Nonlinear Sequential Caputo Fractional Differential Systems: Existence and Hyers–Ulam Stability Under Coupled Mixed Boundary Constraints

Abstract

In this paper, we study a nonlinear system of sequential Caputo fractional differential equations equipped with coupled mixed multi-point boundary conditions. In particular, the boundary conditions involve the values of the unknown functions at the endpoints expressed as linear combinations of their values at several interior points, forming a closed system of relations. The existence of solutions is established by applying the Leray–Schauder alternative, while uniqueness is proved using Banach’s contraction principle. In addition, we investigate the Hyers–Ulam stability of the proposed system. Several examples are included to demonstrate the applicability of the theoretical results. Some special cases of the general problem are also discussed.

1. Introduction

Sequential Caputo fractional differential equations refer to systems in which derivatives of fractional order are applied successively, typically involving Caputo’s definition due to its compatibility with initial conditions of classical type. These equations are crucial in modeling complex dynamical systems that exhibit memory and hereditary properties, such as viscoelastic materials, diffusion processes, and biological systems. The sequential aspect enables the capture of layered temporal effects by combining multiple fractional orders, often leading to more accurate and flexible representations of real-world phenomena. Unlike classical integer-order models, sequential Caputo systems offer refined control over system dynamics, making them valuable for theoretical analysis and applications where multi-scale temporal behavior is significant [1,2,3,4,5,6].

Coupled systems consist of two or more interdependent differential equations in which the solution of one equation directly influences the others. These systems naturally arise in modeling phenomena involving multiple interacting components, such as in mechanical structures, electrical circuits, or biological networks. In the context of fractional calculus, coupling adds complexity by integrating memory effects across interconnected variables. Studying such systems is essential for understanding the joint dynamics of processes in which the isolation of individual components fails to capture the full behavior of the system [7,8,9,10,11,12,13].

Ulam–Hyers stability is a concept in the qualitative theory of differential equations that addresses the stability of approximate solutions. Specifically, it ensures that if a function nearly satisfies a given differential equation, then there exists an exact solution nearby. This type of stability is particularly important in applied mathematics, where exact solutions are often inaccessible, and numerical or approximate methods are used. In the context of fractional differential equations, Ulam–Hyers stability provides a robust framework for assessing the reliability and resilience of solutions under small perturbations or modeling inaccuracies [14,15,16,17].

In recent years, discrete fractional models involving the nabla fractional operator have received significant attention owing to their capability to effectively describe memory and hereditary characteristics in discrete-time dynamical systems. These nabla fractional difference equations serve as discrete analogues of continuous fractional differential equations and have been widely applied in the qualitative analysis of memory-driven processes [18]. In particular, several recent contributions have addressed coupled nabla fractional systems and investigated the existence, nonexistence, uniqueness, multiplicity, and stability of solutions for problems subject to various types of nonlocal boundary conditions, including multi-point summation boundary conditions [19,20]. The obtained results highlight the importance of discrete fractional operators in modeling systems where past states influence present dynamics. Motivated by these recent developments, the present study focuses on a coupled system of continuous fractional differential equations with multi-point summation boundary conditions, thereby extending the applicability of memory-dependent fractional models from discrete to continuous settings.

Let us now recall some works on closed boundary conditions. In ref. [21], Bashir Ahmad studied mixed nonlinearities with nonlocal closed boundary conditions.

$${}^C{\mathcal{D}}_{{\mathcal{T}}^{-}}^{\mu }\mathfrak{Y}\left(\tau \right)+\lambda I_{{\mathcal{T}}^{-}}^{\rho }{I}_{0^{+}}^{\sigma }{\mathcal{K}}_1(\tau ,\mathfrak{Y}\left(\tau \right))={\mathcal{K}}_2(\tau ,\mathfrak{Y}\left(\tau \right)),\tau \in J:=[0,\mathcal{T}],$$

(1)

$$\mathfrak{Y}\left(\tau \right)=p_1\mathfrak{Y}\left(\xi \right)+p_2\mathcal{T}{\mathfrak{Y}}^{\prime }\left(\xi \right),\mathcal{T}{\mathfrak{Y}}^{\prime }\left(\xi \right)=q_1\mathfrak{Y}\left(\xi \right)+q_2\mathcal{T}{\mathfrak{Y}}^{\prime }\left(\xi \right),0<\xi <\mathcal{T},$$

(2)

where ${}^C{D}_{{\mathcal{T}}^{\_}}^{\mu }$ represents the right Caputo fractional derivative operator of order $\mu \in (1,2]$, $I_{{\mathcal{T}}^{-}}^{\rho }$ and $I_{0^{+}}^{\sigma }$, respectively, designate the right and left Riemann–Liouville operators of fractional orders $\rho ,\sigma >0$. ${\phi }_1,{\phi }_2:[0,\mathcal{T}]\times \mathbb{R}\to \mathcal{R}$ are continuous functions and $\lambda ,p_1,p_2,q_1,q_2\in \mathcal{R}$. Recently, ref. [22] investigated the existence and uniqueness of solutions for a nonlinear system of fractional differential equations with coupled closed boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^c{\mathcal{D}}^{\alpha })\mathfrak{Y}\left(\tau \right)=\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right)),\hfill \\ {(}^c{\mathcal{D}}^{\beta })\mathfrak{Z}\left(\tau \right)=\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right)),\hfill \end{array}\right.\end{array}$$

with closed boundary conditions;

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \mathfrak{Y}\left(\mathcal{T}\right)={\mathfrak{r}}_1\mathfrak{Z}\left(0\right)+{\mathfrak{r}}_2\mathcal{T}{\mathfrak{Z}}^{\prime }\left(0\right),\mathcal{T}{\mathfrak{Y}}^{\prime }\left(\mathcal{T}\right)={\mathfrak{s}}_1\mathfrak{Z}\left(0\right)+{\mathfrak{s}}_2\mathcal{T}{\mathfrak{Z}}^{\prime }\left(0\right),\hfill \\ & \mathfrak{Z}\left(\mathcal{T}\right)={\mathfrak{p}}_1\mathfrak{Y}\left(0\right)+{\mathfrak{p}}_2\mathcal{T}{\mathfrak{Y}}^{\prime }\left(0\right),\mathcal{T}{\mathfrak{Z}}^{\prime }\left(\mathcal{T}\right)={\mathfrak{q}}_1\mathfrak{Y}\left(0\right)+{\mathfrak{q}}_2\mathcal{T}{\mathfrak{Y}}^{\prime }\left(0\right),\hfill \end{array}\right.\end{array}$$

where ${}^c\mathcal{D}^{\alpha }$ and ${}^c\mathcal{D}^{\beta }$ denote Caputo fractional derivatives of order $\alpha ,\beta ,1<\alpha ,\beta <2$, respectively, ${\mathfrak{r}}_1,{\mathfrak{r}}_2,{\mathfrak{s}}_1,{\mathfrak{s}}_2,{\mathfrak{p}}_1$,${\mathfrak{p}}_2,{\mathfrak{q}}_1,{\mathfrak{q}}_2,\in \mathcal{R},\mathcal{T}>0$ and $\mathcal{K},\mathcal{L}\in \mathcal{C}(\mathcal{J}\times \mathcal{R}\times \mathcal{R}\times \mathcal{R})$. Motivated by the mentioned work, in this paper, we propose a nonlocal variant of closed boundary conditions and study a coupled sequential fractional differential equation containing a right Caputo fractional derivative. Specifically, we explore the criteria that ensure the existence and stability of solutions for the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^c{\mathcal{D}}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1})\mathfrak{Y}\left(\tau \right)=\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right)),\hfill \\ {(}^c{\mathcal{D}}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1})\mathfrak{Z}\left(\tau \right)=\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right)),\hfill \end{array}\right.\end{array}$$

(3)

with multi-point boundary conditions.

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \mathfrak{Y}\left(\mathcal{T}\right)=\sum _{i=1}^m{\mathfrak{r}}_{1i}\mathfrak{Z}\left({\mathfrak{k}}_1\right)+\sum _{i=1}^m\mathfrak{r}{2}_i\mathcal{T}{\mathfrak{Z}}^{\prime }\left({\mathfrak{k}}_2\right),\mathcal{T}{\mathfrak{Y}}^{\prime }\left(\mathcal{T}\right)=\sum _{i=1}^m{{\mathfrak{s}}_1}_i\mathfrak{Z}\left({\mathfrak{u}}_1\right)+\sum _{i=1}^m{{\mathfrak{s}}_2}_i\mathcal{T}{\mathfrak{Z}}^{\prime }\left({\mathfrak{u}}_2\right),}\hfill \\ & {\displaystyle \mathfrak{Z}\left(\mathcal{T}\right)=\sum _{j=1}^n{{\mathfrak{p}}_1}_j\mathfrak{Y}\left({\zeta }_1\right)+\sum _{j=1}^n{{\mathfrak{p}}_2}_j\mathcal{T}{\mathfrak{Y}}^{\prime }\left({\zeta }_2\right),\mathcal{T}{\mathfrak{Z}}^{\prime }\left(\mathcal{T}\right)=\sum _{j=1}^n{{\mathfrak{q}}_1}_j\mathfrak{Y}\left({\mathfrak{v}}_1\right)+\sum _{j=1}^n{{\mathfrak{q}}_2}_j\mathcal{T}{\mathfrak{Y}}^{\prime }\left({\mathfrak{v}}_2\right),}\hfill \end{array}\right.\end{array}$$

(4)

where ${}^c\mathcal{D}^{\alpha }$ and ${}^c\mathcal{D}^{\beta }$ denote the Caputo fractional derivatives of order $\alpha ,\beta ,1<\alpha ,\beta <2$, respectively, ${\mathfrak{r}}_i,{\mathfrak{s}}_i,{\mathfrak{p}}_i,{\mathfrak{q}}_i,\in \mathcal{R}, i\in \{1,2\}$, and $\mathcal{K},\mathcal{L}\in \mathcal{C}(\mathcal{J}\times \mathcal{R}\times \mathcal{R}\times \mathcal{R})$ and ${\mathfrak{k}}_i,{\mathfrak{u}}_i,{\zeta }_i,{\mathfrak{v}}_i\in [0,\mathcal{T}], i\in \{1,2\}$. The boundary conditions (4) can be interpreted as the values of the unknown function and its derivative at the right end point $\mathcal{T}$ of the interval $[0,\mathcal{T}]$ are proportional to a linear combination of these values at arbitrary non-local positions ${\mathfrak{k}}_i,{\mathfrak{u}}_i,{\zeta }_i,{\mathfrak{v}}_i\in [0,\mathcal{T}], i\in \{1,2\}$. Physically, the nonlocal multi-point closed boundary conditions provide a flexible mechanism to close the boundary at arbitrary positions in the given domain instead of the left end point of the domain.

The organization of this paper is summarized as follows:

• Section 2 (Preliminaries): This section presents the fundamental definitions, auxiliary lemmas, and supporting results required for the subsequent analysis.
• Section 3 (Existence and Uniqueness Results): In this section, we derive the integral equivalent formulation of the proposed problem and establish the main existence and uniqueness results using appropriate fixed-point techniques.
• Section 4 (Hyers–Ulam Stability): This section is devoted to the study of the Hyers–Ulam stability of the considered system and provides sufficient conditions ensuring stability.
• Section 5 (Examples): Two illustrative examples are presented to demonstrate the applicability of the theoretical results.
• Section 6 (Conclusions): The final section summarizes the main findings and discusses possible directions for future research.

2. Auxiliary Results

In this section, we present several definitions and auxiliary results that will be used in the proofs of the main theorems.

Definition 1

([23]). Let $\alpha >0$ and let $\mathfrak{Y}:[0,\infty )\to \mathbb{R}$ be a function such that the following integral exists. The Riemann–Liouville fractional integral of order α is defined by

$$\begin{array}{c}\hfill {\mathcal{I}}^{\alpha }\mathfrak{Y}\left(\tau \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\tau }{(\tau -s)}^{\alpha -1}\mathfrak{Y}\left(s\right)ds,\tau >0,\end{array}$$

where $\mathrm{\Gamma }(·)$ denotes the Gamma function given by

$$\mathrm{\Gamma }\left(p\right)={\int }_0^{\infty }{t}^{p-1}{e}^{-t}dt,p>0.$$

Definition 2

([23]). Let $n\in \mathbb{N}$ and let α satisfy $n-1<\alpha <n$. If $\mathfrak{Y}:[0,\infty )\to \mathbb{R}$ is $(n-1)$ times continuously differentiable and its n-th derivative belongs to $L_{\mathrm{loc}}^1\left([0,\infty )\right)$, then the Caputo fractional derivative of order α is defined by

$$\begin{array}{c}\hfill {}^c\mathcal{D}^{\alpha }\mathfrak{Y}\left(\tau \right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\int }_0^{\tau }{(\tau -s)}^{n-\alpha -1}{\mathfrak{Y}}^{\left(n\right)}\left(s\right)ds,\tau >0.\end{array}$$

In the following, we study the associated linear fractional system as a preliminary step toward the nonlinear analysis.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^c{\mathcal{D}}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1})\mathfrak{Y}\left(\tau \right)={\mathcal{H}}_1\left(\tau \right),\hfill \\ {(}^c{\mathcal{D}}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1})\mathfrak{Z}\left(\tau \right)={\mathcal{H}}_2\left(\tau \right),\hfill \end{array}\right.\end{array}$$

(5)

supplemented with the boundary conditions (4), with ${\mathcal{H}}_1,{\mathcal{H}}_2\in \mathcal{C}[0,\mathcal{T}]$. We denote by

$$\begin{array}{cc}\hfill & {\displaystyle {\mathcal{A}}_1=e^{-{\lambda }_1\mathcal{T}},{\mathcal{A}}_2=\frac{(1-e^{-{\lambda }_1\mathcal{T}})}{{\lambda }_1},{\mathcal{A}}_3=\sum _{i=1}^m\left\{-{\mathfrak{r}}_1{e}^{-{\lambda }_2{\mathfrak{k}}_1}+{\mathfrak{r}}_2\mathcal{T}{\lambda }_2{e}^{-{\lambda }_2{\mathfrak{k}}_2}\right\}}\hfill \\ & {\displaystyle {\mathcal{A}}_4=\sum _{i=1}^m\left\{-{\mathfrak{r}}_1\frac{(1-e^{-{\lambda }_2{\mathfrak{k}}_1})}{{\lambda }_2}-{\mathfrak{r}}_2\mathcal{T}\frac{\left({\lambda }_2{e}^{-{\lambda }_2{\mathfrak{k}}_2}\right)}{{\lambda }_2}\right\},{\mathcal{A}}_5=\mathcal{T}(-{\lambda }_1{e}^{-{\lambda }_1\mathcal{T}}),{\mathcal{A}}_6=\mathcal{T}\left({\lambda }_1{e}^{-{\lambda }_1\mathcal{T}}\right)}\hfill \\ & {\displaystyle {\mathcal{A}}_7=\sum _{i=1}^m\left\{-{\mathfrak{s}}_1{e}^{-{\lambda }_2{\mathfrak{u}}_1}+{\mathfrak{s}}_2\mathcal{T}{\lambda }_2{e}^{-{\lambda }_2{\mathfrak{u}}_2}\right\},{\mathcal{A}}_8=\sum _{i=1}^m\left\{-{\mathfrak{s}}_1\frac{(1-e^{-{\lambda }_2{\mathfrak{u}}_1})}{{\lambda }_2}-{\mathfrak{s}}_2\mathcal{T}\frac{\left({\lambda }_2{e}^{-{\lambda }_2{\mathfrak{u}}_2}\right)}{{\lambda }_2}\right\}}\hfill \\ & {\displaystyle {\mathcal{A}}_9=\sum _{j=1}^n\left\{-{\mathfrak{p}}_1{e}^{-{\lambda }_1{\zeta }_1}+{\mathfrak{p}}_2\mathcal{T}{\lambda }_1{e}^{-{\lambda }_1{\zeta }_2}\right\},{\mathcal{A}}_{10}=\sum _{j=1}^n\left\{-{\mathfrak{p}}_1\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1{\zeta }_1})-{\mathfrak{p}}_2\mathcal{T}\frac{1}{{\lambda }_1}\left({\lambda }_1{e}^{-{\lambda }_1{\zeta }_2}\right)\right\}}\hfill \\ & {\displaystyle {\mathcal{A}}_{11}=e^{-{\lambda }_2\mathcal{T}},{\mathcal{A}}_{12}=\frac{1}{{\lambda }_2}(1-e^{-{\lambda }_2\mathcal{T}}),{\mathcal{A}}_{13}=\sum _{j=1}^n\left\{-{\mathfrak{q}}_1{e}^{-{\lambda }_1\left({\mathfrak{v}}_1\right)}+{\mathfrak{q}}_2\mathcal{T}{e}^{-{\lambda }_1{\mathfrak{v}}_2}\right\}}\hfill \\ & {\displaystyle {\mathcal{A}}_{14}=\sum _{j=1}^n\left\{-{\mathfrak{q}}_1\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\mathfrak{v}})-{\mathfrak{q}}_2\frac{\mathcal{T}}{{\lambda }_1}\left({\lambda }_1{e}^{-{\lambda }_1{\mathfrak{v}}_2}\right)\right\},{\mathcal{A}}_{15}=\left[-{\lambda }_2\mathcal{T}{e}^{-{\lambda }_2\mathcal{T}}\right],{\mathcal{A}}_{16}=\frac{\mathcal{T}}{{\lambda }_2}\left({\lambda }_2{e}^{-{\lambda }_2\mathcal{T}}\right).}\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle {\mathcal{E}}_1=\sum _{i=1}^m\{}& {\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}+{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\}\hfill \\ & -{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\hfill \\ \hfill {\displaystyle {\mathcal{E}}_2=\sum _{i=1}^m\{}& {\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\}+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s},\hfill \\ \hfill {\displaystyle {\mathcal{E}}_3=\sum _{i=1}^n\{}& {\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\}-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\hfill \\ \hfill {\displaystyle {\mathcal{E}}_4=\sum _{i=1}^n\{}& {\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\}+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s},\hfill \\ \hfill {\mathrm{\Delta }}_1=& ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{14})({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & -({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\hfill \\ \hfill {\mathrm{\Delta }}_2=& ({\mathcal{A}}_4{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{11}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{15})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & -({\mathcal{A}}_{12}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{16})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_3{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_7)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14}).\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\displaystyle \mathfrak{Y}\left(\tau \right)=}& \left[{\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\right(({\mathcal{E}}_3{\mathbb{A}}_{15}-{\mathcal{E}}_4{\mathbb{A}}_{11})({\mathbb{A}}_4{\mathbb{A}}_7-{\mathbb{A}}_3{\mathbb{A}}_8)({\mathbb{A}}_2{\mathbb{A}}_7-{\mathbb{A}}_3{\mathbb{A}}_6)({\mathbb{A}}_{12}{\mathbb{A}}_{15}-{\mathbb{A}}_{11}{\mathbb{A}}_{16})\hfill \\ & -({\mathcal{E}}_1{\mathbb{A}}_7-{\mathcal{E}}_2{\mathbb{A}}_3)({\mathbb{A}}_{12}{\mathbb{A}}_{15}-{\mathbb{A}}_{11}{\mathbb{A}}_{16})({\mathbb{A}}_4{\mathbb{A}}_7-{\mathbb{A}}_3{\mathbb{A}}_8)({\mathbb{A}}_{10}{\mathbb{A}}_{15}-{\mathbb{A}}_{11}{\mathbb{A}}_{16})\left)\right]e^{-{\lambda }_1\tau }\hfill \\ & +\left[{\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\right(({\mathcal{E}}_3{\mathbb{A}}_{15}-{\mathcal{E}}_4{\mathbb{A}}_{11})({\mathbb{A}}_4{\mathbb{A}}_7-{\mathbb{A}}_3{\mathbb{A}}_8)({\mathbb{A}}_1{\mathbb{A}}_7-{\mathbb{A}}_3{\mathbb{A}}_5)({\mathbb{A}}_{12}{\mathbb{A}}_{15}-{\mathbb{A}}_{11}{\mathbb{A}}_{16})\hfill \\ & -({\mathcal{E}}_1{\mathbb{A}}_7-{\mathcal{E}}_2{\mathbb{A}}_3)({\mathbb{A}}_{12}{\mathbb{A}}_{15}-{\mathbb{A}}_{11}{\mathbb{A}}_{16})({\mathbb{A}}_9{\mathbb{A}}_{15}-{\mathbb{A}}_{11}{\mathbb{A}}_{13})({\mathbb{A}}_4{\mathbb{A}}_7-{\mathbb{A}}_3{\mathbb{A}}_8)\left)\right]{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1\tau })\hfill \\ & +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)d\mathfrak{s},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\displaystyle \mathfrak{Z}\left(\tau \right)}& =\left[{\displaystyle \frac{1}{{\mathrm{\Delta }}_2}}\right(({\mathcal{E}}_1{\mathbb{A}}_5-{\mathcal{E}}_2{\mathbb{A}}_1)({\mathbb{A}}_{10}{\mathbb{A}}_{13}-{\mathbb{A}}_9{\mathbb{A}}_{14})({\mathbb{A}}_{12}{\mathbb{A}}_{13}-{\mathbb{A}}_9{\mathbb{A}}_{16})({\mathbb{A}}_2{\mathbb{A}}_5-{\mathbb{A}}_1{\mathbb{A}}_6)\hfill \\ & -({\mathcal{E}}_3{\mathbb{A}}_{13}-{\mathcal{E}}_4{\mathbb{A}}_9)({\mathbb{A}}_2{\mathbb{A}}_5-{\mathbb{A}}_1{\mathbb{A}}_6)({\mathbb{A}}_4{\mathbb{A}}_5-{\mathbb{A}}_1{\mathbb{A}}_8)({\mathbb{A}}_{10}{\mathbb{A}}_{13}-{\mathbb{A}}_9{\mathbb{A}}_{14})\left)\right]e^{-{\lambda }_2\tau }\hfill \\ & +\left[{\displaystyle \frac{1}{{\mathrm{\Delta }}_2}}\right(({\mathcal{E}}_1{\mathbb{A}}_5-{\mathcal{E}}_2{\mathbb{A}}_1)({\mathbb{A}}_{10}{\mathbb{A}}_{13}-{\mathbb{A}}_9{\mathbb{A}}_{14})({\mathbb{A}}_{11}{\mathbb{A}}_{13}-{\mathbb{A}}_9{\mathbb{A}}_{15})({\mathbb{A}}_2{\mathbb{A}}_5-{\mathbb{A}}_1{\mathbb{A}}_6)\hfill \\ & -({\mathcal{E}}_3{\mathbb{A}}_{13}-{\mathcal{E}}_4{\mathbb{A}}_9)({\mathbb{A}}_2{\mathbb{A}}_5-{\mathbb{A}}_1{\mathbb{A}}_6)({\mathbb{A}}_3{\mathbb{A}}_5-{\mathbb{A}}_1{\mathbb{A}}_7)({\mathbb{A}}_{10}{\mathbb{A}}_{13}-{\mathbb{A}}_9{\mathbb{A}}_{14})\left)\right]{\displaystyle \frac{1}{{\lambda }_2}}(1-e^{-{\lambda }_2\tau })\hfill \\ & +{\int }_0^{\tau }{e}^{-{\lambda }_2(\tau -\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)d\mathfrak{s},\hfill \end{array}$$

(8)

Lemma 1.

If ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2\ne 0$, then the solution $(\mathfrak{Y},\mathfrak{Z})\in {\left(\mathcal{C}[0,\mathcal{T}]\right)}^2$ of the boundary value problem (3) and (4) is given by

$$\begin{array}{cc}\hfill {\displaystyle \mathfrak{Y}\left(\tau \right)=}& {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left[\right(\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{15}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n\left(\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\right)\left\}{\mathcal{A}}_{11}\right)}\hfill \\ & \times ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)}\hfill \\ & \times ({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left[\right(\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{15}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)}\hfill \\ & \times ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)}\hfill \\ & {\displaystyle \times ({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)]+{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)d\mathfrak{s},}\hfill \\ \hfill {\displaystyle \mathfrak{Z}\left(\tau \right)=}& {\displaystyle \frac{e^{-{\lambda }_2\tau }}{{\mathrm{\Delta }}_2}\left[\right(\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_5}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_1\right)}\hfill \\ & \times ({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{12}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{16})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & {\displaystyle -\left(\right\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{13}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\left\}{\mathcal{A}}_9\right)}\hfill \\ & \times ({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_4{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_2}(1-e^{-{\lambda }_2\tau })\frac{1}{{\mathrm{\Delta }}_2}\left[\right(\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_5}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_1\right)}\hfill \\ & \times ({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{11}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{15})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & {\displaystyle -\left(\right\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{13}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\left\}{\mathcal{A}}_9\right)}\hfill \\ & {\displaystyle \times ({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_3{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_7)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})]+{\int }_0^{\tau }{e}^{-{\lambda }_2(\tau -\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)d\mathfrak{s}.}\hfill \end{array}$$

Proof. 

The system (3) can be equivalently written as the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^c{\mathcal{D}}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1})\mathfrak{Y}\left(\tau \right)={\mathcal{H}}_1\left(\tau \right),\tau \in (0,1),\hfill \\ {(}^c{\mathcal{D}}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1})\mathfrak{Z}\left(\tau \right)={\mathcal{H}}_2\left(\tau \right),\tau \in (0,1),\hfill \end{array}\right.\end{array}$$

(9)

where ${}^c\mathcal{D}^{-1}$ is the integral operator $\mathcal{I}$. The general solution of the system (9) is

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \mathfrak{Y}\left(\tau \right)={\mathfrak{c}}_0{e}^{-{\lambda }_1\tau }+\frac{{\mathfrak{c}}_1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })+{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)d\mathfrak{s},}\hfill \\ & {\displaystyle \mathfrak{Z}\left(\tau \right)={\mathfrak{d}}_0{e}^{-{\lambda }_2\tau }+\frac{{\mathfrak{d}}_1}{{\lambda }_2}(1-e^{-{\lambda }_2\tau })+{\int }_0^{\tau }{e}^{-{\lambda }_2(\tau -\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)d\mathfrak{s},}\hfill \end{array}\right.\end{array}$$

(10)

we differentiate the system (10) and we find

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle {\mathfrak{Y}}^{\prime }\left(\tau \right)=-{\lambda }_1{\mathfrak{c}}_0{e}^{-{\lambda }_1\tau }+\frac{{\mathfrak{c}}_1}{{\lambda }_1}\left({\lambda }_1{e}^{-{\lambda }_1\tau }\right)-{\lambda }_1{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)d\mathfrak{s}+{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\tau \right),}\hfill \\ & {\displaystyle {\mathfrak{Z}}^{\prime }\left(\tau \right)=-{\lambda }_2{\mathfrak{d}}_0{e}^{-{\lambda }_2\tau }+\frac{{\mathfrak{d}}_1}{{\lambda }_2}\left({\lambda }_2{e}^{-{\lambda }_2\tau }\right)-{\lambda }_2{\int }_0^{\tau }{e}^{-{\lambda }_2(\tau -\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)d\mathfrak{s}+{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\tau \right),}\hfill \end{array}\right.\end{array}$$

(11)

where ${\mathfrak{c}}_0,{\mathfrak{c}}_1,{\mathfrak{d}}_0$ and ${\mathfrak{d}}_1$ are unknown arbitrary constants. Using (11) in the nonlocal closed boundary conditions: $\mathfrak{Y}\left(\mathcal{T}\right)={\mathfrak{r}}_1\mathfrak{Z}\left({\mathfrak{k}}_1\right)+{\mathfrak{r}}_2\mathcal{T}{\mathfrak{Z}}^{\prime }\left({\mathfrak{k}}_2\right), \mathcal{T}{\mathfrak{Y}}^{\prime }\left(\mathcal{T}\right)={\mathfrak{s}}_1\mathfrak{Z}\left({\mathfrak{u}}_1\right)+{\mathfrak{s}}_2\mathcal{T}{\mathfrak{Z}}^{\prime }\left({\mathfrak{u}}_2\right), \mathfrak{Z}\left(\mathcal{T}\right)={\mathfrak{p}}_1\mathfrak{Y}\left({\zeta }_1\right)+{\mathfrak{p}}_2\mathcal{T}{\mathfrak{Y}}^{\prime }\left({\zeta }_2\right),$ and $\mathcal{T}{\mathfrak{Z}}^{\prime }\left(\mathcal{T}\right)={\mathfrak{q}}_1\mathfrak{Y}\left({\mathfrak{v}}_1\right)+{\mathfrak{q}}_2\mathcal{T}{\mathfrak{Y}}^{\prime }\left({\mathfrak{v}}_2\right),$ we find a system of algebraic equations in ${\mathfrak{c}}_0,{\mathfrak{c}}_1,{\mathfrak{d}}_0$ and ${\mathfrak{d}}_1$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathcal{A}}_1{\mathfrak{c}}_0+{\mathcal{A}}_2{\mathfrak{c}}_1+{\mathcal{A}}_3{\mathfrak{d}}_0+{\mathcal{A}}_4{\mathfrak{d}}_1-{\mathcal{E}}_1=0,\hfill \\ & {\mathcal{A}}_5{\mathfrak{c}}_0+{\mathcal{A}}_6{\mathfrak{c}}_1+{\mathcal{A}}_7{\mathfrak{d}}_0+{\mathcal{A}}_8{\mathfrak{d}}_1-{\mathcal{E}}_2=0,\hfill \\ & {\mathcal{A}}_9{\mathfrak{c}}_0+{\mathcal{A}}_{10}{\mathfrak{c}}_1+{\mathcal{A}}_{11}{\mathfrak{d}}_0+{\mathcal{A}}_{12}{\mathfrak{d}}_1-{\mathcal{E}}_3=0,\hfill \\ & {\mathcal{A}}_{13}{\mathfrak{c}}_0+{\mathcal{A}}_{14}{\mathfrak{c}}_1+{\mathcal{A}}_{15}{\mathfrak{d}}_0+{\mathcal{A}}_{16}{\mathfrak{d}}_1-{\mathcal{E}}_4=0,\hfill \end{array}\right.\end{array}$$

(12)

$$\begin{array}{cc}\hfill {\mathfrak{c}}_0=& {\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}[({\mathcal{E}}_3{\mathcal{A}}_{15}-{\mathcal{E}}_4{\mathcal{A}}_{11})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & -({\mathcal{E}}_1{\mathcal{A}}_7-{\mathcal{E}}_2{\mathcal{A}}_3)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})],\hfill \\ \hfill {\mathfrak{c}}_1=& {\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}[({\mathcal{E}}_3{\mathcal{A}}_{15}-{\mathcal{E}}_4{\mathcal{A}}_{11})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & -({\mathcal{E}}_1{\mathcal{A}}_7-{\mathcal{E}}_2{\mathcal{A}}_3)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)],\hfill \\ \hfill {\mathfrak{d}}_0=& {\displaystyle \frac{1}{{\mathrm{\Delta }}_2}}[({\mathcal{E}}_1{\mathcal{A}}_5-{\mathcal{E}}_2{\mathcal{A}}_1)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{12}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{16})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & -({\mathcal{E}}_3{\mathcal{A}}_{13}-{\mathcal{E}}_4{\mathcal{A}}_9)({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_4{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})],\hfill \\ \hfill {\mathfrak{d}}_1=& {\displaystyle \frac{1}{{\mathrm{\Delta }}_2}}[({\mathcal{E}}_1{\mathcal{A}}_5-{\mathcal{E}}_2{\mathcal{A}}_1)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{11}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{15})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & -({\mathcal{E}}_3{\mathcal{A}}_{13}-{\mathcal{E}}_4{\mathcal{A}}_9)({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_3{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_7)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})].\hfill \end{array}$$

Substituting the values of ${\mathfrak{c}}_0,{\mathfrak{c}}_1,{\mathfrak{d}}_0$, and ${\mathfrak{d}}_1$ in (10) yields the solution. This completes the proof. □

3. Main Results

Let us introduce the space $\mathcal{X}=\left\{\mathfrak{Y}\right(\tau \left)\right|\mathfrak{Y}\left(\tau \right)\in \mathcal{C}\left(\right[0,\mathcal{T}\left]\right)\}$ endowed with the norm $\left|\right|\mathfrak{Y}\left|\right|=max\left\{\right|\mathfrak{Y}\left(\tau \right),\tau \in [0,\mathcal{T}]\}$. Obviously $(\mathfrak{Y},||·|\left|\right)$ is a Banach space. Also let $\mathcal{Y}=\left\{\mathfrak{Z}\right(\tau \left)\right|\mathfrak{Z}\left(\tau \right)\in \mathcal{C}\left(\right[0,\mathcal{T}\left]\right)\}$ endowed with the norm $\left|\right|\mathfrak{Z}\left|\right|=max\left\{\right|\mathfrak{Z}\left(\tau \right),\tau \in [0,\mathcal{T}]\}$.The product space $(\mathcal{X}\times \mathcal{Y},|\left|\right(\mathfrak{Y},\mathfrak{Z}\left)\right|\left|\right)$ is also a Banach space with norm $\left|\right|\mathfrak{Y},\mathfrak{Z}\left|\right|=\left|\right|\mathfrak{Y}\left|\right|+\left|\right|\mathfrak{Z}\left|\right|.$ Given Lemma 1, we define the operator $\mathcal{T}:\mathcal{X}\times \mathcal{Y}\to \mathcal{X}\times \mathcal{Y}$ by

$$\begin{array}{c}\hfill \mathcal{T}(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)=\left[\begin{array}{c}{\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)\\ {\mathcal{T}}_2(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)\end{array}\right],\end{array}$$

where

$$\begin{array}{cc}\hfill & {\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)\hfill \\ \hfill {\displaystyle =}& {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left[\right(\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{15}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n\left(\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\right)\left\}{\mathcal{A}}_{11}\right)}\hfill \\ & \times ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)}\hfill \\ & \times ({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left[\right(\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{15}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)}\hfill \\ & \times ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)}\hfill \\ & {\displaystyle \times ({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)]+{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)d\mathfrak{s},}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\mathcal{T}}_2(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)\hfill \\ \hfill {\displaystyle =}& {\displaystyle \frac{e^{-{\lambda }_2\tau }}{{\mathrm{\Delta }}_2}\left[\right(\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_5}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_1\right)}\hfill \\ & \times ({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{12}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{16})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & {\displaystyle -\left(\right\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{13}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\left\}{\mathcal{A}}_9\right)}\hfill \\ & \times ({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_4{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_2}(1-e^{-{\lambda }_2\tau })\frac{1}{{\mathrm{\Delta }}_2}\left[\right(\{\sum _{i=1}^m\left({\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_1({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{k}}_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_5}\hfill \\ & {\displaystyle -\{\sum _{i=1}^m\left({\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{i=1}^m\left({\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{i=1}^m\left({\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left({\mathfrak{u}}_2\right)\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathcal{T}\right)\left\}{\mathcal{A}}_1\right)}\hfill \\ & \times ({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{11}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{15})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\hfill \\ & {\displaystyle -\left(\right\{\sum _{j=1}^n\left({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{p}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\zeta }_2\right)\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{13}}\hfill \\ & {\displaystyle -\{\sum _{j=1}^n\left({\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)-\sum _{j=1}^n\left({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}\right)}\hfill \\ & {\displaystyle +\sum _{j=1}^n\left({\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}{\mathcal{H}}_1\left({\mathfrak{v}}_2\right)\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)\mathfrak{d}\mathfrak{s}-\mathcal{T}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathcal{T}\right)\left\}{\mathcal{A}}_9\right)}\hfill \\ & {\displaystyle \times ({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_3{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_7)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})]+{\int }_0^{\tau }{e}^{-{\lambda }_2(\tau -\mathfrak{s})}{\mathcal{I}}^{\beta -1}{\mathcal{H}}_2\left(\mathfrak{s}\right)d\mathfrak{s}.}\hfill \end{array}$$

(14)

We need the following hypotheses in the sequel:

• $\left({\mathcal{H}}_1\right)$ Assume that there exist real constant ${\mathfrak{m}}_i,{\mathfrak{n}}_i>0,i=1,2$ and ${\mathfrak{m}}_0>0,{\mathfrak{n}}_0>0$ such that, for all $\tau \in [0,\mathcal{T}],{\mathcal{U}}_i\in {\mathcal{R}}_e,i=1,2$,

  $$\begin{array}{cc}\hfill & |\mathcal{K}(\tau ,{\mathcal{U}}_1,{\mathcal{U}}_2)|\le {\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2\left|{\mathcal{U}}_2\right|,\hfill \\ & |\mathcal{L}(\tau ,{\mathcal{U}}_1,{\mathcal{U}}_2)|\le {\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2\left|{\mathcal{U}}_2\right|.\hfill \end{array}$$
• $\left({\mathcal{H}}_2\right)$ Assume that $\mathcal{K},\mathcal{L}:{\mathcal{R}}_e^2\to {\mathcal{R}}_e$ are continuous functions and there exist constants ${\mathcal{M}}_i,{\mathcal{N}}_i,i=1,2$ such that for all $\tau \in [0,\mathcal{T}]$ and ${\mathfrak{Y}}_i,{\mathfrak{Z}}_i,i=1,2,$

  $$\begin{array}{cc}\hfill & |\mathcal{K}(\tau ,{\mathfrak{Y}}_1,{\mathfrak{Y}}_2)-\mathcal{K}(\tau ,{\mathfrak{Z}}_1,{\mathfrak{Z}}_2)|\le {\mathcal{M}}_1|{\mathfrak{Y}}_1-{\mathfrak{Z}}_1|+{\mathcal{M}}_2|{\mathfrak{Y}}_2-{\mathfrak{Z}}_2|\hfill \\ & |\mathcal{L}(\tau ,{\mathfrak{Y}}_1,{\mathfrak{Y}}_2)-\mathcal{L}(\tau ,{\mathfrak{Z}}_1,{\mathfrak{Z}}_2)|\le {\mathcal{N}}_1|{\mathfrak{Y}}_1-{\mathfrak{Z}}_1|+{\mathcal{N}}_2|{\mathfrak{Y}}_2-{\mathfrak{Z}}_2|.\hfill \end{array}$$

• For convenience, we set

$$\begin{array}{cc}\hfill {\mathfrak{M}}_1=& {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\left\{\right(\sum _{j=1}^n\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{15}\hfill \\ & +\sum _{j=1}^n\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{11})\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_7+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_3\right)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\left\{\right(\sum _{j=1}^n\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_2\hfill \\ & +\sum _{j=1}^n\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{11})\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_7+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_3\right)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right]\}+{\displaystyle \frac{1}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1}),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mathfrak{M}}_2=& {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_2+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_{11}\right)\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +(\sum _{i=1}^m\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_7\hfill \\ & +\sum _{i=1}^m\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_3)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_{15}+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_{11}\right)\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +(\sum _{i=1}^m\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_1{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_7\hfill \\ & +\sum _{i=1}^m\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_3)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right]\},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{M}}}_1=& {\displaystyle \frac{e^{-{\lambda }_2\tau }}{{\mathrm{\Delta }}_2}}\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_5+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_1\right)\hfill \\ & \times \left[({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{12}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{16})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\right]\hfill \\ & +(\sum _{j=1}^n\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{13}\hfill \\ & +\sum _{j=1}^n\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_9)\hfill \\ & \times \left[({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_4{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_5+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_1\right)\hfill \\ & \times \left[({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{11}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{15})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\right]\hfill \\ & +(\sum _{j=1}^n\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{13}\hfill \\ & +\sum _{j=1}^n\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_9)\hfill \\ & \times \left[({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_3{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_7)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})\right]\}+{\displaystyle \frac{1}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1}),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\widehat{\mathfrak{M}}}_2=& {\displaystyle \frac{e^{-{\lambda }_2\tau }}{{\mathrm{\Delta }}_2}}\left\{\right(\sum _{i=1}^m\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_1{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_5\hfill \\ & +\sum _{i=1}^m\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_1)\hfill \\ & \times \left[({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{12}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{16})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_{13}+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_9\right)\hfill \\ & \times \left[({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_4{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\left\{\right(\sum _{i=1}^m\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_1{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_5\hfill \\ & +\sum _{i=1}^m\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_1)\hfill \\ & \times \left[({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})({\mathcal{A}}_{11}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{15})({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_{13}+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_9\right)\hfill \\ & \times \left[({\mathcal{A}}_2{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_6)({\mathcal{A}}_3{\mathcal{A}}_5-{\mathcal{A}}_1{\mathcal{A}}_7)({\mathcal{A}}_{10}{\mathcal{A}}_{13}-{\mathcal{A}}_9{\mathcal{A}}_{14})\right]\}+{\displaystyle \frac{1}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2}),\hfill \end{array}$$

(18)

$$\begin{array}{c}\hfill \mathrm{\Phi }=min\{1-[({\mathfrak{M}}_1+\widehat{{\mathfrak{M}}_1}){\kappa }_1+({\mathfrak{M}}_2+\widehat{{\mathfrak{M}}_2})\widehat{{\kappa }_1}],1-[({\mathfrak{M}}_1+\widehat{{\mathfrak{M}}_1}){\kappa }_2+({\mathfrak{M}}_2+\widehat{{\mathfrak{M}}_2})\widehat{{\kappa }_2}]\}.\end{array}$$

(19)

We apply the following known result to prove the existence of solutions for the problems (3) and (4).

Lemma 2.

Leray–Schauder alternative. Let $\mathcal{K}\left(X\right)=\{x\in \mathcal{D}:x=kX(x)for some0<k<1\},$ where $X:\mathcal{D}\to \mathcal{D}$ is a completely continuous operator. Then either the set $F\left(X\right)$ is unbounded or there exists at least one fixed point of the operator X.

Existence Results via Leray–Schauder Alternative

In this section, we prove an existence result based on the Leray–Schauder alternative.

Theorem 1.

Assume that $\left({\mathcal{H}}_1\right)$ holds. In addition, it is assumed that

$$\begin{array}{c}\hfill ({\mathfrak{M}}_1+{\mathfrak{M}}_2){\kappa }_1+(\widehat{{\mathfrak{M}}_1}+\widehat{{\mathfrak{M}}_2})\widehat{{\kappa }_1}<1,\end{array}$$

(20)

$$\begin{array}{c}\hfill ({\mathfrak{M}}_1+{\mathfrak{M}}_2){\kappa }_2+(\widehat{{\mathfrak{M}}_1}+\widehat{{\mathfrak{M}}_2})\widehat{{\kappa }_2}<1,\end{array}$$

(21)

then system (3) and (4) has at least one solution on $[0,\mathcal{T}].$

Proof. 

First, we show that $\mathcal{T}$ is continuous, continuity of functions $\mathfrak{Y}$ and $\mathfrak{Z}$. Let $\mathrm{\Omega }\subset 𝒳\times 𝒴$ be bounded. Then there exist positive constants ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ such that

$$\begin{array}{c}\hfill \left|\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))\right|\le {\mathcal{L}}_1,\left|\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))\right|\le {\mathcal{L}}_2,\end{array}$$

$\forall (\mathfrak{Y},\mathfrak{Z})\in \mathrm{\Omega }$. Then for any $(\mathfrak{Y},\mathfrak{Z})\in \mathrm{\Omega }$, we have

$$\begin{array}{cc}\hfill & \parallel {\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)\parallel \hfill \\ \hfill \le |& {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left[\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\zeta }_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{k}}_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\zeta }_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{k}}_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8))\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)d\mathfrak{s}|,}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \le & {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\left\{\right(\sum _{j=1}^n\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{15}\hfill \\ & +\sum _{j=1}^n\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{11})\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_7+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_3\right)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\left\{\right(\sum _{j=1}^n\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_2\hfill \\ & +\sum _{j=1}^n\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{11})\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_7+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_3\right)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right]\}+{\displaystyle \frac{1}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1})\hfill \\ & +{\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_2+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_{11}\right)\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +(\sum _{i=1}^m\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_7\hfill \\ & +\sum _{i=1}^m\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_3)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_{15}+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_{11}\right)\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +(\sum _{i=1}^m\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_1{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_7\hfill \\ & +\sum _{i=1}^m\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_3)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right]\},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill |{𝒯}_1(\mathfrak{Y},\mathfrak{Z})|={\mathfrak{M}}_1{\mathcal{L}}_1+{\mathfrak{M}}_2{\mathcal{L}}_2.\end{array}$$

Similarly, we get

$$\begin{array}{cc}\hfill |{𝒯}_2(\mathfrak{Y},\mathfrak{Z})|=& {\mathcal{L}}_1{\mathfrak{M}}_3+{\mathcal{L}}_2{\mathfrak{M}}_4,\hfill \end{array}$$

which, in view of the notation (15) and (16) yields

$$\begin{array}{c}\hfill \left|\right|{𝒯}_1(\mathfrak{Y},\mathfrak{Z})\left|\right|\le {\mathfrak{M}}_1{\mathcal{L}}_1+{\mathfrak{M}}_2{\mathcal{L}}_2.\end{array}$$

(23)

Likewise, using the notation (17) and (18), we have

$$\begin{array}{c}\hfill \left|\right|{𝒯}_2(\mathfrak{Y},\mathfrak{Z})\left|\right|\le \widehat{{\mathfrak{M}}_1}{\mathcal{L}}_1+\widehat{{\mathfrak{M}}_2}{\mathcal{L}}_2.\end{array}$$

(24)

Then it follows from (23) and (24) that, from the above inequalities, we conclude that the operator $𝒯$ is uniformly bounded, since

$$\begin{array}{c}\hfill \left|\right|𝒯(\mathfrak{Y},\mathfrak{Z})\left|\right|\le ({\mathfrak{M}}_1+\widehat{{\mathfrak{M}}_1}){\mathcal{L}}_1+({\mathfrak{M}}_2+\widehat{{\mathfrak{M}}_2}){\mathcal{L}}_2,\end{array}$$

(25)

which shows that the operator $𝒯$ is uniformly bounded.

Next, we show that $𝒯$ is equi-continuous. Let ${\tau }_1,{\tau }_2\in [0,\mathcal{T}]$ with ${\tau }_1<{\tau }_2$. Then we have

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1(\mathfrak{Y}\left({\tau }_2\right),\mathfrak{Y}\left({\tau }_2\right))-{\mathcal{T}}_1(\mathfrak{Y}\left({\tau }_1\right),\mathfrak{Y}\left({\tau }_1\right))|\hfill \\ & \le \left|{\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\right[\left(\right\{\sum _{j=1}^n({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\zeta }_2\right)+{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & +\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle +\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{k}}_2\right)+{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & +\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\zeta }_2\right)+{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & +\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle +\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{k}}_2\right)+{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & +\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\left)\right|\hfill \\ & {\displaystyle +\left|{\int }_0^{\tau }[e^{-{\lambda }_1({\tau }_2-\mathfrak{s})}-e^{-{\lambda }_1({\tau }_1-\mathfrak{s})}]{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)d\mathfrak{s}\right|+\left|{\int }_{{\tau }_2}^{{\tau }_1}{e}^{-{\lambda }_1({\tau }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)d\mathfrak{s}\right|\to 0,}\hfill \end{array}$$

clearly, $|{\mathcal{T}}_1(\mathfrak{Y}\left({\tau }_2\right),\mathfrak{Y}\left({\tau }_2\right))-{\mathcal{T}}_1(\mathfrak{Y}\left({\tau }_1\right),\mathfrak{Y}\left({\tau }_1\right))|\to 0$, and similarly $|{\mathcal{T}}_2(\mathfrak{Y}\left({\tau }_2\right),\mathfrak{Y}\left({\tau }_2\right))-{\mathcal{T}}_2(\mathfrak{Y}\left({\tau }_1\right),$$\mathfrak{Y}\left({\tau }_1\right)\left)\right|\to 0$ as ${\tau }_2\to {\tau }_1$ independent of $(\mathfrak{Y},\mathfrak{Z})\in \mathrm{\Omega }$. In consequence, the operator $\mathcal{T}(\mathfrak{Y},\mathfrak{Z})$ is equi-continuous. Hence, it follows, according to Arzelá Ascoli theorem, that $\mathcal{T}(\mathfrak{Y},\mathfrak{Z})$ is completely continuous.

Finally, it will be established that the set $\mathcal{W}=\left\{\right(\mathfrak{Y},\mathfrak{Z})\in 𝒳\times 𝒴|(\mathfrak{Y},\mathfrak{Z})=\sigma \mathcal{T}(\mathfrak{Y},\mathfrak{Z}),0<\sigma <\mathcal{T}\}$ and show that it is bounded. Let $(\mathfrak{Y},\mathfrak{Z})\in \mathcal{W}$, then $(\mathfrak{Y},\mathfrak{Z})=\sigma \mathcal{T}(\mathfrak{Y},\mathfrak{Z})$ and for any $\tau \in [0,\mathcal{T}]$, we have

$$\begin{array}{c}\hfill (\mathfrak{Y},\mathfrak{Z})\left(\tau \right)=\left[\begin{array}{c}\sigma {\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\left(\tau \right),\\ \sigma {\mathcal{T}}_2(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)\end{array}\right].\end{array}$$

Then, given the assumption, we obtain

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)|\hfill \\ & {\displaystyle \le \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}}\hfill \\ & -{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)({\zeta }_2)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)({\mathfrak{v}}_2)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)(\mathcal{T})\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -(\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)({\mathfrak{k}}_2)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)({\mathfrak{u}}_2)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)(\mathcal{T})\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}((\{\sum _{j=1}^n({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)({\zeta }_2)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)({\mathfrak{v}}_2)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)(\mathcal{T})\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)({\mathfrak{k}}_2)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathfrak{n}}_0+{\mathfrak{n}}_1|{\mathcal{U}}_1|+{\mathfrak{n}}_2|{\mathcal{U}}_2|)({\mathfrak{u}}_2)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8))\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathfrak{m}}_0+{\mathfrak{m}}_1|{\mathcal{U}}_1|+{\mathfrak{m}}_2|{\mathcal{U}}_2|)d\mathfrak{s}.}\hfill \end{array}$$

Hence, we have

$$\begin{array}{c}\hfill \left|\right|\mathfrak{Y}\left|\right|\le ({\mathfrak{m}}_0+{\mathfrak{m}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{m}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_1+({\mathfrak{n}}_0+{\mathfrak{n}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{n}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_2.\end{array}$$

(26)

Similarly, one can obtain

$$\begin{array}{c}\hfill \left|\right|\mathfrak{Z}\left|\right|\le ({\mathfrak{m}}_0+{\mathfrak{m}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{m}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_3+({\mathfrak{n}}_0+{\mathfrak{n}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{n}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_4.\end{array}$$

(27)

From (26) and (27), we obtain

$$\begin{array}{cc}\hfill \left|\right|\mathfrak{Y}\left|\right|+\left|\right|\mathfrak{Z}\left|\right|\le & ({\mathfrak{m}}_0+{\mathfrak{m}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{m}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_1+({\mathfrak{n}}_0+{\mathfrak{n}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{n}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_2\hfill \\ & +({\mathfrak{m}}_0+{\mathfrak{m}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{m}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_3+({\mathfrak{n}}_0+{\mathfrak{n}}_1\left|\right|{\mathcal{U}}_1\left|\right|+{\mathfrak{n}}_2\left|\right|{\mathcal{U}}_2\left|\right|){\mathfrak{M}}_4,\hfill \end{array}$$

Consequently, we get

$$\begin{array}{c}\hfill \left|\right|(\mathfrak{Y},\mathfrak{Z})\left|\right|\le {\displaystyle \frac{\left({\mathfrak{m}}_0({\mathfrak{M}}_1+{\mathfrak{M}}_3)\right)+\left({\mathfrak{n}}_0({\mathfrak{M}}_2+{\mathfrak{M}}_4)\right)}{\mathrm{\Phi }}}\end{array}$$

which shows that the set $\mathcal{W}$ is bounded. Thus, by Lemma 2, the operator $\mathcal{T}$ has at least one fixed point. In consequence, the problem (3) and (4) has at least one solution on $[0,\mathcal{T}]$. The proof is complete. □

We make use of Banach’s contraction mapping principle to establish the existence and uniqueness of solutions for the problem (3) and (4).

Theorem 2.

Assume that $\mathfrak{Y},\mathfrak{Z}:[0,\mathcal{T}]\times {\mathcal{R}}^2\to \mathcal{R}$ are continuous functions and there exist constants ${\mathcal{M}}_i,\mathcal{N},i=1,2$ such that for all $\tau \in [0,\mathcal{T}]$ and ${\mathfrak{Y}}_i,{\mathfrak{Z}}_i\in \mathcal{R},i=1,2$.

$$\begin{array}{cc}\hfill & |\mathcal{K}(\tau ,{\mathfrak{Y}}_1,{\mathfrak{Y}}_2)-\mathcal{K}(\tau ,{\mathfrak{Z}}_1,{\mathfrak{Z}}_2)|\le {\mathcal{M}}_1|{\mathfrak{Y}}_1-{\mathfrak{Z}}_1|+{\mathcal{M}}_2|{\mathfrak{Y}}_2-{\mathfrak{Y}}_2|\hfill \\ & |\mathcal{L}(\tau ,{\mathfrak{Y}}_1,{\mathfrak{Y}}_2)-\mathcal{L}(\tau ,{\mathfrak{Z}}_1,{\mathfrak{Z}}_2)|\le {\mathcal{N}}_1|{\mathfrak{Y}}_1-{\mathfrak{Z}}_1|+{\mathcal{N}}_2|{\mathfrak{Y}}_2-{\mathfrak{Z}}_2|.\hfill \end{array}$$

In addition, assume that

$$\begin{array}{c}\hfill ({\mathfrak{M}}_1+{\mathfrak{M}}_3)({\mathcal{M}}_1+{\mathcal{M}}_2)+({\mathfrak{M}}_2+{\mathfrak{M}}_4)({\mathcal{N}}_1+{\mathcal{N}}_2)<1,\end{array}$$

where ${\mathfrak{M}}_1,{\mathfrak{M}}_2,{\mathfrak{M}}_{31},{\mathfrak{M}}_4$ are given by (15)–(18). Then the boundary value problem (4) has a unique solution.

Proof. 

Fixing ${\displaystyle \underset{\tau \in [0,\mathcal{T}]}{sup}\mathcal{K}(\tau ,0,0)={\mathcal{Q}}_1<\infty }$ and ${\displaystyle \underset{\tau \in [0,\mathcal{T}]}{sup}\mathcal{L}(\tau ,0,0)={\mathcal{Q}}_2<\infty }$ and using the assumption $\left({\mathcal{H}}_2\right)$, we obtain the following:

$$\begin{array}{cc}\hfill & \left|\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))\right|=|\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))-\mathcal{K}(\tau ,0,0)+\mathcal{K}(\tau ,0,0)|\le {\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1,\hfill \\ \hfill & \left|\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))\right|=|\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))-\mathcal{L}(\tau ,0,0)+\mathcal{L}(\tau ,0,0)|\le {\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2.\hfill \end{array}$$

Now, we consider the closed ball ${\mathcal{B}}_r=\{(\mathfrak{Y},\mathfrak{Z})\in 𝒳\times 𝒴:|\left|(\mathfrak{Y},\mathfrak{Z})\right||\le r\}$ and show that $\mathcal{T}\left({\mathcal{B}}_r\right)\subset {\mathcal{B}}_r$, where

$$\begin{array}{c}\hfill {\displaystyle \frac{{\mathcal{Q}}_1({\mathfrak{M}}_1+{\mathfrak{M}}_3)+{\mathcal{Q}}_2({\mathfrak{M}}_2+{\mathfrak{M}}_4)}{1-[({\mathfrak{M}}_1+{\mathfrak{M}}_3)({\mathcal{M}}_1+{\mathcal{M}}_2)+({\mathfrak{M}}_2+{\mathfrak{M}}_4)({\mathcal{N}}_1+{\mathcal{N}}_2)]}}<r.\end{array}$$

Let $(\mathfrak{Y},\mathfrak{Z})\in {\mathcal{B}}_r$. We obtain,

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\left(\tau \right)|\hfill \\ & {\displaystyle \le \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}}\hfill \\ & -{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\left({\zeta }_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\left(\mathcal{T}\right)\}{\mathcal{A}}_{11})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\left({\mathfrak{k}}_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1\left)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1\left)\left(\mathcal{T}\right)\right\}{\mathcal{A}}_3\left)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\left({\zeta }_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\left(\mathcal{T}\right)\}{\mathcal{A}}_{11})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\left({\mathfrak{k}}_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1\left)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}({\mathcal{N}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{N}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_2)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1\left)\left(\mathcal{T}\right)\right\}{\mathcal{A}}_3\left)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right)\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}({\mathcal{M}}_1\left|\right|\mathfrak{Y}\left|\right|+{\mathcal{M}}_2\left|\right|\mathfrak{Z}\left|\right|+{\mathcal{Q}}_1)d\mathfrak{s},}\hfill \end{array}$$

which, on taking the norm for $\tau \in [0,\mathcal{T}]$, yields

$$\begin{array}{c}\hfill \left|\right|\left({\mathcal{T}}_1(\mathfrak{Y},\mathfrak{Z})\right)\left|\right|\le [({\mathcal{M}}_1+{\mathcal{M}}_2)r+{\mathcal{Q}}_1]{\mathfrak{M}}_1+[({\mathcal{N}}_1+{\mathcal{N}}_2)r+{\mathcal{Q}}_2]{\mathfrak{M}}_2.\end{array}$$

In the same way, we obtain

$$\begin{array}{c}\hfill \left|\right|\left({\mathcal{T}}_2(\mathfrak{Y},\mathfrak{Z})\right)\left|\right|\le [({\mathcal{M}}_1+{\mathcal{M}}_2)r+{\mathcal{Q}}_1]{\mathfrak{M}}_3+[({\mathcal{N}}_1+{\mathcal{N}}_2)r+{\mathcal{Q}}_2]{\mathfrak{M}}_4.\end{array}$$

Consequently, it follows that $\left|\right|\mathcal{T}(\mathfrak{Y},\mathfrak{Z})\left|\right|\le r$, that is, $(\mathfrak{Y},\mathfrak{Z})\in {\mathcal{B}}_r$. Hence, $\mathcal{T}\left({\mathcal{B}}_r\right)\subset {\mathcal{B}}_r$. Next, we show that the operator $\mathcal{T}$ is a contraction. For that, let $({\mathfrak{Y}}_1,{\mathfrak{Z}}_1),({\mathfrak{Y}}_2,{\mathfrak{Z}}_2)\in \mathcal{X}\times \mathcal{Y}$. Then, for any $\tau \in [0,\mathcal{T}]$, we get

$$\begin{array}{cc}\hfill & \left|\right|{\mathcal{T}}_1({\mathfrak{Y}}_2,{\mathfrak{Z}}_2)\left(\tau \right)-{\mathcal{T}}_1({\mathfrak{Y}}_1,{\mathfrak{Z}}_1)\left|\right|\hfill \\ & {\displaystyle \le \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left[\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\zeta }_2\right)\hfill \\ & -{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\mathfrak{v}}_2\right)\hfill \\ & +{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)\hfill \\ & \times ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}}\hfill \\ & -{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\mathfrak{k}}_2\right)\hfill \\ & -{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\mathfrak{u}}_2\right)\hfill \\ & +{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)\hfill \\ & \times ({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\zeta }_2\right)\hfill \\ & -{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\mathfrak{v}}_2\right)\hfill \\ & +{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)\hfill \\ & \times ({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}}\hfill \\ & -{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\mathfrak{k}}_2\right)\hfill \\ & -{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}|\mathcal{L}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{L}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left({\mathfrak{u}}_2\right)\hfill \\ & +{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)\hfill \\ & \times ({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8))\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}|\mathcal{K}(\tau ,{\mathfrak{Y}}_2\left(\tau \right),{\mathfrak{Z}}_2\left(\tau \right))-\mathcal{K}(\tau ,{\mathfrak{Y}}_1\left(\tau \right),{\mathfrak{Z}}_1\left(\tau \right))|d\mathfrak{s},}\hfill \\ \hfill \le & ({\mathcal{M}}_1\left|\right|{\mathfrak{Y}}_2-{\mathfrak{Y}}_1\left|\right|+{\mathcal{M}}_2\left|\right|{\mathfrak{Z}}_2-{\mathfrak{Z}}_1\left|\right|){\mathfrak{M}}_1+({\mathcal{N}}_1\left|\right|{\mathfrak{Y}}_2-{\mathfrak{Y}}_1\left|\right|+{\mathcal{N}}_2\left|\right|{\mathfrak{Z}}_2-{\mathfrak{Z}}_1\left|\right|){\mathfrak{M}}_2.\hfill \end{array}$$

In consequence, we get

$$\begin{array}{c}\hfill \left|\right|{\mathcal{T}}_1({\mathfrak{Y}}_2,{\mathfrak{Z}}_2)\left(\tau \right)-{\mathcal{T}}_1({\mathfrak{Y}}_1,{\mathfrak{Z}}_1)\left(\tau \right)\left|\right|\le [({\mathcal{M}}_1+{\mathcal{M}}_2){\mathfrak{M}}_1+({\mathcal{N}}_1+{\mathcal{N}}_2){\mathfrak{M}}_2]\left(\right||{\mathfrak{Y}}_2-{\mathfrak{Y}}_1\left|\right|+\left|\right|{\mathfrak{Z}}_2-{\mathfrak{Z}}_1\left|\right|).\end{array}$$

(28)

Similarly,

$$\begin{array}{c}\hfill \left|\right|{\mathcal{T}}_2({\mathfrak{Y}}_2,{\mathfrak{Z}}_2)\left(\tau \right)-{\mathcal{T}}_2({\mathfrak{Y}}_1,{\mathfrak{Z}}_1)\left(\tau \right)\left|\right|\le [({\mathcal{M}}_1+{\mathcal{M}}_2){\mathfrak{M}}_3+({\mathcal{N}}_1+{\mathcal{N}}_2){\mathfrak{M}}_4]\left(\right||{\mathfrak{Y}}_2-{\mathfrak{Y}}_1\left|\right|+\left|\right|{\mathfrak{Z}}_2-{\mathfrak{Z}}_1\left|\right|).\end{array}$$

(29)

It follows from (28) and (29) that

$$\begin{array}{cc}\hfill \left|\right|\mathcal{T}({\mathfrak{Y}}_2,{\mathfrak{Z}}_2)\left(\tau \right)-& \mathcal{T}({\mathfrak{Y}}_1,{\mathfrak{Z}}_1)\left(\tau \right)\left|\right|\le [({\mathcal{M}}_1+{\mathcal{M}}_2)({\mathfrak{M}}_1+{\mathfrak{M}}_3)+({\mathcal{N}}_1+{\mathcal{N}}_2)({\mathfrak{M}}_2+{\mathfrak{M}}_4)]\left(\right||{\mathfrak{Y}}_2-{\mathfrak{Y}}_1\left|\right|+\left|\right|{\mathfrak{Z}}_2-{\mathfrak{Z}}_1\left|\right|).\hfill \end{array}$$

which, given (19), shows that the operator $\mathcal{T}$ is a contraction. Hence, it follows by Banach’s contraction mapping principle that the operator $\mathcal{T}$ has a unique fixed point, which corresponds to the unique solution of the problem (3) and (4). This completes the proof. □

4. Hyers–Ulam Stability of the System

In this section, we investigate the Hyers–Ulam stability of the proposed coupled fractional system. Consider the following perturbed system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left({}^c\mathcal{D}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1}\right)\mathfrak{Y}\left(\tau \right)-\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))\le {ϵ}_1,\hfill \\ \left({}^c\mathcal{D}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1}\right)\mathfrak{Z}\left(\tau \right)-\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))\le {ϵ}_2,\hfill \end{array}\right.\tau \in [0,\mathcal{T}],\end{array}$$

(30)

where ${ϵ}_1,{ϵ}_2>0$ are given constants.

Definition 3

([24]). The boundary value problem (3) is said to be Hyers–Ulam stable if there exist positive constants ${\mathfrak{M}}_1,{\mathfrak{M}}_2,{\mathfrak{M}}_3,{\mathfrak{M}}_4$ such that for every ${ϵ}_1,{ϵ}_2>0$, and for every pair $(\mathfrak{Y},\mathfrak{Z})\in \mathcal{C}\left(\right[0,\mathcal{T}],\mathbb{R})\times \mathcal{C}\left(\right[0,\mathcal{T}],\mathbb{R})$ satisfying (30), there exists an exact solution $({\mathfrak{Y}}^{*},{\mathfrak{Z}}^{*})$ of problem (3) such that

$$\begin{array}{cc}\hfill |\mathfrak{Y}\left(\tau \right)-{\mathfrak{Y}}^{*}\left(\tau \right)|& \le {\mathfrak{M}}_1{ϵ}_1+{\mathfrak{M}}_2{ϵ}_2,\hfill \\ \hfill |\mathfrak{Z}\left(\tau \right)-{\mathfrak{Z}}^{*}\left(\tau \right)|& \le {\mathfrak{M}}_3{ϵ}_1+{\mathfrak{M}}_4{ϵ}_2,\hfill \end{array}$$

for all $\tau \in [0,\mathcal{T}]$.

Remark 1.

A pair $(\mathfrak{Y},\mathfrak{Z})$ is said to satisfy the inequality (30) if there exist functions ${\mathcal{Z}}_1,{\mathcal{Z}}_2\in \mathcal{C}([0,\mathcal{T}],\mathbb{R})$ such that

$$\begin{array}{cc}\hfill \left({}^c\mathcal{D}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1}\right)\mathfrak{Y}\left(\tau \right)& =\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))+{\mathcal{Z}}_1\left(\tau \right),\hfill \\ \hfill \left({}^c\mathcal{D}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1}\right)\mathfrak{Z}\left(\tau \right)& =\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))+{\mathcal{Z}}_2\left(\tau \right),\hfill \end{array}$$

for all $\tau \in [0,\mathcal{T}]$, where

$$|{\mathcal{Z}}_1\left(\tau \right)|\le {ϵ}_1,|{\mathcal{Z}}_2\left(\tau \right)|\le {ϵ}_2,\tau \in [0,\mathcal{T}].$$

Remark 2.

If $(\mathfrak{Y},\mathfrak{Z})$ represent a solution of inequality (30), then $(\mathfrak{Y},\mathfrak{Z})$ is a solution of the following inequality

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & |\mathfrak{Y}\left(\tau \right)-{\mathfrak{Y}}^{*}\left(\tau \right)|\le {\mathfrak{M}}_1{ϵ}_1+{\mathfrak{M}}_2{ϵ}_2,\tau \in [0,\mathcal{T}],\hfill \\ & |\mathfrak{Z}\left(\tau \right)-{\mathfrak{Z}}^{*}\left(\tau \right)|\le {\mathfrak{M}}_3{ϵ}_1+{\mathfrak{M}}_4{ϵ}_2,\tau \in [0,\mathcal{T}],\hfill \end{array}\right.\end{array}$$

as from Remark 1, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(}^c{\mathcal{D}}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1})\mathfrak{Y}\left(\tau \right)=\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))+{\mathcal{Z}}_1\left(\tau \right),\tau \in [0,\mathcal{T}],\hfill \\ {(}^c{\mathcal{D}}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1})\mathfrak{Z}\left(\tau \right)=\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))+{\mathcal{Z}}_2\left(\tau \right),\tau \in [0,\mathcal{T}],\hfill \end{array}\right.\end{array}$$

with the help of Definition 3 and Remark 1, we verified Remark 2, in the following lines

$$\begin{array}{cc}\hfill & |\mathfrak{Y}\left(\tau \right)-{\mathfrak{Y}}^{*}\left(\tau \right)|\hfill \\ & {\displaystyle \le \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}\left[\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\zeta }_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{k}}_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})]\hfill \\ & {\displaystyle +\frac{1}{{\lambda }_1}(1-e^{-{\lambda }_1\tau })\frac{1}{{\mathrm{\Delta }}_1}\left(\right(\left\{\sum _{j=1}^n\right({\mathfrak{p}}_1{\int }_0^{{\zeta }_1}{e}^{-{\lambda }_1({\zeta }_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & -\sum _{j=1}^n({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}{e}^{-{\lambda }_1({\zeta }_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +\sum _{j=1}^n\left({\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\int }_0^{{\zeta }_2}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\zeta }_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\right\}{\mathcal{A}}_{15}\hfill \\ & -\{{\mathfrak{q}}_1{\int }_0^{{\mathfrak{v}}_1}{e}^{-{\lambda }_1({\mathfrak{v}}_1-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-\sum _{j=1}^n({\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\int }_0^{{\mathfrak{v}}_2}{e}^{-{\lambda }_1({\mathfrak{v}}_2-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{q}}_2\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{v}}_2\right)+{\lambda }_2\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_2(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_{11}\right)({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\hfill \\ & {\displaystyle -\left(\right\{{\mathfrak{r}}_1{\int }_0^{{\mathfrak{k}}_1}{e}^{-{\lambda }_2({\mathfrak{k}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\int }_0^{{\mathfrak{k}}_2}{e}^{-{\lambda }_2({\mathfrak{k}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}}\hfill \\ & +{\mathfrak{r}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{k}}_2\right)-{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\}{\mathcal{A}}_7\hfill \\ & -\{{\mathfrak{s}}_1{\int }_0^{{\mathfrak{u}}_1}{e}^{-{\lambda }_2({\mathfrak{u}}_1-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}-{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\int }_0^{{\mathfrak{u}}_2}{e}^{-{\lambda }_2({\mathfrak{u}}_2-\mathfrak{s})}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & +{\mathfrak{s}}_2\mathcal{T}{\mathcal{I}}^{\beta -1}\left(\mathcal{L}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left({\mathfrak{u}}_2\right)+{\lambda }_1\mathcal{T}{\int }_0^{\mathcal{T}}{e}^{-{\lambda }_1(\mathcal{T}-\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\mathfrak{d}\mathfrak{s}\hfill \\ & -\mathcal{T}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)\left(\mathcal{T}\right)\left\}{\mathcal{A}}_3\right)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8))\hfill \\ & {\displaystyle +{\int }_0^{\tau }{e}^{-{\lambda }_1(\tau -\mathfrak{s})}{\mathcal{I}}^{\alpha -1}\left(\mathcal{K}(\mathfrak{s},\mathfrak{Y}\left(\mathfrak{s}\right),\mathfrak{Z}\left(\mathfrak{s}\right))\right)d\mathfrak{s},}\hfill \end{array}$$

$$\begin{array}{cc}\le {ϵ}_1[\hfill & {\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\left\{\right(\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_2\hfill \\ & +\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{11})\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_7+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_3\right)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\left\{\right(\left[{\mathfrak{p}}_1{\displaystyle \frac{{\zeta }_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_1})+{\lambda }_1{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\zeta }_2})+{\mathfrak{p}}_2\mathcal{T}{\displaystyle \frac{{\zeta }_2^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_2\hfill \\ & +\left[{\mathfrak{q}}_1{\displaystyle \frac{{\mathfrak{v}}_1^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_1})+{\lambda }_1{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1{\mathfrak{v}}_2})+{\mathfrak{q}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{v}}_2^{\alpha -1}}{{\lambda }_1\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_{11})\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})\right]{\mathcal{A}}_7+\left[{\lambda }_1\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\right]{\mathcal{A}}_3\right)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right]\}+{\displaystyle \frac{1}{{\lambda }_1\Gamma \left(\alpha \right)}}(1-e^{-{\lambda }_1})]\hfill \\ & +{ϵ}_2\left[{\displaystyle \frac{e^{-{\lambda }_1\tau }}{{\mathrm{\Delta }}_1}}\right\{\left(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_2+\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_{11}\right)\hfill \\ & \times \left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_2{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_6)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +(\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_2{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_7\hfill \\ & +\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_3)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_{10}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\}\hfill \\ & +{\displaystyle \frac{1}{{\lambda }_1}}(1-e^{-{\lambda }_1}\tau ){\displaystyle \frac{1}{{\mathrm{\Delta }}_1}}\left\{\right(\left[{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})\right]{\mathcal{A}}_2\hfill \\ & +\left[{\lambda }_2\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2\mathcal{T}})+\mathcal{T}{\displaystyle \frac{{\mathcal{T}}^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_{11})\left[({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)({\mathcal{A}}_1{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_5)({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})\right]\hfill \\ & +(\left[{\mathfrak{r}}_1{\displaystyle \frac{{\mathfrak{k}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_1})+{\lambda }_1{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{k}}_2})+{\mathfrak{r}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{k}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_7\hfill \\ & +\left[{\mathfrak{s}}_1{\displaystyle \frac{{\mathfrak{u}}_1^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_1})+{\lambda }_2{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta }}{{\lambda }_2\Gamma \left(\beta \right)}}(1-e^{-{\lambda }_2{\mathfrak{u}}_2})+{\mathfrak{s}}_2\mathcal{T}{\displaystyle \frac{{\mathfrak{u}}_2^{\beta -1}}{\Gamma \left(\beta \right)}}\right]{\mathcal{A}}_3)\hfill \\ & \times \left[({\mathcal{A}}_{12}{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{16})({\mathcal{A}}_9{\mathcal{A}}_{15}-{\mathcal{A}}_{11}{\mathcal{A}}_{13})({\mathcal{A}}_4{\mathcal{A}}_7-{\mathcal{A}}_3{\mathcal{A}}_8)\right]\left\}\right]\hfill \\ \hfill \le & {\mathfrak{M}}_1{ϵ}_1+{\mathfrak{M}}_2{ϵ}_2.\hfill \end{array}$$

(31)

By the same method, we can obtain that

$$\begin{array}{c}\hfill |\mathfrak{Z}\left(\tau \right)-{\mathfrak{Z}}^{*}\left(\tau \right)|\le {\mathfrak{M}}_3{ϵ}_1+{\mathfrak{M}}_4{ϵ}_2,\tau \in [0,\mathcal{T}],\end{array}$$

(32)

where ${\mathfrak{M}}_1,{\mathfrak{M}}_2,{\mathfrak{M}}_3,{\mathfrak{M}}_4$ are given by (15)–(18). Hence, Remark 2 is verified, with the help of (31) and (32). Thus, the nonlinear sequential coupled system of Caputo fractional differential equations is Hyers–Ulam stable, and consequently, the system (3) is Hyers–Ulam stable.

5. Examples

Consider the following coupled system of sequential Caputo fractional differential equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left({}^c\mathcal{D}^{\alpha }+{\lambda }_1{}^c\mathcal{D}^{\alpha -1}\right)\mathfrak{Y}\left(\tau \right)=\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right)),\hfill \\ \left({}^c\mathcal{D}^{\beta }+{\lambda }_2{}^c\mathcal{D}^{\beta -1}\right)\mathfrak{Z}\left(\tau \right)=\mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right)).\hfill \end{array}\right.\end{array}$$

(33)

The system is supplemented with the coupled multi-point boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \mathfrak{Y}\left(\mathcal{T}\right)=\sum _{i=1}^m\mathfrak{r}{\mathfrak{1}}_i\mathfrak{Z}\left({\mathfrak{k}}_1\right)+\sum _{i=1}^m\mathfrak{r}{\mathfrak{2}}_i\mathcal{T}{\mathfrak{Z}}^{\prime }\left({\mathfrak{k}}_2\right),\mathcal{T}{\mathfrak{Y}}^{\prime }\left(\mathcal{T}\right)=\sum _{i=1}^m{{\mathfrak{s}}_1}_i\mathfrak{Z}\left({\mathfrak{u}}_1\right)+\sum _{i=1}^m{{\mathfrak{s}}_2}_i\mathcal{T}{\mathfrak{Z}}^{\prime }\left({\mathfrak{u}}_2\right),}\hfill \\ & {\displaystyle \mathfrak{Z}\left(\mathcal{T}\right)=\sum _{j=1}^n{{\mathfrak{p}}_1}_j\mathfrak{Y}\left({\zeta }_1\right)+\sum _{j=1}^n{{\mathfrak{p}}_2}_j\mathcal{T}{\mathfrak{Y}}^{\prime }\left({\zeta }_2\right),\mathcal{T}{\mathfrak{Z}}^{\prime }\left(\mathcal{T}\right)=\sum _{j=1}^n{{\mathfrak{q}}_1}_j\mathfrak{Y}\left({\mathfrak{v}}_1\right)+\sum _{j=1}^n{{\mathfrak{q}}_2}_j\mathcal{T}{\mathfrak{Y}}^{\prime }\left({\mathfrak{v}}_2\right),}\hfill \end{array}\right.\end{array}$$

(34)

for the above system, the associated constants were computed as

$${\mathbb{A}}_1=0.090718,{\mathbb{A}}_2=0.757735,{\mathbb{A}}_3=0.636377,{\mathbb{A}}_4=0.856517,$$

$${\mathbb{A}}_5=0.217723,{\mathbb{A}}_6=0.108862,{\mathbb{A}}_7=0.00633549,{\mathbb{A}}_8=0.472758,$$

$${\mathbb{A}}_9=0.399548,{\mathbb{A}}_{10}=0.832956,{\mathbb{A}}_{11}=0.0333733,{\mathbb{A}}_{12}=0.568604,$$

$${\mathbb{A}}_{13}=0.146026,{\mathbb{A}}_{14}=0.786147,{\mathbb{A}}_{15}=0.113469,{\mathbb{A}}_{16}=0.0667465.$$

Moreover,

$${\mathfrak{M}}_1\approx 0.472758,{\mathfrak{M}}_2\approx 0.000125368,{\widehat{\mathfrak{M}}}_1\approx 0.000405607,{\widehat{\mathfrak{M}}}_2=0.354819.$$

Example 1.

To illustrate Theorem 1, consider the nonlinear functions

$$\begin{array}{cc}\hfill \mathcal{K}(\tau ,\mathfrak{Y}(\tau ),\mathfrak{Z}(\tau \left)\right)& ={\displaystyle \frac{1}{3({\tau }^2+7)}}{\displaystyle \frac{{\left|\mathfrak{Y}\left(\tau \right)\right|}^2}{1+\left|\mathfrak{Y}\right(\tau \left)\right|}}+{\displaystyle \frac{1}{27}}sin\left(\mathfrak{Z}\left(\tau \right)\right)+{\displaystyle \frac{1}{8\sqrt{2}+9}},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \mathcal{L}(\tau ,\mathfrak{Y}(\tau ),\mathfrak{Z}(\tau \left)\right)& ={\displaystyle \frac{1}{30(1+|\mathfrak{Z}\left(\tau \right)\left|\right)}}{\displaystyle \frac{{\left|\mathfrak{Y}\left(\tau \right)\right|}^2}{{(9+{\tau }^2)}^2}}+{\displaystyle \frac{\mathfrak{Z}\left(\tau \right)cos\left(\mathfrak{Y}\left(\tau \right)\right)}{2{(\tau +4)}^2}}.\hfill \end{array}$$

(36)

From (35), the constants were determined as

$${\kappa }_0={\displaystyle \frac{1}{24}},{\kappa }_1={\displaystyle \frac{1}{21}},{\kappa }_2={\displaystyle \frac{1}{27}},$$

$${\widehat{\kappa }}_0={\displaystyle \frac{1}{32}},{\widehat{\kappa }}_1={\displaystyle \frac{1}{30}},{\widehat{\kappa }}_2={\displaystyle \frac{1}{81}}.$$

Furthermore,

$$\begin{array}{cc}\hfill ({\mathfrak{M}}_1+{\mathfrak{M}}_2){\kappa }_1+({\widehat{\mathfrak{M}}}_1+{\widehat{\mathfrak{M}}}_2){\widehat{\kappa }}_1& \approx 0.034359075<1,\hfill \\ \hfill ({\mathfrak{M}}_1+{\mathfrak{M}}_2){\kappa }_2+({\widehat{\mathfrak{M}}}_1+{\widehat{\mathfrak{M}}}_2){\widehat{\kappa }}_2& \approx 0.02189968<1.\hfill \end{array}$$

Therefore, all the assumptions of Theorem 1 were satisfied. Consequently, the boundary value problem (33) and (34), with nonlinearities defined in (35), admitted at least one solution on the interval $[0,2]$.

Example 2.

To illustrate the applicability of Theorem 2, we consider the nonlinear functions defined by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathcal{K}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))={\displaystyle \frac{e^{-{\tau }^2}}{{\tau }^2+40}}{tan}^{-1}\left(\mathfrak{Y}\left(\tau \right)\right)+{\displaystyle \frac{1}{(35+{\tau }^3)\left(1+\left|\mathfrak{Z}\right(\tau \left)\right|\right)}}+{\displaystyle \frac{cos\tau }{8\sqrt{{\tau }^2+1}}},\hfill \\ \mathcal{L}(\tau ,\mathfrak{Y}\left(\tau \right),\mathfrak{Z}\left(\tau \right))={\displaystyle \frac{1}{({\tau }^4+36)\left(1+\left|\mathfrak{Y}\right(\tau \left)\right|\right)}}+{\displaystyle \frac{\left|\mathfrak{Y}\right(\tau \left)\right|}{\sqrt{{\tau }^2+625}}}cos\left(\mathfrak{Z}\left(\tau \right)\right)+{\displaystyle \frac{e^{\tau }}{2({sin}^2\tau +3)}}.\hfill \end{array}\right.\end{array}$$

(37)

From (37), we obtained the constants

$${\mathcal{M}}_1={\displaystyle \frac{1}{40}},{\mathcal{M}}_2={\displaystyle \frac{1}{35}},{\mathcal{N}}_1={\displaystyle \frac{1}{36}},{\mathcal{N}}_2={\displaystyle \frac{1}{25}}.$$

A direct computation yielded

$$({\mathfrak{M}}_1+{\mathfrak{M}}_3)({\mathcal{M}}_1+{\mathcal{M}}_2)+({\mathfrak{M}}_2+{\mathfrak{M}}_4)({\mathcal{N}}_1+{\mathcal{N}}_2)\approx 0.0494939148<1.$$

Therefore, all the hypotheses of Theorem 2 were satisfied. Consequently, the boundary value problem (33), with the functions $\mathcal{K}$ and $\mathcal{L}$ defined in (37), admitted a unique solution on the interval $[0,2]$.

6. Conclusions

In this paper, we investigated a nonlinear system of sequential Caputo fractional differential equations subject to coupled mixed multi-point boundary conditions. We established the existence of solutions by applying the Leray–Schauder alternative and proved uniqueness using Banach’s contraction principle. Furthermore, we examined the Hyers–Ulam stability of the system and derived sufficient conditions ensuring its stability.

The results showed that fixed-point techniques provided an effective framework for analyzing coupled fractional systems with nonlocal boundary constraints. The theoretical findings were supported by illustrative examples, which confirmed the validity and applicability of the main results. Several special cases were also presented to demonstrate the flexibility of the proposed framework.

Future research could have addressed more general nonlinear terms or extended the analysis to broader classes of multi-point boundary conditions. The construction of appropriate numerical methods for approximating solutions could also have offered further insight into the qualitative behavior and potential applications of such coupled fractional systems.