Representation of Solutions and Ulam–Hyers Stability of the Two-Sided Fractional Matrix Delay Differential Equations

Abstract

This paper investigates linear two-sided fractional matrix delay differential equations (TSFMDDE). Firstly, the two-sided fractional delayed Mittag-Leffler matrix functions (TSFDMLMF) are constructed. Further, the representation of solutions of two-sided homogeneous and nonhomogeneous problems are studied, and Ulam–Hyers (UH) stability of a two-sided nonhomogeneous problem is discussed. Lastly, we provide a numerical example to demonstrate our results. In the numerical example, the fractional order $\beta =0.6$, delay $\varrho =2$, UH constant $u_h\approx 5.92479$, $n=2$, and $s\in [-2,4]$.

1. Introduction

Fractional differential equations (FDEs) serve as a generalized mathematical framework compared to integer counterparts, offering a powerful tool for characterizing memory effects and hereditary traits in diverse materials and dynamical processes [1,2,3,4,5,6]. Many researchers have conducted extensive research on the fundamental theoretical analysis of FDEs [7,8,9,10]. For representation of solutions and stability of FDEs, we refer to [11,12,13,14]. Wang and Li [11] utilized the Laplace transform method to prove the UH stability of linear FDEs. Liang [12] gave a representation of a solution to the fractional linear system with pure delay. Li [13] studied the existence of solutions and the UH stability of the conformable FDEs with constant coefficients. Xiao and Wang [14] investigated the stability of Caputo-type fractional stochastic differential equations.

Delay differential equations (DDEs) represent crucial mathematical models that depict phenomena featuring historical dependencies or propagation delays. These equations find extensive applications across diverse fields, including neural networks, ecological dynamics, and control systems. In recent years, exploring the representation of solutions has become a hot topic in DDEs. For the recent works of delay systems in application fields, e.g., control theory, we refer to [15,16,17,18]. Khusainov and Shuklin [15] considered relative controllability for the linear control system with pure delay. Futher, Pospíšil [16] investigated the relative controllability for linear systems of neutral differential equations with a delay. Wang et al. [17] analyzed relative controllability of semilinear delay differential systems with permutable matrices. Further, Wang et al. [18] also studied the relative controllability of a fractional stochastic system with pure delay.

It is noteworthy that Khusainov and Shuklin [19] proposed the conception of delayed exponential matrix function (DEMF). Subsequently, based on the above concept, Li and Wang [20] and Mahmudov [21] studied DEMF via Mittag-Leffler functions and applied it to the construction of solutions for Caputo fractional DDEs. Further, Li and Wang [22] constructed the solutions to nonhomogeneous fractional DDEs and showed the existence and uniqueness of solutions to nonlinear fractional DDEs. In particular, Diblík [23] extended these results to the two-side linear matrix delay discrete equations and constructed their solutions. Ulam stability originated from the exploration of system stability in [24], Hyers [25] further developed this theory, and it was later termed UH stability by researchers. Soon after, UH stability of FDEs became widely studied [26,27,28,29,30]. Ibrahim [26] considered the generalized UH stability for fractional differential equations in a complex Banach space. Al-khateeb et al. [27] discussed UH stability for nonlinear sequential FDEs involving integral boundary conditions. Wang et al. [28] studied UH stability of Caputo-type stochastic FDEs with time delays. Derakhshan [29] investigated UH stability for variable order FDEs in fluid mechanics. Girgin [30] analyzed Caputo-type non-linear FDEs. However, the above research did not study UH stability for matrix delay FDEs, especially two-sided matrix delay FDEs.

Inspired by the above results, we investigate the representation of solutions and UH stability for linear TSFMDDE

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right),s\in \mathcal{P}:=[0,\mathcal{T}],\varrho >0,\hfill \\ \hfill & \mathcal{X}\left(s\right)=\Phi \left(s\right),-\varrho \le s\le 0,\hfill \end{array}\right.\end{array}$$

(1)

where ${{}^c\mathcal{D}}_0^{\beta }$ represents the Caputo derivative of $0<\beta <1$ (see Definition 2), $\mathcal{X}(·)$, $\mathcal{C}(·)$, $\Phi (·)$ are continuously differentiable $n\times n$ matrix functions (n is positive integer) in $[-\varrho ,\mathcal{T}]$, $\mathcal{A},\mathcal{B}$ are given real constant $n\times n$ matrices, and $\mathcal{T}=l\varrho$ for fixed $l\in {\mathbb{N}}^{*}$.

The present paper is structured as follows. In Section 2, TSFDMLMF are constructed, and some essential definitions and lemmas are presented. In Section 3, the representation of the solutions and the UH stability of the linear problem are explored. In Section 4, an example is given to validate our results.

2. Preliminaries

In this paper, the matrix norm is denoted by $\parallel \mathcal{A}\parallel ={max}_{1\le i\le n}{\sum }_{j=1}^n\left|a_{ij}\right|$. Additionally, we note ${\parallel \mathcal{A}\parallel }_{\mathcal{P}}={max}_{s\in \mathcal{P}}\parallel \mathcal{A}\left(s\right)\parallel$. The methodological structure of this paper is shown by the following Figure 1:

Definition 1 

(see [1]). The Riemann–Liouville derivative of $0<\beta <1$ for $g:[0,\infty )\to \mathbb{R}$ is defined by $\left({{}^{RL}\mathcal{D}}_0^{\beta }g\right)\left(s\right)={\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{\displaystyle \frac{g\left(z\right)}{{(s-z)}^{\beta }}}dz,s>0,$ where $\Gamma (·)$ is the Gamma function.

Definition 2 

(see [1]). The Caputo derivative of $0<\beta <1$ for $g:[0,\infty )\to \mathbb{R}$ is defined by $\left({{}^c\mathcal{D}}_0^{\beta }g\right)\left(s\right)=\left({{}^{RL}\mathcal{D}}_0^{\beta }g\right)\left(s\right)-{\displaystyle \frac{g\left(0\right)}{\Gamma (1-\beta )}}{s}^{-\beta },s>0.$

Proposition 1 

(see [1]). For $0<\beta <1$, $\left({{}^c\mathcal{D}}_0^{\beta }g\right)\left(s\right)={\displaystyle \frac{1}{\Gamma (1-\beta )}}{\int }_0^s{\displaystyle \frac{g^{\prime }\left(z\right)}{{(s-z)}^{\beta }}}dz$, $s>0$.

Definition 3. 

If for arbitrary δ and a solution $\mathcal{Y}:\mathcal{P}\to {\mathbb{R}}^{n\times n}$ of

$$∥\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{Y}\right)\left(s\right)-\mathcal{A}\mathcal{Y}(s-\varrho )-\mathcal{Y}(s-\varrho )\mathcal{B}-\mathcal{C}\left(s\right)∥\le \delta ,$$

(2)

one can find a solution $\mathcal{X}\left(s\right)$ of (1) and a positive constant $u_h>0$ such that

$$\parallel \mathcal{Y}\left(s\right)-\mathcal{X}\left(s\right)\parallel \le u_h\delta ,$$

then (1) is Ulam–Hyers stable.

Definition 4 

(see [19]). The DEMF $e_{\varrho }^{As}$ is defined by

$$\begin{array}{c}\hfill e_{\varrho }^{Bs}=\left\{\begin{array}{cc}\hfill & \Theta ,-\infty <s<-\varrho ,\hfill \\ \hfill & \mathbb{I},-\varrho \le s\le 0,\hfill \\ \hfill & \mathbb{I}+A{\displaystyle \frac{s}{1!}}+A^2{\displaystyle \frac{{(s-\varrho )}^2}{2!}}+\dots +A^l{\displaystyle \frac{{(s-(l-1)\varrho )}^l}{l!}},(l-1)\varrho \le s\le l\varrho ,l\in {\mathbb{N}}^{*},\hfill \end{array}\right.\end{array}$$

where Θ and $\mathbb{I}$ represent zero matrix and identity matrix, respectively.

Figure 1. The flowchart of methodological structure in this paper.

Next, inspired by Definition 4, we introduce TSFDMLMF, which generalizes the concept of DEMF $e_{\varrho }^{As}$ of [19].

Definition 5. 

The TSFDMLMF ${\mathcal{X}}_{\beta }^{\varrho }\left(s\right)$ is defined as

$$\begin{array}{c}\hfill {\mathcal{X}}_{\beta }^{\varrho }\left(s\right)=\left\{\begin{array}{cc}\hfill & \Theta ,-\infty <s<-\varrho ,\hfill \\ \hfill & \mathbb{I},-\varrho \le s\le 0,\hfill \\ \hfill & \mathbb{I}+{\displaystyle \frac{s^{\beta }}{\Gamma (\beta +1)}}(\mathcal{A}+\mathcal{B}),0\le s\le \varrho ,\hfill \\ \hfill & \dots \hfill \\ \hfill & \mathbb{I}+{\displaystyle \frac{s^{\beta }}{\Gamma (\beta +1)}}(\mathcal{A}+\mathcal{B})+{\displaystyle \frac{{(s-\varrho )}^{2\beta }}{\Gamma (2\beta +1)}}({\mathcal{A}}^2+2\mathcal{A}\mathcal{B}+{\mathcal{B}}^2)\hfill \\ \hfill & +\dots +\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right){\mathcal{A}}^{l-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-(l-1)\varrho )}^{l\beta }}{\Gamma (l\beta +1)}},(l-1)\varrho \le s\le l\varrho ,l\in {\mathbb{N}}^{*}.\hfill \end{array}\right.\end{array}$$

(3)

Proposition 2. 

For any $s\in (-\infty ,\mathcal{T}]$, the following inequality holds

$$\begin{array}{c}\hfill ∥{\mathcal{X}}_{\beta }^{\varrho }\left(s\right)∥\le \sum _{k=0}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^k{\displaystyle \frac{{(\mathcal{T}-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}.\end{array}$$

(4)

Proof. 

For $s\in (-\infty ,-\varrho )$, $∥{\mathcal{X}}_{\beta }^{\varrho }\left(s\right)∥=0\le {\mathcal{X}}_0^{\parallel \mathcal{A}\parallel ,\parallel \mathcal{B}\parallel }\left(\mathcal{T}\right)$, thus (4) holds. For $s\in [-\varrho ,\mathcal{T}]$, by (3), we have

$$\begin{array}{cc}\hfill ∥{\mathcal{X}}_{\beta }^{\varrho }\left(s\right)∥=& ∥\sum _{k=0}^l\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\mathcal{A}}^{k-j}{\mathcal{B}}^j{\displaystyle \frac{{(s-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}∥\hfill \\ \hfill \le & \sum _{k=0}^l\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)∥{\mathcal{A}}^{k-j}∥∥{\mathcal{B}}^j∥{\displaystyle \frac{{(s-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}\hfill \\ \hfill \le & \sum _{k=0}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^k{\displaystyle \frac{{(\mathcal{T}-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}.\hfill \end{array}$$

□

Now, we introduce a two-parameter TSFDMLMF, which is a generalization of two-parameter Mittag-Leffler matrix function ${\mathbb{E}}_{\beta ,\alpha }\left(A\right)={\sum }_{k=0}^{\infty }{\displaystyle \frac{A^k}{\Gamma (k\beta +\alpha )}}$ and DEMF $e_{\varrho }^{As}$.

Definition 6. 

The two-parameter TSFDMLMF ${\mathcal{X}}_{\beta ,\alpha }^{\varrho }\left(s\right)$ is defined as, for $l\in {\mathbb{N}}^{*}$ and $\alpha \in \mathbb{R}$,

$${\mathcal{X}}_{\beta ,\alpha }^{\varrho }\left(s\right)=\left\{\begin{array}{cc}\hfill & \Theta ,-\infty <s\le -\varrho ,\hfill \\ \hfill & \mathbb{I}{\displaystyle \frac{{(s+\varrho )}^{\beta -1}}{\Gamma \left(\alpha \right)}},-\varrho <s\le 0,\hfill \\ \hfill & \dots \hfill \\ & \mathbb{I}{\displaystyle \frac{{(s+\varrho )}^{\beta -1}}{\Gamma \left(\alpha \right)}}+{\displaystyle \frac{s^{2\beta -1}}{\Gamma (\beta +\alpha )}}(\mathcal{A}+\mathcal{B})+{\displaystyle \frac{{(s-\varrho )}^{3\beta -1}}{\Gamma (2\beta +\alpha )}}({\mathcal{A}}^2+2\mathcal{A}\mathcal{B}+{\mathcal{B}}^2)\hfill \\ \hfill & +\dots +\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right){\mathcal{A}}^{l-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-(l-1)\varrho )}^{(l+1)\beta -1}}{\Gamma (l\beta +\alpha )}},(l-1)\varrho <s\le l\varrho .\hfill \end{array}\right.$$

(5)

Lemma 1. 

For any $s\in \left(\right(l-1)\varrho ,l\varrho ]$, $l\in \mathbb{N}$, the following inequality holds

$$\begin{array}{c}\hfill ∥{\mathcal{X}}_{\beta ,\alpha }^{\varrho }\left(s\right)∥\le \sum _{k=0}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^k{\displaystyle \frac{{(s-(k-1)\varrho )}^{(k+1)\beta -1}}{\Gamma (k\beta +\alpha )}}.\end{array}$$

Proof. 

For $s\in \left(\right(l-1)\varrho ,l\varrho ]$, $l\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill ∥{\mathcal{X}}_{\beta ,\alpha }^{\varrho }\left(s\right)∥=& ∥\sum _{k=0}^l\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\mathcal{A}}^{k-j}{\mathcal{B}}^j{\displaystyle \frac{{(s-(k-1)\varrho )}^{(k+1)\beta -1}}{\Gamma (k\beta +\alpha )}}∥\hfill \\ \hfill \le & \sum _{k=0}^l\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)∥{\mathcal{A}}^{k-j}∥∥{\mathcal{B}}^j∥{\displaystyle \frac{{(s-(k-1)\varrho )}^{(k+1)\beta -1}}{\Gamma (k\beta +\alpha )}}\hfill \\ \hfill \le & \sum _{k=0}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^k{\displaystyle \frac{{(s-(k-1)\varrho )}^{(k+1)\beta -1}}{\Gamma (k\beta +\alpha )}}.\hfill \end{array}$$

This proof is completed. □

Lemma 2. 

For $s\in \left(\right(l-1)\varrho ,l\varrho ]$, $0\le t\le s$, and $l\in {\mathbb{N}}^{*}$, we have

$$\begin{array}{cc}\hfill & {\int }_t^s{(s-h)}^{-\beta }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(h-\varrho -t)dh\hfill \\ \hfill =& {\int }_t^s{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dh+{\int }_{t+\varrho }^s{(s-h)}^{-\beta }(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(h-\varrho -t)}^{2\beta -1}}{\Gamma \left(2\beta \right)}}dh\hfill \\ \hfill & +\dots +{\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }\left(\sum _{k=0}^{l-1}\left({\displaystyle \genfrac{}{}{0pt}{}{l-1}{k}}\right){\mathcal{A}}^{l-1-k}{\mathcal{B}}^k{\displaystyle \frac{{(h-(l-1)\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}\right)dh.\hfill \end{array}$$

Proof. 

For $s\in \left(\right(l-1)\varrho ,l\varrho ]$, $0\le t\le s$, and $l\in \mathbb{N}$, by (5), we have

$$\begin{array}{cc}\hfill & {\int }_t^s{(s-h)}^{-\beta }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(h-\varrho -t)dh\hfill \\ \hfill =& {\int }_t^{t+\varrho }{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dh\hfill \\ \hfill & +{\int }_{t+\varrho }^{t+2\varrho }{(s-h)}^{-\beta }\left(\sum _{j=0}^1\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){\mathcal{A}}^{j-k}{\mathcal{B}}^k{\displaystyle \frac{{(h-j\varrho -t)}^{(j+1)\beta -1}}{\Gamma \left(\right(j+1\left)\beta \right)}}\right)dh\hfill \\ \hfill & +\dots +{\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }\left(\sum _{j=0}^{l-1}\sum _{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){\mathcal{A}}^{j-k}{\mathcal{B}}^k{\displaystyle \frac{{(h-j\varrho -t)}^{(j+1)\beta -1}}{\Gamma \left(\right(j+1\left)\beta \right)}}\right)dh\hfill \\ \hfill =& {\int }_t^{t+\varrho }{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dh+{\int }_{t+\varrho }^{t+2\varrho }{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dh\hfill \\ \hfill & +\dots +{\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dh+{\int }_{t+\varrho }^{t+2\varrho }{(s-h)}^{-\beta }(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(h-\varrho -t)}^{2\beta -1}}{\Gamma \left(2\beta \right)}}dh\hfill \\ \hfill & +\dots +{\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(h-\varrho -t)}^{2\beta -1}}{\Gamma \left(2\beta \right)}}dh\hfill \\ \hfill & +\dots +{\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }\left(\sum _{k=0}^{l-1}\left({\displaystyle \genfrac{}{}{0pt}{}{l-1}{k}}\right){\mathcal{A}}^{l-1-k}{\mathcal{B}}^k{\displaystyle \frac{{(h-(l-1)\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}\right)dh\hfill \\ \hfill =& {\int }_t^s{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dh+{\int }_{t+\varrho }^s{(s-h)}^{-\beta }(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(h-\varrho -t)}^{2\beta -1}}{\Gamma \left(2\beta \right)}}dh\hfill \\ \hfill & +\dots +{\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }\left(\sum _{k=0}^{l-1}\left({\displaystyle \genfrac{}{}{0pt}{}{l-1}{k}}\right){\mathcal{A}}^{l-1-k}{\mathcal{B}}^k{\displaystyle \frac{{(h-(l-1)\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}\right)dh.\hfill \end{array}$$

The proof is completed. □

Lemma 3. 

(see [22]). For $s\in \left(\right(l-1)\varrho ,l\varrho ]$, $0\le t\le s$, and $l\in {\mathbb{N}}^{*}$, we have

$$\begin{array}{c}\hfill {\int }_{t+(l-1)\varrho }^s{(s-h)}^{-\beta }{(h-(l-1)\varrho -t)}^{l\beta -1}dh={(s-(l-1)\varrho -t)}^{(l-1)\beta }\mathbb{B}[1-\beta ,l\beta ],\end{array}$$

where $\mathbb{B}[\alpha ,\beta ]={\int }_0^1{t}^{\alpha -1}{(1-t)}^{\beta -1}ds$ is a Beta function.

3. Representation of Solutions

Here, we explore the representation of solutions for linear TSFMDDE.

Theorem 1. 

${\mathcal{X}}_{\beta }^{\varrho }\left(s\right)$, defined by (3), is a solution of the equation

$$\begin{array}{c}\hfill \left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B},s\in \mathcal{P},\end{array}$$

(6)

with initial condition $\mathcal{X}\left(s\right)=\mathbb{I}$, $-\varrho \le s\le 0$.

Proof. 

From Definition 5, ${\mathcal{X}}_{\beta }^{\varrho }\left(s\right)$ satisfies the initial condition. For $s\in (-\infty ,-\varrho )$, ${\mathcal{X}}_{\beta }^{\varrho }\left(s\right)={\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )=\Theta$, thus, (6) holds. For arbitrary $s\in [-\varrho ,0]$, ${\mathcal{X}}_{\beta }^{\varrho }\left(s\right)=\mathbb{I}$ and ${\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )=\Theta$. Note that $\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\Theta =\mathcal{A}\Theta +\Theta \mathcal{B}$. Hence, (6) holds.

For arbitrary $s\in \left[\right(l-1)\varrho ,l\varrho ]$, $l\in {\mathbb{N}}^{*}$, we illustrate the result by using mathematical induction.

(i) For $l=1$, $0\le s\le \varrho$, we get

$$\begin{array}{c}\hfill {\mathcal{X}}_{\beta }^{\varrho }\left(s\right)=\mathbb{I}+{\displaystyle \frac{s^{\beta }}{\Gamma (\beta +1)}}(\mathcal{A}+\mathcal{B}),{\left({\mathcal{X}}_{\beta }^{\varrho }\right)}^{\prime }\left(s\right)={\displaystyle \frac{\beta s^{\beta -1}}{\Gamma (\beta +1)}}(\mathcal{A}+\mathcal{B}),{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )=\mathbb{I}.\end{array}$$

Next, by Proposition 1 and Lemma 3, we obtain

$$\begin{array}{cc}\hfill \left({{}^c\mathcal{D}}_0^{\beta }{\mathcal{X}}_{\beta }^{\varrho }\right)\left(s\right)=& {\displaystyle \frac{\beta (\mathcal{A}+\mathcal{B})}{\Gamma (\beta +1)\Gamma (1-\beta )}}{\int }_0^s{t}^{\beta -1}{(s-t)}^{-\beta }dt={\displaystyle \frac{\beta \Gamma (1-\beta )\Gamma \left(\beta \right)}{\Gamma (\beta +1)\Gamma (1-\beta )}}(\mathcal{A}+\mathcal{B})\hfill \\ \hfill & =\mathcal{A}+\mathcal{B}=A{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )+{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )B.\hfill \end{array}$$

(ii) Assume $l=\mathcal{M}$, $(\mathcal{M}-1)\varrho \le s\le \mathcal{M}\varrho$, $\mathcal{M}\in {\mathbb{N}}^{*}$, then

$$\begin{array}{cc}\hfill \left({{}^c\mathcal{D}}_0^{\beta }{\mathcal{X}}_{\beta }^{\varrho }\right)\left(s\right)& =(\mathcal{A}+\mathcal{B})+\dots +\sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right){\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^k{\displaystyle \frac{\mathcal{M}\beta {(s-(\mathcal{M}-1)\varrho )}^{\mathcal{M}\beta -1}}{\Gamma (\mathcal{M}\beta +1)}}\hfill \\ \hfill & =\mathcal{A}{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )+{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\mathcal{B}.\hfill \end{array}$$

Next, for $l=\mathcal{M}+1$, $\mathcal{M}\varrho \le s\le (\mathcal{M}+1)\varrho$, we obtain

$$\begin{array}{cc}\hfill {\left({\mathcal{X}}_{\beta }^{\varrho }\right)}^{\prime }\left(s\right)=& {\displaystyle \frac{\beta s^{(\beta -1)}}{\Gamma (\beta +1)}}(\mathcal{A}+\mathcal{B})+\dots +\sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right){\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^k{\displaystyle \frac{\mathcal{M}\beta {(s-(\mathcal{M}-1)\varrho )}^{\mathcal{M}\beta -1}}{\Gamma (\mathcal{M}\beta +1)}}\hfill \\ \hfill & +\sum _{k=0}^{\mathcal{M}+1}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}+1}{k}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k{\displaystyle \frac{(\mathcal{M}+1)\beta {(s-\mathcal{M}\varrho )}^{(\mathcal{M}+1)\beta -1}}{\Gamma \left(\right(\mathcal{M}+1)\beta +1)}}.\hfill \end{array}$$

By Proposition 1 and Lemma 3, the left side of (6) is

$$\begin{array}{cc}\hfill & \left({{}^c\mathcal{D}}_0^{\beta }{\mathcal{X}}_{\beta }^{\varrho }\right)\left(s\right)\hfill \\ \hfill =& {\displaystyle \frac{\beta (\mathcal{A}+\mathcal{B}){\int }_0^s{t}^{\beta -1}{(s-t)}^{-\beta }dt}{\Gamma (\beta +1)\Gamma (1-\beta )}}\hfill \\ \hfill & +\dots +{\displaystyle \frac{\mathcal{M}\beta {\sum }_{k=0}^{\mathcal{M}}\left(\genfrac{}{}{0pt}{}{\mathcal{M}}{k}\right){\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^k}{\Gamma (\mathcal{M}\beta +1)\Gamma (1-\beta )}}{\int }_{(\mathcal{M}-1)\varrho }^s{(t-(\mathcal{M}-1)\varrho )}^{\mathcal{M}\beta -1}{(s-t)}^{-\beta }dt\hfill \\ \hfill & +{\displaystyle \frac{(\mathcal{M}+1)\beta {\sum }_{k=0}^{\mathcal{M}+1}\left(\genfrac{}{}{0pt}{}{\mathcal{M}+1}{k}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k}{\Gamma \left(\right(\mathcal{M}+1)\beta +1)\Gamma (1-\beta )}}{\int }_{\mathcal{M}\varrho }^s{(t-\mathcal{M}\varrho )}^{(\mathcal{M}+1)\beta -1}{(s-t)}^{-\beta }dt\hfill \\ \hfill =& (\mathcal{A}+\mathcal{B})+\dots +\sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right){\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-(\mathcal{M}-1)\varrho )}^{(\mathcal{M}-1)\varrho }}{\Gamma \left(\right(\mathcal{M}-1)\varrho +1)}}\hfill \\ \hfill & +\sum _{k=0}^{\mathcal{M}+1}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}+1}{k}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-\mathcal{M}\varrho )}^{\mathcal{M}\varrho }}{\Gamma (\mathcal{M}\varrho +1)}}.\hfill \end{array}$$

(7)

Additionally, the right side of (6) is

$$\begin{array}{cc}\hfill & \mathcal{A}{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )+{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\mathcal{B}\hfill \\ \hfill =& \mathcal{A}\left(\mathbb{I}+(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(s-\varrho )}^{\beta }}{\Gamma (\beta +1)}}+\dots +\sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right){\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-\mathcal{M}\varrho )}^{\mathcal{M}\varrho }}{\Gamma (\mathcal{M}\varrho +1)}}\right)\hfill \\ \hfill & +\left(\mathbb{I}+(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(s-\varrho )}^{\beta }}{\Gamma (\beta +1)}}+\dots +\sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right){\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-\mathcal{M}\varrho )}^{\mathcal{M}\varrho }}{\Gamma (\mathcal{M}\varrho +1)}}\right)\mathcal{B}\hfill \\ \hfill =& (\mathcal{A}+\mathcal{B})+({\mathcal{A}}^2+2\mathcal{A}\mathcal{B}+{\mathcal{B}}^2){\displaystyle \frac{{(s-\varrho )}^{\beta }}{\Gamma (\beta +1)}}\hfill \\ \hfill & +\dots +\sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right)\left({\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k+{\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^{k+1}\right){\displaystyle \frac{{(s-\mathcal{M}\varrho )}^{\mathcal{M}\varrho }}{\Gamma (\mathcal{M}\varrho +1)}}.\hfill \end{array}$$

(8)

Since (6) holds for $l=\mathcal{M}$, the first $\mathcal{M}$ terms of (7) and (8) are equal. Now, we only need to verify that the last terms of (7) and (8) are equal. Rearranging the coefficient of the last term in (8), one has

$$\begin{array}{cc}\hfill & \sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right)\left({\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k+{\mathcal{A}}^{\mathcal{M}-k}{\mathcal{B}}^{k+1}\right)\hfill \\ \hfill =& \sum _{k=0}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k+\sum _{k=1}^{\mathcal{M}+1}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k-1}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k\hfill \\ \hfill =& \sum _{k=1}^{\mathcal{M}}\left(\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}}{k-1}}\right)\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k+{\mathcal{A}}^{\mathcal{M}+1}+{\mathcal{B}}^{\mathcal{M}+1}\hfill \\ \hfill =& \sum _{k=1}^{\mathcal{M}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}+1}{k}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k+{\mathcal{A}}^{\mathcal{M}+1}+{\mathcal{B}}^{\mathcal{M}+1}=\sum _{k=0}^{\mathcal{M}+1}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}+1}{k}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k.\hfill \end{array}$$

Hence, the following equation holds,

$$\begin{array}{cc}\hfill & \mathcal{A}{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )+{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\mathcal{B}\hfill \\ \hfill =& (\mathcal{A}+\mathcal{B})+({\mathcal{A}}^2+2\mathcal{A}\mathcal{B}+{\mathcal{B}}^2){\displaystyle \frac{{(s-\varrho )}^{\beta }}{\Gamma (\beta +1)}}+\dots +\sum _{k=0}^{\mathcal{M}+1}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{M}+1}{k}}\right){\mathcal{A}}^{\mathcal{M}+1-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-\mathcal{M}\varrho )}^{\mathcal{M}\varrho }}{\Gamma (\mathcal{M}\varrho +1)}}\hfill \\ \hfill =& \left({{}^c\mathcal{D}}_0^{\beta }{\mathcal{X}}_{\beta }^{\varrho }\right)\left(s\right),\hfill \end{array}$$

which shows that for any $(l-1)\varrho \le s\le l\varrho$ and $l\in {\mathbb{N}}^{*}$, (6) holds. □

Now, we study the solution for the homogeneous problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B},s\in \mathcal{P},\varrho >0,\hfill \\ \hfill & \mathcal{X}\left(s\right)=\Phi \left(s\right),-\varrho \le s\le 0.\hfill \end{array}\right.\end{array}$$

(9)

Theorem 2. 

If $\mathcal{B}\Phi \left(s\right)=\Phi \left(s\right)\mathcal{B}$, $s\in [-\varrho ,0]$, then a solution $\mathcal{X}\left(s\right)$ of (9) can be given by

$$\begin{array}{c}\hfill \mathcal{X}\left(s\right)={\mathcal{X}}_{\beta }^{\varrho }\left(s\right)\Phi (-\varrho )+{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho -t){\Phi }^{\prime }\left(t\right)dt,-\varrho \le s\le \mathcal{T}.\end{array}$$

(10)

Proof. 

For $s\in [-\varrho ,0]$, we have ${\mathcal{X}}_{\beta }^{\varrho }(s-\varrho -t)=\mathbb{I}$, $-\varrho \le t\le s$, ${\mathcal{X}}_{\beta }^{\varrho }(s-\varrho -t)=\Theta$, $-s\le t\le 0$, and

$$\begin{array}{c}\hfill \mathcal{X}=\Phi (-\varrho )+{\int }_{-\varrho }^s{\Phi }^{\prime }\left(t\right)dt=\Phi (-\varrho )+\Phi \left(s\right)-\Phi (-\varrho )=\Phi \left(s\right),\end{array}$$

thus, the initial condition is satisfied.

For $s\in \left[\right(l-1)\varrho ,l\varrho ]$, $l\in {\mathbb{N}}^{*}$, we obtain

$$\begin{array}{c}\hfill \mathcal{X}(s-\varrho )={\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\Phi (-\varrho )+{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-2\varrho -t){\Phi }^{\prime }\left(t\right)dt,\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{X}}^{\prime }\left(s\right)={\left({\mathcal{X}}_{\beta }^{\varrho }\right)}^{\prime }\left(s\right)\Phi (-\varrho )+{\int }_{-\varrho }^0{\left({\mathcal{X}}_{\beta }^{\varrho }\right)}^{\prime }(s-\varrho -t){\Phi }^{\prime }\left(t\right)dt.\end{array}$$

From Proposition 1 and Theorem 1, we obtain the Caputo derivative of $\mathcal{X}\left(s\right)$,

$$\begin{array}{cc}\hfill \left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=& \left({{}^c\mathcal{D}}_0^{\beta }{\mathcal{X}}_{\beta }^{\varrho }\right)\left(s\right)\Phi (-\varrho )+{\displaystyle \frac{1}{\Gamma (1-\beta )}}{\int }_0^s{(s-h)}^{-\beta }{\int }_{-\varrho }^0{\left({\mathcal{X}}_{\beta }^{\varrho }\right)}^{\prime }(h-\varrho -t){\Phi }^{\prime }\left(t\right)dtdh\hfill \\ \hfill =& \left(\mathcal{A}{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )+{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\mathcal{B}\right)\Phi (-\varrho )\hfill \\ \hfill & +{\int }_{-\varrho }^0\left({\displaystyle \frac{1}{\Gamma (1-\beta )}}{\int }_0^s{(s-h)}^{-\beta }{\left({\mathcal{X}}_{\beta }^{\varrho }\right)}^{\prime }(h-\varrho -t)dh\right)dt\hfill \\ \hfill =& \mathcal{A}{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\Phi (-\varrho )+{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\mathcal{B}\Phi (-\varrho )+{\int }_{-\varrho }^0\left({{}^c\mathcal{D}}_0^{\beta }{\mathcal{X}}_{\beta }^{\varrho }\right)(s-\varrho -t){\Phi }^{\prime }\left(t\right)dt\hfill \\ \hfill =& \mathcal{A}{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\Phi (-\varrho )+\mathcal{A}{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-2\varrho -t){\Phi }^{\prime }\left(t\right)dt\hfill \\ \hfill & +{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho )\Phi (-\varrho )\mathcal{B}+{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-2\varrho -t){\Phi }^{\prime }\left(t\right)dt\mathcal{B}\hfill \\ \hfill =& \mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B}.\hfill \end{array}$$

The proof is completed. □

Next, we give the solution for nonhomogeneous problem (1). Denote $Y(·)$, given by (10), is a solution of (9), then, the solution of (1) is given by

$$\mathcal{X}\left(s\right)=Y\left(s\right)+\overline{\mathcal{X}}\left(s\right),-\varrho \le s\le \mathcal{T},$$

where $\overline{\mathcal{X}}\left(s\right)$ is a solution of the following nonhomogeneous equation

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right),s\in \mathcal{P},\hfill \\ \hfill & \mathcal{X}\left(s\right)=\Theta ,-\varrho \le s\le 0.\hfill \end{array}\right.\end{array}$$

(11)

Theorem 3. 

If $\mathcal{BC}\left(s\right)=\mathcal{C}\left(s\right)\mathcal{B}$, for $s\in \mathcal{P}$, then a solution $\overline{\mathcal{X}}\left(s\right)$ of (11) is given by

$$\overline{\mathcal{X}}\left(s\right)={\int }_0^s{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)\mathcal{C}\left(t\right)dt,-\varrho \le s\le \mathcal{T}.$$

(12)

Proof. 

By using the method of variation of constants, with $\mathcal{X}\left(s\right)=\Theta ,-\varrho \le s\le 0$, the solution of (11) satisfies

$$\overline{\mathcal{X}}\left(s\right)={\int }_0^s{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)H\left(t\right)dt,-\varrho \le s\le \mathcal{T},$$

where $H\left(s\right)$, $-\varrho \le s\le \mathcal{T}$ is an unknown matrix function, and $\mathcal{B}H\left(s\right)=H\left(s\right)\mathcal{B}$, for $s\in \mathcal{P}$. Obviously, for $-\varrho \le s\le 0$, $\overline{\mathcal{X}}\left(s\right)=\Theta$ satisfies the initial condition. Now, we consider the case $0<s\le \mathcal{T}$.

(i) For $0<s\le \varrho$, the right side of (11) is

$$\begin{array}{cc}\hfill & \mathcal{A}\overline{\mathcal{X}}(s-\varrho )+\overline{\mathcal{X}}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right)\hfill \\ \hfill =& \mathcal{A}{\int }_0^{s-\varrho }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-2\varrho -t)H\left(t\right)dt+{\int }_0^{s-\varrho }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-2\varrho -t)H\left(t\right)dt\mathcal{B}+H\left(s\right)\hfill \\ \hfill =& \mathcal{C}\left(s\right),\hfill \end{array}$$

where we note that ${\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-2\varrho -·)=\Theta$ due to (5).

From Definition 2 and Lemma 3, the left side of (11) is

$$\begin{array}{cc}\hfill \left({{}^c\mathcal{D}}_0^{\beta }\overline{\mathcal{X}}\right)\left(s\right)=& \left({{}^{RL}\mathcal{D}}_0^{\beta }\overline{\mathcal{X}}\right)\left(s\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{(s-h)}^{-\beta }{\int }_0^h{\mathcal{X}}_{\beta ,\beta }^{\varrho }(h-2\varrho -t)H\left(t\right)dtdh\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{\int }_t^s{(s-h)}^{-\beta }\mathbb{I}{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dhH\left(t\right)dt\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{\displaystyle \frac{\mathbb{B}[1-\beta ,\beta ]}{\Gamma \left(\beta \right)}}H\left(t\right)dt=H\left(s\right).\hfill \end{array}$$

Thus, $H\left(s\right)=\mathcal{C}\left(s\right)$.

(ii) For $l\varrho <s\le (l+1)\varrho$, $l\in {\mathbb{N}}^{*}$, the right side of (11) is

$$\begin{array}{cc}\hfill & \mathcal{A}\overline{\mathcal{X}}(s-\varrho )+\overline{\mathcal{X}}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right)\hfill \\ \hfill =& \mathcal{A}{\int }_0^{s-\varrho }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-2\varrho -t)H\left(t\right)dt+{\int }_0^{s-\varrho }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-2\varrho -t)H\left(t\right)dt\mathcal{B}+\mathcal{C}\left(s\right)\hfill \\ \hfill =& \mathcal{A}\left({\int }_0^{s-\varrho }\mathbb{I}{\displaystyle \frac{{(s-\varrho -t)}^{\beta -1}}{\Gamma \left(\beta \right)}}H\left(t\right)dt\right.\hfill \\ \hfill & \left.+\dots +{\int }_0^{s-l\varrho }\sum _{k=0}^{l-1}\left({\displaystyle \genfrac{}{}{0pt}{}{l-1}{k}}\right){\mathcal{A}}^{l-1-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-l\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}H\left(t\right)dt\right)\hfill \\ \hfill & +\left({\int }_0^{s-\varrho }\mathbb{I}{\displaystyle \frac{{(s-\varrho -t)}^{\beta -1}}{\Gamma \left(\beta \right)}}H\left(t\right)dt\right.\hfill \\ \hfill & \left.+\dots +{\int }_0^{s-l\varrho }\sum _{k=0}^{l-1}\left({\displaystyle \genfrac{}{}{0pt}{}{l-1}{k}}\right){\mathcal{A}}^{l-1-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-l\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}H\left(t\right)dt\right)\mathcal{B}+\mathcal{C}\left(s\right)\hfill \\ \hfill =& {\int }_0^{s-\varrho }(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(s-\varrho -t)}^{\beta -1}}{\Gamma \left(\beta \right)}}H\left(t\right)dt\hfill \\ \hfill & +\dots +{\int }_0^{s-l\varrho }\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right){\mathcal{A}}^{l-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-l\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}H\left(t\right)dt+\mathcal{C}\left(s\right).\hfill \end{array}$$

By Definition 2, Lemmas 2 and 3, the left side of (11) is

$$\begin{array}{cc}\hfill & \left({{}^c\mathcal{D}}_0^{\beta }\overline{\mathcal{X}}\right)\left(s\right)=\left({{}^{RL}\mathcal{D}}_0^{\beta }\overline{\mathcal{X}}\right)\left(s\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{(s-h)}^{-\beta }{\int }_0^h{\mathcal{X}}_{\beta ,\beta }^{\varrho }(h-2\varrho -t)H\left(t\right)dtdh\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{\int }_t^s{(s-h)}^{-\beta }{\mathcal{X}}_{\beta ,\beta }^{\varrho }(h-2\varrho -t)dhH\left(t\right)dt\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{ds}}{\int }_0^s{\int }_t^s{(s-h)}^{-\beta }{\displaystyle \frac{{(h-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dhH\left(t\right)dt\hfill \\ \hfill & +\dots +\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right){\mathcal{A}}^{l-k}{\mathcal{B}}^k{\displaystyle \frac{d}{ds}}{\int }_0^{s-l\varrho }{\int }_{t+l\varrho }^s{(s-h)}^{-\beta }{\displaystyle \frac{{(h-l\varrho -t)}^{\beta -1}}{\Gamma (1-\beta )\Gamma \left(\right(l+1\left)\beta \right)}}dhH\left(t\right)dt\hfill \\ \hfill =& H\left(s\right)+{\int }_0^{s-\varrho }(\mathcal{A}+\mathcal{B}){\displaystyle \frac{{(s-\varrho -t)}^{\beta -1}}{\Gamma \left(\beta \right)}}H\left(t\right)dt\hfill \\ \hfill & +\dots +{\int }_0^{s-l\varrho }\sum _{k=0}^l\left({\displaystyle \genfrac{}{}{0pt}{}{l}{k}}\right){\mathcal{A}}^{l-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-l\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}H\left(t\right)dt\hfill \\ \hfill =& \mathcal{A}\overline{\mathcal{X}}(s-\varrho )+\overline{\mathcal{X}}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right).\hfill \end{array}$$

Thus, we have $H\left(s\right)=\mathcal{C}\left(s\right)$. Consequently, (12) is achieved. □

From Theorems 2 and 3, the following result holds.

Theorem 4. 

If $\mathcal{B}\Phi \left(s\right)=\Phi \left(s\right)\mathcal{B}$, $s\in [-\varrho ,0]$, and $\mathcal{BC}\left(s\right)=\mathcal{C}\left(s\right)\mathcal{B}$, $s\in \mathcal{P}$, then a solution $\mathcal{X}\left(s\right)$ of (1) can be expressed by

$$\begin{array}{c}\hfill \mathcal{X}\left(s\right)={\mathcal{X}}_{\beta }^{\varrho }\left(s\right)\Phi (-\varrho )+{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho -t){\Phi }^{\prime }\left(t\right)dt+{\int }_0^s{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)\mathcal{C}\left(t\right)dt,s\in [-\varrho ,\mathcal{T}].\end{array}$$

4. Ulam–Hyers Result

Theorem 5. 

If $\mathcal{B}\Phi \left(s\right)=\Phi \left(s\right)\mathcal{B}$, $s\in [-\varrho ,0]$, and $\mathcal{BC}\left(s\right)=\mathcal{C}\left(s\right)\mathcal{B}$, $s\in \mathcal{P}$, then (1) is Ulam–Hyers stable.

Proof. 

Suppose $\mathcal{Y}\left(s\right)$ is the solution to (2), that is,

$$∥\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{Y}\right)\left(s\right)-\mathcal{A}\mathcal{Y}(s-\varrho )-\mathcal{Y}(s-\varrho )\mathcal{B}-\mathcal{C}\left(s\right)∥\le \delta .$$

Let $\mathcal{X}:\mathcal{P}\to {\mathbb{R}}^{n\times n}$ is a solution to (1), satisfying

$$\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right),$$

for each $s\in \mathcal{P}$ and $0<\beta <1$; $\mathcal{Y}\left(s\right)=\mathcal{X}\left(s\right)$, for $s\in [-\varrho ,0]$. We assume

$$Q\left(s\right)=\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{Y}\right)\left(s\right)-\mathcal{A}\mathcal{Y}(s-\varrho )-\mathcal{Y}(s-\varrho )\mathcal{B}-\mathcal{C}\left(s\right),$$

(13)

where $\parallel Q\left(s\right)\parallel \le \delta$ and $\mathcal{B}Q\left(s\right)=Q\left(s\right)\mathcal{B}$, for $s\in \mathcal{P}$. Further, we obtain the solution $\mathcal{Y}\left(s\right)$ from Theorem 4 and (13):

$$\begin{array}{c}\hfill \mathcal{Y}\left(s\right)={\mathcal{X}}_{\beta }^{\varrho }\left(s\right)\Phi (-\varrho )+{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho -t){\Phi }^{\prime }\left(t\right)dt+{\int }_0^s{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)(\mathcal{C}\left(t\right)+Q\left(t\right))dt,\end{array}$$

for $s\in \mathcal{P}$. Similarly, for $s\in \mathcal{P}$, $\mathcal{X}\left(s\right)$ is represented by

$$\begin{array}{c}\hfill \mathcal{X}\left(s\right)={\mathcal{X}}_{\beta }^{\varrho }\left(s\right)\Phi (-\varrho )+{\int }_{-\varrho }^0{\mathcal{X}}_{\beta }^{\varrho }(s-\varrho -t){\Phi }^{\prime }\left(t\right)dt+{\int }_0^s{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)\mathcal{C}\left(t\right)dt.\end{array}$$

Then, by employing Lemma 1, we obtain

$$\begin{array}{cc}\hfill \parallel \mathcal{Y}\left(s\right)-\mathcal{X}\left(s\right)\parallel =& ∥{\int }_0^s{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)Q\left(t\right)dt∥\hfill \\ \hfill \le & {\int }_0^s\parallel Q\left(t\right)\parallel ∥{\mathcal{X}}_{\beta ,\beta }^{\varrho }(s-\varrho -t)∥dt\hfill \\ \hfill \le & \delta {\int }_0^s\sum _{k=0}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^k{\displaystyle \frac{{(s-k\varrho -t)}^{(k+1)\beta -1}}{\Gamma (k\beta +\alpha )}}dt\hfill \\ \hfill \le & \delta \left({\int }_0^s{\displaystyle \frac{{(s-t)}^{\beta -1}}{\Gamma \left(\beta \right)}}dt+{\int }_0^{s-\varrho }(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel ){\displaystyle \frac{{(s-\varrho -t)}^{2\beta -1}}{\Gamma \left(2\beta \right)}}dt\right.\hfill \\ \hfill & \left.+\dots +{\int }_0^{s-(l-1)\varrho }{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^{l-1}{\displaystyle \frac{{(s-(l-1)\varrho -t)}^{l\beta -1}}{\Gamma \left(l\beta \right)}}dt\right)\hfill \\ \hfill \le & \delta \sum _{k=1}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^{k-1}{\displaystyle \frac{{(s-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}\hfill \\ \hfill \le & \delta \sum _{k=1}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^{k-1}{\displaystyle \frac{{\mathcal{T}}^{k\beta }}{\Gamma (k\beta +1)}}.\hfill \end{array}$$

Thus, $u_h={\sum }_{k=1}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^{k-1}{\displaystyle \frac{{\mathcal{T}}^{k\beta }}{\Gamma (k\beta +1)}}>0$, according to Definition 3, (1) is Ulam–Hyers stable. □

Remark 1. 

From the proof of Theorem 5, we have $u_h={\sum }_{k=1}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^{k-1}{\displaystyle \frac{{\mathcal{T}}^{k\beta }}{\Gamma (k\beta +1)}}>0$, thus, the UH stability of (1) is mainly influenced by the fractional order β.

To more clearly illustrate the theoretical findings, Table 1 contrasts representative solutions and stability of delay FDEs with our proposed result.

Table 1. Comparisons of solutions and stability.

Study	Equation Type	Delayed Matrix Function	Result Type	Comparisons of Conditions
[20]	$\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )$, $\mathcal{X}\in {\mathbb{R}}^n$, $s\in \mathcal{P}$ $\mathcal{X}\left(s\right)=\Phi \left(s\right)$, $s\in [-\varrho ,0]$	${\sum }_{k=0}^l{\mathcal{A}}^k{\displaystyle \frac{{(s-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}$, $l\in \mathbb{N}$	1. Representation of solutions 2. Finite time stability	1. $\mathcal{X}$ and $\Phi$ are vector functions 2. ${\mathbb{E}}_{\beta }\left(\parallel \mathcal{A}\parallel s^{\beta }\right)\le {\displaystyle \frac{\alpha }{\gamma +M}}$,  $M={\int }_{-\varrho }^0\parallel {\Phi }^{\prime }\left(t\right)\parallel dt<\infty$ 3. ${\mathbb{E}}_{\beta }\left(\parallel \mathcal{A}\parallel s^{\beta }\right)+{\mathbb{E}}_{\beta }\left(\parallel \mathcal{A}\parallel s^{\beta }\right)<{\displaystyle \frac{\alpha }{\gamma }}$,  $\beta \le {\displaystyle \frac{1}{k}}$, $k\in {\mathbb{N}}^{*}$
[22]	1. $\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{C}\left(s\right)$ 2.$\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{C}(s,\mathcal{X}\left(s\right))$,  $\mathcal{X}\in {\mathbb{R}}^n$, $s\in \mathcal{P}$ $\mathcal{X}\left(s\right)=\Phi \left(s\right)$, $s\in [-\varrho ,0]$	${\sum }_{k=0}^l{\mathcal{A}}^k{\displaystyle \frac{{(s-(k-1)\varrho )}^{(k+1)\beta -1}}{\Gamma (k\beta +\alpha )}}$, $l\in \mathbb{N}$	1. Representation of solutions for nonhomogeneous problem 2. Existence of solutions for nonlinear problem 3. Finite time stability for nonlinear problem	1. $\mathcal{X}$ and $\Phi$ are vector functions 2. $\parallel \mathcal{C}(s,x)-\mathcal{C}(s,y)\parallel \le K\parallel x-y\parallel$ K$\left({\sum }_{k=1}^l{\displaystyle \frac{{\parallel \mathcal{A}\parallel }^{k-1}{(\mathcal{T}-(k-1)\varrho )}^{k\beta }}{\Gamma (k\beta +1)}}\right)<1$ $\parallel \mathcal{C}(s,y)\parallel \le L\parallel y\parallel$ ${\mathbb{E}}_{\beta }\left(\parallel \mathcal{A}{s}^{\beta }\parallel \right){\mathbb{E}}_{\beta }\left(L\Gamma \left(\beta \right){\mathbb{E}}_{\beta ,\beta }\left(\parallel \mathcal{A}\parallel s^{\beta }\right)s^{\beta }\right)$ $<{\displaystyle \frac{\eta }{\delta +M}}$
[23]	$\Delta \mathcal{X}\left(l\right)=\mathcal{A}\mathcal{X}(l-\varrho )+\mathcal{X}(l-\varrho )\mathcal{B}+\mathcal{C}\left(l\right)$, $l\in {\mathcal{Z}}_0^{\infty }$ $\mathcal{X}\left(l\right)=\Phi \left(l\right)$, $l\in {\mathcal{Z}}_{-\varrho }^0$ $\Delta$ is the first order forward difference	${\sum }_{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{l-(k-1)\varrho }{k}}\right){\sum }_{i=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\mathcal{A}}^{k-i}{\mathcal{B}}^i$, $l\in {\mathcal{Z}}_{(j-1)(\varrho +1)+1}^{j(\varrho +1)}$	1. Representation of solutions for homogeneous problem 2. Representation of solutions for nonhomogeneous problem	1. $\mathcal{X}$ is variable matrix 2. $\mathcal{B}\Phi \left(l\right)=\Phi \left(l\right)\mathcal{B}$, $l\in {\mathcal{Z}}_{-\varrho }^0$ 3. $\mathcal{B}\mathcal{C}\left(l\right)=\mathcal{C}\left(l\right)\mathcal{B}$, $l\in {\mathcal{Z}}_0^{\infty }$
This paper	$\left({{}^c\mathcal{D}}_0^{\beta }\mathcal{X}\right)\left(s\right)=\mathcal{A}\mathcal{X}(s-\varrho )+\mathcal{X}(s-\varrho )\mathcal{B}+\mathcal{C}\left(s\right)$, $s\in \mathcal{P},$ $\mathcal{X}\left(s\right)=\Theta ,-\varrho \le s\le 0$	${\sum }_{j=0}^l{\sum }_{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){\mathcal{A}}^{j-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-(j-1)\varrho )}^{j\beta }}{\Gamma (j\beta +1)}}$, ${\sum }_{j=0}^l{\sum }_{k=0}^j\left({\displaystyle \genfrac{}{}{0pt}{}{j}{k}}\right){\mathcal{A}}^{j-k}{\mathcal{B}}^k{\displaystyle \frac{{(s-(j-1)\varrho )}^{(j+1)\beta -1}}{\Gamma (j\beta +\alpha )}}$, $l\in \mathbb{N}$	1. Representation of solutions for homogeneous problem 2. Representation of solutions for nonhomogeneous problem 3. UH stability for nonhomogeneous problem	1. $\mathcal{X}$ is matrix function 2. $\mathcal{B}\Phi \left(s\right)=\Phi \left(s\right)\mathcal{B}$, $s\in [-\varrho ,0]$, 3. $\mathcal{BC}\left(s\right)=\mathcal{C}\left(s\right)\mathcal{B}$, $s\in \mathcal{P}$

As demonstrated in Table 1, the existing results mainly consist of two-sided matrix discrete equations [23] and one-sided vector fractional differential equations [20,22]. We retained the two-sided coefficient and firstly studied the matrix delay fractional differential equation with the two-sided coefficient. We provided the expressions of the solutions and also analyzed the UH stability. Our results do not require the kind of constraints on the delayed matrix function as in the finite-time stability in [20,22]. Of course, our results also have a research gap. Firstly, we need some matrix functions to be commutative, e.g., $\mathcal{B}\Phi \left(s\right)=\Phi \left(s\right)\mathcal{B}$, $s\in [-\varrho ,0]$, and $\mathcal{BC}\left(s\right)=\mathcal{C}\left(s\right)\mathcal{B}$, $s\in \mathcal{P}$. Additionally, this article only considers the response of system (1) to its own structural perturbations, i.e., UH stability, and does not analyze the asymptotic response of system (1) to initial perturbation, i.e., Lyapunov stability, or the instantaneous response of system (1) to initial perturbation within a finite time interval, i.e., finite-time stability.

5. Numerical Example

Here, we provide a numerical example to demonstrate our results.

Example 1. 

Consider the following linear TSFMDDE

$$\left\{\begin{array}{cc}\hfill & \left({{}^c\mathcal{D}}_0^{0.6}\mathcal{X}\right)\left(s\right)=\left[\begin{array}{cc}0.4& 0\\ 0& 0.3\end{array}\right]\mathcal{X}(s-2)+\mathcal{X}(s-2)\left[\begin{array}{cc}0.2& 0\\ 0& 0.3\end{array}\right]+\left[\begin{array}{cc}s& 0\\ 0& 2s\end{array}\right],s\in [0,4],\hfill \\ \hfill & \mathcal{X}\left(s\right)=\Phi \left(s\right)=\left[\begin{array}{cc}{s}^2& 0\\ 0& 3s^2\end{array}\right],-2\le s\le 0,\mathcal{X}\left(s\right)\in {\mathbb{R}}^{2\times 2}.\hfill \end{array}\right.$$

(14)

By Theorem 4, for $-2\le s\le 4$, we have

$$\begin{array}{cc}\hfill \mathcal{X}\left(s\right)=& {\mathcal{X}}_{0.6}^2\left(s\right)\left[\begin{array}{cc}4& 0\\ 0& 12\end{array}\right]+{\int }_{-2}^0{\mathcal{X}}_{0.6}^2(s-2-t)\left[\begin{array}{cc}2t& 0\\ 0& 6t\end{array}\right]dt+{\int }_0^s{\mathcal{X}}_{0.6,0.6}^2(s-2-t)\left[\begin{array}{cc}t& 0\\ 0& 2t\end{array}\right]dt\hfill \\ \hfill =& \left[\begin{array}{cc}{x}_{11}\left(s\right)& x_{12}\left(s\right)\\ x_{21}\left(s\right)& x_{22}\left(s\right)\end{array}\right],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill x_{11}\left(s\right)& =\left\{\begin{array}{cc}\hfill & s^2,s\in [-2,0],\hfill \\ \hfill & {\displaystyle \frac{12}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{25}{24\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{6}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)s^{{\displaystyle \frac{8}{5}}}-{\displaystyle \frac{1}{13}}{s}^{{\displaystyle \frac{13}{5}}}\right),s\in [0,2],\hfill \\ \hfill & {\displaystyle \frac{12}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{36}{25\Gamma \left(2.2\right)}}{(s-2)}^{{\displaystyle \frac{6}{5}}}+{\displaystyle \frac{25}{24\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{5}{22\Gamma \left(1.2\right)}}{(s-2)}^{{\displaystyle \frac{11}{5}}}\hfill \\ \hfill & +{\displaystyle \frac{18}{5\Gamma \left(2.2\right)}}\left({\displaystyle \frac{1}{11}}(s-4){(s-2)}^{{\displaystyle \frac{11}{5}}}-{\displaystyle \frac{1}{16}}{(s-2)}^{{\displaystyle \frac{16}{5}}}\right)\hfill \\ \hfill & +{\displaystyle \frac{6}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)\left(s^{{\displaystyle \frac{8}{5}}}-{(s-2)}^{{\displaystyle \frac{8}{5}}}\right)-{\displaystyle \frac{1}{13}}\left(s^{{\displaystyle \frac{13}{5}}}-{(s-2)}^{{\displaystyle \frac{13}{5}}}\right)\right),s\in [2,4],\hfill \end{array}\right.\hfill \\ \hfill x_{22}\left(s\right)& =\left\{\begin{array}{cc}\hfill & 3s^2,s\in [-2,0],\hfill \\ \hfill & {\displaystyle \frac{36}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{25}{12\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{18}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)s^{{\displaystyle \frac{8}{5}}}-{\displaystyle \frac{1}{13}}{s}^{{\displaystyle \frac{13}{5}}}\right),s\in [0,2],\hfill \\ \hfill & {\displaystyle \frac{36}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{108}{25\Gamma \left(2.2\right)}}{(s-2)}^{{\displaystyle \frac{6}{5}}}+{\displaystyle \frac{25}{12\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{5}{11\Gamma \left(1.2\right)}}{(s-2)}^{{\displaystyle \frac{11}{5}}}\hfill \\ \hfill & +{\displaystyle \frac{54}{5\Gamma \left(2.2\right)}}\left({\displaystyle \frac{1}{11}}(s-4){(s-2)}^{{\displaystyle \frac{11}{5}}}-{\displaystyle \frac{1}{16}}{(s-2)}^{{\displaystyle \frac{16}{5}}}\right)\hfill \\ \hfill & +{\displaystyle \frac{18}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)\left(s^{{\displaystyle \frac{8}{5}}}-{(s-2)}^{{\displaystyle \frac{8}{5}}}\right)-{\displaystyle \frac{1}{13}}\left(s^{{\displaystyle \frac{13}{5}}}-{(s-2)}^{{\displaystyle \frac{13}{5}}}\right)\right),s\in [2,4],\hfill \end{array}\right.\hfill \\ \hfill x_{12}\left(s\right)& =x_{21}\left(s\right)=0,s\in [-2,4].\hfill \end{array}$$

Now, we verify UH stability of (14). Let $\delta =4$ and $Q\left(s\right)$ is given by

$$\begin{array}{c}\hfill Q\left(s\right)=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right].\end{array}$$

Obviously,

$$\begin{array}{c}\hfill \parallel Q\left(s\right)\parallel =s\le 4,s\in [0,4].\end{array}$$

Hence, the solution $\mathcal{Y}\left(s\right)$ of (13) is

$$\begin{array}{cc}\hfill \mathcal{Y}\left(s\right)=& {\mathcal{X}}_{0.6}^2\left(s\right)\left[\begin{array}{cc}4& 0\\ 0& 12\end{array}\right]+{\int }_{-2}^0{\mathcal{X}}_{0.6}^2(s-2-t)\left[\begin{array}{cc}2t& 0\\ 0& 6t\end{array}\right]dt+{\int }_0^s{\mathcal{X}}_{0.6,0.6}^2(s-2-t)\left[\begin{array}{cc}2t& 0\\ 0& 3t\end{array}\right]dt\hfill \\ \hfill =& \left[\begin{array}{cc}{y}_{11}\left(s\right)& y_{12}\left(s\right)\\ y_{21}\left(s\right)& y_{22}\left(s\right)\end{array}\right],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill y_{11}\left(s\right)& =\left\{\begin{array}{cc}\hfill & s^2,s\in [-2,0],\hfill \\ \hfill & {\displaystyle \frac{12}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{25}{12\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{6}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)s^{{\displaystyle \frac{8}{5}}}-{\displaystyle \frac{1}{13}}{s}^{{\displaystyle \frac{13}{5}}}\right),s\in [0,2],\hfill \\ \hfill & {\displaystyle \frac{12}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{36}{25\Gamma \left(2.2\right)}}{(s-2)}^{{\displaystyle \frac{6}{5}}}+{\displaystyle \frac{25}{12\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{5}{11\Gamma \left(1.2\right)}}{(s-2)}^{{\displaystyle \frac{11}{5}}}\hfill \\ \hfill & +{\displaystyle \frac{18}{5\Gamma \left(2.2\right)}}\left({\displaystyle \frac{1}{11}}(s-4){(s-2)}^{{\displaystyle \frac{11}{5}}}-{\displaystyle \frac{1}{16}}{(s-2)}^{{\displaystyle \frac{16}{5}}}\right)\hfill \\ \hfill & +{\displaystyle \frac{6}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)\left(s^{{\displaystyle \frac{8}{5}}}-{(s-2)}^{{\displaystyle \frac{8}{5}}}\right)-{\displaystyle \frac{1}{13}}\left(s^{{\displaystyle \frac{13}{5}}}-{(s-2)}^{{\displaystyle \frac{13}{5}}}\right)\right),s\in [2,4],\hfill \end{array}\right.\hfill \\ \hfill y_{22}\left(s\right)& =\left\{\begin{array}{cc}\hfill & 3s^2,s\in [-2,0],\hfill \\ \hfill & {\displaystyle \frac{36}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{25}{8\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{18}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)s^{{\displaystyle \frac{8}{5}}}-{\displaystyle \frac{1}{13}}{s}^{{\displaystyle \frac{13}{5}}}\right),s\in [0,2],\hfill \\ \hfill & {\displaystyle \frac{36}{5\Gamma \left(1.6\right)}}{s}^{{\displaystyle \frac{3}{5}}}+{\displaystyle \frac{108}{25\Gamma \left(2.2\right)}}{(s-2)}^{{\displaystyle \frac{6}{5}}}+{\displaystyle \frac{25}{8\Gamma \left(0.6\right)}}{s}^{{\displaystyle \frac{8}{5}}}+{\displaystyle \frac{15}{22\Gamma \left(1.2\right)}}{(s-2)}^{{\displaystyle \frac{11}{5}}}\hfill \\ \hfill & +{\displaystyle \frac{54}{5\Gamma \left(2.2\right)}}\left({\displaystyle \frac{1}{11}}(s-4){(s-2)}^{{\displaystyle \frac{11}{5}}}-{\displaystyle \frac{1}{16}}{(s-2)}^{{\displaystyle \frac{16}{5}}}\right)\hfill \\ \hfill & +{\displaystyle \frac{18}{\Gamma \left(1.6\right)}}\left({\displaystyle \frac{1}{8}}(s-2)\left(s^{{\displaystyle \frac{8}{5}}}-{(s-2)}^{{\displaystyle \frac{8}{5}}}\right)-{\displaystyle \frac{1}{13}}\left(s^{{\displaystyle \frac{13}{5}}}-{(s-2)}^{{\displaystyle \frac{13}{5}}}\right)\right),s\in [2,4],\hfill \end{array}\right.\hfill \\ \hfill y_{12}\left(s\right)& =y_{21}\left(s\right)=0,s\in [-2,4].\hfill \end{array}$$

Further, according to Theorem 5, one gets

$$\begin{array}{cc}\hfill u_h=\sum _{k=1}^l{(\parallel \mathcal{A}\parallel +\parallel \mathcal{B}\parallel )}^{k-1}{\displaystyle \frac{{\mathcal{T}}^{k\beta }}{\Gamma (k\beta +1)}}=& \sum _{k=1}^{2}0.7^{k-1}{\displaystyle \frac{4^{0.6k}}{\Gamma (0.6k+1)}}\hfill \\ \hfill =& {\displaystyle \frac{4^{0.6}}{\Gamma \left(1.6\right)}}+0.7{\displaystyle \frac{4^{1.2}}{\Gamma \left(2.2\right)}}\approx 5.92479>0,\hfill \end{array}$$

thus the system (14) is Ulam–Hyers stable.

Finally, we visualize the solutions and UH stability features of (14) (see Figure 2) and simultaneously present the variation patterns of the error $u_h\delta$ between the approximate solution $\mathcal{Y}\left(s\right)$ and the exact solution $\mathcal{X}\left(s\right)$ as the fractional order β increases (see Figure 3). In Figure 2, Curves $y_{12}\left(s\right)$, $y_{21}\left(s\right)$, $x_{12}\left(s\right)$, and $x_{21}\left(s\right)$ overlap each other. Curves $x_{11}\left(s\right)$ and $y_{11}\left(s\right)$, as well as curves $x_{22}\left(s\right)$ and $y_{22}\left(s\right)$, are close to each other. Figure 3 shows that the error $u_h\delta$ increases as fractional order β increases.

Figure 2. Images of the solutions and UH stability features of (14).

Figure 3. The variation patterns of the error $u_h\delta$ as the fractional order $\beta$ increases.

6. Conclusions

This paper studies the representation of solutions and UH stability of linear TSFMDDE. By constructing a special TSFDMLMF, we construct the solutions for homogeneous and nonhomogeneous problems and discuss the UH stability of nonhomogeneous equations through the representation of the solutions. For future work, it is interesting to investigate the relative controllability of linear TSFMDDE, based on previous results [15,16,31].