Fractional p-Laplacian Coupled Systems with Multi-Point Boundary Conditions

Abstract

This article is allocated to the existence and uniqueness of solutions for a system of nonlinear differential equations consisting of the Caputo fractional-order derivatives. Our main results are proved via standard tools of fixed point theory. Finally, the presented results are clarified by constructing some examples.

1. Introduction

Fractional differential equations (FDEs) have been applied to qualify multiplex problems in various fields of natural sciences, as physics, biology, chemistry, image processing and so on, see [1,2,3,4,5]. In the literature, different types of fractional derivatives and fractional integrals have been presented, where many of those definitions deal with the Riemann–Liouville and Caputo fractional derivatives. These definitions have been applied in many nonlocal boundary problems to present some physical phenomena (see [6,7,8,9]). Additionally, FDEs with a p-Laplacian operator appeared when Leibenson [10] was working to deduce an exact formula to construct a turbulent flow. Recently, many applications of FDEs with p-Laplacian operator have been obtained in the various areas such as glaciology and dynamics, see [11,12,13,14,15] and references therein. Hence, it is important to consider FDEs consisting of the p-Laplacian operator in different spaces. Recently, Srivastava et al. [16] have applied the p-Laplacian operator to consider the existence of solutions for a class of nonlinear differential equations of the form

$${\displaystyle {}^C{\mathcal{D}}^{\rho }\left({\theta }_P\left[{}^C{\mathcal{D}}^{\mu }\sigma \left(w\right)\right]\right)+\kappa (w,\sigma \left(w\right))=\theta ,w\in [c,d],}$$

(1)

supplemented with coupled nonlocal boundary conditions

$${\displaystyle \sigma \left(c\right)={}^C{\mathcal{D}}^{\mu }\left(c\right)=\theta ,\sigma \left(d\right)=\sum _{i=1}^m{\lambda }_i\sigma \left({\eta }_i\right),c<{\eta }_i<d,}$$

(2)

in which $\kappa :[c,d]\times E⟶E$ is a given function, E is a Banach space with zero element $\theta$, $0<\rho \le 1$, $1<\mu \le 2$, ${\lambda }_i,i=1,2,\dots ,m$ are real constants, ${\displaystyle {}^C{\mathcal{D}}^{\rho }}$, denote the Caputo fractional derivatives, ${\theta }_p$ indicates a p-Laplacian operator, that is

$$\begin{array}{c}\hfill {\theta }_p\left(s\right)={\left|s\right|}^{p-2}s,p>1,{\theta }_p^{-1}={\theta }_q,\frac{1}{p}+\frac{1}{q}=1.\end{array}$$

The aim of this work is to extend the problem (1) and (2) on coupled system of differential equations by introducing a new coupled system of FDEs. The significance of coupled systems of FDEs is that such systems have appeared in various fields of sciences, see [17,18,19]. In this paper, the existence of solutions for a system of nonlinear differential equations consisting of the Caputo fractional-order derivatives of the form

$$\left\{\begin{array}{c}{\displaystyle {}^C{\mathcal{D}}^{{\rho }_1}\left({\theta }_{p_1}\left[{}^C{\mathcal{D}}^{{\mu }_1}{\sigma }_1\left(w\right)\right]\right)={\kappa }_1(w,{\sigma }_1\left(w\right),{\sigma }_2\left(w\right)),w\in [c_1,d_1],}\hfill \\ {\displaystyle {}^C{\mathcal{D}}^{{\rho }_2}\left({\theta }_{p_2}\left[{}^C{\mathcal{D}}^{{\mu }_2}{\sigma }_2\left(w\right)\right]\right)={\kappa }_2(w,{\sigma }_1\left(w\right),{\sigma }_2\left(w\right)),}\hfill \end{array}\right.$$

(3)

supplemented with coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {\sigma }_1\left(c_1\right)={}^C{\mathcal{D}}^{{\mu }_1}{\sigma }_1\left(c_1\right)=0,{\sigma }_1\left(d_1\right)=\sum _{i=1}^m{\lambda }_i{\sigma }_2\left({\eta }_i\right),}\hfill \\ {\displaystyle {\sigma }_2\left(c_1\right)={}^C{\mathcal{D}}^{{\mu }_2}{\sigma }_2\left(c_1\right)=0,{\sigma }_2\left(d_1\right)=\sum _{j=1}^n{\overline{\lambda }}_j{\sigma }_1\left({\overline{\eta }}_j\right)}\hfill \end{array}\right.$$

(4)

is investigated, in which $0<{\rho }_1,{\rho }_2\le 1$, $1<{\mu }_1,{\mu }_2\le 2$, ${\displaystyle {}^C{\mathcal{D}}^{\delta }}$ denotes the Caputo fractional derivative of order $\delta \in \{{\rho }_1,{\rho }_2,{\mu }_1,{\mu }_2\}$, ${\theta }_p$ $(p=p_1,p_2)$ indicates a p-Laplacian operator, ${\kappa }_1,{\kappa }_2:[c_1,d_1]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ are continuous functions with some conditions, ${\lambda }_i,{\overline{\lambda }}_j,$ $i=1,2,\dots ,m,$ $j=1,2,\dots ,n$ are real constants and $c_1<{\eta }_i,{\overline{\eta }}_j<d_1$.

The tools of fixed point theory will be applied to develop the existence theory for the problem (3) and (4). In fact, the Banach contraction mapping principle is applied to prove a uniqueness result, while two existence results are derived via Leray–Schauder alternative and Krasnosel’skiĭ fixed point theorem. The presented system in this article is not only new, but some other systems are special cases of this system, for example, by taking ${\mu }_1={\mu }_2=p_1=p_2=2$ and ${\rho }_1={\rho }_2=1$, we have the third order system of the form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\sigma }_1^{‴}\left(w\right)=\kappa (w,{\sigma }_1\left(w\right),{\sigma }_2\left(w\right)),w\in [c_1,d_1],\hfill \\ {\sigma }_2^{‴}\left(w\right)=\kappa (w,{\sigma }_1\left(w\right),{\sigma }_2\left(w\right)),\hfill \\ {\displaystyle {\sigma }_1\left(c_1\right)={\sigma }_1^{″}\left(c_1\right)=0,{\sigma }_1\left(d_1\right)=\sum _{i=1}^m{\lambda }_i{\sigma }_2\left({\eta }_i\right),c_1<{\eta }_i<d_1,}\hfill \\ {\displaystyle {\sigma }_2\left(c_1\right)={\sigma }_2^{″}\left(c_1\right)=0,{\sigma }_2\left(d_1\right)=\sum _{j=1}^n{\overline{\lambda }}_j{\sigma }_1\left({\overline{\eta }}_j\right),c_1<{\overline{\eta }}_j<d_1.}\hfill \end{array}\right.\end{array}$$

(5)

The structure of this article is as follows: In the next section, some definitions and basic Lemmas are collected, which will be required to prove the main results. In Section 3, an auxiliary lemma is proved to convert the problem (3) and (4) into a fixed point problem. The existence and uniqueness results are considered in Section 4. In Section 5, some examples are included to illustrate the obtained results.

2. Preliminaries

In this section, some results and concepts concerning the fractional calculus are presented.

Definition 1 

([2,20]). The fractional integral of Riemann–Liouville type and order $\eta >0$ of a function $\sigma \in L^1[c_1,d_1]$ is defined by

$$\begin{array}{c}\hfill {\displaystyle {}^{RL}{\mathbb{I}}^{\eta }\sigma \left(w\right)=\frac{1}{\Gamma \left(\eta \right)}{\int }_{c_1}^w{(w-s)}^{\eta -1}\sigma \left(s\right)ds,(w>c_1,\eta >0),}\end{array}$$

where Γ is the Gamma function.

Definition 2 

([2]). The fractional derivative of Caputo type and order η of a function $\sigma \in A{C}^n[c_1,d_1]$ is represented by

$${\displaystyle {}^C{\mathcal{D}}^{\eta }\sigma \left(w\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma (n-\eta )}{\int }_{c_1}^w{(w-s)}^{n-\eta -1}{\sigma }^{\left(n\right)}\left(s\right)ds,\eta \notin \mathbb{N},}\hfill \\ {\sigma }^{\left(n\right)}\left(w\right),\eta \in \mathbb{N},\hfill \end{array}\right.}$$

(6)

where ${\sigma }^{\left(n\right)}\left(w\right)=\frac{d^n\sigma \left(w\right)}{d{w}^n}$, $\eta >0$ and $n=\left[\eta \right]+1$.

Lemma 1 

([2]). Let $\eta >\rho >0$ and $\sigma \in L^1[c_1,d_1]$. Then,

(i) 

${\displaystyle {}^{RL}{\mathbb{I}}^{\eta }{}^C{\mathcal{D}}^{\eta }\sigma \left(w\right)=\sigma \left(w\right)+\sum _{i=0}^{n-1}{b}_i{(w-c_1)}^i}$, for some $b_i\in \mathbb{R}(i=0,1,2,\dots ,n-1)$, where $n=\left[\eta \right]+1$,

(ii) 

${\displaystyle {}^C{\mathcal{D}}^{\eta }{}^{RL}{\mathbb{I}}^{\eta }\sigma \left(w\right)=\sigma \left(w\right)}$,

(iii) 

${\displaystyle {}^C{\mathcal{D}}^{\rho }{}^{RL}{\mathbb{I}}^{\eta }\sigma \left(w\right)={}^{RL}{\mathbb{I}}^{\eta -\rho }\sigma \left(w\right).}$

3. An Auxiilliary Result

Lemma 2. 

Let $h_1,h_2\in C^2([c_1,d_1],\mathbb{R})$. Then, the unique solution of the system

$$\left\{\begin{array}{c}{\displaystyle {}^C{\mathcal{D}}^{{\rho }_1}\left({\theta }_{P_1}\left[{}^C{\mathcal{D}}^{{\mu }_1}{\sigma }_1\left(w\right)\right]\right)=h_1\left(w\right),w\in [c_1,d_1],}\hfill \\ {\displaystyle {}^C{\mathcal{D}}^{{\rho }_2}\left({\theta }_{P_2}\left[{}^C{\mathcal{D}}^{{\mu }_2}{\sigma }_2\left(w\right)\right]\right)=h_2\left(w\right),}\hfill \end{array}\right.$$

(7)

supplemented with coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {\sigma }_1\left(c_1\right)={}^C{\mathcal{D}}^{{\mu }_1}{\sigma }_1\left(c_1\right)=0,{\sigma }_1\left(d_1\right)=\sum _{i=1}^m{\lambda }_i{\sigma }_2\left({\eta }_i\right),}\hfill \\ {\displaystyle {\sigma }_2\left(c_1\right)={}^C{\mathcal{D}}^{{\mu }_2}{\sigma }_2\left(c_1\right)=0,{\sigma }_2\left(d_1\right)=\sum _{j=1}^n{\overline{\lambda }}_j{\sigma }_2\left({\overline{\eta }}_j\right),}\hfill \end{array}\right.$$

(8)

is given by

$$\begin{array}{ccc}\hfill {\sigma }_1\left(w\right)& =& {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(w\right)\right)\right)+\frac{w-c_1}{\mathbb{A}}[(d_1-c_1)\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left({\eta }_i\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(d_1\right)\right)\right)+{\mathbb{A}}_1(\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left({\overline{\eta }}_j\right)\right)\right)}\hfill \\ & & {\displaystyle -{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(d_1\right)\right)\right)\left)\right],}\hfill \\ \hfill {\sigma }_2\left(w\right)& =& {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_2\left(w\right)\right)\right)+\frac{w-c_1}{\mathbb{A}}[{\mathbb{A}}_2\left(\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left({\eta }_i\right)\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(d_1\right)\right)\right)+(d_1-c_1)\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left({\overline{\eta }}_j\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(d_1\right)\right)\right)],}\hfill \\ \hfill \end{array}$$

(9)

where

$$\begin{array}{c}\hfill {\mathbb{A}}_1=\sum _{i=1}^m{\lambda }_i({\eta }_i-c_1),{\mathbb{A}}_2=\sum _{j=1}^n{\overline{\lambda }}_j({\overline{\eta }}_j-c_1),\frac{1}{p_1}+\frac{1}{q_1}=1,\frac{1}{p_2}+\frac{1}{q_2}=1,\end{array}$$

(10)

with

$$\begin{array}{c}\hfill \mathbb{A}={(d_1-c_1)}^2-{\mathbb{A}}_1{\mathbb{A}}_2.\end{array}$$

(11)

Proof. 

Let ${\displaystyle {\varphi }_1\left(w\right)={\theta }_{p_1}\left[{}^C{\mathcal{D}}^{{\mu }_1}{\sigma }_1\left(w\right)\right]}$ and ${\displaystyle {\varphi }_2\left(w\right)={\theta }_{p_2}\left[{}^C{\mathcal{D}}^{{\mu }_2}{\sigma }_2\left(w\right)\right]}$. Then, the system (7) can be divided into two problems:

$$\left\{\begin{array}{c}{\displaystyle {}^C{\mathcal{D}}^{{\rho }_1}\left[{\varphi }_1\left(w\right)\right]=h_1\left(w\right),w\in [c_1,d_1],}\hfill \\ {\displaystyle {}^C{\mathcal{D}}^{{\rho }_2}\left[{\varphi }_2\left(w\right)\right]=h_2\left(w\right),}\hfill \\ {\varphi }_1\left(c_1\right)=0,{\varphi }_2\left(c_1\right)=0.\hfill \end{array}\right.$$

(12)

and

$$\left\{\begin{array}{c}{\displaystyle {}^C{\mathcal{D}}^{{\mu }_1}{\sigma }_1\left(w\right)={\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(w\right)\right),w\in [c_1,d_1],}\hfill \\ {\displaystyle {}^C{\mathcal{D}}^{{\mu }_2}{\sigma }_2\left(w\right)={\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(w\right)\right),}\hfill \end{array}\right.$$

(13)

supplemented with coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle {\sigma }_1\left(c_1\right)=0,{\sigma }_1\left(d_1\right)=\sum _{i=1}^m{\lambda }_i{\sigma }_2\left({\eta }_i\right),}\hfill \\ {\displaystyle {\sigma }_2\left(c_1\right)=0,{\sigma }_2\left(d_1\right)=\sum _{j=1}^n{\overline{\lambda }}_j{\sigma }_1\left({\overline{\eta }}_j\right).}\hfill \end{array}\right.$$

(14)

The solution of the system (12) can be written as ${\displaystyle {\varphi }_1\left(w\right)={}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(w\right)}$ and ${\displaystyle {\varphi }_2\left(w\right)={}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(w\right),}$ respectively. On the other hand, by applying the Lemma 1 and the fractional integrals ${\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}}$ and ${\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_2}}$ on both sides of the equations in (13), we obtain

$$\begin{array}{ccc}\hfill & & {\displaystyle {\sigma }_1\left(w\right)={}^{RL}{\mathbb{I}}^{{\mu }_1}({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(w\right)\right)+e_0+e_1(w-c_1),}\hfill \\ & & {\displaystyle {\sigma }_2\left(w\right)={}^{RL}{\mathbb{I}}^{{\mu }_2}{\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(w\right)\right)+f_0+f_1(w-c_1).}\hfill \end{array}$$

(15)

Now, by applying the boundary conditions ${\sigma }_1\left(c_1\right)={\sigma }_2\left(c_1\right)=0$ in (15), we obtain $e_0=f_0=0$. Hence, we have

$$\begin{array}{ccc}\hfill & & {\displaystyle {\sigma }_1\left(w\right)={}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(w\right)\right)\right)+e_1(w-c_1),}\hfill \\ & & {\displaystyle {\sigma }_2\left(w\right)={}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(w\right)\right)\right)+f_1(w-c_1).}\hfill \end{array}$$

(16)

By the boundary conditions ${\sigma }_1\left(d_1\right)={\sum }_{i=1}^m{\lambda }_i{\sigma }_2\left({\eta }_i\right)$, ${\sigma }_2\left(d_1\right)={\sum }_{j=1}^n{\overline{\lambda }}_j{\sigma }_1\left({\overline{\eta }}_j\right)$ and (16), we obtain

$$\begin{array}{ccc}\hfill & & {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(d_1\right)\right)\right)+e_1(d_1-c_1)=\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left({\eta }_i\right)\right)\right)+f_1\sum _{i=1}^m{\lambda }_i({\eta }_i-c_1),}\hfill \\ & & {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(d_1\right)\right)\right)+f_1(d_1-c_1)=\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left({\overline{\eta }}_j\right)\right)\right)+e_1\sum _{j=1}^n{\overline{\lambda }}_j({\overline{\eta }}_j-c_1).}\hfill \end{array}$$

(17)

Consequently,

$$\left\{\begin{array}{c}{\displaystyle e_1(d_1-c_1)-f_1{\mathbb{A}}_1=\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left({\eta }_i\right)\right)\right)-{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(d_1\right)\right)\right),}\hfill \\ {\displaystyle -e_1{\mathbb{A}}_2+f_1(d_1-c_1)=\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left({\overline{\eta }}_j\right)\right)\right)-{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(d_1\right)\right)\right).}\hfill \end{array}\right.$$

(18)

By solving the above system, we obtain

$$\begin{array}{ccc}\hfill e_1& =& {\displaystyle \frac{1}{\mathbb{A}}[(d_1-c_1)\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left({\eta }_i\right)\right)\right)-(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(d_1\right)\right)\right)}\hfill \\ & & {\displaystyle +{\mathbb{A}}_1\left(\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left({\overline{\eta }}_j\right)\right)\right)-{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(d_1\right)\right)\right)\right)],}\hfill \\ \hfill f_1& =& {\displaystyle \frac{1}{\mathbb{A}}[{\mathbb{A}}_2\left(\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left({\eta }_i\right)\right)\right)\right)-(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left(d_1\right)\right)\right)}\hfill \\ & & {\displaystyle +(d_1-c_1)\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{h}_1\left({\overline{\eta }}_j\right)\right)\right)-(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{h}_2\left(d_1\right)\right)\right)].}\hfill \end{array}$$

Replacing $e_1$ and $f_1$ in (16), we have the solution (9). The converse can be proved by direct computation. This completes the proof. □

4. Existence and Uniqueness Results

Let ${\mathbb{X}}^{\ast }=\{w\left(q\right):w\left(q\right)\in C([c_1,d_1],\mathbb{R})\}$ be the Banach space of all continuous functions from $[c_1,d_1]$ into $\mathbb{R}$, which has been equipped with the norm

$$\parallel v\parallel =\underset{q\in [c_1,d_1]}{sup}\left|v\left(q\right)\right|.$$

The space $({\mathbb{X}}^{\ast },\parallel .\parallel )$ is a Banach space. In addition, the product space $({\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast },\parallel .\parallel )$ is also a Banach space with the norm $\parallel (v,w)\parallel =\parallel v\parallel +\parallel w\parallel$ for $(v,w)\in {\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast }.$ In view of Lemma 2, we define an operator $\mathbb{B}:{\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast }⟶{\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast }$ as $\mathbb{B}(v,w)\left(q\right)=({\mathbb{B}}_1(v,w)\left(q\right),{\mathbb{B}}_2(v,w)\left(q\right))$ where

$$\begin{array}{ccc}\hfill {\mathbb{B}}_1(v,w)\left(q\right)& =& {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1(q,v\left(q\right),w\left(q\right))\right)\right)}\hfill \\ & & {\displaystyle +\frac{q-c_1}{\mathbb{A}}[(d_1-c_1)\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2({\eta }_i,v\left({\eta }_i\right),w\left({\eta }_i\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)}\hfill \\ & & {\displaystyle +{\mathbb{A}}_1(\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1({\overline{\eta }}_j,v\left({\overline{\eta }}_j\right),w\left({\overline{\eta }}_j\right))\right)\right)}\hfill \\ & & {\displaystyle -{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right))],q\in [c_1,d_1],}\hfill \end{array}$$

(19)

and

$$\begin{array}{ccc}\hfill {\mathbb{B}}_2(v,w)\left(q\right)& =& {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2(q,v\left(q\right),w\left(q\right))\right)\right)}\hfill \\ & & {\displaystyle +\frac{q-c_1}{\mathbb{A}}[{\mathbb{A}}_2\left(\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2({\eta }_i,v\left({\eta }_i\right),w\left({\eta }_i\right))\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)}\hfill \\ & & {\displaystyle +(d_1-c_1)\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1({\overline{\eta }}_j,v\left({\eta }_j\right),w\left({\eta }_j\right))\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)].}\hfill \end{array}$$

(20)

For computational convenience, we set

$$\begin{array}{ccc}\hfill {\Phi }_1& =& \frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{{(d_1-c_1)}^2\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{|{\mathbb{A}}_1|\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{({\overline{\eta }}_j-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1},\hfill \\ \hfill {\Phi }_2& =& \frac{\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}\sum _{i=1}^m\left|{\lambda }_i\right|{({\eta }_i-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\hfill \\ & & +\frac{|{\mathbb{A}}_1|{(d_1-c_1)}^2\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2},\hfill \\ \hfill {\Phi }_3& =& \frac{\Gamma ({\rho }_2(q_2-1)+1)}{\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\hfill \\ & & +\frac{\Gamma ({\rho }_2(q_2-1)+1)\left|{\mathbb{A}}_2\right|}{\left|\mathbb{A}\right|\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma \left({\mu }_2\right)\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}\sum _{i=1}^m\left|{\lambda }_i\right|{({\eta }_i-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\hfill \\ & & +\frac{{(d_1-c_1)}^2\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma \left({\mu }_2\right)\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2},\hfill \\ \hfill {\Phi }_4& =& \frac{|{\mathbb{A}}_2|{(d_1-c_1)}^2\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{({\overline{\eta }}_j-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1},\hfill \\ \hfill {\Phi }_1^{\ast }& =& {\Phi }_1-\frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1},\hfill \\ \hfill {\Phi }_3^{\ast }& =& {\Phi }_3-\frac{\Gamma ({\rho }_2(q_2-1)+1)}{\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}.\hfill \end{array}$$

(21)

The following lemma is used in the sequel.

Lemma 3. 

It holds:

$$\frac{1}{\Gamma \left(\mu \right)}{\int }_a^t{(t-s)}^{\mu -1}{(s-a)}^{\rho }ds=\frac{\Gamma (\rho +1)}{\Gamma (\mu +\rho +1)}{(t-a)}^{\mu +\rho }.$$

4.1. Uniqueness Result

In the following theorem, the Banach contraction mapping principle is applied to establish the existence and uniqueness result for the system (3) and (4).

Theorem 1. 

Assume that the functions ${\kappa }_1,{\kappa }_2:[c_1,d_1]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ satisfies the condition

• $\left(H_1\right)$ There exist real constants $m_i,n_i(i=1,2)$such that for all $q\in [c_1,d_1]$ and $v_i,w_i\in \mathbb{R},i=1,2$,

we have

$$\begin{array}{ccc}& & |{\kappa }_1(q,v_1,w_1)-{\kappa }_1(q,v_2,w_2)|\le m_1|v_1-v_2|+m_2|w_1-w_2|,\hfill \\ & & |{\kappa }_2(q,v_1,w_1)-{\kappa }_2(q,v_2,w_2)|\le n_1|v_1-v_2|+n_2|w_1-w_2|.\hfill \end{array}$$

Then, the system (3) and (4) has a unique solution on $[c_1,d_1]$ if

$$\begin{array}{c}\hfill ({\Phi }_1+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3)(n_1+n_2)<1,\end{array}$$

(22)

where ${\Phi }_i(i=1,2,3,4)$ are given in (21).

Proof. 

The hypotheses of Banach’s contraction mapping principle will be considered in the following steps:

Step 1. $\mathbb{B}\left({\mathfrak{B}}_x\right)\subseteq {\mathfrak{B}}_x$ where ${\mathfrak{B}}_x=\{(v,w)\in {\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast }:\parallel (v,w)\parallel \le x\}$ with

$$x\ge \frac{({\Phi }_1+{\Phi }_4)M+({\Phi }_2+{\Phi }_3)N}{1-[({\Phi }_1+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3)(n_1+n_2)]},$$

$M={sup}_{q\in [c_1,d_1]}\left|{\kappa }_1(q,0,0)\right|,$ $N={sup}_{q\in [c_1,d_1]}\left|{\kappa }_2(q,0,0)\right|$.

Step 2. $\mathbb{B}$ is a contraction.

For Step 1, let $(v,w)\in {\mathfrak{B}}_x.$ Then, we obtain

$$\begin{array}{ccc}& & |{\mathbb{B}}_1(v,w)\left(q\right)|\hfill \\ & \le & {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}|{\kappa }_1(q,v\left(q\right),w\left(q\right))-{\kappa }_1(q,0,0)|+|{\kappa }_1(q,0,0)|\right)\right)}\hfill \\ & & {\displaystyle +\frac{q-c_1}{\left|\mathbb{A}\right|}[(d_1-c_1)\sum _{i=1}^m|{\lambda }_i|{}^{RL}{\mathbb{I}}^{{\mu }_2}({\theta }_{q_2}({}^{RL}{\mathbb{I}}^{{\rho }_2}|{\kappa }_2({\eta }_i,v\left({\eta }_i\right),w\left({\eta }_i\right))-{\kappa }_1({\eta }_i,0,0)|}\hfill \\ & & {\displaystyle +|{\kappa }_2({\eta }_i,0,0)\left|\right))+(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\right({}^{RL}{\mathbb{I}}^{{\rho }_1}|{\kappa }_1(d_1,v\left(d_1\right),w\left(d_1\right))-{\kappa }_1(d_1,0,0)|}\hfill \\ & & {\displaystyle +|{\kappa }_1(d_1,0,0)\left|\right))+{\mathbb{A}}_1\left(\sum _{j=1}^n|{\overline{\lambda }}_j|{}^{RL}{\mathbb{I}}^{{\mu }_1}\right({\theta }_{q_1}({}^{RL}{\mathbb{I}}^{{\rho }_1}|{\kappa }_1({\overline{\eta }}_j,v\left({\overline{\eta }}_j\right),w\left({\overline{\eta }}_j\right))-{\kappa }_1({\overline{\eta }}_j,0,0)|}\hfill \\ & & {\displaystyle +|{\kappa }_1({\overline{\eta }}_j,0,0)\left|\right))+{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\right({}^{RL}{\mathbb{I}}^{{\rho }_2}|{\kappa }_2(d_1,v\left(d_1\right),w\left(d_1\right))-{\kappa }_2(d_1,0,0)|}\hfill \\ & & +|{\kappa }_2(d_1,0,0)\left|\right)\left)\right)]\hfill \\ & \le & \frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^q{(q-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s(m_1\parallel v\parallel +m_2\parallel w\parallel +M)\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{(d_1-c_1)}^2|}{\Gamma \left({\mu }_2\right)\left|\mathbb{A}\right|}\hfill \\ & & \times \sum _{i=1}^m\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}(n_1\parallel v\parallel +n_2\parallel w\parallel +N)d\tau \right)ds\hfill \\ & & +\frac{{(d_1-c_1)}^2}{\Gamma \left({\mu }_1\right)\left|\mathbb{A}\right|}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}(m_1\parallel v\parallel +m_2\parallel w\parallel +M)d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|(d_1-c_1)}{\mathbb{A}}\left(\sum _{j=1}^n|{\overline{\lambda }}_j|\frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_1-1}{\theta }_{q_2}\right({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}(m_1\parallel v\parallel \hfill \\ & & +m_2\parallel w\parallel +M\left)d\tau \right)ds\hfill \\ & & +\frac{1}{\Gamma \left({\mu }_2\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}(n_1\parallel v\parallel +n_2\parallel w\parallel +N)d\tau \right)ds)\hfill \\ & \le & (m_1\parallel v\parallel +m_2\parallel w\parallel +M\left)\right\{\frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^q{(q-s)}^{{\mu }_1-1}{\theta }_{q_1}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\hfill \\ & & +\frac{|d_1-c_1{|}^2}{\left|\mathbb{A}\right|}\frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_1-1}{\theta }_{q_1}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1\left|\right|d_1-c_1|}{\left|\mathbb{A}\right|}\sum _{j=1}^n\frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_2-1}{\theta }_{q_2}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\}\hfill \\ & & +(n_1\parallel v\parallel +n_2\parallel w\parallel +N\left)\right\{\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}\sum _{i=1}^m\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{\theta }_{q_2}\left(\frac{{(s-c_1)}^{{\rho }_2}}{\Gamma ({\rho }_2+1)}\right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1\left|\right|d_1-c_1|}{\left|A\right|\Gamma \left({\mu }_2\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{\theta }_{q_2}\left(\frac{{(s-c_1)}^{{\rho }_2}}{\Gamma ({\rho }_2+1)}\right)ds\}\hfill \\ & \le & (m_1\parallel v\parallel +m_2\parallel w\parallel +M\left)\right\{\frac{1}{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{c_1}^q{(q-s)}^{{\mu }_1-1}{(s-c_1)}^{{\rho }_1(q_1-1)}ds\hfill \\ & & +\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)}{\int }_a^b{(b-s)}^{{\mu }_1-1}{(s-c_1)}^{{\rho }_1(q_1-1)}ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_1(q_2-1)}ds\}\hfill \\ & & +(n_1\parallel v\parallel +n_2\parallel w\parallel +N\left)\right\{\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}}\hfill \\ & & \times \sum _{i=1}^n\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_2(q_2-1)}ds\hfill \\ & & +\frac{|A_1|(d_1-c_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_2(q_2-1)}ds\}\hfill \\ & \le & (m_1\parallel v\parallel +m_2\parallel w\parallel +M\left)\right\{\frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{{(d_1-c_1)}^2\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{|A_1|\Gamma ({\rho }_1(q_2-1)+1)}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)\Gamma ({\rho }_1(q_2-1)+{\mu }_2)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{({\overline{\eta }}_j-c_1)}^{{\rho }_1(q_2-1)+{\mu }_2}\}\hfill \\ & & +(n_1\parallel v\parallel +n_2\parallel w\parallel +N)\hfill \\ & & \times \{\frac{{(d_1-c_1)}^2\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{({\eta }_i-c_1)}^{{\mu }_2+{\rho }_2(q_2-1)}\hfill \\ & & +\frac{|{\mathbb{A}}_1|(d_1-c_1)\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\}\hfill \\ & =& (m_1\parallel v\parallel +m_2\parallel w\parallel +M){\Phi }_1+(n_1\parallel v\parallel +n_2\parallel w\parallel +N){\Phi }_2\hfill \\ & =& (m_1{\Phi }_1+n_1{\Phi }_2)\parallel v\parallel +(m_2{\Phi }_1+n_2{\Phi }_2)\parallel w\parallel +{\Phi }_{1}M+{\Phi }_{2}N\hfill \\ & \le & (m_1{\Phi }_1+n_1{\Phi }_2+m_2{\Phi }_1+n_2{\Phi }_2)x+{\Phi }_{1}M+{\Phi }_{2}N.\hfill \end{array}$$

Similarly, we can show that

$$\begin{array}{ccc}& & |{\mathbb{B}}_2(v,w)\left(q\right)|\le (m_1{\Phi }_4+n_1{\Phi }_3+m_2{\Phi }_4+n_2{\Phi }_3)x+{\Phi }_{4}M+{\Phi }_{3}N.\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{ccc}\hfill \parallel \mathbb{B}(v,w)\parallel & =& \parallel {\mathbb{B}}_1(v,w)\parallel +\parallel {\mathbb{B}}_2(v,w)\parallel \hfill \\ & \le & \left[({\Phi }_1+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3)(n_1+n_2)\right]x\hfill \\ & & +({\Phi }_1+{\Phi }_4)M+({\Phi }_2+{\Phi }_3)N\hfill \\ & \le & x,\hfill \end{array}$$

which implies that $\mathbb{B}\left({\mathfrak{B}}_x\right)\subseteq {\mathfrak{B}}_x.$

Now, we prove Step 2, that is, the operator $\mathbb{B}$ is a contraction. Let $(v_1,w_1),(v_2,w_2)\in {\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast }$ and $q\in [c_1,d_1]$. Then, we have

$$\begin{array}{ccc}& & |{\mathbb{B}}_1(v_1,w_1)\left(q\right)-{\mathbb{B}}_1(v_2,w_2)\left(q\right)|\hfill \\ & \le & {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}|{\kappa }_1(q,v_1\left(q\right),w_1\left(q\right))-{\kappa }_1(q,v_2\left(q\right),w_2\left(q\right))|\right)\right)}\hfill \\ & & {\displaystyle +\frac{q-c_1}{\left|\mathbb{A}\right|}[(d_1-c_1)\sum _{i=1}^m|{\lambda }_i\left|{}^{RL}{\mathbb{I}}^{{\mu }_2}\right({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}\right|{\kappa }_2({\eta }_i,v_1\left({\eta }_i\right),w_1\left({\eta }_i\right))}\hfill \\ & & -{\kappa }_2({\eta }_i,v_2\left({\eta }_i\right),w_2\left({\eta }_i\right))\left)\right)\hfill \\ & & {\displaystyle +(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}|{\kappa }_1(d_1,v_1\left(d_1\right),w_1\left(d_1\right))-{\kappa }_1(d_1,v_2\left(d_1\right),w_2\left(d_1\right))|\right)\right)}\hfill \\ & & {\displaystyle +{\mathbb{A}}_1(\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}|{\kappa }_1({\overline{\eta }}_j,v_1\left({\overline{\eta }}_j\right),w_1\left({\overline{\eta }}_j\right))-{\kappa }_1({\overline{\eta }}_j,v_2\left({\overline{\eta }}_j\right),w_2\left({\overline{\eta }}_j\right))\right)\right)}\hfill \\ & & {\displaystyle +{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}|{\kappa }_2(d_1,v_1\left(d_1\right),w_1\left(d_1\right))-{\kappa }_2(d_1,v_2\left(d_1\right),w_2\left(d_1\right))|\right)\right)\left)\right]}\hfill \\ & \le & \frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^q{(q-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s(m_1\parallel v_1-w_1\parallel +m_2\parallel w_1-w_2\parallel )\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{(d_1-c_1)}^2|}{\Gamma \left({\mu }_2\right)\left|\mathbb{A}\right|}\hfill \\ & & \times \sum _{i=1}^m\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}(n_1\parallel v_1-v_2\parallel +n_2\parallel w_1-w_2\parallel )d\tau \right)ds\hfill \\ & & +\frac{{(d_1-c_1)}^2}{\Gamma \left({\mu }_1\right)\left|\mathbb{A}\right|}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}(m_1\parallel v_1-v_2\parallel +m_2\parallel w_1-w_2\parallel )d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|(d_1-c_1)}{\mathbb{A}}\left(\sum _{j=1}^n|{\overline{\lambda }}_j|\frac{1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_1-1}{\theta }_{q_2}\right({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}(m_1\parallel v_1-v_2\parallel \hfill \\ & & +m_2\parallel w_1-w_2\parallel )d\tau )ds\hfill \\ & & +\frac{1}{\Gamma \left({\mu }_2\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}(n_1\parallel v_1-v_2\parallel +n_2\parallel w_1-w_2\parallel )d\tau \right)ds\left)\right)\hfill \\ & \le & (m_1\parallel v_1-v_2\parallel +m_2\parallel w_1-w_2\parallel )\{\frac{1}{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{c_1}^q{(q-s)}^{{\mu }_1-1}{(s-a)}^{{\rho }_1(q_1-1)}ds\hfill \\ & & +\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)}{\int }_a^b{(b-s)}^{{\mu }_1-1}{(s-c_1)}^{{\rho }_1(q_1-1)}ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_1(q_2-1)}ds\}\hfill \\ & & +(n_1\parallel v_1-v_2\parallel +n_2\parallel w_1-w_2\parallel )\{\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}}\hfill \\ & & \times \sum _{i=1}^n\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_2(q_2-1)}ds\hfill \\ & & +\frac{|A_1|(d_1-c_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_2(q_2-1)}ds\}\hfill \\ & \le & (m_1\parallel v_1-v_2\parallel +m_2\parallel w_1-w_2\parallel )\hfill \\ & & \times \{\frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{{(d_1-c_1)}^2\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{|A_1|\Gamma ({\rho }_1(q_2-1)+1)}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)\Gamma ({\rho }_1(q_2-1)+{\mu }_2)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{({\overline{\eta }}_j-c_1)}^{{\rho }_1(q_2-1)+{\mu }_2}\}\hfill \\ & & +(n_1\parallel v_1-v_2\parallel +n_2\parallel w_1-w_2\parallel )\hfill \\ & & \times \{\frac{{(d_1-c_1)}^2\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{({\eta }_i-c_1)}^{{\mu }_2+{\rho }_2(q_2-1)}\hfill \\ & & +\frac{|{\mathbb{A}}_1|(d_1-c_1)\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\}\hfill \\ & =& (m_1\parallel v_1-v_2\parallel +m_2\parallel w_1-w_2\parallel ){\Phi }_1+(n_1\parallel v_1-v_2\parallel +n_2\parallel w_1-w_2\parallel ){\Phi }_2\hfill \\ & =& (m_1{\Phi }_1+n_1{\Phi }_2)\parallel v_1-v_2\parallel +(m_2{\Phi }_1+n_2{\Phi }_2)\parallel w_1-w_2\parallel .\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \parallel {\mathbb{B}}_1(v_2,w_2)-{\mathbb{B}}_1(v_1,w_1)\parallel \le (m_1{\Phi }_1+n_1{\Phi }_2+m_2{\Phi }_1+n_2{\Phi }_2)(\parallel v_2-v_1\parallel +\parallel w_2-w_1\parallel ).\end{array}$$

(23)

Similarly, one can find that

$$\begin{array}{c}\hfill \parallel {\mathbb{B}}_2(v_2,w_2)-{\mathbb{B}}_2(v_1,w_1)\parallel \le (m_1{\Phi }_4+n_1{\Phi }_3+m_2{\Phi }_4+n_2{\Phi }_3)(\parallel v_2-v_1\parallel +\parallel w_2-w_1\parallel ).\end{array}$$

(24)

By (23) and (24), we infer that

$$\begin{array}{ccc}\hfill & & \parallel \mathbb{B}(v_2,w_2)-\mathbb{B}(v_1,w_1)\parallel \hfill \\ & \le & (({\Phi }_1+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3)(n_1+n_2))(\parallel v_2-v_1\parallel +\parallel w_2-w_1\parallel ),\hfill \end{array}$$

(25)

which implies that $\mathbb{B}$ is a contraction. Thus, by applying the Banach contraction mapping principle, the system (3) and (4) has a unique solution on $[c_1,d_1]$. The proof is completed. □

4.2. Existence Results

Now, two existence results for the system (3) and (4) are proved via Leray–Schauder alternative [21] and Krasnosel’skiĭ fixed point theorem [22].

Theorem 2. 

Let ${\kappa }_1,{\kappa }_2:[c_1,d_1]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ be two continuous functions such that for all $q\in [c_1,d_1]$ and $v_i,w_i\in \mathbb{R}$, we have

$$\begin{array}{ccc}& & |{\kappa }_1(q,v_1,w_1)|\le t_0+t_1|v_1|+t_2\left|w_1\right|,\hfill \\ & & |{\kappa }_2(q,v_2,w_2)|\le u_0+u_1|v_2|+u_2\left|w_2\right|,\hfill \end{array}$$

where $t_i,u_i$ are real constants with $t_0,u_0>0$. Then, the system (3) and (4) has at least one solution on $[c_1,d_1],$ provided that

$$\begin{array}{c}\hfill ({\Phi }_1+{\Phi }_4)t_1+({\Phi }_2+{\Phi }_3)u_1<1,and({\Phi }_1+{\Phi }_4)t_2+({\Phi }_2+{\Phi }_3)u_2<1,\end{array}$$

(26)

where ${\Phi }_i$, $i=1,2,3,4$ are defined in (21).

Proof. 

In view of the continuity property of the functions ${\kappa }_1$ and ${\kappa }_2$, we conclude that the operator $\mathbb{B}$ is continuous. Next, the completely continuous property of the operator $\mathbb{B}$ is showed. Let S be a bounded set of ${\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast }$. Then, there exist positive constant ${\mathbb{D}}_1$ and ${\mathbb{D}}_2$ such that for all $(v,w)\in S$ we have $|{\kappa }_1(q,v\left(q\right),w\left(q\right))|\le {\mathbb{D}}_1$ and $|{\kappa }_2(q,v\left(q\right),w\left(q\right))|\le {\mathbb{D}}_2.$ In consequence, for all $(u,v)\in S$, we have

$$\begin{array}{ccc}& & |{\mathbb{B}}_1(v,w)\left(q\right)|\hfill \\ & \le & {\mathbb{D}}_1\{\frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{{(d_1-c_1)}^2\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{|A_1|\Gamma ({\rho }_1(q_2-1)+1)}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)\Gamma ({\rho }_1(q_2-1)+{\mu }_2)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{({\overline{\eta }}_j-c_1)}^{{\rho }_1(q_2-1)+{\mu }_2}\}\hfill \\ & & +{\mathbb{D}}_2\{\frac{{(d_1-c_1)}^2\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{({\eta }_i-c_1)}^{{\mu }_2+{\rho }_2(q_2-1)}\hfill \\ & & +\frac{|{\mathbb{A}}_1|(d_1-c_1)\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\},\hfill \end{array}$$

which yields

$$\begin{array}{c}\hfill \parallel {\mathbb{B}}_1(v,w)\parallel \le {\Phi }_1{\mathbb{D}}_1+{\Phi }_2{\mathbb{D}}_2.\end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill \parallel {\mathbb{B}}_2(v,w)\parallel \le {\Phi }_3{\mathbb{D}}_1+{\Phi }_4{\mathbb{D}}_2.\end{array}$$

Hence, we obtain

$$\begin{array}{c}\hfill \parallel \mathbb{B}(v,w)\parallel =\parallel {\mathbb{B}}_1(v,w)\parallel +\parallel {\mathbb{B}}_2(v,w)\parallel \le ({\Phi }_1+{\Phi }_3){\mathbb{D}}_1+({\Phi }_2+{\Phi }_4){\mathbb{D}}_2.\end{array}$$

Consequently, the uniformly boundedness property of the operator $\mathbb{B}$ is obtained. Now the equicontinuous property of the operator $\mathbb{B}$ is verified. Let ${\overline{q}}_1,{\overline{q}}_2\in [c_1,d_1]$ with ${\overline{q}}_1<{\overline{q}}_2$, Then, we obtain

$$\begin{array}{ccc}& & |{\mathbb{B}}_1(v,w)\left({\overline{q}}_2\right)-{\mathbb{B}}_1(v,w)\left({\overline{q}}_1\right)|\hfill \\ & \le & \frac{{\mathbb{D}}_1}{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{{\overline{q}}_1}|{({\overline{q}}_2-s)}^{{\mu }_1-1}-{({\overline{q}}_1-s)}^{{\mu }_1-1}|{\theta }_{q_1}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{{\mathbb{D}}_1}{\Gamma \left({\mu }_1\right)}{\int }_{{\overline{q}}_1}^{{\overline{q}}_2}\left|{({\overline{q}}_2-s)}^{{\mu }_1-1}\right|{\theta }_{q_1}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}(d_1-c_1){\mathbb{D}}_1\sum _{i=1}^m\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}d\tau \right)ds\hfill \\ & & +\frac{({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_1\right)}\sum _{j=1}^n\left|{\lambda }_j\right|{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}d\tau \right)ds\hfill \\ & \le & \frac{{\mathbb{D}}_1}{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{c_1}^{{\overline{q}}_1}|{({\overline{q}}_2-s)}^{{\mu }_1-1}-{({\overline{q}}_1-s)}^{{\mu }_1-1}|{\theta }_{q_1}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\hfill \\ & & +\frac{{\mathbb{D}}_1}{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{{\overline{q}}_1}^{{\overline{q}}_2}\left|{({\overline{q}}_2-s)}^{{\mu }_1-1}\right|{\theta }_{q_1}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\hfill \\ & & +\frac{({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}(d_1-c_1){\mathbb{D}}_1\sum _{i=1}^m\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}d\tau \right)ds\hfill \\ & & +\frac{({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_1\right)}\sum _{j=1}^n\left|{\lambda }_j\right|{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}d\tau \right)ds\hfill \\ & \le & \frac{{\mathbb{D}}_1}{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{c_1}^{{\overline{q}}_1}|{({\overline{q}}_2-s)}^{{\mu }_1-1}-{({\overline{q}}_1-s)}^{{\mu }_1-1}|{(s-c_1)}^{{\rho }_1}ds\hfill \\ & & +\frac{{\mathbb{D}}_1}{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{{\overline{q}}_1}^{{\overline{q}}_2}\left|{({\overline{q}}_2-s)}^{{\mu }_1-1}\right|{(s-c_1)}^{{\rho }_1}ds\hfill \\ & & +\frac{({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}(d_1-c_1){\mathbb{D}}_1\sum _{i=1}^m\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}d\tau \right)ds\hfill \\ & & +\frac{({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_1\right)}\sum _{j=1}^n\left|{\lambda }_j\right|{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_1-1}{\theta }_{q_1}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|({\overline{q}}_2-{\overline{q}}_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{\theta }_{q_2}\left({\int }_{c_1}^s{\mathbb{D}}_1\frac{{(s-\tau )}^{{\rho }_2-1}}{\Gamma \left({\rho }_2\right)}d\tau \right)ds\to 0,\hfill \end{array}$$

as ${\overline{q}}_1\to {\overline{q}}_2,$ independently of $(v,w)\in S$. Hence, ${\mathbb{B}}_1(v,w)$ is equicontinuous. Similarly, we can show that ${\mathbb{B}}_2(v,w)$ is equicontinuous. Consequently, $\mathbb{B}(v,w)$ is equicontinuous.

Finally, it will be indicated that the set $\mathcal{E}=\{(v,w)\in {\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast };\parallel (v,w)\parallel =\lambda \mathbb{B}(v,w)\},0\le \lambda \le 1\}$ is bounded. Let $(v,w)\in \mathcal{E}$, then $\parallel (v,w)\parallel =\lambda \mathbb{B}(v,w)$ and for all $q\in [c_1,d_1]$ we have $v\left(q\right)=\lambda {\mathbb{B}}_1(v,w)\left(q\right)$ and $w\left(q\right)=\lambda {\mathbb{B}}_2(v,w)\left(q\right).$ Thus, we have

$$\begin{array}{ccc}& & \left|v\right(q\left)\right|\hfill \\ & \le & (t_0+t_1\parallel v\parallel +t_2\parallel w\parallel \left)\right\{\frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{{(d_1-c_1)}^2\Gamma ({\rho }_1(q_1-1)+1)}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\hfill \\ & & +\frac{|A_1|\Gamma ({\rho }_1(q_2-1)+1)}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)\Gamma ({\rho }_1(q_2-1)+{\mu }_2)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{({\overline{\eta }}_j-c_1)}^{{\rho }_1(q_2-1)+{\mu }_2}\}\hfill \\ & & +(u_0+u_1\parallel v\parallel +u_2\parallel w\parallel )\hfill \\ & & \times \{\frac{{(d_1-c_1)}^2\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{({\eta }_i-c_1)}^{{\mu }_2+{\rho }_2(q_2-1)}\hfill \\ & & +\frac{|{\mathbb{A}}_1|(d_1-c_1)\Gamma ({\rho }_2(q_2-1)+1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\}.\hfill \end{array}$$

and consequently

$$\parallel v\parallel \le (t_0+t_1\parallel v\parallel +t_2\parallel w\parallel ){\Phi }_1+(u_0+u_1\parallel v\parallel +u_2\parallel w\parallel ){\Phi }_2.$$

Similarly, we obtain

$$\begin{array}{c}\hfill \parallel w\parallel \le (t_0+t_1\parallel v\parallel +t_2\parallel w\parallel ){\Phi }_4+(u_0+u_1\parallel v\parallel +u_2\parallel w\parallel ){\Phi }_3.\end{array}$$

Hence, we have

$$\begin{array}{ccc}\hfill \parallel v\parallel +\parallel w\parallel & \le & ({\Phi }_1+{\Phi }_4)t_0+({\Phi }_2+{\Phi }_3)u_0+[(({\Phi }_1+{\Phi }_4)t_1+({\Phi }_2+{\Phi }_3)u_1]\parallel v\parallel \hfill \\ & & +[(({\Phi }_1+{\Phi }_4)t_2+({\Phi }_2+{\Phi }_3)u_2]\parallel w\parallel ,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel (v,w)\parallel \le \frac{({\Phi }_1+{\Phi }_4)t_0+({\Phi }_2+{\Phi }_3)u_0}{{\mathrm{N}}_0},\end{array}$$

where

$${\mathrm{N}}_0=min\{1-[({\Phi }_1+{\Phi }_4)t_1+({\Phi }_2+{\Phi }_3)u_1],1-[({\Phi }_1+{\Phi }_4)t_2+({\Phi }_2+{\Phi }_3)u_2]\}.$$

Thus, the Leray–Schauder alternative implies that the operator $\mathbb{B}$ has at least one fixed point. Hence, the system (3) and (4) has at least one solution on $[c_1,d_1]$. The proof is completed. □

Now, Krasnosel’skiĭ’s fixed point theorem [22] is applied to prove the second existence result.

Theorem 3. 

Let ${\kappa }_1,{\kappa }_2:[c_1,d_1]\times \mathbb{R}⟶\mathbb{R}$ be two continuous functions satisfying the condition

• $\left(H_1\right)$ of Theorem 1. Moreover, suppose that
• $\left(H_2\right)$ There exist $\mathcal{R},\mathcal{S}\in C([c_1,d_1],{\mathbb{R}}_{+})$ such that

  $$\begin{array}{c}\hfill |{\kappa }_1(q,v,w)|\le \mathcal{R}\left(q\right),|{\kappa }_2(q,v,w)|\le \mathcal{S}\left(q\right),foreach(q,v,w)\in [c_1,d_1]\times \mathbb{R}\times \mathbb{R}.\end{array}$$

Then, the problem (3) and (4) has at least one solution on $[c_1,d_1],$ provided that

$$\begin{array}{c}\hfill [{\Phi }_1^{\ast }+{\Phi }_4](m_1+m_2)+[{\Phi }_3^{\ast }+{\Phi }_2](n_1+n_2)<1.\end{array}$$

(27)

Proof. 

We split the operator $\mathbb{B}$ into four operator ${\mathbb{B}}_{1,1}$, ${\mathbb{B}}_{1,2}$, ${\mathbb{B}}_{2,1}$, ${\mathbb{B}}_{2,2}$ as

$$\begin{array}{ccc}\hfill {\mathbb{B}}_{1,1}(v,w)\left(q\right)& =& {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1(q,v\left(q\right),w\left(q\right))\right)\right),q\in [c_1,d_1],}\hfill \\ \hfill {\mathbb{B}}_{1,2}(v,w)\left(q\right)& =& {\displaystyle \frac{q-c_1}{\mathbb{A}}[(d_1-c_1)\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2({\eta }_i,v\left({\eta }_i\right),w\left({\eta }_i\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)}\hfill \\ & & {\displaystyle +{\mathbb{A}}_1(\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1({\overline{\eta }}_j,v\left({\overline{\eta }}_j\right),w\left({\overline{\eta }}_j\right))\right)\right)}\hfill \\ & & {\displaystyle -{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)\left)\right],}\hfill \\ \hfill {\mathbb{B}}_{2,1}(v,w)\left(q\right)& =& {\displaystyle {}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2(q,v\left(q\right),w\left(q\right))\right)\right),}\hfill \\ \hfill {\mathbb{B}}_{2,2}(v,w)\left(q\right)& =& {\displaystyle \frac{q-c_1}{\mathbb{A}}[{\mathbb{A}}_2\left(\sum _{i=1}^m{\lambda }_i{}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2({\eta }_i,v\left({\eta }_i\right),w\left({\eta }_i\right))\right)\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)}\hfill \\ & & {\displaystyle +(d_1-c_1)\sum _{j=1}^n{\overline{\lambda }}_j{}^{RL}{\mathbb{I}}^{{\mu }_1}\left({\theta }_{q_1}\left({}^{RL}{\mathbb{I}}^{{\rho }_1}{\kappa }_1({\overline{\eta }}_j,v\left({\eta }_j\right),w\left({\eta }_j\right))\right)\right)}\hfill \\ & & {\displaystyle -(d_1-c_1){}^{RL}{\mathbb{I}}^{{\mu }_2}\left({\theta }_{q_2}\left({}^{RL}{\mathbb{I}}^{{\rho }_2}{\kappa }_2(d_1,v\left(d_1\right),w\left(d_1\right))\right)\right)].}\hfill \end{array}$$

Obviously, ${\mathbb{B}}_1={\mathbb{B}}_{1,1}+{\mathbb{B}}_{1,2}$ and ${\mathbb{B}}_2={\mathbb{B}}_{2,1}+{\mathbb{B}}_{2,2}$. Let ${\mathbb{B}}_x=\{(v,w)\in {\mathbb{X}}^{\ast }\times {\mathbb{X}}^{\ast };\parallel (v,w)\parallel \le x\}$ with $x\ge ({\Phi }_1+{\Phi }_3)\parallel \mathcal{R}\parallel +({\Phi }_2+{\Phi }_4)\parallel \mathcal{S}\parallel .$ According to the proof of Theorem 2, we have

$$\begin{array}{c}\hfill \parallel {\mathbb{B}}_{1,1}(v_1,v_2)+{\mathbb{B}}_{1,2}(w_1,w_2)\parallel \le {\Phi }_1\parallel \mathcal{R}\parallel +{\Phi }_2\parallel \mathcal{S}\parallel ,\end{array}$$

and

$$\begin{array}{c}\hfill \parallel {\mathbb{B}}_{2,1}(v_1,v_2)+{\mathbb{B}}_{2,2}(w_1,w_2)\parallel \le {\Phi }_3\parallel \mathcal{R}\parallel +{\Phi }_4\parallel \mathcal{S}\parallel .\end{array}$$

Hence, ${\mathbb{B}}_1(v_1,v_2)+{\mathbb{B}}_2(w_1,w_2)\in {\mathbb{B}}_x$. Next, it will be proved that the operator $({\mathbb{B}}_{1,2},{\mathbb{B}}_{2,2})$ is a contraction. As in the proof of Theorem 1, for $(v_1,w_1),(v_2,w_2)\in B_x$ we have

$$\begin{array}{ccc}\hfill & & |{\mathbb{B}}_{1,2}(v_1,v_2)\left(q\right)-{\mathbb{B}}_{1,2}(w_1,w_2)\left(q\right)|\hfill \\ & \le & (m_1\parallel v_1-w_1\parallel +m_2\parallel v_2-w_2\parallel )\hfill \\ & & \times \{\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma {({\rho }_1+1)}^{q_1-1}\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{d_1}{(b-s)}^{{\mu }_1-1}{(s-c_1)}^{{\rho }_1(q_1-1)}ds\hfill \\ & & +\frac{|{\mathbb{A}}_1|}{\Gamma {({\rho }_1+1)}^{q_2-1}\Gamma \left({\mu }_2\right)}\sum _{j=1}^n\left|{\overline{\lambda }}_j\right|{\int }_{c_1}^{{\overline{\eta }}_j}{({\overline{\eta }}_j-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_1(q_2-1)}ds\}\hfill \\ & & +(n_1\parallel v_1-v_2\parallel +n_2\parallel w_1-w_2\parallel )\{\frac{{(d_1-c_1)}^2}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}}\hfill \\ & & \times \sum _{i=1}^n\left|{\lambda }_i\right|{\int }_{c_1}^{{\eta }_i}{({\eta }_i-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_2(q_2-1)}ds\hfill \\ & & +\frac{|A_1|(d_1-c_1)}{\left|\mathbb{A}\right|\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+1)}^{q_2-1}}{\int }_{c_1}^{d_1}{(d_1-s)}^{{\mu }_2-1}{(s-c_1)}^{{\rho }_2(q_2-1)}ds\}\hfill \\ & \le & ({\Phi }_1^{\ast }(m_1\parallel v_1-w_1\parallel +\parallel v_2-w_2\parallel )+{\Phi }_2(n_1\parallel v_1-w_1\parallel +\parallel v_2-w_2\parallel )\hfill \\ & =& [{\Phi }_1^{\ast }{m}_1+{\Phi }_2{n}_1]\parallel v_1-w_1\parallel +[{\Phi }_1^{\ast }{m}_2+{\Phi }_2{n}_2]\parallel v_2-w_2\parallel ,\hfill \end{array}$$

(28)

and consequently

$$\begin{array}{ccc}& & \parallel {\mathbb{B}}_{1,2}(v_1,v_2)-{\mathbb{B}}_{1,2}(w_1,w_2)\parallel \hfill \\ & \le & [{\Phi }_1^{\ast }{m}_1+{\Phi }_2{n}_1]\parallel v_1-w_1\parallel +[{\Phi }_1^{\ast }{m}_2+{\Phi }_2{n}_2]\parallel v_2-w_2\parallel .\hfill \end{array}$$

In a similar way, we obtain

$$\begin{array}{ccc}\hfill & & \parallel {\mathbb{B}}_{2,2}(v_1,v_2)-{\mathbb{B}}_{2,2}(w_1,w_2)\parallel \hfill \\ & \le & [{\Phi }_4{m}_1+{\Phi }_3^{\ast }{n}_1]\parallel v_1-w_1\parallel +[{\Phi }_4{m}_2+{\Phi }_3^{\ast }{n}_2]\parallel v_2-w_2\parallel .\hfill \end{array}$$

(29)

By (28) and (29), we obtain

$$\begin{array}{ccc}\hfill & & \parallel ({\mathbb{B}}_{1,2},{\mathbb{B}}_{2,2})(v_1,v_2)-({\mathbb{B}}_{1,2},{\mathbb{B}}_{2,2})(w_1,w_2)\parallel \hfill \\ & \le & \left\{[{\Phi }_1^{\ast }+{\Phi }_4](m_1+m_2)+[{\Phi }_2+{\Phi }_3^{\ast }](n_1+n_2)\right\}(\parallel v_1-w_1\parallel +\parallel v_2-w_2\parallel ),\hfill \end{array}$$

(30)

which, by applying the condition (27), the contraction property of the operator $({\mathbb{B}}_{1,2},{\mathbb{B}}_{2,2})$ is obtained. On the other hand, in view of the continuity property of ${\kappa }_1$ and ${\kappa }_2$, the operator $({\mathbb{B}}_{1,1},{\mathbb{B}}_{2,1})$ is continuous. In addition,

$$\begin{array}{ccc}& & \parallel ({\mathbb{B}}_{1,2},{\mathbb{B}}_{2,1})(v,w)\parallel \hfill \\ & \le & \frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\parallel \mathcal{R}\parallel \hfill \\ & & +\frac{\Gamma ({\rho }_2(q_2-1)+1)}{\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\parallel \mathcal{S}\parallel ,\hfill \end{array}$$

as

$$\parallel {\mathbb{B}}_{1,2}(v,w)\parallel \le \frac{\Gamma ({\rho }_1(q_1-1)+1)}{\Gamma \left({\mu }_1\right)\Gamma {({\rho }_1+)}^{q_1-1}\Gamma ({\rho }_1(q_1-1)+{\mu }_1)}{(d_1-c_1)}^{{\rho }_1(q_1-1)+{\mu }_1}\parallel \mathcal{R}\parallel ,$$

and

$$\parallel {\mathbb{B}}_{2,1}(v,w)\parallel \le \frac{\Gamma ({\rho }_2(q_2-1)+1)}{\Gamma \left({\mu }_2\right)\Gamma {({\rho }_2+)}^{q_2-1}\Gamma ({\rho }_2(q_2-1)+{\mu }_2)}{(d_1-c_1)}^{{\rho }_2(q_2-1)+{\mu }_2}\parallel \mathcal{S}\parallel .$$

Thus, $({\mathbb{B}}_{1,1},{\mathbb{B}}_{2,1})$ is uniformly bounded. In the final step, we prove that the operator $({\mathbb{B}}_{1,1},{\mathbb{B}}_{2,1}){\mathbb{B}}_x$ is equicontinuous. For ${\overline{q}}_1,{\overline{q}}_2\in [c_1,d_1]$ with ${\overline{q}}_1<{\overline{q}}_2$ and for all $(v,w)\in {\mathbb{B}}_x$, we have

$$\begin{array}{ccc}& & |{\mathbb{B}}_{1,1}(v,w)\left({\overline{q}}_2\right)-{\mathbb{B}}_{1,1}(v,w)\left({\overline{q}}_1\right)|\hfill \\ & \le & \frac{\parallel \mathcal{R}\parallel }{\Gamma \left({\mu }_1\right)}{\int }_{c_1}^{{\overline{q}}_1}|{({\overline{q}}_2-s)}^{{\mu }_1-1}-{({\overline{q}}_1-s)}^{{\mu }_1-1}|{\theta }_{q_1}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & & +\frac{\parallel \mathcal{R}\parallel }{\Gamma \left({\mu }_1\right)}{\int }_{{\overline{q}}_1}^{{\overline{q}}_2}\left|{({\overline{q}}_2-s)}^{{\mu }_1-1}\right|{\theta }_{q_1}\left({\int }_{c_1}^s\frac{{(s-\tau )}^{{\rho }_1-1}}{\Gamma \left({\rho }_1\right)}d\tau \right)ds\hfill \\ & \le & \frac{\parallel \mathcal{R}\parallel }{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{c_1}^{{\overline{q}}_1}|{({\overline{q}}_2-s)}^{{\mu }_1-1}-{({\overline{q}}_1-s)}^{{\mu }_1-1}|{\theta }_{q_1}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\hfill \\ & & +\frac{\parallel \mathcal{R}\parallel }{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{{\overline{q}}_1}^{{\overline{q}}_2}\left|{({\overline{q}}_2-s)}^{{\mu }_1-1}\right|{\theta }_{q_1}\left(\frac{{(s-c_1)}^{{\rho }_1}}{\Gamma ({\rho }_1+1)}\right)ds\hfill \\ & \le & \frac{\parallel \mathcal{R}\parallel }{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{c_1}^{{\overline{q}}_1}|{({\overline{q}}_2-s)}^{{\mu }_1-1}-{({\overline{q}}_1-s)}^{{\mu }_1-1}|{(s-c_1)}^{{\rho }_1}ds\hfill \\ & & +\frac{\parallel \mathcal{R}\parallel }{\Gamma \left({\mu }_1\right){(\Gamma ({\rho }_1+1))}^{q_1-1}}{\int }_{{\overline{q}}_1}^{{\overline{q}}_2}\left|{({\overline{q}}_2-s)}^{{\mu }_1-1}\right|{(s-c_1)}^{{\rho }_1}ds\to 0,\hfill \end{array}$$

independently of $(v,w)\in {\mathbb{B}}_x$. Similarly, one can find that

$$|{\mathbb{B}}_{2,1}(v,w)\left({\overline{q}}_2\right)-{\mathbb{B}}_{2,1}(v,w)\left({\overline{q}}_1\right)|⟶0as{\overline{q}}_1⟶{\overline{q}}_2.$$

Thus, $|({\mathbb{B}}_{1,1},{\mathbb{B}}_{2,1})\left({\overline{q}}_2\right)-({\mathbb{B}}_{1,1},{\mathbb{B}}_{2,1})\left({\overline{q}}_1\right)|⟶0$ as ${\overline{q}}_1⟶{\overline{q}}_2$. Hence, the operator $({\mathbb{B}}_{1,1},{\mathbb{B}}_{1,2})$ is equicontinuous. Arzelá–Ascoli theorem implies that the operator $({\mathbb{B}}_{1,1},{\mathbb{B}}_{2,1})$ is compact on ${\mathbb{B}}_x$ and hence by applying Krasnosel’skiĭ’s fixed point theorem, we conclude that the operator $\mathbb{B}$ has at least one fixed point, which is a solution of the system (3) and (4). The proof is finished. □

5. Examples

In this section, we illustrate our main results by considering the following system of nonlinear differential equations consisting of the Caputo fractional-order derivatives of the form

$$\left\{\begin{array}{c}{\displaystyle {}^C{\mathcal{D}}^{\frac{1}{2}}\left({\theta }_2\left[{}^C{\mathcal{D}}^{\frac{4}{3}}{\sigma }_1\left(w\right)\right]\right)={\kappa }_1(w,{\sigma }_1\left(w\right),{\sigma }_2\left(w\right)),w\in [1/4,11/4],}\hfill \\ {\displaystyle {}^C{\mathcal{D}}^{\frac{1}{3}}\left({\theta }_3\left[{}^C{\mathcal{D}}^{\frac{5}{3}}{\sigma }_2\left(w\right)\right]\right)={\kappa }_2(w,{\sigma }_1\left(w\right),{\sigma }_2\left(w\right)),}\hfill \end{array}\right.$$

(31)

with coupled boundary conditions as

$$\left\{\begin{array}{c}{\displaystyle {\sigma }_1\left(\frac{1}{4}\right)={}^C{\mathcal{D}}^{\frac{4}{3}}{\sigma }_1\left(\frac{1}{4}\right)=0,}\hfill \\ {\displaystyle {\sigma }_1\left(\frac{11}{4}\right)=\frac{1}{11}{\sigma }_2\left(\frac{1}{2}\right)+\frac{2}{13}{\sigma }_2\left(\frac{3}{4}\right)+\frac{3}{17}{\sigma }_2\left(\frac{5}{4}\right),}\hfill \\ {\displaystyle {\sigma }_2\left(\frac{1}{4}\right)={}^C{\mathcal{D}}^{\frac{5}{3}}{\sigma }_2\left(\frac{1}{4}\right)=0,}\hfill \\ {\displaystyle {\sigma }_2\left(\frac{11}{4}\right)=\frac{4}{19}{\sigma }_1\left(\frac{3}{2}\right)+\frac{5}{23}{\sigma }_1\left(\frac{7}{4}\right)+\frac{6}{29}{\sigma }_1\left(\frac{9}{4}\right)+\frac{7}{31}{\sigma }_1\left(\frac{5}{2}\right).}\hfill \end{array}\right.$$

(32)

Comparing problem (31) and (32) with (3) and (4), we have ${\rho }_1=1/2$, ${\rho }_2=1/3$, ${\mu }_1=4/3$, ${\mu }_2=5/3$, $p_1=2$, $p_2=3,$$q_1=2,$$q_2=3/2,$$c_1=1/4$, $d_1=11/4$, $m=3$, ${\lambda }_1=1/11$, ${\lambda }_2=2/13$, ${\lambda }_3=3/17$, ${\eta }_1=1/2$, ${\eta }_2=3/4$, ${\eta }_3=5/4$, $n=4$, ${\overline{\lambda }}_1=4/19$, ${\overline{\lambda }}_2=5/23$, ${\overline{\lambda }}_3=6/29$, ${\overline{\lambda }}_4=7/31$,${\overline{\eta }}_1=3/2$, ${\overline{\eta }}_2=7/4$, ${\overline{\eta }}_3=9/4$, ${\overline{\eta }}_4=5/2$. By using the Maple program, we obtain ${\mathbb{A}}_1\approx 0.2761209379$, ${\mathbb{A}}_2\approx 1.511102471$, $\mathbb{A}\approx 5.832752968$, ${\Phi }_1\approx 13.37197426$, ${\Phi }_2\approx 1.880065187$, ${\Phi }_3\approx 12.91632487$, ${\Phi }_4\approx 10.85384001$, ${\Phi }_1^{\ast }\approx 6.985157994$ and ${\Phi }_3^{\ast }\approx 6.713970944$.

(i) Assume that two nonlinear functions are given by

$${\kappa }_1(w,{\sigma }_1,{\sigma }_2)=\frac{2}{5(4w+63)}\left(\frac{{\sigma }_1^2+2\left|{\sigma }_1\right|}{1+|{\sigma }_1|}\right)+\frac{256}{{(4w+11)}^4}\left(\frac{|{\sigma }_2|}{1+|{\sigma }_2|}\right)+\frac{1}{4},$$

(33)

and

$${\kappa }_2(w,{\sigma }_1,{\sigma }_2)=\frac{1}{2(8w+35)}|sin{\sigma }_1|+\frac{1}{38{(4w+1)}^2}\left(\frac{{\sigma }_2^2+2\left|{\sigma }_2\right|}{1+|{\sigma }_2|}\right)+\frac{1}{5}.$$

(34)

Observe that both of them are unbounded and also satisfy the Lipschitz condition as

$$|{\kappa }_1(w,{\sigma }_1,{\sigma }_2)-{\kappa }_1(w,{\delta }_1,{\delta }_2)|\le \frac{1}{80}|{\sigma }_1-{\delta }_1|+\frac{1}{81}|{\sigma }_2-{\delta }_2|,$$

and

$$|{\kappa }_2(w,{\sigma }_1,{\sigma }_2)-{\kappa }_2(w,{\delta }_1,{\delta }_2)|\le \frac{1}{74}|{\sigma }_1-{\delta }_1|+\frac{1}{76}|{\sigma }_2-{\delta }_2|,$$

with Lipschitz constants $m_1=1/80$, $m_2=1/81$, $n_1=1/74$ and $n_2=1/76$. Since

$$({\Phi }_1+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3)(n_1+n_2)\approx 0.9965473651<1.$$

by Theorem 1, the problem (31) and (32), with functions ${\kappa }_1,{\kappa }_1$ given in (33) and (34), has a unique solution on the interval $[1/4,11/4]$.

(ii) If two nonlinear functions are defined by

$${\kappa }_1(w,{\sigma }_1,{\sigma }_2)=\frac{1}{2}+\frac{1}{2(4w+20)}\left(\frac{|{\sigma }_1{|}^{2023}}{1+{\sigma }_1^{2022}}\right)+\frac{1}{5(4w+7)}\left|{\sigma }_2\right|e^{-{\sigma }_1^2},$$

(35)

and

$${\kappa }_2(w,{\sigma }_1,{\sigma }_2)=\frac{1}{3}+\frac{1}{28(w+1)}|{\sigma }_1|{sin}^{26}\left(\pi {\sigma }_2^4\right)+\frac{1}{2\pi (2w+9)}\left|{\sigma }_2\right|{tan}^{-1}{\sigma }_1^8,$$

(36)

then they are bounded, since

$$|{\kappa }_1(w,{\sigma }_1,{\sigma }_2)|\le \frac{1}{2}+\frac{1}{42}|{\sigma }_1|+\frac{1}{40}\left|{\sigma }_2\right|,$$

and

$$|{\kappa }_2(w,{\sigma }_1,{\sigma }_2)|\le \frac{1}{3}+\frac{1}{35}|{\sigma }_1|+\frac{1}{38}\left|{\sigma }_2\right|.$$

By setting constants $t_0=1/2$, $t_1=1/42$, $t_2=1/40$, $u_0=1/3$, $u_1=1/35$, $u_2=1/38$, the inequalities (26) are satisfied as

$$({\Phi }_1+{\Phi }_4)t_1+({\Phi }_2+{\Phi }_3)u_1\approx 0.9995591034<1$$

and

$$({\Phi }_1+{\Phi }_4)t_2+({\Phi }_2+{\Phi }_3)u_2\approx 0.9950240426<1.$$

By Theorem 2, the fractional p-Laplacian coupled system with multi-point boundary conditions (31) and (32), with ${\kappa }_1,{\kappa }_2$ given in (35) and (36), has at least one solution on the interval $[1/4,11/4]$.

(iii) Consider the nonlinear functions

$${\kappa }_1(w,{\sigma }_1,{\sigma }_2)=\frac{1}{6}+\frac{1}{8(w+6)}\left(\frac{|{\sigma }_1|}{1+|{\sigma }_1|}\right)+\frac{1}{13{(4w+1)}^2}sin\left|{\sigma }_2\right|,$$

(37)

and

$${\kappa }_2(w,{\sigma }_1,{\sigma }_2)=\frac{1}{8}+\frac{1}{48(w+1)}{tan}^{-1}\left|{\sigma }_1\right|+\frac{1}{29(4w+1)}\left(\frac{|{\sigma }_2|}{1+|{\sigma }_2|}\right).$$

(38)

It is clear that both of them are bounded by

$$|{\kappa }_1(w,{\sigma }_1,{\sigma }_2)|\le \frac{1}{6}+\frac{1}{8(w+6)}+\frac{1}{13{(4w+1)}^2}:=\mathcal{R}\left(w\right),$$

and

$$|{\kappa }_2(w,{\sigma }_1,{\sigma }_2)|\le \frac{1}{8}+\frac{\pi }{96(w+1)}+\frac{1}{29(4w+1)}:=\mathcal{S}\left(w\right).$$

In addition, they satisfy the Lipschitz condition as

$$|{\kappa }_1(w,{\sigma }_1,{\sigma }_2)-{\kappa }_1(w,{\delta }_1,{\delta }_2)|\le \frac{1}{50}|{\sigma }_1-{\delta }_1|+\frac{1}{52}|{\sigma }_2-{\delta }_2|,$$

and

$$|{\kappa }_2(w,{\sigma }_1,{\sigma }_2)-{\kappa }_2(w,{\delta }_1,{\delta }_2)|\le \frac{1}{60}|{\sigma }_1-{\delta }_1|+\frac{1}{58}|{\sigma }_2-{\delta }_2|,$$

with Lipschitz constants $m_1=1/50$, $m_2=1/52$, $n_1=1/60$, $n_2=1/58$. Unfortunately, the inequality

$$({\Phi }_1+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3)(n_1+n_2)\approx 1.452114004>1,$$

is failed and cannot be used to guarantee the uniqueness result. However, we have

$$({\Phi }_1^{\ast }+{\Phi }_4)(m_1+m_2)+({\Phi }_2+{\Phi }_3^{\ast })(n_1+n_2)\approx 0.9912445862<1,$$

which means that the inequality in (27) holds. Consequently, by Theorem 3, the fractional p-Laplacian system (31) and (32), with two nonlinear bounded Lipschitz functions (37) and (38), has at least one solution on the interval $[1/4,11/4]$.

6. Conclusions

In this paper, the existence and uniqueness results have been established for a fractional p-Laplacian coupled system of nonlinear differential equations involving the Caputo fractional-order derivatives, subjected to multi-point boundary conditions. The fixed point technique has been applied to the given system by transforming it into a fixed point problem. Our problem is novel and some other problems are special cases of our problem. For example, by taking ${\mu }_1={\mu }_2=p_1=p_2=2$ and ${\rho }_1={\rho }_2=1$, our problem corresponds to the problem (5). For future work, we plan to extend the results of this paper to other kinds of boundary value problems.