Study of a Coupled System of Integro-Differential Equations Involving the Generalized Proportional Caputo Derivatives

Abstract

This paper deals with a new coupled system of integro-differential equations involving the generalized proportional Caputo derivatives equipped with nonlocal four-point boundary conditions. Sufficient criteria for the existence and uniqueness of solutions for the studied system are derived based on Krasnoselskii’s and Banach fixed-point theorems, respectively. Applications are constructed with three different cases to illustrate the main results.

1. Introduction

Over the past few years, topics related to fractional calculus have consistently drawn the attention of many researchers—not only mathematicians, but also researchers from other specialities like science, engineering, and medicine. This great attention did not come out of nowhere; it began with the theory of fractional calculus, which involves many different types of fractional operators that are scalable, generalizable, and integrable between each other, as well as useful in modelling for multiple aspects of life [1,2,3,4]. The precise and scientifically interpretable results that fractional operators provide when used in modelling make them essential tools that any scientific researcher should be keen to understand.

This close relationship encourages fractional calculus specialists to more deeply study fractional calculus and its operators [5,6,7,8,9]. Consequently, ever more significant efforts have been made to develop these operators. Many researchers are concerned with introducing new fractional derivatives with nonsingularity on their kernels, such as Caputo–Fabrizio and Atangana–Balenue. For recent studies on such derivatives, we refer the reader to [10,11,12] and the references therein. In another line, some researchers are concerned with generalizing the traditional operators of fractional orders. This field of study began in [13,14,15], the authors of which proposed new fractional operators that combine Riemann–Liouville and Hadamard fractional operators side by side by combining Caputo and Caputo–Hadamard fractional derivatives. For recent studies on these operators, we refer to [16,17,18,19,20] and the references therein.

In [21], Khalil et al. introduced a new type of derivative known as the conformable derivative, which has attracted considerable attention from researchers, who have obtained some valuable results; see, for example [22,23,24]. Unfortunately, this derivative does not tend to the original function when its order tends to zero, which constitutes an obstacle to the application of this definition. In order to overcome this obstacle, Anderson et al., in [25], proposed a proportional derivative of fractional order $\rho \in [0,1]$ which is defined for a differentiable function f as

$${\mathbb{D}}^{\rho }f\left(t\right)={\chi }_1(\rho ,t)f\left(t\right)+{\chi }_0(\rho ,t){\displaystyle \frac{d}{dt}}f\left(t\right)$$

(1)

where ${\chi }_0,{\chi }_1:[0,1]\times \mathbb{R}\to [0,\infty )$ are continuous functions and satisfy the following conditions for all $t\in \mathbb{R}$:

$$\begin{array}{cc}\hfill & \underset{\rho \to 0^{+}}{lim}{\chi }_0(\rho ,t)=0,\underset{\rho \to 1^{-}}{lim}{\chi }_0(\rho ,t)=1,{\chi }_0(\rho ,t)\ne 0,\rho \in (0,1],\hfill \\ \hfill & \underset{\rho \to 0^{+}}{lim}{\chi }_1(\rho ,t)=1,\underset{\rho \to 1^{-}}{lim}{\chi }_1(\rho ,t)=0,{\chi }_1(\rho ,t)\ne 0,\rho \in [0,1).\hfill \end{array}$$

(2)

The fractional derivative in (1) expands the theory of conformable derivatives and arises generally in control theory. In [26], the combination of the proportional derivative and Caputo fractional derivative was studied when ${\chi }_0(\rho ,t)=\rho t^{1-\rho },{\chi }_1(\rho ,t)=(1-\rho )t^{\rho }$, while in [27,28], Jarad et al. studied the properties of the proportional derivatives involving exponential functions in their kernels when ${\chi }_0(\rho ,t)=\rho$ and ${\chi }_1(\rho ,t)=1-\rho$. The importance and the advantages of the generalized proportional Caputo derivatives were demonstrated in [29]. As a result, this novel derivative has been shown to offer more flexibility in mathematical modelling than the traditional fractional derivatives. For instance, in [30], the authors introduced a comparative study of the entropy generation for the natural convection flow of Caputo-type proportional fractional derivatives and Atangana–Baleanu derivatives. Similarly, Günerhan et al. [31] also introduced a comparative study of an HIV epidemic model in the sense of a Caputo derivative and generalized proportional Caputo fractional derivative. In [32], the authors discuss gene regulatory networks modeled by generalized proportional Caputo fractional derivatives. Further, three neural network models have been investigated in the sense of the proportional Caputo derivative in [33]. For recent applications using this novel fractional derivative, we refer the reader to [34,35,36,37] and references therein.

Investigations into the existence and uniqueness of solutions to differential equations, which comprise the generalized proportional fractional derivatives and initial/boundary conditions, have been established by several mathematicians. Hristova and Abass considered the Riemann–Liouville-type generalized proportional fractional equation to investigate a nonlinear initial value problem in [38] and a linear scalar boundary value problem in [39]. Additionally, Abbas, in [40], introduced a coupled system involving the generalized proportional Riemann–Liouville fractional derivatives. Simultaneously, the authors of [41] used a Caputo-type generalized proportional fractional derivative to study the initial value problem for nonlinear vector-order differential equations. Shammakh and Alzumi in [42] introduced a nonlinear fractional boundary value problem including Caputo-type generalized proportional derivatives with nonlocal multipoint and substrip boundary conditions. After that, studies were conducted on this type of derivative in various fields; for instance, we refer to [43,44] and references therein.

In this work, we establish the existence and uniqueness of solutions for an integro-differential system of generalized proportional Caputo differential equations:

$$\left\{\begin{array}{c}{C}_{{\mathbb{D}}^{{\alpha }_1,\rho }}\left[u_1\left(t\right)+{\lambda }_1{I}^{{\beta }_1,\rho }{\varphi }_1(t,u_1\left(t\right),u_2\left(t\right))\right]={\psi }_1(t,u_1\left(t\right),u_2\left(t\right)),1<{\alpha }_1\le 2,\hfill \\ C_{{\mathbb{D}}^{{\alpha }_2,\rho }}\left[u_2\left(t\right)+{\lambda }_2{I}^{{\beta }_2,\rho }{\varphi }_2(t,u_1\left(t\right),u_2\left(t\right))\right]={\psi }_2(t,u_1\left(t\right),u_2\left(t\right)),2<{\alpha }_2\le 3,\hfill \\ t\in \mathcal{J}:=[0,a],\hfill \end{array}\right.$$

(3)

supplemented with coupled four-point boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_1\left(0\right)=0,\hfill & u_1\left(a\right)={\gamma }_1{u}_2\left({\theta }_2\right),\hfill \\ u_2\left(0\right)=0,\hfill & u_2\left({\theta }_1\right)=0,u_2\left(a\right)={\gamma }_2{u}_1\left({\theta }_2\right),0<{\theta }_1<{\theta }_2<a,\hfill \end{array}\right.\end{array}$$

(4)

where $\rho \in (0,1],C_{{\mathbb{D}}^{\delta ,\rho }}$ denotes the Caputo-type generalized proportional fractional derivatives of order $\delta \in \{{\alpha }_1,{\alpha }_2\}$, while $I^{\kappa ,\rho }$ denotes the generalized proportional fractional integrals of real positive order $\kappa \in \{{\beta }_1,{\beta }_2\}$; ${\gamma }_i$ are positive real constants; ${\lambda }_i$ are non-negative real constants; and ${\varphi }_i,{\psi }_i;(i=1,2):\mathcal{J}\times \mathbb{R}\times \mathbb{R}⟶$$\mathbb{R}$ are continuous functions.

Based on this motivation, our study has the following comparative advantages:

1.

Bridging the research gap in the topic of generalized proportional Caputo fractional derivatives, considering that most of the previous theoretical studies on this type of derivative are limited to studying value problems that consist of one fractional differential equation.

2.

Formulating a new coupled system of integro-differential equations, considering different fractional orders for the system.

3.

Possibility of obtaining different special systems by giving specific values of the parameters of (3). Indeed, the system (3) is converted to be integro-differential system in the classical Caputo sense. For a similar system, we refer to the work [45]. In addition, when letting both ${\lambda }_1$ and ${\lambda }_2$ be equal to zero (or one of them in (3), one gets a new boundary value problem in the gerneralized proportional Caputo derivatives.

4.

Introducing a scalability study by considering more complex boundary conditions than introduced and (4).

5.

Enhancing future studies in generalization fractional operators by considering more complicated proportional derivatives such as $\psi -$Caputo-type [46] and (k, $\psi$)-Caputo-type [47].

The paper is structured as follows: Section 2 focuses on all basic concepts related to the generalized proportional Caputo fractional derivatives. In Section 3, we present our contributions to the field. Section 4 provides an application of the studied problem. In the Section 5, the results are discussed.

2. Preliminaries

In this section, we revisit some important definitions and properties related to the generalized fractional proportional derivative and integral.

Definition 1

([48]). The left-sided Riemann–Liouville fractional integral operator of the order $\delta \in \mathbb{C},\mathfrak{R}\left(\delta \right)>0$ is defined by

$$\begin{array}{c}\hfill {(}_a{I}^{\delta }f)\left(\tau \right)={\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\mathit{\int }}_a^{\tau }{(\tau -s)}^{\delta -1}f\left(s\right)ds,(\tau >a),\end{array}$$

(5)

where $\Gamma \left(\delta \right)$ denotes the Gamma function.

Definition 2

([48]). Let $f\left(\tau \right)$ be a function from the space $A{C}^n[a,b]$. Then, the left-sided Caputo fractional derivative operator of order $\delta \in \mathbb{C},\mathfrak{R}\left(\delta \right)⩾0$ is defined by

$$\begin{array}{c}\hfill {(}_a^C{D}^{\delta }f)\left(\tau \right)={(}_a{I}^{n-\delta }{D}^{n}f)\left(\tau \right)={\displaystyle \frac{1}{\Gamma (n-\delta )}}{\int }_a^{\tau }{\displaystyle \frac{f^{\left(n\right)}\left(s\right)}{{(\tau -s)}^{\delta -n+1}}}ds,\end{array}$$

(6)

where n is the smallest integer bigger than the real part of δ.

Definition 3

([27,28]). Let us consider $\delta ⩾0$ with $\rho \in (0,1].$ Then, the left-sided generalized proportional fractional integral of a function $f\in L^1[a,b]$ is defined by

$$\left\{\begin{array}{c}{(}_a{I}^{0,\rho }f)\left(\tau \right)=f\left(\tau \right),\hfill \\ {(}_a{I}^{\delta ,\rho }f)\left(\tau \right)={\displaystyle \frac{1}{{\rho }^{\delta }\Gamma \left(\delta \right)}}{\int }_a^{\tau }{e}^{{\displaystyle \frac{\rho -1}{\rho }}(\tau -s)}{(\tau -s)}^{\delta -1}f\left(s\right)ds,(\delta >0).\hfill \end{array}\right.$$

(7)

Remark 1.

Clearly, if we take $\rho =1$ in Definition 3, we obtain the left-sided Riemann–Liouville fractional integral operator (5).

Definition 4

([27]). Let us consider the order $\delta ⩾0$ with $\rho \in (0,1]$. Then, the left-sided generalized proportional Caputo fractional derivative of a function $f\in C^{\left(n\right)}[a,b]$ is defined by

$$\left\{\begin{array}{c}({}_a{}^C{\mathbb{D}}^{0,\rho }f)\left(\tau \right)={}_{a}I^{n,\rho }\left({\mathbb{D}}^{n,\rho }f\right)\left(\tau \right)=f\left(\tau \right)\hfill \\ {}_a{}^C{\mathbb{D}}^{\mathrm{\delta },\rho }\left(\tau \right)={}_{a}I^{n-\delta ,\rho }\left({\mathbb{D}}^{n,\rho }f\right)\left(\tau \right)\hfill \\ ={\displaystyle \frac{1}{{\rho }^{n-\delta }\Gamma (n-\delta )}}{\int }_a^{\tau }{e}^{{\displaystyle \frac{\rho -1}{\rho }}(\tau -s)}{(\tau -s)}^{n-\delta -1}{\mathbb{D}}^{n,\rho }f\left(s\right)ds,(n=\left[\delta \right]+1,\delta >0),\hfill \end{array}\right.$$

(8)

where ${\mathbb{D}}^{1,\rho }$ is the proportional derivative satisfying ${\mathbb{D}}^{1,\rho }f\left(\tau \right)={\mathbb{D}}^{\rho }f\left(\tau \right)=(1-\rho )f\left(t\right)+\rho f^{\prime }\left(\tau \right)$, and the operator ${\mathbb{D}}^{n,\rho }$ just means applying the proportional derivative n times: ${\mathbb{D}}^{n,\rho }f\left(\tau \right)=\left(\underset{n \mathrm{times}}{\underbrace{{\mathbb{D}}^{\rho }{\mathbb{D}}^{\rho }\dots {\mathbb{D}}^{\rho }}}f\right)\left(\tau \right)$.

Remark 2.

Again, if $\rho =1$, Definition 4 reduces to the standard left-sided Caputo fractional derivative operator (6).

Remark 3.

The generalized proportional Caputo fractional derivative of a constant is not zero for $\rho \in (0,1)$.

Proposition 1

([27]). Let $\delta ,\gamma \in \mathbb{C}$ be such that $Re\left(\delta \right),Re\left(\gamma \right)>0$. Then, for any $\rho \in (0,1]$ and $n=[Re(\delta \left)\right]+1$, we have

i.

$\left({}_a{}^C\mathbb{D}^{\delta ,\rho }{e}^{{\displaystyle \frac{\rho -1}{\rho }}\tau }{(\tau -a)}^{\gamma -1}\right)\left(s\right)={\displaystyle \frac{{\rho }^{\delta }\Gamma \left(\gamma \right)}{\Gamma (\gamma -\delta )}}{e}^{{\displaystyle \frac{\rho -1}{\rho }}s}{(s-a)}^{\gamma -\delta -1},Re\left(\delta \right)>n$

ii.

$\left({}_a{}^C\mathbb{D}^{\delta ,\rho }{e}^{{\displaystyle \frac{\rho -1}{\rho }}\tau }{(\tau -a)}^k\right)\left(s\right)=0$, for $k=0,1,\dots ,n-1$,

iii.

$\left({}_a{}^C\mathbb{D}^{\delta ,\rho }{e}^{{\displaystyle \frac{\rho -1}{\rho }}(-)}\right)\left(\tau \right)=0\mathrm{for}\tau >a,$ for $\rho \in (0,1),$

iv.

$\left({}_{a}I^{\delta ,\rho }{e}^{{\displaystyle \frac{\rho -1}{\rho }}\tau }{(\tau -a)}^{\gamma -1}\right)\left(s\right)={\displaystyle \frac{\Gamma \left(\gamma \right)}{{\rho }^{\delta }\Gamma (\gamma +\delta )}}{e}^{{\displaystyle \frac{\rho -1}{\rho }}s}{(s-a)}^{\gamma +\delta -1}\gamma >0.$

Theorem 1

([27]). For $\rho \in (0,1]$ and $n=[Re(\delta \left)\right]+1$, we have

$$\begin{array}{c}\hfill \left({}_{a}I^{\delta ,\rho }{}_a{}^C{\mathbb{D}}^{\mathrm{\delta },\rho }f\right)\left(\tau \right)=f\left(\tau \right)-\sum _{k=0}^{n-1}{\displaystyle \frac{\left({\mathbb{D}}^{k,\rho }f\right)\left(a\right)}{{\rho }^k\Gamma (k+1)}}{(\tau -a)}^k{e}^{{\displaystyle \frac{\rho -1}{\rho }}(\tau -a)},(f\in C^n[a,b]).\end{array}$$

(9)

In the current work, we write the generalized proportional fractional integral ${}_{a}I^{\delta ,\rho }$ and the generalized proportional Caputo fractional derivative ${}_a{}^C{\mathbb{D}}^{\mathrm{\delta },\rho }$ as $I^{\delta ,\rho }$ and ${}^C{\mathbb{D}}^{\mathrm{\delta },\rho }$, respectively.

In the following lemma, we present the solution of the linear version of the system (3) complemented with boundary data (4), which plays a key role in our main results.

Lemma 1.

Let

$$\Delta ={\theta }_{1}K\left({\theta }_1\right)\left(a^2({\theta }_1-a)K^2\left(a\right)+{\gamma }_1{\gamma }_2{\theta }_2^2({\theta }_2-{\theta }_1)K^2\left({\theta }_2\right)\right)\ne 0.$$

(10)

For $\rho \in (0,1]$ and $f_i,g_i;(i=1,2)\in C(\mathcal{J},\mathbb{R})$. Then, the solution of the linear system

$$\left\{\begin{array}{c}{}^C{\mathbb{D}}^{\alpha 1,\rho }\left[u_1\left(t\right)+{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(t\right)\right]=g_1\left(t\right),1<{\alpha }_1\le 2,\hfill \\ \\ {}^C{\mathbb{D}}^{\alpha 2,\rho }\left[u_2\left(t\right)+{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(t\right)\right]=g_2\left(t\right),2<{\alpha }_2\le 3,\hfill \\ u_1\left(0\right)=0,u_1\left(a\right)={\gamma }_1{u}_2\left({\theta }_2\right),\hfill \\ u_2\left(0\right)=0,u_2\left({\theta }_1\right)=0,u_2\left(a\right)={\gamma }_2{u}_1\left({\theta }_2\right),0<{\theta }_1<{\theta }_2<a,\hfill \end{array}\right.$$

(11)

is given by

$$\begin{array}{cc}\hfill u_1\left(t\right)& ={\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K(t-w)dw\hfill \\ & -{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}{f}_1\left(w\right)K(t-w)dw\hfill \\ & +{\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{A_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{f}_2\left(w\right)K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{A_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{A_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{f}_2\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K(a-w)dw\hfill \\ & +{\displaystyle \frac{A_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{A_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{f}_1\left(w\right)K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{f}_1\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{f}_2\left(w\right)K({\theta }_2-w)dw\},\hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill u_2\left(t\right)& ={\displaystyle \frac{1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^t{(t-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K(t-w)dw\hfill \\ & -{\displaystyle \frac{{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^t{(t-w)}^{{\beta }_2-1}{f}_2\left(w\right)K(t-w)dw\hfill \\ & +{\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{B_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{f}_2\left(w\right)K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{B_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{B_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{f}_2\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K(a-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K({\theta }_2-w)dw\hfill \\ \hfill & -{\displaystyle \frac{B_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{f}_1\left(w\right)K({\theta }_2-w)dw\hfill \\ \hfill & -{\displaystyle \frac{B_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K(a-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{f}_1\left(w\right)K(a-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{f}_2\left(w\right)K({\theta }_2-w)dw\hfill \\ \hfill & +t[{\displaystyle \frac{B_4{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{f}_2\left(w\right)K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{B_4}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{B_5}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_5{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{f}_1\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_5{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{B_5{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{f}_2\left(w\right)K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_6{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{f}_2\left(w\right)K(a-w)dw\hfill \\ & +{\displaystyle \frac{B_6}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{g}_2\left(w\right)K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_6{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{g}_1\left(w\right)K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{B_6{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{f}_1\left(w\right)K({\theta }_2-w)dw\left]\right\},\hfill \end{array}$$

(13)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}K\left(t\right)=e^{\frac{\rho -1}{\rho }t},A_1=a{\gamma }_1{\theta }_2({\theta }_2-a)K(a+{\theta }_2),A_2={\gamma }_1{\theta }_2{\theta }_1({\theta }_1-{\theta }_2)K({\theta }_2+{\theta }_1),\hfill \\ A_3=a{\theta }_1(a-{\theta }_1)K(a+{\theta }_1),B_1={\gamma }_1{\gamma }_2{\theta }_2^3{K}^2\left({\theta }_2\right)-a^3{K}^2\left(a\right),\hfill \\ B_2=a{\theta }_1^{2}K(a+{\theta }_1),B_3={\theta }_1^2{\gamma }_2{\theta }_{2}K({\theta }_2+{\theta }_1),B_4=a^2{K}^2\left(a\right)-{\gamma }_1{\gamma }_2{\theta }_2^2{K}^2\left({\theta }_2\right),\hfill \\ B_5={\gamma }_2{\theta }_1{\theta }_{2}K({\theta }_2+{\theta }_1),B_6=a{\theta }_{1}K(a+{\theta }_1).\hfill \end{array}\right.\end{array}$$

(14)

Proof. 

To begin, we apply the generalized proportional fractional integrals $I^{{\alpha }_1,\rho }$ and $I^{{\alpha }_2,\rho }$ to both sides of the first and second equations in (11), respectively, with the aid of (9) to get

$$\left\{\begin{array}{cc}{u}_1\left(t\right)\hfill & =I^{{\alpha }_1,\rho }{g}_1\left(t\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(t\right)+(c_0+c_{1}t)K\left(t\right),\hfill \\ u_2\left(t\right)\hfill & =I^{{\alpha }_2,\rho }{g}_2\left(t\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(t\right)+(d_0+d_{1}t+d_2{t}^2)K\left(t\right),\hfill \end{array}\right.$$

(15)

where $c_k={\sum }_{k=0}^1{\displaystyle \frac{{\mathbb{D}}^{k,\rho }{u}_1\left(0\right)}{{\rho }^{k}k!}}$ and $d_i={\sum }_{i=0}^2{\displaystyle \frac{{\mathbb{D}}^{i,\rho }{u}_2\left(0\right)}{{\rho }^{i}i!}}$ are arbitrary real numbers. Now, combining the conditions $u_1\left(0\right)=0=u_2\left(0\right)$ with (15), we find that $c_0=0=d_0.$ Thus, we have

$$\begin{array}{c}\hfill u_1\left(t\right)=I^{{\alpha }_1,\rho }{g}_1\left(t\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(t\right)+c_{1}tK\left(t\right).\end{array}$$

(16)

and

$$\begin{array}{c}\hfill u_2\left(t\right)=I^{{\alpha }_2,\rho }{g}_2\left(t\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(t\right)+(d_{1}t+d_2{t}^2)K\left(t\right).\end{array}$$

(17)

Now, applying $u_2\left({\theta }_1\right)=0$ to Equation (17) gives

$$\begin{array}{c}\hfill d_1{\theta }_{1}K\left({\theta }_1\right)+d_2{\theta }_1^{2}K\left({\theta }_1\right)={\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)-I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right).\end{array}$$

(18)

Next, we plug in the boundary conditions $u_1\left(a\right)={\gamma }_1{u}_2\left({\theta }_2\right)$ and $u_2\left(a\right)={\gamma }_2{u}_1\left({\theta }_2\right)$, into (16) and (17). This leads to

$$\begin{array}{c}\hfill I^{{\alpha }_1,\rho }{g}_1\left(a\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(a\right)+c_{1}aK\left(a\right)={\gamma }_1(I^{{\alpha }_2,\rho }{g}_2\left({\theta }_2\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_2\right)+(d_1{\theta }_2+d_2{\theta }_2^2)K\left({\theta }_2\right)),\end{array}$$

(19)

and

$$\begin{array}{c}\hfill I^{{\alpha }_2,\rho }{g}_2\left(a\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(a\right)+(d_{1}a+d_2{a}^2)K\left(a\right)={\gamma }_2(I^{{\alpha }_1,\rho }{g}_1\left({\theta }_2\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left({\theta }_2\right)+c_1{\theta }_{2}K\left({\theta }_2\right)).\end{array}$$

(20)

Thus, we now have the following system of three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{d}_1{\theta }_{1}K\left({\theta }_1\right)+d_2{\theta }_1^{2}K\left({\theta }_1\right)={\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)-I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right),\hfill \\ d_1{\gamma }_1{\theta }_{2}K\left({\theta }_2\right)+d_2{\gamma }_1{\theta }_2^{2}K\left({\theta }_2\right)-c_{1}aK\left(a\right)=I^{{\alpha }_1,\rho }{g}_1\left(a\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(a\right)\hfill \\ -{\gamma }_1(I^{{\alpha }_2,\rho }{g}_2\left({\theta }_2\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_2\right)),\hfill \\ d_{1}aK\left(a\right)+d_2{a}^{2}K\left(a\right)-c_1{\gamma }_2{\theta }_{2}K\left({\theta }_2\right)={\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(a\right)-I^{{\alpha }_2,\rho }{g}_2\left(a\right)\hfill \\ +{\gamma }_2(I^{{\alpha }_1,\rho }{g}_1\left({\theta }_2\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left({\theta }_2\right)).\hfill \end{array}\right.\end{array}$$

(21)

By solving the system, we find the values of the constants as follows:

$$\begin{array}{cc}\hfill d_1& ={\displaystyle \frac{1}{\Delta }}\{a{\theta }_1^2\left[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(a\right)-I^{{\alpha }_2,\rho }{g}_2\left(a\right)+{\gamma }_2\left(I^{{\alpha }_1,\rho }{g}_1\left({\theta }_2\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left({\theta }_2\right)\right)\right]K(a+{\theta }_1)\hfill \\ & -a^3\left[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)-I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right)\right]K^2\left(a\right)+{\gamma }_2{\gamma }_1{\theta }_2^3\left[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)-I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right)\right]K^2\left({\theta }_2\right)\hfill \\ & -{\gamma }_2{\theta }_2{\theta }_1^2\left[I^{{\alpha }_1,\rho }{g}_1\left(a\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(a\right)-{\gamma }_1\left(I^{{\alpha }_2,\rho }{g}_2\left({\theta }_2\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_2\right)\right)\right]K({\theta }_2+{\theta }_1)\},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill d_2& ={\displaystyle \frac{1}{\Delta }}\{a^2\left[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)-I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right)\right]K^2\left(a\right)-a{\theta }_1[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(a\right)-I^{{\alpha }_2,\rho }{g}_2\left(a\right)\hfill \\ & +{\gamma }_2\left(I^{{\alpha }_1,\rho }{g}_1\left({\theta }_2\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left({\theta }_2\right)\right)]K(a+{\theta }_1)+{\gamma }_2{\theta }_2[{\theta }_1(I^{{\alpha }_1,\rho }{g}_1\left(a\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(a\right)\hfill \\ & -{\gamma }_1\left(I^{{\alpha }_2,\rho }{g}_2\left({\theta }_2\right)-{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_2\right)\right))K({\theta }_2+{\theta }_1)-{\gamma }_1{\theta }_2({\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)\hfill \\ \hfill & -I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right)\left)K^2\left({\theta }_2\right)\right]\},\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill c_1& ={\displaystyle \frac{1}{\Delta }}\left\{{\gamma }_1{\theta }_1{\theta }_2({\theta }_1-{\theta }_2)\right[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left(a\right)+{\gamma }_2\left(I^{{\alpha }_1,\rho }{g}_1\left({\theta }_2\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left({\theta }_2\right)\right)\hfill \\ & -I^{{\alpha }_2,\rho }{g}_2\left(a\right)]K({\theta }_2+{\theta }_1)+a(a-{\theta }_1){\theta }_1[I^{{\alpha }_1,\rho }{g}_1\left(a\right)-{\lambda }_1{I}^{{\beta }_1,\rho }{f}_1\left(a\right)-{\gamma }_1(I^{{\alpha }_2,\rho }{g}_2\left({\theta }_2\right)\hfill \\ & -{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_2\right)\left)\right]K(a+{\theta }_1)+a{\gamma }_1{\theta }_2({\theta }_2-a)\left[{\lambda }_2{I}^{{\beta }_2,\rho }{f}_2\left({\theta }_1\right)-I^{{\alpha }_2,\rho }{g}_2\left({\theta }_1\right)\right]K(a+{\theta }_2)\},\hfill \end{array}$$

(24)

where $\Delta$ is given in (10). By inserting Equation (24) into (16), and substituting Equations (22) and (23) into (17), we obtain solutions (12) and (13). With some straightforward calculations, we can also verify the converse of the lemma, which completes the proof. □

3. Main Results

As a result of Lemma (1), problems (3) and (4) can be transformed into a fixed-point problem as $(u_1,u_2)=\mathcal{Q}(u_1,u_2),$ where $\mathcal{Q}:E\times E\to E\times E$ is an operator defined by

$$\mathcal{Q}(u_1,u_2)\left(t\right)=\left(\begin{array}{c}{\mathcal{Q}}_1(u_1,u_2)\left(t\right)\\ {\mathcal{Q}}_2(u_1,u_2)\left(t\right),\end{array}\right)$$

(25)

with

$$\begin{array}{cc}\hfill {\mathcal{Q}}_1(u_1,u_2)\left(t\right)& ={\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & -{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & +{\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{A_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{A_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{A_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & +{\displaystyle \frac{A_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{A_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\},\hfill \end{array}$$

(26)

and

$$\begin{array}{cc}\hfill {\mathcal{Q}}_2(u_1,u_2)\left(t\right)& ={\displaystyle \frac{1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^t{(t-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ \hfill & -{\displaystyle \frac{{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^t{(t-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & +{\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{B_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{B_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ \hfill & -{\displaystyle \frac{B_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & +{\displaystyle \frac{B_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +t[{\displaystyle \frac{B_4{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{B_4}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{B_5}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_5{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_5{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{B_5{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_6{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & +{\displaystyle \frac{B_6}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_6{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{B_6{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\left]\right\}.\hfill \end{array}$$

(27)

Here, $E=C(\mathcal{J},\mathbb{R})$ is a Banach space with norm $\left|\right|x\left|\right|={sup}_{t\in \mathcal{J}}\left|x\left(t\right)\right|$. Then, $(E\times E,|\left|\right(x,y\left)\right|\left|\right)$ is a Banach space with the norm defined as $\left|\right|(x,y)\left|\right|=\left|\right|x\left|\right|+\left|\right|y\left|\right|$.

3.1. Notations and Hypotheses

For simplicity, we adopted the following notations:

$$\left\{\begin{array}{cc}\hfill {\Xi }_1& =({\Lambda }_1+{\Lambda }_5),{\Xi }_2=({\Lambda }_2+{\Lambda }_6),{\Xi }_3=({\Lambda }_3+{\Lambda }_7),{\Xi }_4=({\Lambda }_4+{\Lambda }_8),\hfill \\ \hfill {\Lambda }_1& ={\displaystyle \frac{a^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right\},\hfill \\ \hfill {\Lambda }_2& ={\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{\left|A_2\right|a^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right\},\hfill \\ \hfill {\Lambda }_3& ={\displaystyle \frac{{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right\},\hfill \\ \hfill {\Lambda }_4& ={\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{\left|A_2\right|{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right\},\hfill \\ \hfill {\Lambda }_5& ={\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{B_2{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{B_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+a\left({\displaystyle \frac{B_5{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{B_6{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right)\right\},\hfill \\ \hfill {\Lambda }_6& ={\displaystyle \frac{a^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{a}{|\Delta |}}\{{\displaystyle \frac{\left|B_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{B_2{a}^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{{\gamma }_1{B}_3{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\hfill \\ \hfill & +a\left({\displaystyle \frac{\left|B_4\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{B_5{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{B_6{a}^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right)\},\hfill \\ \hfill {\Lambda }_7& ={\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{B_2{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{B_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+a\left({\displaystyle \frac{B_5{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{B_6{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right)\right\},\hfill \\ \hfill {\Lambda }_8& ={\displaystyle \frac{{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{a}{|\Delta |}}\{{\displaystyle \frac{\left|B_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_2{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{B_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+a\left({\displaystyle \frac{\left|B_4\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_5{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_6{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right)\}.\hfill \end{array}\right.$$

(28)

To reach the main results, we rely on the following assumptions:

$\left({\mathbf{T}}_{\mathbf{1}}\right)$ The functions ${\varphi }_i,{\psi }_i;i=1,2:\mathcal{J}\times \mathbb{R}\times \mathbb{R}⟶$$\mathbb{R}$ are continuous.

$\left({\mathbf{T}}_{\mathbf{2}}\right)$ There exist positive constants $L_i$ and ${\overline{L}}_i;(i=1,2)$ such that for all $t\in \mathcal{J}$ and $u_i,v_i\in \mathbb{R}$, $i=1,2$, we have

$$\begin{array}{c}\hfill |{\psi }_i(t,u_1,u_2)-{\psi }_i(t,{\overline{u}}_1,{\overline{u}}_2)|\le L_i\left(\right|u_1-{\overline{u}}_1|+|u_2-{\overline{u}}_2\left|\right),\end{array}$$

(29)

$$\begin{array}{c}\hfill |{\varphi }_i(t,u_1,u_2)-{\varphi }_i(t,{\overline{u}}_1,{\overline{u}}_2)|\le {\overline{L}}_i\left(\right|u_1-{\overline{u}}_1|+|u_2-{\overline{u}}_2\left|\right).\end{array}$$

(30)

3.2. Existence Result via Krasnoselskii’s Fixed-Point Theorem

We now present the first existence result of the solution of the coupled system (3) and (4) using Krasnoselskii’s fixed-point theorem [49].

Theorem 2.

Suppose that $\left({\mathbf{T}}_{\mathbf{1}}\right)$ and $\left({\mathbf{T}}_{\mathbf{2}}\right)$ hold. In addition, we impose the following assumption:

$\left({\mathbf{T}}_{\mathbf{3}}\right)$ There exist non-negative functions ${\tau }_i\left(t\right)$ and ${\varkappa }_i\left(t\right);(i=1,2)$ belonging to $C(\mathcal{J},{\mathbb{R}}^{+})$ such that

$$\begin{array}{c}\hfill |{\psi }_i(t,u_1\left(t\right),u_2\left(t\right))|\le {\tau }_i\left(t\right),\end{array}$$

(31)

$$\begin{array}{c}\hfill |{\varphi }_i(t,u_1\left(t\right),u_2\left(t\right))|\le {\varkappa }_i\left(t\right),\end{array}$$

(32)

for all $(t,u_1\left(t\right),u_2\left(t\right))\in \mathcal{J}\times \mathbb{R}\times \mathbb{R}.$ Then, the coupled system (3)–(4) admits at least one solution on $\mathcal{J},$ provided that

$$\begin{array}{c}\hfill {\Lambda }_2{L}_2+{\Lambda }_4{\overline{L}}_2+{\Lambda }_5{L}_1+{\Lambda }_7{\overline{L}}_1<1,\end{array}$$

(33)

where ${\Lambda }_i(i=2,4,5,7)$ are presented in (28).

Proof. 

Let us define the ball ${\mathbb{B}}_{ϵ}=\{(u_1,u_2)\in E\times E:\parallel (u_1,u_2)\parallel \le ϵ\},$ where the radius $ϵ$ is chosen such that

$$\begin{array}{c}\hfill ϵ\ge {\Xi }_1\parallel {\tau }_1\parallel +{\Xi }_2\parallel {\tau }_2\parallel +{\Xi }_3\parallel {\varkappa }_1\parallel +{\Xi }_4\parallel {\varkappa }_2\parallel .\end{array}$$

(34)

Now, we decompose the operator $\mathcal{Q}$, defined on the ball ${\mathbb{B}}_{ϵ}$ into four components, as follows:

$$\begin{array}{cc}\hfill {\vartheta }_1(u_1,u_2)\left(t\right)& ={\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & -{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & +{\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{A_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{A_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\vartheta }_2(u_1,u_2)\left(t\right)& ={\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{A_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{A_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{A_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\aleph }_1(u_1,u_2)\left(t\right)& ={\displaystyle \frac{1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^t{(t-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & -{\displaystyle \frac{{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^t{(t-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(t-w)dw\hfill \\ & +{\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{B_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ \hfill & -{\displaystyle \frac{B_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & +{\displaystyle \frac{B_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & +{\displaystyle \frac{B_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ \hfill & +t[{\displaystyle \frac{B_4{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{B_4}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_1-w)dw\hfill \\ & -{\displaystyle \frac{B_5{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{B_5{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_6{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & +{\displaystyle \frac{B_6}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\left]\right\},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\aleph }_2(u_1,u_2)\left(t\right)& ={\displaystyle \frac{tK\left(t\right)}{\Delta }}\{{\displaystyle \frac{B_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & -{\displaystyle \frac{B_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ \hfill & +{\displaystyle \frac{B_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & +t[{\displaystyle \frac{B_5}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ \hfill & -{\displaystyle \frac{B_5{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(a-w)dw\hfill \\ & -{\displaystyle \frac{B_6{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\hfill \\ & +{\displaystyle \frac{B_6{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K({\theta }_2-w)dw\left]\right\}.\hfill \end{array}$$

(38)

Hence, ${\mathcal{Q}}_1(u_1,u_2)\left(t\right)={\vartheta }_1(u_1,u_2)\left(t\right)+{\vartheta }_2(u_1,u_2)\left(t\right)$ and ${\mathcal{Q}}_2(u_1,u_2)\left(t\right)={\aleph }_1(u_1,u_2)\left(t\right)$$+{\aleph }_2(u_1,u_2)\left(t\right).$

Step I: First, let $(u_1,u_2)$ and $({\overline{u}}_1,{\overline{u}}_2)$ be elements in ${\mathbb{B}}_{ϵ}$. Applying $K(\tau -w)<1,\forall \tau >w,$ where $\tau \in \{t,a,{\theta }_1,{\theta }_2\},$ we obtain

$$\begin{array}{cc}\hfill & \parallel {\vartheta }_1(u_1,u_2)\left(t\right)+{\vartheta }_2({\overline{u}}_1,{\overline{u}}_2)\left(t\right)\parallel \hfill \\ & \le \underset{t\in \mathcal{J}}{sup}\left\{{\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}\right|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}\right|{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_1\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ \hfill & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_w\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))\right|dw\left)\right\}\hfill \\ & \le \parallel {\tau }_1\parallel \left({\displaystyle \frac{a^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right\}\right)\hfill \\ & +\parallel {\tau }_2\parallel \left({\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{\left|A_2\right|a^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right\}\right)\hfill \\ & +\parallel {\varkappa }_1\parallel \left({\displaystyle \frac{{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right\}\right)\hfill \\ & +\parallel {\varkappa }_2\parallel \left({\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{\left|A_2\right|{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right\}\right)\hfill \\ & \le \parallel {\tau }_1\parallel {\Lambda }_1+\parallel {\tau }_2\parallel {\Lambda }_2+\parallel {\varkappa }_1\parallel {\Lambda }_3+\parallel {\varkappa }_2\parallel {\Lambda }_4.\hfill \end{array}$$

(39)

In the same way, we derive the following:

$$\begin{array}{c}\hfill \parallel {\aleph }_1(u_1,u_2)\left(t\right)+{\aleph }_2({\overline{u}}_1,{\overline{u}}_2)\left(t\right)\parallel \le \parallel {\tau }_1\parallel {\Lambda }_5+\parallel {\tau }_2\parallel {\Lambda }_6+\parallel {\varkappa }_1\parallel {\Lambda }_7+\parallel {\varkappa }_2\parallel {\Lambda }_8.\end{array}$$

(40)

Thus, by combining both estimates and using (34), we obtain

$$\begin{array}{cc}\hfill \parallel ({\vartheta }_1,{\aleph }_1)(u_1,u_2)+({\vartheta }_2,{\aleph }_2)({\overline{u}}_1,{\overline{u}}_2)\parallel & \le {\Xi }_1\parallel {\tau }_1\parallel +{\Xi }_2\parallel {\tau }_2\parallel +{\Xi }_3\parallel {\varkappa }_1\parallel +{\Xi }_4\parallel {\varkappa }_2\parallel \hfill \\ & \le ϵ.\hfill \end{array}$$

(41)

Hence,

$$\begin{array}{c}\hfill ({\vartheta }_1,{\aleph }_1)(u_1,u_2)+({\vartheta }_2,{\aleph }_2)({\overline{u}}_1,{\overline{u}}_2)\in {\mathbb{B}}_{ϵ}.\end{array}$$

(42)

Step II: Next, we now proceed to illustrate that the operator $({\vartheta }_2,{\aleph }_2)$ is a contraction operator, as follows. Setting $(u_1,u_2)$, $({\overline{u}}_1,{\overline{u}}_2)\in {\mathbb{B}}_{ϵ}$, and using condition $\left({\mathbf{T}}_{\mathbf{2}}\right)$, we obtain

$$\begin{array}{cc}\hfill & \parallel {\vartheta }_2(u_1,u_2)-{\vartheta }_2({\overline{u}}_1,{\overline{u}}_2)\parallel \hfill \\ & =\underset{t\in \mathcal{J}}{sup}|{\vartheta }_2(u_1,u_2)\left(t\right)-{\vartheta }_2({\overline{u}}_1,{\overline{u}}_2)\left(t\right)|\hfill \\ & \le {\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}\right|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_1\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\}\hfill \\ & \le \left(L_2(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel )\right)\left[{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{\left|A_2\right|a^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right)\right]\hfill \\ & +\left({\overline{L}}_2(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel )\right)\left[{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{\left|A_2\right|{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right)\right].\hfill \end{array}$$

(43)

Hence, we obtain from (28)

$$\begin{array}{c}\hfill \parallel {\vartheta }_2(u_1,u_2)-{\vartheta }_2({\overline{u}}_1,{\overline{u}}_2)\parallel \le \left({\Lambda }_2{L}_2+{\Lambda }_4{\overline{L}}_2\right)(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel ).\end{array}$$

(44)

By applying the same methodology, we also deduce

$$\begin{array}{c}\hfill \parallel {\aleph }_2(u_1,u_2)-{\aleph }_2({\overline{u}}_1,{\overline{u}}_2)\parallel \le \left({\Lambda }_5{L}_1+{\Lambda }_7{\overline{L}}_1\right)(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel ).\end{array}$$

(45)

Thus,

$$\begin{array}{cc}\hfill \parallel ({\vartheta }_2,{\aleph }_2)(u_1,u_2)-({\vartheta }_2,{\aleph }_2)({\overline{u}}_1,{\overline{u}}_2)\parallel & \le \left({\Lambda }_2{L}_2+{\Lambda }_4{\overline{L}}_2+{\Lambda }_5{L}_1+{\Lambda }_7{\overline{L}}_1\right)(\parallel u_1-{\overline{u}}_1\parallel \hfill \\ & +\parallel u_2-{\overline{u}}_2\parallel ).\hfill \end{array}$$

(46)

Therefore, by (33), we derive that the operator $({\vartheta }_2,{\aleph }_2)$ is a contraction operator.

Step III: We now show that $({\vartheta }_1,{\aleph }_1)$ is continuous and compact. First, we note that the operator $({\vartheta }_1,{\aleph }_1)$ is continuous by virtue of the assumption $\left({\mathbf{T}}_{\mathbf{1}}\right).$ Now, we have to demonstrate that $({\vartheta }_1,{\aleph }_1)$ is uniformly bounded on the set ${\mathbb{B}}_{ϵ}$ as follows. For each $t\in \mathcal{J}$, $(u_1,u_2)\in {\mathbb{B}}_{ϵ},$ and from the assumption $\left({\mathbf{T}}_{\mathbf{3}}\right)$, one has

$$\begin{array}{cc}\hfill & \left|{\vartheta }_1(u_1,u_2)\left(t\right)\right|\hfill \\ & \le {\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_2\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}\right|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\}\hfill \\ \hfill & \le \parallel {\tau }_1\parallel \left\{{\displaystyle \frac{a^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right)\right\}\hfill \\ & +\parallel {\varkappa }_1\parallel \left\{{\displaystyle \frac{{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right)\right\}\hfill \end{array}$$

Using (28) gives

$$\begin{array}{c}\hfill \parallel {\vartheta }_1(u_1,u_2)\parallel \le \parallel {\tau }_1\parallel {\Lambda }_1+\parallel {\varkappa }_1\parallel {\Lambda }_3.\end{array}$$

(47)

Similarly, we can find that

$$\begin{array}{c}\hfill \parallel {\aleph }_1(u_1,u_2)\parallel \le \parallel {\tau }_2\parallel {\Lambda }_6+\parallel {\varkappa }_2\parallel {\Lambda }_8.\end{array}$$

(48)

Finally, it remains for us to establish that $({\vartheta }_1,{\aleph }_1)$ is equicontinuous. To show this, we define ${sup}_{t\in \mathcal{J}}|{\psi }_i(t,u_1\left(t\right),u_2\left(t\right))|={\overline{\Psi }}_i<\infty ,{sup}_{t\in \mathcal{J}}\left|{\varphi }_i(t,u_1\left(t\right),u_2\left(t\right))\right|={\overline{\Phi }}_i<\infty ;i=1,2$ and for each $t_1,t_2\in \mathcal{J}$ with $t_1<t_2$ and $(u_1,u_2)\in {\mathbb{B}}_{ϵ}$. Consequently, we arrive at the following:

$$\begin{array}{c}|{\vartheta }_1(u_1,u_2)\left(t_2\right)-{\vartheta }_1(u_1,u_2)\left(t_1\right)|\hfill \\ \le {\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}|{\int }_0^{t_2}{(t_2-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t_2-w)dw\hfill \\ -{\int }_0^{t_1}{(t_1-w)}^{{\alpha }_1-1}{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t_1-w)dw|\hfill \\ +{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}|{\int }_0^{t_2}{(t_2-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t_2-w)dw\hfill \\ -{\int }_0^{t_1}{(t_1-w)}^{{\beta }_1-1}{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))K(t_1-w)dw|\hfill \\ +{\displaystyle \frac{|t_{2}K\left(t_2\right)-t_{1}K\left(t_1\right)|}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_2\right|{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}\right|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))|dw\hfill \\ +{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ +{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\},\hfill \\ \le {\displaystyle \frac{{\overline{\Psi }}_1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}\left[{\int }_0^{t_1}\right|{(t_2-w)}^{{\alpha }_1-1}-{(t_1-w)}^{{\alpha }_1-1}|K(t_1-w)dw\hfill \\ +{\int }_0^{t_1}{(t_2-w)}^{{\alpha }_1-1}|K(t_2-w)-K(t_1-w)|dw\hfill \\ +{\int }_{t_1}^{t_2}{(t_2-w)}^{{\alpha }_1-1}K(t_2-w)dw]\hfill \\ +{\displaystyle \frac{{\lambda }_1{\overline{\Phi }}_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}\left[{\int }_0^{t_1}\right|{(t_2-w)}^{{\beta }_1-1}-{(t_1-w)}^{{\beta }_1-1}|K(t_1-w)dw\hfill \\ +{\int }_0^{t_1}{(t_2-w)}^{{\beta }_1-1}|K(t_2-w)-K(t_1-w)|dw\hfill \\ +{\int }_{t_1}^{t_2}{(t_2-w)}^{{\beta }_1-1}K(t_2-w)dw]+{\displaystyle \frac{1}{|\Delta |}}(\left|(t_2-t_1)\right|K\left(t_1\right)\hfill \\ +t_2|K\left(t_2\right)-K\left(t_1\right)|\left)\right\{{\overline{\Psi }}_1\left[{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right]+{\overline{\Phi }}_1[{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\hfill \\ +{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\left]\right\}.\hfill \end{array}$$

(49)

Applying the Mean Value Theorem for the function $K(t-w)=e^{{\displaystyle \frac{\rho -1}{\rho }}(t-w)}$, with ${\sigma }_i$ lying in the interval $(t_1,t_2)$, gives us

$$\begin{array}{cc}\hfill |{\vartheta }_1(u_1,u_2)\left(t_2\right)-{\vartheta }_1(u_1,u_2)\left(t_1\right)|& \le {\displaystyle \frac{{\overline{\Psi }}_1}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\left((t_2^{{\alpha }_1}-t_1^{{\alpha }_1})+(t_2-t_1)(t_2^{{\alpha }_1}-{(t_2-t_1)}^{{\alpha }_1})\right)\hfill \\ & +{\displaystyle \frac{{\lambda }_1{\overline{\Phi }}_1}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}((t_2^{{\beta }_1}-t_1^{{\beta }_1})+(t_2-t_1)(t_2^{{\beta }_1}\hfill \\ & -{(t_2-t_1)}^{{\beta }_1})+{\displaystyle \frac{1}{|\Delta |}}\left((t_2-t_1)+t_2(t_2-t_1)\right)\{\hfill \\ & {\overline{\Psi }}_1\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right)\hfill \\ & +{\overline{\Phi }}_1\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right)\}.\hfill \end{array}$$

(50)

Analogously, it can be shown that

$$\begin{array}{cc}\hfill |{\aleph }_1(u_1,u_2)\left(t_2\right)-{\aleph }_1(u_1,u_2)\left(t_1\right)|& \le {\displaystyle \frac{{\overline{\Psi }}_2}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\left((t_2^{{\alpha }_2}-t_1^{{\alpha }_2})+(t_2-t_1)(t_2^{{\alpha }_2}-{(t_2-t_1)}^{{\alpha }_2})\right)\hfill \\ & +{\displaystyle \frac{{\lambda }_2{\overline{\Phi }}_2}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}((t_2^{{\beta }_2}-t_1^{{\beta }_2})+(t_2-t_1)(t_2^{{\beta }_2}\hfill \\ & -{(t_2-t_1)}^{{\beta }_2}\left)\right)+{\displaystyle \frac{1}{|\Delta |}}\left((t_2-t_1)+t_2(t_2-t_1)\right)\{\hfill \\ & {\overline{\Psi }}_2\left({\displaystyle \frac{\left|B_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{B_2{a}^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{B_3{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right)\hfill \\ & +{\overline{\Phi }}_2\left({\displaystyle \frac{\left|B_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_2{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right)\}\hfill \\ & +{\displaystyle \frac{1}{|\Delta |}}\left((t_2^2-t_1^2)+t_2^2(t_2-t_1)\right)\left\{{\overline{\Psi }}_2\right({\displaystyle \frac{\left|B_4\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\hfill \\ & +{\displaystyle \frac{B_5{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{B_6{a}^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}})\hfill \\ & +{\overline{\Phi }}_2\left({\displaystyle \frac{\left|B_4\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_5{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{B_6{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right)\}.\hfill \end{array}$$

(51)

Because the right-hand sides of both the inequalities (50) and (51) tend to zero as $t_1\to t_2$, independently of $(u_1,u_2)\in {\mathbb{B}}_{ϵ},$ it follows from the Arzelà–Ascoli theorem that $({\vartheta }_1,{\aleph }_1)$ is compact on ${\mathbb{B}}_{ϵ}$. Hence, all the conditions of Krasnoselskii’s fixed-point theorem are fulfilled; we conclude that the coupled system (3) and (4) has at least one solution on the interval $\mathcal{J}$. □

3.3. Existence and Uniqueness Result Derived via the Banach Fixed-Point Theorem

The uniqueness of the solutions for the coupled system (3) and (4) is established in the subsequent result. The proof relies on the Banach fixed-point theorem [50].

Theorem 3.

Assume that $\left({\mathbf{T}}_{\mathbf{1}}\right)$ and $\left({\mathbf{T}}_{\mathbf{2}}\right)$ are fulfilled. In addition, if

$$\begin{array}{c}\hfill {\Xi }_1{L}_1+{\Xi }_2{L}_2+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2<1,\end{array}$$

(52)

holds, then the coupled system (3) and (4) admits a unique solution on the interval $\mathcal{J}$ where ${\Xi }_i(i=1,\dots ,4)$ are introduced in (28).

Proof. 

Let ${sup}_{t\in \mathcal{J}}\left|{\psi }_i(t,0,0)\right|={\xi }_i$ and ${sup}_{t\in \mathcal{J}}\left|{\varphi }_i(t,0,0)\right|={\eta }_i;i=1,2$. We consider the closed, bounded, and convex ball ${\mathbb{B}}_r=\{(u_1,u_2)\in E\times E:\parallel (u_1,u_2)\parallel \le r\},$ with

$$\begin{array}{c}\hfill r\ge {\displaystyle \frac{{\Xi }_1{\xi }_1+{\Xi }_2{\xi }_2+{\Xi }_3{\eta }_1+{\Xi }_4{\eta }_2}{1-\left[{\Xi }_1{L}_1+{\Xi }_2{L}_2+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2\right]}}.\end{array}$$

(53)

To begin, we note that the continuity of the operator $\mathcal{Q}$ follows directly from assumption $\left({\mathbf{T}}_{\mathbf{1}}\right)$. Now, to show that $\mathcal{Q}\left({\mathbb{B}}_r\right)\subset {\mathbb{B}}_r$, we start with the triangle inequality and plug in assumption $\left({\mathbf{T}}_{\mathbf{2}}\right)$. This gives us

$$\begin{array}{cc}\hfill |{\psi }_i(t,u_1\left(t\right),u_2\left(t\right))|& \le |{\psi }_i(t,u_1\left(t\right),u_2\left(t\right))-{\psi }_i(t,0,0)|+|{\psi }_i(t,0,0)|\hfill \\ & \le L_i\left(\right|u_1\left(t\right)|+|u_2\left(t\right)\left|\right)+{\xi }_i\hfill \\ & \le L_i(\parallel u_1\parallel +\parallel u_2\parallel )+{\xi }_i\le L_{i}r+{\xi }_i,\hfill \end{array}$$

(54)

Similarly, we get

$$\begin{array}{c}\hfill |{\varphi }_i(t,u_1\left(t\right),u_2\left(t\right))|\le {\overline{L}}_{i}r+{\eta }_i.\end{array}$$

(55)

Then, for any $t\in \mathcal{J}$ and $(u_1,u_2)\in {\mathbb{B}}_r,$ we apply the inequality $K(\tau -w)<1$, for all $\tau >w,$ where $\tau \in \{t,a,{\theta }_1,{\theta }_2\},$ to get

$$\begin{array}{cc}\hfill \parallel {\mathcal{Q}}_1(u_1,u_2)\parallel & \le {\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}\right|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_1\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\}.\hfill \end{array}$$

(56)

We notice that all the integrals in (56) match the structure of the left-sided Riemann–Liouville fractional integral formula (5), and this allows us to apply the power function property of the Riemann–Liouville fractional integral, where $\left|{\psi }_i(w,u_1\left(w\right),u_2\left(w\right))\right|$ and $\left|{\varphi }_i(w,u_1\left(w\right),u_2\left(w\right))\right|$ are considered to be bounded as in (54) and (55). Thus, we obtain

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)}}{\int }_0^{\zeta }{(\zeta -w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\le {\displaystyle \frac{(L_{1}r+{\xi }_1){\zeta }^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}},\hfill \\ {\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)}}{\int }_0^{\tau }{(\tau -w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\le {\displaystyle \frac{(L_{2}r+{\xi }_2){\tau }^{{\alpha }_2}}{\Gamma ({\alpha }_2+1)}},\hfill \\ {\displaystyle \frac{1}{\Gamma \left({\beta }_1\right)}}{\int }_0^{\zeta }{(\zeta -w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\le {\displaystyle \frac{({\overline{L}}_{1}r+{\eta }_1){\zeta }^{{\beta }_1}}{\Gamma ({\beta }_1+1)}},\hfill \\ {\displaystyle \frac{1}{\Gamma \left({\beta }_2\right)}}{\int }_0^{\tau }{(\tau -w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\le {\displaystyle \frac{({\overline{L}}_{2}r+{\eta }_2){\tau }^{{\beta }_2}}{\Gamma ({\beta }_2+1)}},\hfill \end{array}\right.$$

(57)

where $\zeta \in \{t,a,{\theta }_2\},$ and $\tau \in \{t,a,{\theta }_1,{\theta }_2\}$. As a result, we arrive at

$$\begin{array}{cc}\hfill \parallel {\mathcal{Q}}_1(u_1,u_2)\parallel & \le \left(L_{1}r+{\xi }_1\right)\left\{{\displaystyle \frac{a^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right)\right\}\hfill \\ & +\left(L_{2}r+{\xi }_2\right)\left\{{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{\left|A_2\right|a^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right)\right\}\hfill \\ \hfill & +\left({\overline{L}}_{1}r+{\eta }_1\right)\left\{{\displaystyle \frac{{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right)\right\}\hfill \\ & +\left({\overline{L}}_{2}r+{\eta }_2\right)\left\{{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{\left|A_2\right|{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right)\right\}\hfill \\ & \le ({\Lambda }_1{L}_1+{\Lambda }_2{L}_2+{\Lambda }_3{\overline{L}}_1+{\Lambda }_4{\overline{L}}_2)r+({\Lambda }_1{\xi }_1+{\Lambda }_2{\xi }_2+{\Lambda }_3{\eta }_1+{\Lambda }_4{\eta }_2).\hfill \end{array}$$

Similarly, we can find that

$$\begin{array}{cc}\hfill \parallel {\mathcal{Q}}_2(u_1,u_2)\parallel & \le {\displaystyle \frac{1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^t{(t-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^t{(t-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|B_1\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}\right|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|B_1\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_2{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ \hfill & +{\displaystyle \frac{B_2}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_2{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_2{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +a\left[{\displaystyle \frac{\left|B_4\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}\right|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|B_4\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_5}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ \hfill & +{\displaystyle \frac{B_5{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_5{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_5{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ \hfill & +{\displaystyle \frac{B_6{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}\left|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ \hfill & +{\displaystyle \frac{B_6}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}\left|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_6{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}\left|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\hfill \\ & +{\displaystyle \frac{B_6{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}\left|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))\right|dw\left]\right\}\hfill \\ & \le ({\Lambda }_5{L}_1+{\Lambda }_6{L}_2+{\Lambda }_7{\overline{L}}_1+{\Lambda }_8{\overline{L}}_2)r+({\Lambda }_5{\xi }_1+{\Lambda }_6{\xi }_2+{\Lambda }_7{\eta }_1+{\Lambda }_8{\eta }_2).\hfill \end{array}$$

From these inequalities and (53), we obtain

$$\begin{array}{cc}\hfill \parallel \mathcal{Q}(u_1,u_2)\parallel & =\parallel {\mathcal{Q}}_1(u_1,u_2)\parallel +\parallel {\mathcal{Q}}_2(u_1,u_2)\parallel \hfill \\ \hfill \le & \left[{\Xi }_1{L}_1+{\Xi }_2{L}_2+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2\right]r+{\Xi }_1{\xi }_1+{\Xi }_2{\xi }_2+{\Xi }_3{\eta }_1+{\Xi }_4{\eta }_2\le r,\hfill \end{array}$$

which implies that $\mathcal{Q}:{\mathbb{B}}_r\to {\mathbb{B}}_r.$

We now show that the mapping $\mathcal{Q}:E\times E\to E\times E$ is a contraction. Thus, for all $(u_1,u_2),({\overline{u}}_1,{\overline{u}}_2)\in E\times E$ and any $t\in \mathcal{J},$ we find that

$$\begin{array}{cc}\hfill & \parallel {\mathcal{Q}}_1(u_1,v_1)-{\mathcal{Q}}_1(u_2,v_2)\parallel \hfill \\ & \le {\displaystyle \frac{1}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^t{(t-w)}^{{\alpha }_1-1}|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_1(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^t{(t-w)}^{{\beta }_1-1}|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_1(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{a}{|\Delta |}}\left\{{\displaystyle \frac{\left|A_1\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\beta }_2-1}\right|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ \hfill & +{\displaystyle \frac{\left|A_1\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_1}{({\theta }_1-w)}^{{\alpha }_2-1}|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^a{(a-w)}^{{\beta }_2-1}|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^a{(a-w)}^{{\alpha }_2-1}|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_1-1}|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_1(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_1-1}|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_1(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{A_3}{{\rho }^{{\alpha }_1}\Gamma \left({\alpha }_1\right)}}{\int }_0^a{(a-w)}^{{\alpha }_1-1}|{\psi }_1(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_1(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ \hfill & +{\displaystyle \frac{A_3{\lambda }_1}{{\rho }^{{\beta }_1}\Gamma \left({\beta }_1\right)}}{\int }_0^a{(a-w)}^{{\beta }_1-1}|{\varphi }_1(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_1(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1}{{\rho }^{{\alpha }_2}\Gamma \left({\alpha }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\alpha }_2-1}|{\psi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\psi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\hfill \\ & +{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2}{{\rho }^{{\beta }_2}\Gamma \left({\beta }_2\right)}}{\int }_0^{{\theta }_2}{({\theta }_2-w)}^{{\beta }_2-1}|{\varphi }_2(w,u_1\left(w\right),u_2\left(w\right))-{\varphi }_2(w,{\overline{u}}_1\left(w\right),{\overline{u}}_2\left(w\right))|dw\}\hfill \\ \hfill & \le \left(L_1(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel )\right)\left[{\displaystyle \frac{a^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\theta }_2^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}+{\displaystyle \frac{A_3{a}^{{\alpha }_1}}{{\rho }^{{\alpha }_1}\Gamma ({\alpha }_1+1)}}\right)\right]\hfill \\ \hfill & +\left(L_2(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel )\right)\left[{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\theta }_1^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{\left|A_2\right|a^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\theta }_2^{{\alpha }_2}}{{\rho }^{{\alpha }_2}\Gamma ({\alpha }_2+1)}}\right)\right]\hfill \\ & +\left({\overline{L}}_1(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel )\right)\left[{\displaystyle \frac{{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_2\right|{\gamma }_2{\lambda }_1{\theta }_2^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}+{\displaystyle \frac{A_3{\lambda }_1{a}^{{\beta }_1}}{{\rho }^{{\beta }_1}\Gamma ({\beta }_1+1)}}\right)\right]\hfill \\ & +\left({\overline{L}}_2(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel )\right)\left[{\displaystyle \frac{a}{|\Delta |}}\left({\displaystyle \frac{\left|A_1\right|{\lambda }_2{\theta }_1^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{\left|A_2\right|{\lambda }_2{a}^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}+{\displaystyle \frac{A_3{\gamma }_1{\lambda }_2{\theta }_2^{{\beta }_2}}{{\rho }^{{\beta }_2}\Gamma ({\beta }_2+1)}}\right)\right].\hfill \end{array}$$

Hence, using (28) gives

$$\parallel {\mathcal{Q}}_1(u_1,{\overline{u}}_1)-{\mathcal{Q}}_1(u_2,{\overline{u}}_2)\parallel \le ({\Lambda }_1{L}_1+{\Lambda }_2{L}_2+{\Lambda }_3{\overline{L}}_1+{\Lambda }_4{\overline{L}}_2)(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel ).$$

(58)

Likewise, we can find that

$$\parallel {\mathcal{Q}}_2(u_1,{\overline{u}}_1)-{\mathcal{Q}}_2(u_2,{\overline{u}}_2)\parallel \le ({\Lambda }_5{L}_1+{\Lambda }_6{L}_2+{\Lambda }_7{\overline{L}}_1+{\Lambda }_8{\overline{L}}_2)(\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel ).$$

(59)

Combining the inequalities in (58) and (59) and using (28), we find that

$$\begin{array}{cc}\hfill \parallel \mathcal{Q}(u_1,{\overline{u}}_1)-\mathcal{Q}(u_2,{\overline{u}}_2)\parallel & \le \left[{\Xi }_1{L}_1+{\Xi }_2{L}_2+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2\right](\parallel u_1-{\overline{u}}_1\parallel +\parallel u_2-{\overline{u}}_2\parallel ).\hfill \end{array}$$

(60)

In view of (52), this shows that $\mathcal{Q}$ is a contraction operator. Applying Banach’s fixed-point theorem, we deduce the existence of a unique fixed point for the operator. Consequently, the coupled system (3) and (4) has a unique solution on the interval $\mathcal{J}.$ □

4. Applications

In this section, some cases are discussed to demonstrate the correctness of the results of Theorems 2 and 3 by examining the following integro-differential system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C{\mathbb{D}}^{\alpha 1,\rho }\left[u_1\left(t\right)+\frac{1}{5}{I}^{{\beta }_1,\rho }{\varphi }_1(t,u_1\left(t\right),u_2\left(t\right))\right]={\psi }_1(t,u_1\left(t\right),u_2\left(t\right)),t\in [0,1],\hfill \\ {}^C{\mathbb{D}}^{\alpha 2,\rho }\left[u_2\left(t\right)+\frac{1}{50}{I}^{{\beta }_2,\rho }{\varphi }_2(t,u_1\left(t\right),u_2\left(t\right))\right]={\psi }_2(t,u_1\left(t\right),u_2\left(t\right)),\hfill \\ u_1\left(0\right)=0,u_1\left(1\right)=\frac{1}{6}{u}_2\left(\frac{1}{10}\right),\hfill \\ u_2\left(0\right)=0,u_2\left(\frac{1}{20}\right)=0,u_2\left(1\right)=\frac{1}{3}{u}_1\left(\frac{1}{10}\right),0<\frac{1}{20}<\frac{1}{10}<1.\hfill \end{array}\right.\end{array}$$

(61)

Here, ${\lambda }_1=1/5,{\lambda }_2=1/50,{\gamma }_1=1/6,{\gamma }_2=1/3,{\theta }_1=1/20,{\theta }_2=1/10,$ and $t\in [0,1].$ In this application, we take

$$\begin{array}{cc}\hfill {\psi }_1(t,u_1\left(t\right),u_2\left(t\right))& ={\displaystyle \frac{6}{97+t^5}}\left(cos\left(u_1\right)+sin\left(\right|u_2\left|\right)+5t\right),\hfill \\ \hfill {\psi }_2(t,u_1\left(t\right),u_2\left(t\right))& ={\displaystyle \frac{1}{21+t^6}}\left(1+{\displaystyle \frac{|u_1|}{\sqrt{1+u_1^2}}}+{\displaystyle \frac{|u_2|}{\sqrt{1+u_2^2}}}\right),\hfill \\ \hfill {\varphi }_1(t,u_1\left(t\right),u_2\left(t\right))& ={\displaystyle \frac{1}{41}}{e}^{-t}\left({\displaystyle \frac{1}{1+|u_1|}}+{\displaystyle \frac{1}{1+|u_2|}}\right),\hfill \\ \hfill {\varphi }_2(t,u_1\left(t\right),u_2\left(t\right))& ={\displaystyle \frac{1}{11}}\left({\displaystyle \frac{arctan\left(t\right)}{4+t^2}}+{\displaystyle \frac{1}{5}}(cos\left(u_1\right)+sin\left(\right|u_2\left|\right)\right).\hfill \end{array}$$

Obviously, condition $\left({\mathbf{T}}_{\mathbf{1}}\right)$ is satisfied because the functions ${\psi }_i$ and ${\varphi }_i;(i=1,2)$ are composed of continuous functions everywhere in $[0,1]\times \mathbb{R}\times \mathbb{R}$. In addition, for each $t\in [0,1]$, we obtain

$$\left\{\begin{array}{c}|{\psi }_1(t,u_1,u_2)-{\psi }_1(t,{\overline{u}}_1,{\overline{u}}_2)|⩽{\displaystyle \frac{6}{97}}\left(\right|u_1-{\overline{u}}_1|+|u_2-{\overline{u}}_2\left|\right),\hfill \\ |{\psi }_2(t,u_1,u_2)-{\psi }_2(t,{\overline{u}}_1,{\overline{u}}_2)|⩽{\displaystyle \frac{1}{21}}\left(\right|u_1-{\overline{u}}_1|+|u_2-{\overline{u}}_2\left|\right),\hfill \\ |{\varphi }_1(t,u_1,u_2)-{\varphi }_1(t,{\overline{u}}_1,{\overline{u}}_2)|⩽{\displaystyle \frac{1}{41}}\left(\right|u_1-{\overline{u}}_1|+|u_2-{\overline{u}}_2\left|\right),\hfill \\ |{\varphi }_2(t,u_1,u_2)-{\varphi }_2(t,{\overline{u}}_1,{\overline{u}}_2)|⩽{\displaystyle \frac{1}{55}}\left(\right|u_1-{\overline{u}}_1|+|u_2-{\overline{u}}_2\left|\right),\hfill \end{array}\right.$$

which confirms the verification of the condition $\left({\mathbf{T}}_{\mathbf{2}}\right)$. Moreover, the condition $\left({\mathbf{T}}_{\mathbf{3}}\right)$ is satisfied as

$$\begin{array}{cc}\hfill & |{\psi }_1(t,u_1\left(t\right),u_2\left(t\right))|⩽{\displaystyle \frac{6(2+5t)}{97+t^5}}={\tau }_1\left(t\right),\left|{\psi }_2(t,u_1\left(t\right),u_2\left(t\right))\right|⩽{\displaystyle \frac{3}{21+t^6}}={\tau }_2\left(t\right),\hfill \\ \hfill & |{\varphi }_1(t,u_1\left(t\right),u_2\left(t\right))|⩽{\displaystyle \frac{2e^{-t}}{41}}={\varkappa }_1\left(t\right),\left|{\varphi }_2(t,u_1\left(t\right),u_2\left(t\right))\right|⩽{\displaystyle \frac{1}{11}}\left({\displaystyle \frac{arctan\left(t\right)}{4+t^2}}+{\displaystyle \frac{2}{5}}\right)={\varkappa }_2\left(t\right).\hfill \end{array}$$

Now, we will prove the results of Theorems 2 and 3 by considering the following cases:

Case 1. If we allow $\rho$ to change while fixing the values of ${\alpha }_i$ and ${\beta }_i;i=1,2$, both existence criteria (33) and uniqueness criteria (52) are satisfied as presented in Table 1.

Table 1. Numerical results under the assumed values for $\rho$.

${\mathit{\alpha }}_1=3/2,{\mathit{\alpha }}_2=5/2,{\mathit{\beta }}_1=1/2,$${\mathit{\beta }}_2=1/3$	$\mathit{\rho }=\frac{7}{10}$	$\mathit{\rho }=\frac{4}{5}$	$\mathit{\rho }=\frac{23}{25}$
${\Lambda }_1$	$3.3426539281$	$2.4354598895$	$1.7905613065$
${\Lambda }_2$	$0.0022681397$	$0.0012239156$	$0.0006695735$
${\Lambda }_3$	$0.7020080029$	$0.5845441384$	$0.4942180187$
${\Lambda }_4$	$0.0075863003$	$0.0060507398$	$0.0048940232$
${\Lambda }_5$	$0.1298115101$	$0.0780126976$	$0.0478849541$
${\Lambda }_6$	$1.9970113631$	$1.2842248266$	$0.8262981590$
${\Lambda }_7$	$0.0706791011$	$0.0526950029$	$0.0401289726$
${\Lambda }_8$	$0.4676977431$	$0.4372626205$	$0.4095177064$
${\Xi }_1{L}_1+{\Xi }_2{L}_2$$+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2$	$0.3374830174$	$0.2402870370$	$0.1736653534$
${\Lambda }_2{L}_2+{\Lambda }_4{\overline{L}}_2+{\Lambda }_5{L}_1+{\Lambda }_7{\overline{L}}_1$	$0.0099993978$	$0.0062790668$	$0.0040615780$

Case 2. We examine the results when ${\alpha }_1$ and ${\alpha }_2$ change. We consider ${\alpha }_1\in \{5/4,7/5,9/5\},{\alpha }_2\in \{11/5,12/5,14/5\}$, and $\rho \in \{3/4,22/25,19/20\}$, with fixed values ${\beta }_1=1/2,$ and ${\beta }_2=1/3$. Thus, one can see from Table 2 that both criteria (33) and (52) are ascending to be less than 1.

Table 2. Numerical results under the assumed values for ${\alpha }_1,{\alpha }_2$, and $\rho$.

${\mathit{\beta }}_1=1/2,$ and ${\mathit{\beta }}_2=1/3$	${\mathit{\alpha }}_1=\frac{5}{4}$, ${\mathit{\alpha }}_2=\frac{11}{5}$, $\mathit{\rho }=\frac{3}{4}$.	${\mathit{\alpha }}_1=\frac{7}{5}$, ${\mathit{\alpha }}_2=\frac{12}{5}$, $\mathit{\rho }=\frac{22}{25}$.	${\mathit{\alpha }}_1=\frac{9}{5}$, ${\mathit{\alpha }}_2=\frac{14}{5}$, $\mathit{\rho }=\frac{19}{20}$.
${\Lambda }_1$	$3.0893718595$	$2.0815056650$	$1.3473741555$
${\Lambda }_2$	$0.0029250728$	$0.0009932381$	$0.0003258431$
${\Lambda }_3$	$0.6366602647$	$0.5201144298$	$0.4769090827$
${\Lambda }_4$	$0.0067289808$	$0.0052225214$	$0.0046762295$
${\Lambda }_5$	$0.1243392014$	$0.0621503521$	$0.0306632887$
${\Lambda }_6$	$2.0209325297$	$1.0479673992$	$0.5348144762$
${\Lambda }_7$	$0.0604601270$	$0.0436086309$	$0.0378619010$
${\Lambda }_8$	$0.4514200659$	$0.4179552478$	$0.4036274104$
${\Xi }_1{L}_1+{\Xi }_2{L}_2$$+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2$	$0.3204933418$	$0.2039912702$	$0.1307013943$
${\Lambda }_2{L}_2+{\Lambda }_4{\overline{L}}_2+{\Lambda }_5{L}_1+{\Lambda }_7{\overline{L}}_1$	$0.0094273561$	$0.0050502288$	$0.0029206980$

Case 3. Finally, in Table 3, we show that the existence criteria (33) and the uniqueness criteria (52) are satisfied for ${\beta }_1\in \{8/5,27/10,7/2\},{\beta }_2\in \{11/10,16/5,19/5\}$, and $\rho \in \{3/4,22/25,19/20\}$, while keeping ${\alpha }_1=3/2,{\alpha }_2=5/2$.

Table 3. Numerical results under the assumed values for ${\beta }_1,{\beta }_2$ and $\rho$.

${\mathit{\alpha }}_1=3/2,{\mathit{\alpha }}_2=5/2$	${\mathit{\beta }}_1=\frac{8}{5}$, ${\mathit{\beta }}_2=\frac{11}{10}$, $\mathit{\rho }=\frac{3}{4}$.	${\mathit{\beta }}_1=\frac{27}{10}$, ${\mathit{\beta }}_2=\frac{16}{5}$, $\mathit{\rho }=\frac{22}{25}$.	${\mathit{\beta }}_1=\frac{7}{2}$, ${\mathit{\beta }}_2=\frac{19}{5}$, $\mathit{\rho }=\frac{19}{20}$.
${\Lambda }_1$	$2.8294195978$	$1.9700321519$	$1.6732904822$
${\Lambda }_2$	$0.0016402446$	$0.0008076970$	$0.0005861524$
${\Lambda }_3$	$0.5415452979$	$0.1464043958$	$0.0423783399$
${\Lambda }_4$	$0.0009656002$	$0.0000049821$	$0.0000013642$
${\Lambda }_5$	$0.0992206060$	$0.0556713920$	$0.0430569282$
${\Lambda }_6$	$1.5853379664$	$0.9487417795$	$0.7486691860$
${\Lambda }_7$	$0.0182493156$	$0.0032900533$	$0.0008403280$
${\Lambda }_8$	$0.1082924054$	$0.0088100423$	$0.0029505164$
${\Xi }_1{L}_1+{\Xi }_2{L}_2$$+{\Xi }_3{\overline{L}}_1+{\Xi }_4{\overline{L}}_2$	$0.2723634297$	$0.1743292492$	$0.1429524292$
${\Lambda }_2{L}_2+{\Lambda }_4{\overline{L}}_2+{\Lambda }_5{L}_1+{\Lambda }_7{\overline{L}}_1$	$0.0066781256$	$0.0035623888$	$0.0027117478$

From the previous cases, the results show the extent to which the conditions presented in Theorems 2 and 3 are met in the different cases studied in these applications.

5. Conclusions

This study provides an in-depth analysis of a novel coupled system of integro-differential equations supplemented with nonlocal four-point boundary conditions. The generalized proportional Caputo derivatives are used according to their importance in both theoretical and applied aspects. The first existence result is proved by using Krasnoselskii’s fixed-point theorem. The Banach contraction mapping principle is used to emphasize the existence and uniqueness of the results. The validity of the results has been shown firstly when the parameter $\rho$ is changed while the fractional orders ${\alpha }_i,{\beta }_i;i=1,2$ are fixed, and secondly, when ${\alpha }_i,{\beta }_i$ are changed.

We hope that our study has succeeded in enriching the capabilities of fractional-order systems by using the generalized proportional Caputo derivatives. In future, we look forward to improving our study with the aid of different types of generalized operators.