Leray–Schauder Alternative for the Existence of Solutions of a Modified Coupled System of Caputo Fractional Differential Equations with Two Point’s Integral Boundary Conditions

Abstract

In this paper, a coupled system of differential equations involving fractional order with integral boundary conditions is discussed. In the problem at hand, three main aspects that are existence, uniqueness, and stability have been investigated. Firstly, the contraction mapping principle is used to discuss the uniqueness of solutions for the proposed fractional system, and secondly, the existence of solutions for the problem is investigated based on Leray–Schauder’s alternative. Thirdly, the stability of the presented coupled system is discussed based on the Hyers–Ulam stability method. Finally, some examples have been given to confirm and illustrate the conclusion. The comparison between the current symmetrical results and the existing literature is deemed satisfactory. It was found that the presented fractional coupled system with two with integral boundary conditions is existent, unique, and stable.

1. Introduction

In the last decades, many researchers have discussed the use of fractional differential equations for modeling various real-life problems. These equations (Eq) are commonly found in engineering and scientific fields such as physics, chemistry, aerodynamics, and biophysics, among others [1,2,3,4,5]. They are also considered to be a more effective tool for describing the hereditary properties of materials and processes compared to traditional integer-order differential equations. As a result, fractional-order models are becoming increasingly popular for their ability to account for effects that are not captured by classical models [6,7,8,9,10,11,12,13,14,15,16,17,18]. Additionally, the study of coupled systems of fractional order is also of great significance as they are often encountered in various applications. Many mathematicians have focused on systems of fractional differential equations with different types of boundary conditions. Research in this area typically examines the existence, uniqueness, and stability for the solutions of these fractional coupled systems. For example, Ahmad et al. [19] examined nonlinear differential equations with Caputo fractional order involving coupled Riemann–Liouville fractional integrals boundary conditions, while Baitiche et al. [20] examined the uniqueness and existence of solutions to certain nonlinear fractional differential equations involving the ψ-Caputo fractional derivative with multipoint boundary conditions. Moreover, the existence and uniqueness of solution for coupled system of Caputo fractional differential equations is discussed by many researchers [21,22,23,24,25,26,27,28,29,30,31,32,33].

The boundary value problem (BVP) of first-order fractional differential equations with Riemann–Liouville integral boundary conditions of different order given by

$$\left\{\begin{array}{c}{}_{ }{}^{c}D{}^{\alpha }u\left(t\right)=f\left(t,u\left(t\right),v\left(t\right)\right), t\in \left. [0,T\right], 1<\alpha \le 2,\\ {}_{ }{}^{c}D{}^{\beta }v\left(t\right)=g\left(t,u\left(t\right),v\left(t\right)\right), t\in \left. [0,T\right], 1<\beta \le 2,\end{array}\right.$$

(1)

with integral boundary conditions is given in the following form:

$$\left\{\begin{array}{c}{{\displaystyle \int }}_0^{T}u\left(s\right)ds=\xi v\left(\psi \right),{{\displaystyle \int }}_0^{T}v\left(s\right)ds=\gamma u\left(\phi \right), \\ u^{\prime }\left(0\right)=v^{\prime }\left(0\right)=0, \psi ,\phi \in \left. [0,T\right], \end{array}\right.$$

(2)

where the Caputo fractional derivatives of order k are denoted ${ }^c{D}^k$, $k=\alpha ,\beta ,$ and $f,g\in C\left(\left. [0,T\right],{ℝ}^2,ℝ\right)$ where $\xi ,\gamma$ are real constants.

In summary, discussing the uniqueness, existence, and stability of solutions of fractional systems of differential equations has several advantages, including verification of solutions, understanding the behavior of systems, design of control strategies, and applications in engineering. Therefore, in this paper, a couple of systems of fractional differential equations involving Caputo integral boundary conditions of different fractional order are analyzed and examined in detail. In particular, the uniqueness, existence, and stability of the presented coupled system in Equations (1) and (2) are discussed and illustrated based on fixed point theorem, Leray–Schauder’s alternative and Hyers–Ulam stability respectively.

2. Preliminaries

To begin with, we revisit the definitions of fractional derivative and integral as outlined in references [34,35,36].

Definition 1. 

The Riemann–Liouville fractional integral of order $\theta$ for a continuous function $z$ is given by

$$I^{\theta }z\left(t\right)=\frac{1}{\Gamma \left(\theta \right)}{{\displaystyle \int }}_0^t\frac{z\left(s\right)}{{\left(t-s\right)}^{1-\theta }}ds, \theta >0$$

Definition 2. 

The Caputo fractional derivatives of order $\theta$ for $\left(z-1\right)-$ times absolutely continuous function $h: [0,\left.\infty \right)\to ℝ$ is defined as

$${ }^c{D}^{\theta }h\left(s\right)=\frac{1}{\Gamma \left(z-\theta \right)}{{\displaystyle \int }}_0^s{\left(s-t\right)}^{z-\theta -1}{h}^{\left(z\right)}\left(t\right)dt, z-1<\theta <z, z=\left. [\theta \right]+1,$$

where $\left. [\theta \right]$ is the integer part of real number $\theta$.

To define the solution for the problems (1) and (2), we prove the following auxiliary lemma.

Lemma 1. 

Let $x,y\in \left. [C(\left. [0,T\right],ℝ\right]$ then the unique solution for the problem

$$\left\{\begin{array}{c}{}_{ }{}^{c}D{}^{\alpha }u\left(t\right)=x\left(t\right), t\in \left. [0,T\right], 1<\alpha \le 2, \\ {}_{ }{}^{c}D{}^{\beta }v\left(t\right)=y\left(t\right), t\in \left. [0,T\right], 1<\beta \le 2, \\ {{\displaystyle \int }}_0^{T}u\left(s\right)ds=\xi v\left(\psi \right),{{\displaystyle \int }}_0^{T}v\left(s\right)ds=\gamma u\left(\phi \right), \\ u^{\prime }\left(0\right)=v^{\prime }\left(0\right)=0, \psi ,\phi \in \left. [0,T\right], \end{array}\right.$$

(3)

is

$$\begin{gathered} u\left(t\right)=\frac{1}{\mathsf{\Theta }}\left(\xi T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds-T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds+\xi \gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds \\ -\xi {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds\right)+{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds, \end{gathered}$$

(4)

and

$$\begin{gathered} v\left(t\right)=\frac{1}{\mathsf{\Theta }}\left(\xi \gamma {{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds-\gamma {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds+T\gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds \\ -T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds\right)+{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds, \end{gathered}$$

(5)

where $\mathsf{\Theta }=T^2-\xi \gamma \ne 0$.

Proof. 

The general solutions for the coupled fractional differential equations in Equation (3) are referred to as [6].

$$u\left(t\right)=a_{0}t+a_1+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{{\displaystyle \int }}_0^t{\left(t-s\right)}^{\alpha -1}x\left(s\right)ds,$$

(6)

$$v\left(t\right)=b_{0}t+b_1+\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{{\displaystyle \int }}_0^t{\left(t-s\right)}^{\beta -1}y\left(s\right)ds,$$

(7)

where $a_0,a_1,b_0,b_1$ are arbitrary constants.

Here

$$u^{\prime }\left(t\right)=a_0+\frac{1}{\mathsf{\Gamma }\left(\alpha -1\right)}{{\displaystyle \int }}_0^t{\left(t-s\right)}^{\alpha -2}x\left(s\right)ds,$$

$$v^{\prime }\left(t\right)=b_0+\frac{1}{\mathsf{\Gamma }\left(\beta -1\right)}{{\displaystyle \int }}_0^t{\left(t-s\right)}^{\beta -2}y\left(s\right)ds.$$

Now, the condition $u^{\prime }\left(0\right)=v^{\prime }\left(0\right)=0$ is applied to get the values $a_0=b_0=0$.

In view of the conditions:

$${{\displaystyle \int }}_0^{T}u\left(s\right)ds=\xi v\left(\psi \right),{{\displaystyle \int }}_0^{T}v\left(s\right)ds=\gamma u\left(\phi \right),$$

we get:

$$a_{1}T+{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds=\xi b_1+\xi {{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds,$$

and

$$b_{1}T+{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds=a_1\gamma +\gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds,$$

so

$$a_1=\frac{1}{T}\left(\xi b_1+\xi {{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds-{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds\right),$$

$$b_1=\frac{1}{T}\left(a_1\gamma +\gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds-{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds\right).$$

Then, after replacing the value of $a_1$ into $b_1$, we get the values of these constants as follows:

$$\begin{gathered} b_1=\frac{1}{T^2-\xi \gamma }\left(\xi \gamma {{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds-\gamma {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds+T\gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds \\ -T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds\right), \end{gathered}$$

$$\begin{gathered} b_1=\frac{1}{\mathsf{\Theta }}\left(\xi \gamma {{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds-\gamma {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds+T\gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds \\ -T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds\right), \end{gathered}$$

and

$$\begin{gathered} a_1=\frac{1}{\mathsf{\Theta }}\left(\xi T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(s\right)ds-T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(\eta \right)d\eta ds+\xi \gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}x\left(s\right)ds \\ -\xi {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}y\left(\eta \right)d\eta ds\right). \end{gathered}$$

By plugging in the values of $a_0,a_1,b_0,b_1$ into Equations (6) and (7), we obtain Equations (4) and (5). Additionally, by performing calculations, we can confirm that the reverse is also true. This concludes the proof. □

3. Existence Results

For the sake of convenience, we set

$${\mathsf{\Delta }}_1=\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{T^{\alpha +2}}{\mathsf{\Gamma }\left(\alpha +2\right)}+\frac{\left|\xi \gamma \right|{\phi }^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right)+\frac{T^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)},$$

$${\mathsf{\Delta }}_2=\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\xi \right|T{\psi }^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}+\frac{\left|\xi \right|T^{\beta +1}}{\mathsf{\Gamma }\left(\beta +2\right)}\right),$$

$${\mathsf{\Delta }}_3=\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\gamma \right|T^{\alpha +1}}{\mathsf{\Gamma }\left(\alpha +2\right)}+\frac{T\left|\gamma \right|{\phi }^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right),$$

(8)

$${\mathsf{\Delta }}_4=\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\xi \gamma \right|{\psi }^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}+\frac{T^{\beta +2}}{\mathsf{\Gamma }\left(\beta +2\right)}\right)+\frac{T^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)},$$

and

$${\mathsf{\Delta }}_0=min\left\{1-\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_1-\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_1,1-\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_2-\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_2\right\}.$$

(9)

The space is considered as follows:

$$S=\left\{u\left(t\right), u\left(t\right)\in C(\left. [0,T\right]\right\},$$

$$V=\left\{v\left(t\right), v\left(t\right)\in C(\left. [0,T\right]\right\}.$$

Endowed with norm $\Vert u\Vert =\begin{array}{c}sup\\ 0\le t\le T\end{array}\left|u\left(t\right)\right|$ and $\Vert v\Vert =\begin{array}{c}sup\\ 0\le t\le T\end{array}\left|v\left(t\right)\right|$, respectively.

It is clear that both $\left(S, \Vert .\Vert \right)$ and $\left(V, \Vert .\Vert \right)$ are Banach Spaces.

Consequently, the product space $\left(S\times V,\Vert \left(u,v\right)\Vert \right)$ is a Banach Space as well (endowed with $\Vert \left(u,v\right)\Vert =\Vert u\Vert +\Vert v\Vert$).

In view of Lemma 1, we define the operator $M:S\times V\to S\times V$ as:

$$M\left(u,v\right)\left(t\right)=\left(M_1\left(u,v\right)\left(t\right),M_2\left(u,v\right)\left(t\right)\right),$$

where

$$\begin{gathered} \begin{array}{c}{M}_1\left(u,v\right)\left(t\right)=\frac{1}{\mathsf{\Theta }}\left(\xi T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}g\left(s,u\left(s\right),v\left(s\right)\right)ds-T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}f\left(\eta ,u\left(\eta \right),v\left(\eta \right)\right)d\eta ds \\ +\xi \gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),v\left(s\right)\right)ds-\xi {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}g\left(\eta ,u\left(\eta \right),v\left(\eta \right)\right)d\eta ds\right)\\ +{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),v\left(s\right)\right)ds,\end{array} \end{gathered}$$

(10)

and

$$\begin{gathered} \begin{array}{c}{M}_2\left(u,v\right)\left(t\right)=\frac{1}{\mathsf{\Theta }}\left(\xi \gamma {{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}g\left(s,u\left(s\right),v\left(s\right)\right)ds-\gamma {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}f\left(\eta ,u\left(\eta \right),v\left(\eta \right)\right)d\eta ds \\ +T\gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}f\left(s,u\left(s\right),v\left(s\right)\right)ds-T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}g\left(\eta ,u\left(\eta \right),v\left(\eta \right)\right)d\eta ds\right)\\ +{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}g\left(s,u\left(s\right),v\left(s\right)\right)ds.\end{array} \end{gathered}$$

(11)

In the first result, the existence of the solutions for the system (1–2) is based on Leray–Schauder’s alternative.

Lemma 2 

(Leray-Schauder alternative [7], p. 4). Let $G:E\to E$ be a completely continuous operator (i.e., a map restricted to any bounded set in $E$ is compact). Let $E\left(G\right)=\left\{u\in E:u=\lambda G\left(u\right) for some 0<\lambda <1\right\}$. Then either the set $E\left(G\right)$ is unbounded or $G$ has at least one fixed point).

Theorem 1. 

Assume $f,g:C\left(\left. [0,T\right],{ℝ}^2\right)\to ℝ$ are continuous function and there exist $m_1,m_2,n_1,n_2\ge 0$ where $m_1,m_2,n_1,n_2$ are real constants and $m_0,n_0>0$ such that $\forall u_i,v_i\in ℝ, \left(i=1,2\right)$, we have

$$\left|f\left(t,u_1,u_2\right)\right|\le m_0+m_1\left|u_1\right|+m_2\left|u_2\right|,$$

$$\left|g\left(t,u_1,u_2\right)\right|\le n_0+n_1\left|u_1\right|+n_2\left|u_2\right|.$$

If

$$\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_1+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_1<1,$$

$$\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_2+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_2<1,$$

where ${\Delta }_i,i=1,2,3,4$ are given by $\left(8\right)$, then the system (1–2) has at least one solution.

Proof. 

The proof will be divided into two steps.

Step 1: show that $M:S\times V\to S\times V$ is completely continuous. The continuity of the operator $M$ holds by the continuity of the functions $f,g.$

Let $Q\subseteq S\times V$ be a bounded. Then, there exists positive constants ${\varphi }_1,{\varphi }_2$ such that

$$\left|f\left(t,u\left(t\right),v\left(t\right)\right)\right|\le {\varphi }_1, \left|g\left(t,u\left(t\right),v\left(t\right)\right)\right|\le {\varphi }_2, \forall t\in \left. [0,T\right].$$

Then, $\forall \left(u,v\right)\in Q,$ we have

$$\left|M_1\left(u,v\right)\left(t\right)\right|\le {\mathsf{\Delta }}_1{\varphi }_1+{\mathsf{\Delta }}_2{\varphi }_2,$$

which implies that

$$\Vert M_1\left(u,v\right)\Vert \le {\mathsf{\Delta }}_1{\varphi }_1+{\mathsf{\Delta }}_2{\varphi }_2.$$

Similarly, we get

$$\Vert M_2\left(u,v\right)\Vert \le {\mathsf{\Delta }}_3{\varphi }_1+{\mathsf{\Delta }}_4{\varphi }_2.$$

Thus, from the above inequalities, it follows that the operator $M$ is uniformly bounded, since

$$\left|\right|M\left(u,v\right)\left|\right|\le \left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right){\varphi }_1+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right){\varphi }_2.$$

Further, we show that the operator $M$ is equicontinuous. Let $t_1,t_2\in \left. [0,T\right]$ with $t_1<t_2.$ This yields

$$\left|M_1\left(u,v\right)\left(t_2\right)-M_1\left(u,v\right)\left(t_1\right)\right|\le \frac{{\varphi }_1}{\mathsf{\Gamma }\left(\alpha \right)}\left({{\displaystyle \int }}_0^{t_2}{\left|t_2-s\right|}^{\alpha -1}ds+{{\displaystyle \int }}_0^{t_1}{\left|t_1-s\right|}^{\alpha -1}ds\right),$$

$$\le \frac{{\varphi }_1}{\mathsf{\Gamma }\left(\alpha \right)}\left({{\displaystyle \int }}_0^{t_1}\left|{\left(t_2-s\right)}^{\alpha -1}-{\left(t_1-s\right)}^{\alpha -1}\right|ds+{{\displaystyle \int }}_{t_1}^{t_2}{\left|t_2-s\right|}^{\alpha -1}ds\right).$$

We can obtain

$$\left|M_1\left(u,v\right)\left(t_2\right)-M_1\left(u,v\right)\left(t_1\right)\right|\le \frac{{\varphi }_1}{\mathsf{\Gamma }\left(\alpha +1\right)}\left. [\left|{\left(t_2-t_1\right)}^{\alpha }-t_2{}^{\alpha }\right|+t_1{}^{\alpha }+{\left|t_2-t_1\right|}^{\alpha }\right].$$

Hence, we have $\left|\right|M_1\left(u,v\right)\left(t_2\right)-M_1\left(u,v\right)\left(t_1\right)\left|\right|\to 0$ independent of $u$ and $v$ as $t_2\to t_1.$ Also, we can obtain

$$\left|M_2\left(u,v\right)\left(t_2\right)-M_2\left(u,v\right)\left(t_1\right)\right|\le \frac{{\varphi }_2}{\mathsf{\Gamma }\left(\beta +1\right)}\left. [\left|{\left(t_2-t_1\right)}^{\beta }-t_2{}^{\beta }\right|+t_1{}^{\beta }+{\left|t_2-t_1\right|}^{\beta }\right],$$

which implies that $\Vert M_2\left(u,v\right)\left(t_2\right)-M_2\left(u,v\right)\left(t_1\right)\Vert \to 0$ independent of $u$ and $v$ as $t_2\to t_1.$

Therefore, the operator $M\left(u,v\right)$ is equicontinuous, and thus the operator $M\left(u,v\right)$ is completely continuous.

Step 2: (Boundedness of operator)

Finally, show that $G=\left\{\left(u,v\right)\in S\times V:\left(u,v\right)=qM\left(u,v\right),q\in \left. [0,1\right]\right\}$ is bounded. Let $\left(u,v\right)\in ℝ,$ with $\left(u,v\right)=qM\left(u,v\right)$ for any $t\in \left. [0,T\right]$, we have

$$u\left(t\right)=q{M}_1\left(u,v\right)\left(t\right), v\left(t\right)=q{M}_2\left(u,v\right)\left(t\right).$$

Then

$$\left|u\left(t\right)\right|\le {\mathsf{\Delta }}_1\left(m_0+m_1\left|u\right|+m_2\left|v\right|\right)+{\mathsf{\Delta }}_2\left(n_0+n_1\left|u\right|+n_2\left|v\right|\right),$$

and

$$\left|v\left(t\right)\right|\le {\mathsf{\Delta }}_3\left(m_0+m_1\left|u\right|+m_2\left|v\right|\right)+{\mathsf{\Delta }}_4\left(n_0+n_1\left|u\right|+n_2\left|v\right|\right).$$

So, we get

$$\Vert u\Vert \le {\mathsf{\Delta }}_1\left(m_0+m_1\Vert u\Vert +m_2\Vert v\Vert \right)+{\mathsf{\Delta }}_2\left(n_0+n_1\Vert u\Vert +n_2\Vert v\Vert \right),$$

and

$$\Vert v\Vert \le {\mathsf{\Delta }}_3\left(m_0+m_1\Vert u\Vert +m_2\Vert v\Vert \right)+{\mathsf{\Delta }}_4\left(n_0+n_1\Vert u\Vert +n_2\Vert v\Vert \right),$$

which imply that

$$\begin{gathered} \Vert u\Vert +\Vert v\Vert \le \left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_0+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_0+\left(\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_1+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_1\right)\Vert u\Vert \\ +\left(\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_2+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_2\right)\Vert v\Vert . \end{gathered}$$

Therefore,

$$\Vert \left(u,v\right)\Vert \le \frac{\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)m_0+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)n_0}{{\mathsf{\Delta }}_0},$$

where ${\mathsf{\Delta }}_0$ is given by $\left(9\right)$,which proves that $G$ is bounded. By Leray–Schauder theorem, the operator $M$ has at least one fixed point. Therefore, system (1–2) has at least one solution on $\left. [0,T\right]$. The proof is complete. □

In the second result, the existence and the uniqueness are established for the solutions of the system (1–2) by utilizing Banach’s contraction mapping principle.

Theorem 2. 

Assume $f,g:C\left(\left. [0,T\right],{ℝ}^2\right)\to ℝ$ are jointly continuous functions and there exist constants $\lambda ,\mu \in ℝ$, such that $\forall u_1,u_2,v_1,v_2\in ℝ,\forall t\in \left. [0,T\right]$, we have

$$\left|f\left(t,u_1,u_2\right)-f\left(t,v_1,v_2\right)\right|\le \lambda \left(\left|u_2-u_1\right|+\left|v_2-v_1\right|\right),$$

$$\left|g\left(t,u_1,u_2\right)-f\left(t,v_1,v_2\right)\right|\le \mu \left(\left|u_2-u_1\right|+\left|v_2-v_1\right|\right).$$

If

$$\lambda \left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)+\mu \left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)<1,$$

then the system (1–2) has a unique solution on $\left. [0,T\right].$.

Proof. 

Define $\begin{array}{c}sup\\ 0\le t\le T\end{array}f\left(t,0,0\right)=k_1<\infty ,\begin{array}{c}sup\\ 0\le t\le T\end{array}g\left(t,0,0\right)=k_2<\infty$ and ${\mathrm{B}}_{\epsilon }=\left\{\left(u,v\right)\in S\times V:\Vert \left(u,v\right)\Vert \le \epsilon \right\}$, and $\epsilon >0$, such that

$$\epsilon \ge \frac{\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)k_1+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)k_2}{1-\lambda \left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)-\mu \left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)}.$$

Firstly, we show that $M{\mathrm{B}}_{\epsilon }\subseteq {\mathrm{B}}_{\epsilon }$.

By our assumption, for $\left(u,v\right)\in {\mathrm{B}}_{\epsilon },t\in \left. [0,T\right],$ we have

$$\left|f\left(t,u\left(t\right),v\left(t\right)\right)\right|\le \left|f\left(t,u\left(t\right),v\left(t\right)\right)-f\left(t,0,0\right)\right|+\left|f\left(t,0,0\right)\right|,$$

$$\le \lambda \left(\left|u\left(t\right)\right|+\left|v\left(t\right)\right|\right)+k_1\le \lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1,$$

$$\le \lambda \epsilon +k_1,$$

and

$$\left|g\left(t,u\left(t\right),v\left(t\right)\right)\right|\le \mu \left(\left|u\left(t\right)\right|+\left|v\left(t\right)\right|\right)+k_2\le \mu \left(\Vert u\Vert +\Vert v\Vert \right)+k_2,$$

$$\le \mu \epsilon +k_2.$$

Which lead to

$$\begin{gathered} \begin{array}{c}\left|M_1\left(u,v\right)\left(t\right)\right|\le \frac{1}{\left|\mathsf{\Theta }\right|}\left(\left|\xi \right|T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}ds\left(\mu \left(\Vert u\Vert +\Vert v\Vert \right)+k_2\right) \\ +T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}d\eta ds\left(\lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1\right)+\left|\xi \gamma \right|{{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}ds\left(\lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1\right) \\ +\left|\xi \right|{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}d\eta ds\left(\mu \left(\Vert u\Vert +\Vert v\Vert \right)+k_2\right)\right)\\ +\begin{array}{c}sup\\ 0\le t\le T\end{array}{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}ds\left(\lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1\right), \end{array} \end{gathered}$$

$$\begin{gathered} \le \left(\lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1\right)\left. [\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{T^{\alpha +2}}{\mathsf{\Gamma }\left(\alpha +2\right)}+\frac{\left|\xi \gamma \right|{\phi }^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right)+\frac{T^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right] \\ +\left(\mu \left(\Vert u\Vert +\Vert v\Vert \right)+k_2\right)\left. [\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\xi \right|T{\psi }^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}+\frac{\left|\xi \right|T^{\beta +1}}{\mathsf{\Gamma }\left(\beta +2\right)}\right)\right], \end{gathered}$$

$$\le \left(\lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1\right){\mathsf{\Delta }}_1+\left(\mu \left(\Vert u\Vert +\Vert v\Vert \right)+k_2\right){\mathsf{\Delta }}_2,$$

$$\le \left(\lambda \epsilon +k_1\right){\mathsf{\Delta }}_1+\left(\mu \epsilon +k_2\right){\mathsf{\Delta }}_2.$$

In a like manner,

$$\left|M_2\left(u,v\right)\left(t\right)\right|\le \left(\lambda \left(\Vert u\Vert +\Vert v\Vert \right)+k_1\right){\mathsf{\Delta }}_3+\left(\mu \left(\Vert u\Vert +\Vert v\Vert \right)+k_2\right){\mathsf{\Delta }}_4$$

$$\le \left(\lambda \epsilon +k_1\right){\mathsf{\Delta }}_3+\left(\mu \epsilon +k_2\right){\mathsf{\Delta }}_4.$$

Hence,

$$\Vert M_1\left(u,v\right)\Vert \le \left(\lambda +k_1\right){\mathsf{\Delta }}_1+\left(\mu \epsilon +k_2\right){\mathsf{\Delta }}_2,$$

and

$$\Vert M_2\left(u,v\right)\Vert \le \left(\lambda \epsilon +k_1\right){\mathsf{\Delta }}_3+\left(\mu \epsilon +k_2\right){\mathsf{\Delta }}_4.$$

Consequently,

$$\Vert M\left(u,v\right)\Vert \le \left(\lambda \epsilon +k_1\right)\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)+\left(\mu \epsilon +k_2\right)\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)\le \epsilon .$$

So, we get $\Vert M\left(u,v\right)\Vert \le \epsilon .$

Now let $\left(u_1,v_1\right),\left(u_2,v_2\right)\in S\times V,\forall t\in \left. [0,T\right]$; then, we get

$$\begin{gathered} \begin{array}{c}\left|M_1\left(u_1,v_1\right)\left(t\right)-M_1\left(u_2,v_2\right)\left(t\right)\right|\\ \le \frac{1}{\left|\Theta \right|}\left(\left|\xi \right|T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\Gamma \left(\beta \right)}ds\mu \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right) \\ +T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}d\eta ds\lambda \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right) \\ +\left|\xi \gamma \right|{{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}ds\lambda \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right) \\ +\left|\xi \right|{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\Gamma \left(\beta \right)}d\eta ds\mu \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right)\right)\\ +\begin{array}{c}sup\\ 0\le t\le T\end{array}{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)}ds\lambda \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right),\end{array} \end{gathered}$$

$$\Vert M_1\left(u_1,v_1\right)-M_1\left(u_2,v_2\right)\Vert \le {\mathsf{\Delta }}_1\lambda \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right)+{\mathsf{\Delta }}_2\mu \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right).$$

(12)

Similarly,

$$\Vert M_2\left(u_1,v_1\right)-M_2\left(u_2,v_2\right)\Vert \le {\mathsf{\Delta }}_3\lambda \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right)+{\mathsf{\Delta }}_4\mu \left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right).$$

(13)

From (12) and (13), we deduced that

$$\Vert M\left(u_1,v_1\right)-M\left(u_2,v_2\Vert \right)\le \left(\lambda \left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)+\mu \left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)\right)\left(\Vert u_2-u_1\Vert +\Vert v_2-v_1\Vert \right).$$

Since $\lambda \left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)+\mu \left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)<1,$ therefore, the operator $M$ is a contraction operator. Hence, by Banach’s fixed-point theorem, the operator $M$ is has unique fixed point on, which is the unique solution of system (1–2). This completes the proof. □

4. Hyers–Ulam Stability

In this section, the Hyers–Ulam stability for the boundary value problems in system (1–2) is discussed by means of integral representation of its general solution defined by

$$u\left(t\right)=M_1\left(u,v\right)\left(t\right),v\left(t\right)=M_2\left(u,v\right)\left(t\right),$$

where $M_1$ and $M_2$ are defined by (10) and (11).

Define the following nonlinear operators $H_1,H_2\in C\left(\left. [0,\mathrm{T}\right],ℝ\right)\times C\left(\left. [0,\mathrm{T}\right],ℝ\right)\to C\left(\left. [0,\mathrm{T}\right],ℝ\right);$

$$\begin{array}{c}{}_{ }{}^{c}D{}^{\alpha }u\left(t\right)-f\left(t,u\left(t\right),v\left(t\right)\right)=H_1\left(u,v\right)\left(t\right), t\in \left. [0,T\right], \\ {}_{ }{}^{c}D{}^{\beta }v\left(t\right)-g\left(t,u\left(t\right),v\left(t\right)\right)=H_2\left(u,v\right)\left(t\right), t\in \left. [0,T\right]. \end{array}$$

For some $r_1,r_2>0,$ we consider the following inequality:

$$\Vert H_1\left(u,v\right)\Vert \le r_1, \Vert H_2\left(u,v\right)\Vert \le r_2.$$

(14)

Definition 3 

([8,9]). The coupled system (1–2) is said to be Hyers–Ulam stable, if there exist $G_1,G_2>0,$ such that for every solution $\left(u^{*},v^{*}\right)\in C\left(\left. [0,T\right],ℝ\right)\times C\left(\left. [0,T\right],ℝ\right)$ of the inequality $\left(14\right),$ there exists a unique solution $\left(u,v\right)\in C\left(\left. [0,T\right],ℝ\right)\times C\left(\left. [0,T\right],ℝ\right)$ of problems (1) and (2) with

$$\Vert \left(u,v\right)-\left(u^{*},v^{*}\right)\Vert \le G_1{r}_1+G_2{r}_2.$$

Theorem 3. 

Let the assumptions of Theorem 1 hold. Then the system (1–2) is Hyers–Ulam stable. 

Proof. 

Let $\left(u,v\right)\in C\left(\left. [0,T\right],ℝ\right)\times C\left(\left. [0,T\right],ℝ\right)$ be the solution of the system (1–2) satisfying $\left(10\right)$ and $\left(11\right)$. Let $\left(u^{*},v^{*}\right)$ be any solution satisfying $\left(14\right):$

$$\begin{array}{c}{}_{ }{}^{c}D{}^{\alpha }{u}^{*}\left(t\right)=f\left(t,u^{*}\left(t\right),v^{*}\left(t\right)\right)+H_1\left(u^{*},v^{*}\right)\left(t\right), t\in \left. [0,T\right], \\ {}_{ }{}^{c}D{}^{\beta }{v}^{*}\left(t\right)=g\left(t,u^{*}\left(t\right),v^{*}\left(t\right)\right)+H_2\left(u^{*},v^{*}\right)\left(t\right), t\in \left. [0,T\right]. \end{array}$$

So,

$$\begin{gathered} \begin{array}{c}{u}^{*}\left(t\right)=M_1\left(u^{*},v^{*}\right)\left(t\right)\\ +\frac{1}{\mathsf{\Theta }}\left(\xi T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}{H}_2\left(u^{*},v^{*}\right)\left(s\right)ds-T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{N}_1\left(u^{*},v^{*}\right)\left(\eta \right)d\eta ds \\ +\xi \gamma {{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{H}_1\left(u^{*},v^{*}\right)\left(s\right)ds-\xi {{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}{H}_2\left(u^{*},v^{*}\right)\left(\eta \right)d\eta ds\right)\\ +{{\displaystyle \int }}_0^t\frac{{\left(t-s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{H}_1\left(u^{*},v^{*}\right)\left(s\right)ds.\end{array} \end{gathered}$$

It follows that

$$\begin{gathered} \begin{array}{l}\left|M_1\left(u^{*},v^{*}\right)\left(t\right)-u^{*}\left(t\right)\right|\\ \le \frac{1}{\left|\mathsf{\Theta }\right|}\left(\left|\xi \right|T{{\displaystyle \int }}_0^{\psi }\frac{{\left(\psi -s\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}ds{r}_2+T{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}d\eta ds{r}_1+\left|\xi \gamma \right|{{\displaystyle \int }}_0^{\phi }\frac{{\left(\phi -s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}ds{r}_1 \\ +\left|\xi \right|{{\displaystyle \int }}_0^T{{\displaystyle \int }}_0^s\frac{{\left(s-\eta \right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}d\eta ds{r}_2\right)+{{\displaystyle \int }}_0^T\frac{{\left(T-s\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}ds{r}_1, \end{array} \end{gathered}$$

$$\le \left. [\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{T^{\alpha +2}}{\mathsf{\Gamma }\left(\alpha +2\right)}+\frac{\left|\xi \gamma \right|{\phi }^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right)+\frac{T^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right] r_1+\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\xi \right|T{\psi }^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}+\frac{\left|\xi \right|T^{\beta +1}}{\mathsf{\Gamma }\left(\beta +2\right)}\right)r_2,$$

$$\le {\mathsf{\Delta }}_1{r}_1+{\mathsf{\Delta }}_2{r}_{2.}$$

Similarly,

$$\begin{gathered} \left|M_2\left(u^{*},v^{*}\right)\left(t\right)-u^{*}\left(t\right)\right|\le \frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\gamma \right|T^{\alpha +1}}{\mathsf{\Gamma }\left(\alpha +2\right)}+\frac{T\left|\gamma \right|{\phi }^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}\right)r_1+\left. [\frac{1}{\left|\mathsf{\Theta }\right|}\left(\frac{\left|\xi \gamma \right|{\psi }^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}+\frac{T^{\beta +2}}{\mathsf{\Gamma }\left(\beta +2\right)}\right)+\frac{T^{\beta }}{\mathsf{\Gamma }\left(\beta +1\right)}\right]r_{2.} \\ \le {\mathsf{\Delta }}_3{r}_1+{\mathsf{\Delta }}_4{r}_2, \end{gathered}$$

So, we deduce based on using the fixed-point property of the operator $M$, presented by Equations $\left(10\right)$ and $\left(11\right)$, that

$$\left|u\left(t\right)-u^{*}\left(t\right)\right|=\left|u\left(t\right)-M_1\left(u^{*},v^{*}\right)\left(t\right)+M_1\left(u^{*},v^{*}\right)\left(t\right)-u^{*}\left(t\right)\right|$$

$$\le \left|M_1\left(u,v\right)\left(t\right)-M_1\left(u^{*},v^{*}\right)\left(t\right)\right|+\left|M_1\left(u^{*},v^{*}\right)\left(t\right)-u^{*}\left(t\right)\right|$$

$$\le \left({\mathsf{\Delta }}_1\lambda +{\mathsf{\Delta }}_2\mu \right)\Vert \left(u,v\right)-\left(u^{*},v^{*}\right)\Vert +{\mathsf{\Delta }}_1{r}_1+{\mathsf{\Delta }}_2{r}_{2,}$$

(15)

and similarly

$$\left|v\left(t\right)-v^{*}\left(t\right)\right|=\left|v\left(t\right)-M_2\left(u^{*},v^{*}\right)\left(t\right)+M_2\left(u^{*},v^{*}\right)\left(t\right)-v^{*}\left(t\right)\right|$$

$$\le \left|M_2\left(u,v\right)\left(t\right)-M_2\left(u^{*},v^{*}\right)\left(t\right)\right|+\left|M_2\left(u^{*},v^{*}\right)\left(t\right)-v^{*}\left(t\right)\right|$$

$$\le \left({\mathsf{\Delta }}_3\lambda +{\mathsf{\Delta }}_4\mu \right)\Vert \left(u,v\right)-\left(u^{*},v^{*}\right)\Vert +{\mathsf{\Delta }}_3{r}_1+{\mathsf{\Delta }}_4{r}_2.$$

(16)

From $\left(15\right)$ and $\left(16\right)$ it follows that

$$\Vert \left(u,v\right)-\left(u^{*},v^{*}\right)\Vert \le \left({\mathsf{\Delta }}_1\lambda +{\mathsf{\Delta }}_2\mu +{\mathsf{\Delta }}_3\lambda +{\mathsf{\Delta }}_4\mu \right)\Vert \left(u,v\right)-\left(u^{*},v^{*}\right)\Vert +\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)r_1+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)r_2,$$

$$\Vert \left(u,v\right)-\left(u^{*},v^{*}\right)\Vert \le \frac{\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)r_1+\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)r_2}{1-\left(\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)\lambda +\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)\mu \right)},$$

with

$$G_1=\frac{\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)}{1-\left(\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)\lambda +\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)\mu \right)},$$

$$G_2=\frac{\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)}{1-\left(\left({\mathsf{\Delta }}_1+{\mathsf{\Delta }}_3\right)\lambda +\left({\mathsf{\Delta }}_2+{\mathsf{\Delta }}_4\right)\mu \right)}.$$

Thus, we obtain the Hyers–Ulam stability condition. □

5. Examples

Example 1. 

Consider the following coupled system of differential equation with different fractional order

$$\left\{\begin{array}{c}{ }^c{D}^{3/2}u\left(t\right)=\frac{1}{40+t^2}+\frac{4\left|u\left(t\right)\right|}{{\left(t+8\right)}^3\left(1+v^2\left(t\right)\right)}+\frac{8\left|v\left(t\right)\right|}{4\sqrt{1600+t^2}}, t\in \left. [0,1\right] \\ { }^c{D}^{4/3}v\left(t\right)=\frac{1}{\sqrt{9+t^2}}+\frac{\left|v\left(t\right)\right|}{80{\left(t+3\right)}^2}+\frac{1}{200}\left|u\left(t\right)\right|, t\in \left. [0,1\right] \\ {{\displaystyle \int }}_0^{1}u\left(s\right)ds=2v\left(1/3\right),{{\displaystyle \int }}_0^{1}v\left(s\right)ds=u\left(1\right), \\ u^{\prime }\left(0\right)=v^{\prime }\left(0\right)=0. \end{array}\right.$$

(17)

$$\alpha =\frac{3}{2},\beta =\frac{4}{3},T=1,\xi =2,\gamma =1,\phi =1,\psi =1/3.$$

Using the given data, we find that $\mathsf{\Theta }=-1,{\mathsf{\Delta }}_1=2.5577,{\mathsf{\Delta }}_2=1.0711,{\mathsf{\Delta }}_3=1.6302,{\mathsf{\Delta }}_4=1.551.$

It is clear that

$$\left|f\left(t,u,v\right)\right|\le \frac{1}{40}+\frac{1}{128}\Vert u\Vert +\frac{1}{20}\Vert v\Vert ,$$

$$\left|g\left(t,u,v\right)\right|\le \frac{1}{3}+\frac{1}{200}\Vert u\Vert +\frac{1}{720}\Vert v\Vert .$$

Thus, $m_0=\frac{1}{40},m_1=\frac{1}{180},m_2=\frac{1}{20},n_0=\frac{1}{3},n_1=\frac{1}{200},n_2=\frac{1}{720}$.

We found $\left({∆}_1+{∆}_3\right)m_1+\left({∆}_2+{∆}_4\right)n_1=0.0364<1$ and $\left({∆}_1+{∆}_3\right)m_2+\left({∆}_2+{∆}_4\right)n_2=0.213<1,$ then by Theorem 1. the problem (17) has at least one solution on $\left. [0, 1\right]$.

Example 2. 

Consider the following coupled system of differential equation with different fractional order

$$\left\{\begin{array}{c}{ }^c{D}^{6/5}u\left(t\right)=\frac{1}{7\pi \sqrt{64+t^4}}\left(\frac{\left|u\left(t\right)\right|}{12+\left|v\left(t\right)\right|}+\frac{\left|v\left(t\right)\right|}{8+\left|v\left(t\right)\right|}\right),\\ { }^c{D}^{5/4}v\left(t\right)=\frac{1}{16\pi \sqrt{81+t^4}}\left(sin\left(u\left(t\right)\right)+sin\left(v\left(t\right)\right)\right),\\ {{\displaystyle \int }}_0^{1}u\left(s\right)ds=2v\left(1/3\right),{{\displaystyle \int }}_0^{1}v\left(s\right)ds=u\left(1\right), \\ u^{\prime }\left(0\right)=v^{\prime }\left(0\right)=0. \end{array}\right.$$

(18)

$$\alpha =\frac{6}{5},\beta =\frac{5}{4},T=1,\xi =2,\gamma =1,\phi =1,\psi =\frac{1}{3}.$$

Using the given data, we find that $\mathsf{\Delta }=-1,{\mathsf{\Delta }}_1=3.1354,{\mathsf{\Delta }}_2=1.1184,{\mathsf{\Delta }}_3=1.3201,{\mathsf{\Delta }}_4=1.478.$

It’s obvious that

$$f\left(t,u\left(t\right),v\left(t\right)\right)=\frac{1}{7\pi \sqrt{64+t^4}}\left(\frac{\left|u\left(t\right)\right|}{12+\left|v\left(t\right)\right|}+\frac{\left|v\left(t\right)\right|}{8+\left|v\left(t\right)\right|}\right),$$

and

$$g\left(t,u\left(t\right),v\left(t\right)\right)=\frac{1}{16\pi \sqrt{81+t^4}}\left(sin\left(u\left(t\right)\right)+sin\left(v\left(t\right)\right)\right),$$

are jointly continuous functions and Lipschitz function with $\lambda =\frac{1}{56\pi },\mu =\frac{1}{72\pi }$. Moreover,

$$\frac{1}{56\pi }\left(3.1354+1.3201\right)+\frac{1}{72\pi }\left(1.1184+1.478\right)=0.0368<1.$$

Then, the conditions of Theorem 2. are satisfied; therefore, the problem (18) has a unique solution on $\left. [0, 1\right]$.

6. Conclusions

This paper discusses a fractional coupled system of differential equations with integral boundary conditions. There are three primary outcomes of this research: firstly, the use of the contraction mapping principle establishes the uniqueness of solutions for the problem at hand, and secondly, the existence solutions for the problem is examined based on Leray–Schauder’s alternative. Thirdly, the stability of the presented coupled system is discussed and illustrated based on Hyers–Ulam stability method. Finally, the conclusion is confirmed and supported by examples. It was found that the presented fractional coupled system with two with integral boundary conditions is existent, unique, and stable.