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& \parallel B\widehat{y}(t)+\sum _{i=1}^d{B}_i\widehat{y}(t-{\vartheta }_i)+g(t,\widehat{y}(t))\parallel \le \epsilon ,t>0,\hfill \\ & D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}=\phi (0)=y_0,\hfill \\ & \widehat{y}(t)=\phi (t),-\vartheta \le t<0.\hfill \end{array}\right.\end{array}\end{array}$$

(8)

Definition 6.

The system (3) is UHS if there exists a constant $K>0$, such that for some sufficiently small $\epsilon >0$ and for each solution $\widehat{y}\in C(\tilde{J},{\mathbb{R}}^n)$ of inequality (8), there exists a unique solution $y\in C_{\omega }(\tilde{J},{\mathbb{R}}^n)$ of (3) such that

$$\begin{array}{c}\hfill \parallel \widehat{y}{(t)-y(t)\parallel }_{C_{\omega }}\le K\epsilon ,\forall t\in \tilde{J}.\end{array}$$

Remark 2.

If the function $\widehat{y}\in C(\tilde{J},{\mathbb{R}}^n)$ satisfies the inequality (8), then there exists $\mathcal{Z}\in C(J,{\mathbb{R}}^n)$ such that the following holds:

(i)

$\parallel \mathcal{Z}(t)\parallel \le \epsilon ,t>0$;

(ii)

${(}^{RL}{D}_{0^{+}}^{\alpha }\widehat{y})(t)=B\widehat{y}(t)+{\displaystyle \sum _{i=1}^d}{B}_i\widehat{y}(t-{\vartheta }_i)+g(t,\widehat{y}(t))+\mathcal{Z}(t),t>0$;

(iii)

$D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}=y_0$;

(iv)

$\widehat{y}(t)=\phi (t),t\in [-\vartheta ,0)$.

Lemma 6.

If the function $\widehat{y}\in C(\tilde{J},{\mathbb{R}}^n)$ satisfies the inequality (8), then the following inequality holds:

$$\begin{array}{ccc}& & \parallel \widehat{y}(t)-Y_{\alpha ,\alpha }(t)y_0-\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & -{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,\widehat{y}(\ell ))d\ell \parallel \le {\displaystyle \frac{\epsilon }{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t},t\in J.\hfill \end{array}$$

Proof. 

Suppose that $\widehat{y}\in C(J,{\mathbb{R}}^n)$ satisfies inequality (8). By Remark 2, it follows that satisfies the following fractional differential system:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }\widehat{y})(t)=B\widehat{y}(t)+\sum _{i=1}^d{B}_i\widehat{y}(t-{\vartheta }_i)+g(t,\widehat{y}(t))+\mathcal{Z}(t),t>0,\hfill \\ & D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}=\phi (0)=y_0,\hfill \\ & \widehat{y}(t)=\phi (t),-\vartheta \le t<0,\hfill \end{array}\right.\end{array}\end{array}$$

From Equation (4), it follows that the solution of the system is:

$$\begin{array}{ccc}\hfill \widehat{y}(t)& =& Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )(g(\ell ,\widehat{y}(\ell ))+\mathcal{Z}(\ell ))d\ell .\hfill \end{array}$$

By Lemma 5, for $\forall t\in J$, we obtain

$$\begin{array}{ccc}& & \parallel \widehat{y}(t)-Y_{\alpha ,\alpha }(t)y_0-\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & -{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,\widehat{y}(\ell ))d\ell \parallel \hfill \\ & \le & {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\mathcal{Z}(\ell )\parallel d\ell \le {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel \parallel \mathcal{Z}(\ell )\parallel d\ell \hfill \\ & \le & \epsilon {\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })d\ell \hfill \\ & \le & {\displaystyle \frac{\epsilon }{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}.\hfill \end{array}$$

□

Theorem 4.

Assume that $\left[H_3\right]$ and $\left[H_4\right]$ are satisfied, then (3) is UHS on $\tilde{J}$.

Proof. 

Let $\widehat{y}\in C(J,{\mathbb{R}}^n)$ satisfy inequality (8) and $y\in C(\tilde{J},{\mathbb{R}}^n)$ be the unique solution of (3), that is,

$$y(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell .$$

For all $t\in [-\vartheta ,0)$, we have

$$\parallel \widehat{y}(t)-y(t)\parallel =\parallel \phi (t)-\phi (t)\parallel =0<K\epsilon ,$$

and when $t=0$, we have

$$\parallel D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}-D_{0^{+}}^{\alpha -1}{y(t)|}_{t=0}\parallel =\parallel \phi (0)-\phi (0)\parallel =0<K\epsilon .$$

By Lemma 6, for any $t\in J$, one has

$$\begin{array}{ccc}& & \parallel \widehat{y}(t)-y(t)\parallel \hfill \\ & \le & \parallel \widehat{y}(t)-Y_{\alpha ,\alpha }(t)y_0-\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & -{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,\widehat{y}(\ell ))d\ell \parallel +{\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel \parallel g(\ell ,\widehat{y}(\ell ))-g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & \le & \epsilon {\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })d\ell \hfill \\ & & +{\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })e^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }\parallel \widehat{y}(\ell )-y(\ell )\parallel d\ell \hfill \\ & \le & {\displaystyle \frac{\epsilon }{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}+{\displaystyle \frac{L_g}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}{\parallel \widehat{y}-y\parallel }_{C_{\omega }}.\hfill \end{array}$$

Multiply both sides of the above equation by $e^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}$, we obtain

$$e^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\parallel \widehat{y}(t)-y(t)\parallel \le {\displaystyle \frac{\epsilon }{\alpha \lambda -L_g}}\le K\epsilon ,t\in J,$$

where

$$K={\displaystyle \frac{1}{\alpha \lambda -L_g}}.$$

□

4.3. Ulam-Hyers-Rassias Stability

Let $\epsilon >0$, $\tilde{J}=[-\vartheta ,0]\cup J$ and let $\psi \in C(J,{\mathbb{R}}^{+}:=(0,+\infty ))$ is a bounded function. Consider the system (3) and the following inequalities:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& \parallel B\tilde{y}(t)+\sum _{i=1}^d{B}_i\tilde{y}(t-{\vartheta }_i)+g(t,\tilde{y}(t))\parallel \le \epsilon \psi (t),t>0,\hfill \\ & D_{0^{+}}^{\alpha -1}\tilde{y}{(t)|}_{t=0}=\phi (0)=y_0,\hfill \\ & \tilde{y}(t)=\phi (t),-\vartheta \le t<0.\hfill \end{array}\right.\end{array}\end{array}$$

(9)

Definition 7.

The system (3) is UHRS with respect to $\psi (·)$ if there exists $\tilde{K}>0$ and function $\psi (·)$, such that for some sufficiently small $\epsilon >0$ and for each solution $\tilde{y}\in C(\tilde{J},{\mathbb{R}}^n)$ of (9), there exists a unique solution $y\in C_{\omega }(\tilde{J},{\mathbb{R}}^n)$ of (3) such that

$$\begin{array}{c}\hfill \parallel \tilde{y}{(t)-y(t)\parallel }_{C_{\omega }}\le \tilde{K}\epsilon \psi (t),\forall t\in \tilde{J},\end{array}$$

Remark 3.

If the function $\tilde{y}\in C(\tilde{J},{\mathbb{R}}^n)$ satisfies the inequality (9), then there exists $\tilde{\mathcal{Z}}\in C(J,{\mathbb{R}}^n)$ such that the following holds:

(i)

$\parallel \tilde{\mathcal{Z}}(t)\parallel \le \epsilon \psi (t),t>0$;

(ii)

${(}^{RL}{D}_{0^{+}}^{\alpha }\tilde{y})(t)=B\tilde{y}(t)+\sum _{i=1}^d{B}_i\tilde{y}(t-{\vartheta }_i)+g(t,\tilde{y}(t))+\tilde{\mathcal{Z}}(t),t>0$;

(iii)

$D_{0^{+}}^{\alpha -1}\tilde{y}{(t)|}_{t=0}=y_0$;

(iv)

$\tilde{y}(t)=\phi (t),t\in [-\vartheta ,0)$.

Lemma 7.

If the function $\tilde{y}\in C(\tilde{J},{\mathbb{R}}^n)$ satisfies the inequality (9), then the following inequality holds:

$$\begin{array}{ccc}& & \parallel \tilde{y}(t)-Y_{\alpha ,\alpha }(t)y_0-\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & -{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,\tilde{y}(\ell ))d\ell \parallel \le {\displaystyle \frac{{\lambda }^{\frac{1}{\alpha }}}{\alpha }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\epsilon {\int }_0^t{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }\psi (\ell )d\ell ,t\in J.\hfill \end{array}$$

Proof. 

Suppose that $\tilde{y}\in C(J,{\mathbb{R}}^n)$ satisfies inequality (9). By Remark 3, it follows that satisfies the following fractional differential system:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }\tilde{y})(t)=B\tilde{y}(t)+\sum _{i=1}^d{B}_i\tilde{y}(t-{\vartheta }_i)+g(t,\tilde{y}(t))+\tilde{\mathcal{Z}}(t),t>0,\hfill \\ & D_{0^{+}}^{\alpha -1}\tilde{y}{(t)|}_{t=0}=\phi (0)=y_0,\hfill \\ & \tilde{y}(t)=\phi (t),-\vartheta \le t<0,\hfill \end{array}\right.\end{array}\end{array}$$

From Equation (4), we have

$$\begin{array}{ccc}\hfill \tilde{y}(t)& =& Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )(g(\ell ,\tilde{y}(\ell ))+\tilde{\mathcal{Z}}(\ell ))d\ell .\hfill \end{array}$$

For all $t\in J$, we have

$$\begin{array}{ccc}& & \parallel \tilde{y}(t)-Y_{\alpha ,\alpha }(t)y_0-\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & -{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,\tilde{y}(\ell ))d\ell \parallel \hfill \\ & \le & {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\tilde{\mathcal{Z}}(\ell )\parallel d\ell \le {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel \parallel \tilde{\mathcal{Z}}(\ell )\parallel d\ell \hfill \\ & \le & {\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })\epsilon \psi (\ell )d\ell \hfill \\ & \le & {\int }_0^t|{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })-{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}(t-\ell )}+{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}(t-\ell )}|\epsilon \psi (\ell )d\ell \hfill \\ & \le & {\int }_0^t{\displaystyle \frac{\Gamma (\alpha +1){(t-\ell )}^{-\alpha -1}}{{\lambda }^2\pi \alpha sin(\pi \alpha )}}\epsilon \psi (\ell )d\ell +{\int }_0^t{\displaystyle \frac{1}{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}(t-\ell )}\epsilon \psi (\ell )d\ell \hfill \\ & \le & \epsilon {\int }_0^t{\displaystyle \frac{\Gamma (\alpha +1){(t-\ell )}^{-\alpha -1}}{{\lambda }^2\pi \alpha sin(\pi \alpha )}}d\ell \underset{t\in J}{sup}\psi (t)+{\displaystyle \frac{\epsilon }{\alpha }}{\lambda }^{{\displaystyle \frac{1}{\alpha }}-1}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}{\int }_0^t{e}^{{\lambda }^{-{\displaystyle \frac{1}{\alpha }}}\ell }\psi (\ell )d\ell \hfill \\ & \le & {\displaystyle \frac{-\epsilon \Gamma (\alpha +1)t^{-\alpha }}{{\lambda }^2\pi {\alpha }^{2}sin(\pi \alpha )}}\underset{t\in J}{sup}\psi (t)+{\displaystyle \frac{{\lambda }^{\frac{1}{\alpha }-1}}{\alpha }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\epsilon {\int }_0^t{e}^{{\lambda }^{-{\displaystyle \frac{1}{\alpha }}}\ell }\psi (\ell )d\ell \hfill \\ & \le & {\displaystyle \frac{{\lambda }^{\frac{1}{\alpha }-1}}{\alpha }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\epsilon {\int }_0^t{e}^{{\lambda }^{-{\displaystyle \frac{1}{\alpha }}}\ell }\psi (\ell )d\ell .\hfill \end{array}$$

□

$\left[H_5\right]$ For all $t>0$, there exists a function $\psi (·)\in C(J,{\mathbb{R}}^{+})$ such that

$$\begin{array}{c}\hfill {\int }_0^t{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }\psi (\ell )d\ell \le M\psi (t),M>0.\end{array}$$

Theorem 5.

Assume that $\left[H_3\right]-\left[H_5\right]$ are satisfied, then (3) is UHRS on $\tilde{J}$.

Proof. 

Let $\tilde{y}\in C(J,{\mathbb{R}}^n)$ satisfy inequality (9) and $y\in C(\tilde{J},{\mathbb{R}}^n)$ be the unique solution of (3), that is

$$y(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell .$$

Let $t\in [-\vartheta ,0)$, we obtain

$$\parallel \tilde{y}(t)-y(t)\parallel =\parallel \phi (t)-\phi (t)\parallel =0<\tilde{K}\epsilon \psi (t),$$

and when $t=0$, we have

$$\parallel D_{0^{+}}^{\alpha -1}\tilde{y}{(t)|}_{t=0}-D_{0^{+}}^{\alpha -1}{y(t)|}_{t=0}\parallel =\parallel \phi (0)-\phi (0)\parallel =0<\tilde{K}\epsilon \psi (t).$$

By Lemma 7, for any $t\in J$, one has

$$\begin{array}{ccc}& & \parallel \tilde{y}(t)-y(t)\parallel \hfill \\ & \le & \parallel \tilde{y}(t)-Y_{\alpha ,\alpha }(t)y_0-\sum _{i=1}^d{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell \hfill \\ & & -{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,\tilde{y}(\ell ))d\ell \parallel +{\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel \parallel g(\ell ,\tilde{y}(\ell ))-g(\ell ,y(\ell ))\parallel d\ell \hfill \\ & \le & {\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel \parallel \tilde{\mathcal{Z}}(\ell )\parallel d\ell +{\int }_0^t\parallel Y_{\alpha ,\alpha }(t-\ell )\parallel L_g{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }\parallel \tilde{y}(\ell )-y(\ell )\parallel d\ell \hfill \\ & \le & {\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })\epsilon \psi (\ell )d\ell \hfill \\ & & +L_g{\int }_0^t{(t-\ell )}^{\alpha -1}{E}_{\alpha ,\alpha }(\lambda {(t-\ell )}^{\alpha })d\ell {\parallel \tilde{y}-y\parallel }_{C_{\omega }}\hfill \\ & \le & {\displaystyle \frac{{\lambda }^{\frac{1}{\alpha }-1}}{\alpha }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\epsilon {\int }_0^t{e}^{{\lambda }^{-{\displaystyle \frac{1}{\alpha }}}\ell }\psi (\ell )d\ell +{\displaystyle \frac{L_g}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}{\parallel \tilde{y}-y\parallel }_{C_{\omega }}\hfill \\ & \le & {\displaystyle \frac{M{\lambda }^{\frac{1}{\alpha }-1}}{\alpha }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\epsilon \psi (t)+{\displaystyle \frac{L_g}{\alpha \lambda }}{e}^{{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}{\parallel \tilde{y}-y\parallel }_{C_{\omega }},\hfill \end{array}$$

multiplying both sides of the above equation by $e^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}$, we have

$$e^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\parallel \tilde{y}(t)-y(t)\parallel \le {\displaystyle \frac{M{\lambda }^{\frac{1}{\alpha }}}{\alpha \lambda -L_g}}\epsilon \psi (t)\le \tilde{K}\epsilon \psi (t),t\in J,$$

where

$$\tilde{K}={\displaystyle \frac{M{\lambda }^{\frac{1}{\alpha }}}{\alpha \lambda -L_g}}.$$

□

5. An Example

In this section, we give some examples to illustrate the accuracy of our main results.

Example 1.

Set $\alpha =0.6$, ${\vartheta }_1=0.2,{\vartheta }_2=0.3,\vartheta =max\{{\vartheta }_1,{\vartheta }_2\}=0.3$ and $T=0.8$. Consider

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }y)(t)=By(t)+\sum _{i=1}^2{B}_{i}y(t-{\vartheta }_i)+g(t,y(t)),t\in (0,0.8],\hfill \\ & \begin{array}{c}\hfill D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}=y_0={(0,0)}^{\top },\end{array}\hfill \\ & \begin{array}{c}\hfill y(t)=\phi (t)={(t^3,0.5t^2)}^{\top },-0.3\le t<0,\end{array}\hfill \end{array}\right.\end{array}\end{array}$$

(10)

where

$$\begin{array}{c}\hfill B=\left(\begin{array}{c}0.30.2\hfill \\ 0.30.6\hfill \end{array}\right),B_1=\left(\begin{array}{c}0.10.2\hfill \\ 0.30.4\hfill \end{array}\right),B_2=\left(\begin{array}{c}0.4-0.2\hfill \\ -0.30.1\hfill \end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill y(t)=\left(\begin{array}{c}{y}_1(t)\hfill \\ y_2(t)\hfill \end{array}\right),g(t,y(t))=\left(\begin{array}{c}{\displaystyle \frac{t}{80}}{y}_1(t)\hfill \\ {\displaystyle \frac{t}{80}}{y}_2(t)\hfill \end{array}\right).\end{array}$$

Set $\gamma =0.5$. For any $t\in (0,0.8]$ and $y(t), z(t)\in {\mathbb{R}}^2$, one has

$$\parallel g(t,y)-g(t,z)\parallel \le {\displaystyle \frac{t}{80}}(|y_1(t)-z_1(t)|+|y_2(t)-z_2(t)|)\le {\displaystyle \frac{1}{100}}\parallel y-z\parallel .$$

By calculation, one has $\parallel B\parallel =0.8$, $\parallel B_1\parallel =0.6$, $\parallel B_2\parallel =0.7$, $\lambda =\parallel B\parallel +\parallel B_1\parallel +\parallel B_2\parallel =2.1$, $L={\displaystyle \frac{1}{100}}$ and $\eta =L{T}^{\alpha }{E}_{\alpha ,\alpha }(\lambda T^{\alpha })\mathbb{B}[1-\gamma ,\alpha ]=0.9548<1$. Hence, $\left[H_1\right]$ and $\left[H_2\right]$ are satisfied. From Theorem 1, we know that the system (10) has a unique solution $y\in C_{\gamma }((0,0.8],{\mathbb{R}}^2)$ as follows:

$$\begin{array}{ccc}\hfill y(t)& =& Y_{\alpha ,\alpha }(t)y_0+\underset{-0.2}{\overset{min(t-0.2,0)}{\int }}{Y}_{\alpha ,\alpha }(t-0.2-\ell )B_1\phi (\ell )d\ell +\underset{-0.3}{\overset{min(t-0.3,0)}{\int }}{Y}_{\alpha ,\alpha }(t-0.3-\ell )B_2\phi (\ell )d\ell \hfill \\ & & \begin{array}{c}\hfill +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell ,\end{array}\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill Y_{\alpha ,\alpha }(t)& =& \sum _{k=0}^{\infty }{B}^k{\displaystyle \frac{{(t)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}+\sum _{k=1}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{1}\right)B^{k-1}{B}_1{\displaystyle \frac{{(t-0.2)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}\hfill \\ & & +\sum _{k=1}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{1}\right)B^{k-1}{B}_2{\displaystyle \frac{{(t-0.3)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}+\sum _{k=2}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{1}\right)\left(\genfrac{}{}{0pt}{}{k-1}{1}\right)B^{k-2}{B}_1{B}_2{\displaystyle \frac{{(t-0.5)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}\hfill \\ & & +\sum _{k=2}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{2}\right)B^{k-2}{B}_1^2{\displaystyle \frac{{(t-0.4)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}+\sum _{k=2}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{2}\right)B^{k-2}{B}_2^2{\displaystyle \frac{{(t-0.6)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}\hfill \\ & & +\sum _{k=3}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{2}\right)\left(\genfrac{}{}{0pt}{}{k-2}{1}\right)B^{k-3}{B}_1^2{B}_2{\displaystyle \frac{{(t-0.7)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}+\sum _{k=3}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{3}\right)B^{k-3}{B}_1^3{\displaystyle \frac{{(t-0.6)}_{+}^{k\alpha +\alpha -1}}{\Gamma (k\alpha +\alpha )}}.\hfill \end{array}$$

The state of the system (10) solution is shown in Figure 1.

Example 2.

Set $\alpha =0.5$, ${\vartheta }_1=3,{\vartheta }_2=7$ and $\vartheta =max\{{\vartheta }_1,{\vartheta }_2\}=7$. Consider the following fractional differential system with two delays:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }y)(t)=By(t)+\sum _{i=1}^2{B}_{i}y(t-{\vartheta }_i)+f(t,y(t)),t>0,\hfill \\ & \begin{array}{c}\hfill D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}=y_0={(0,1)}^{\top },\end{array}\hfill \\ & \begin{array}{c}\hfill y(t)=\phi (t)={(t^2,t+1)}^{\top },-7\le t<0,\end{array}\hfill \end{array}\right.\end{array}\end{array}$$

(11)

where

$$\begin{array}{c}\hfill y(t)=\left(\begin{array}{c}{y}_1(t)\hfill \\ y_2(t)\hfill \end{array}\right),B=\left(\begin{array}{c}0.10.2\hfill \\ 0.30.4\hfill \end{array}\right),B_1=\left(\begin{array}{c}0.50.1\hfill \\ 0.30.2\hfill \end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B_2=\left(\begin{array}{c}0.2-0.1\hfill \\ -0.30.5\hfill \end{array}\right), f(t,y(t))=\left(\begin{array}{c}{\displaystyle \frac{\sqrt{3}}{3}}{e}^{-3t}sin(y_1(t))\hfill \\ {\displaystyle \frac{\sqrt{3}}{3}}{e}^{-3t}sin(y_2(t))\hfill \end{array}\right).\end{array}$$

For any $t>0$ and $y,z\in {\mathbb{R}}^2$, one has

$$\begin{array}{ccc}\hfill \parallel f(t,y(t))-f(t,z(t))\parallel & \le & {\displaystyle \frac{\sqrt{3}}{3}}{e}^{-3t}(|y_1(t)-z_1(t)|+|y_2(t)-z_2(t)|)\hfill \\ & \le & {\displaystyle \frac{\sqrt{3}}{3}}{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\parallel y-z\parallel ,t\ge 0,\hfill \end{array}$$

Thus, $\left[H_4\right]$ is hold and $L_g={\displaystyle \frac{\sqrt{3}}{3}}$. By calculation, one has $\lambda =\parallel B\parallel +\parallel B_1\parallel +\parallel B_2\parallel =2$. Therefore, $L_g<\alpha \lambda$, i.e., $\left[H_5\right]$ is hold. For $\omega =-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}=-\sqrt{2}$, using Theorem 3, we know that the system (11) has a unique solution $y\in C_{\omega }(J,{\mathbb{R}}^2)$ which has the following from:

$$y(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^2{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell .$$

Obviously, assumptions $\left[H_3\right]$ and $\left[H_4\right]$ hold. According to Theorem 4, the system (11) is UHS. Next, let $\psi (t)=e^t,M=1$, one has

$$\begin{array}{c}\hfill {\int }_0^t{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }{e}^{\ell }d\ell \le {\int }_0^t{e}^{\ell }d\ell <e^t.\end{array}$$

Hence, the $\left[H_5\right]$ is satisfied. Therefore, all the assumptions of Theorem 5 were satisfied, the system (11) is UHRS.

Next, we will present an example of a differential system with three delays to prove the accuracy of the main results of this paper.

Example 3.

Set $\alpha =0.5$, ${\vartheta }_1=2,{\vartheta }_2=4,{\vartheta }_3=9$ and $\vartheta =max\{{\vartheta }_1,{\vartheta }_2,{\vartheta }_3\}=9$. Consider the following fractional multi-delay differential system:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}& {(}^{RL}{D}_{0^{+}}^{\alpha }y)(t)=By(t)+\sum _{i=1}^3{B}_{i}y(t-{\vartheta }_i)+f(t,y(t)),t>0,\hfill \\ & \begin{array}{c}\hfill D_{0^{+}}^{\alpha -1}\widehat{y}{(t)|}_{t=0}=y_0={(0,1,0)}^{\top },\end{array}\hfill \\ & y(t)=\phi (t)={(t^2,t+1,2t)}^{\top },-9\le t<0,\hfill \end{array}\right.\end{array}\end{array}$$

(12)

where

$$\begin{array}{c}\hfill y(t)=\left(\begin{array}{c}{y}_1(t)\hfill \\ y_2(t)\hfill \\ y_3(t)\hfill \end{array}\right),B=\left(\begin{array}{ccc}0.1& 0.1& -0.1\\ 0.2& 0.3& 0.1\\ 0.5& -0.2& 0.3\end{array}\right),B_1=\left(\begin{array}{ccc}0.2& -0.1& 0.1\\ -0.3& 0.3& 0.1\\ 0.2& 0.1& -0.2\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B_2=\left(\begin{array}{ccc}0.2& 0.3& -0.2\\ 0.1& 0.1& 0.4\\ 0.1& -0.1& 0.3\end{array}\right), B_3=\left(\begin{array}{ccc}0.1& -0.1& 0.2\\ 0.1& 0.1& 0.3\\ 0.2& 0.1& 0.1\end{array}\right), f(t,y(t))=\left(\begin{array}{c}\sqrt{2}{e}^{-2t}{y}_1(t)\hfill \\ \sqrt{2}{e}^{-2t}{y}_2(t)\hfill \\ \sqrt{2}{e}^{-2t}{y}_3(t)\hfill \end{array}\right).\end{array}$$

Note that for any $y,z\in {\mathbb{R}}^3$, we have

$$\begin{array}{ccc}\hfill \parallel f(t,y(t))-f(t,z(t))\parallel & \le & \sqrt{2}{e}^{-2t}(|y_1(t)-z_1(t)|+|y_2(t)-z_2(t)|+|y_3(t)-z_3(t)|)\hfill \\ & \le & \sqrt{2}{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}t}\parallel y-z\parallel ,t\ge 0,\hfill \end{array}$$

Thus, $\left[H_4\right]$ is hold and $L_g=\sqrt{2}$. By calculation, one has $\lambda =\parallel B\parallel +\parallel B_1\parallel +\parallel B_2\parallel +\parallel B_3\parallel =3$. Hence, $L_g<\alpha \lambda$, that is $\left[H_5\right]$ is hold. For $\omega =-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}=-\sqrt{3}$, by Theorem 3, we know that the system (12) has a unique solution $y\in C_{\omega }(J,{\mathbb{R}}^3)$ which has the following from:

$$y(t)=Y_{\alpha ,\alpha }(t)y_0+\sum _{i=1}^3{\int }_{-{\vartheta }_i}^{min(t-{\vartheta }_i,0)}{Y}_{\alpha ,\alpha }(t-{\vartheta }_i-\ell )B_i\phi (\ell )d\ell +{\int }_0^t{Y}_{\alpha ,\alpha }(t-\ell )g(\ell ,y(\ell ))d\ell .$$

Obviously, assumptions $\left[H_3\right]$ and $\left[H_4\right]$ hold. According to Theorem 4, the system (12) is UHS. Next, let $\psi (t)=e^t,M=1$, one has

$$\begin{array}{c}\hfill {\int }_0^t{e}^{-{\lambda }^{{\displaystyle \frac{1}{\alpha }}}\ell }{e}^{\ell }d\ell \le {\int }_0^t{e}^{\ell }d\ell <e^t.\end{array}$$

Therefore, the assumption $\left[H_5\right]$ holds. Thus, according to Theorem 5, the system (12) is UHRS.

6. Conclusions

This paper employs the fixed point theorem, fractional calculus, and properties of the multi-delayed perturbation of two parameter Mittag-Leffler type matrix function. Because the solution of the Riemann-Liouville type fractional multi-delay differential system has a singularity, we first prove the existence and uniqueness of system solutions in $C_{\gamma }((0,T],{\mathbb{R}}^n)$. However, since the solution of the system is divergent when the time tends to infinite in $C_{\gamma }$ space, we prove the existence and uniqueness of the solution of the system over the infinite interval in the weighted space $C_{\omega }$. Subsequently, we investigate Ulam-type stability for the system on the infinite time interval in $C_{\omega }$ space. Finally, three examples are provided to demonstrate the applicability of these results. In future, we could explore the extension of the stability theory for nonlinear systems, focusing on key issues like asymptotic stability and stability manifolds, and addressing the practical applications of these theoretical achievements in fields such as engineering control and biological systems.