A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

Abstract

We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

1. Introduction

The Langevin equation provides a decent approach to describe the evolution of fluctuating physical phenomena. Examples include anomalous diffusion [1], time evolution of the velocity of the Brownian motion [2,3], diffusion with inertial effects [4], gait variability [5], harmonization of a many-body problem [6], financial aspects [7], etc. However, the failure of the ordinary Langevin equation for correct description of the dynamical systems in complex media led to its several generalizations. One such example is that of the Langevin equation, involving fractional-order derivative operators, which provides a more flexible model for fractal processes. For some recent results on Langevin equation, see ([8,9,10,11,12]) and the references therein.

The topic of q-difference equations has evolved into an important area of research, as such equations are always completely controllable and appear in the q-optimal control problem [13]. Furthermore, the variational q-calculus is regarded as a generalization of the continuous variational calculus due to the presence of an extra parameter q whose nature may be physical or economical. The variational calculus on the q-uniform lattice is concerned with the study of the q-Euler equation and its applications to commutation equations, and isoperimetric and Lagrange problems. In other words, the q-Euler–Lagrange equation is solved for finding the extremum of the functional involved instead of solving the Euler–Lagrange equation [14]. There do exist q-variants of certain significant concepts, such as q-analogues of fractional operators, q-Laplace transform, q-Taylor’s formula, etc.

Fractional-order operators are found to be of great utility in improving the mathematical modeling of several real-world problems. The variational principles based on fractional derivative operators lead to the class of fractional Euler–Lagrange equations [15]. In addition, one can find some interesting results on optimal control theories for fractional differential systems in the articles [16,17,18,19,20,21].

The popularity of fractional calculus in the recent years led to the birth of the fractional analogue of q-difference equations (fractional q-difference equations), for instance, see [22,23]. One can find interesting results on nonlinear boundary value problems involving fractional q-derivative and q-integral operators, and different kinds of boundary conditions in the articles [24,25,26,27,28,29,30,31,32,33,34,35,36,37]. In a recent work [38], the authors studied the existence of solutions for a nonlinear fractional q-integro-difference equation equipped with q-integral boundary conditions. However, it is observed that there are a few results for coupled systems of fractional q-integro-difference equations [39]. More recently, a coupled system of nonlinear fractional q-integro-difference equations with q-integral coupled boundary conditions was studied in [40].

The objective of the present work is to enrich the literature on boundary value problems of coupled systems of fractional q-integro-difference equations. Keeping in mind the importance of the fractional Langevin equation, we introduce and study a new problem consisting of a coupled system of Langevin-type nonlinear fractional q-integro-difference equations complemented with nonlocal multipoint boundary conditions. The proposed problem is interesting in the sense that it enhances the literature on fractional q-variant of Langevin equations with mixed nonlinearities in terms of the parameter $q.$ On the other hand, the consideration of multipoint non-separated boundary conditions involving the values of the unknown functions together with their q-derivatives at the end points as well as the interior nonlocal positions of given domain extends the scope of the present work to a more general situation (also see Section 5). For the motivation of nonlocal boundary conditions, we recall that nonlocal multipoint boundary conditions appear in feedback controls problems, optimal boundary control of (finite) string vibrations arising from interior arbitrary positions, etc. For more details, see [41,42,43,44]. In precise terms, we investigate the following boundary value problem:

$$\left\{\begin{array}{c}{\displaystyle {}^c{D}_q^{p_1}{(}^c{D}_q^{p_2}+{\lambda }_1)x\left(t\right)={\alpha }_1{f}_1(t,x\left(t\right),y\left(t\right))+{\beta }_1{I}_q^{{\xi }_1}{g}_1(t,x\left(t\right),y\left(t\right)),0\le t\le 1,}\\ {\displaystyle {}^c{D}_q^{r_1}{(}^c{D}_q^{r_2}+{\lambda }_2)y\left(t\right)={\alpha }_2{f}_2(t,x\left(t\right),y\left(t\right))+{\beta }_2{I}_q^{{\xi }_2}{g}_2(t,x\left(t\right),y\left(t\right)),0\le t\le 1,}\end{array}\right.$$

(1)

$$\left\{\begin{array}{c}{\displaystyle {\mu }_{1}x\left(0\right)-{\mu }_2\left(t^{(1-p_2)}{D}_{q}x\left(t\right)\right){|}_{t=0}=\sum _{j=1}^n{a}_{j}y\left({\eta }_j\right),}\\ {\displaystyle {\mu }_{3}y\left(0\right)-{\mu }_4\left(t^{(1-r_2)}{D}_{q}y\left(t\right)\right){|}_{t=0}=\sum _{j=1}^n{b}_{j}x\left({\eta }_j\right),}\\ {\displaystyle {\sigma }_{1}x\left(1\right)+{\sigma }_2{D}_{q}x\left(1\right)=\sum _{j=1}^n{k}_j{D}_{q}y\left({\eta }_j\right),}\\ {\displaystyle {\sigma }_{3}y\left(1\right)+{\sigma }_4{D}_{q}y\left(1\right)=\sum _{j=1}^n{m}_j{D}_{q}x\left({\eta }_j\right),}\end{array}\right.$$

(2)

where ${}^c{D}_q^{p_i}$ and ${}^c{D}_q^{r_i}$ denote the fractional q-derivative operators of the Caputo type, $0<p_i,r_i\le 1$, $0<q<1,$$I_q^{{\xi }_i}(.)$ denotes Riemann–Liouville integral of order ${\xi }_i>0,$$f_i,g_i$ are given continuous functions, ${\lambda }_i\ne 0,{\alpha }_i,{\beta }_i,i=1,2,$ and $a_j,b_j,k_j,m_j,j=1,\cdots ,n$ are real constants and ${\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4,{\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4\in \mathbb{R},{\eta }_j\in (0,1),j=1,\cdots ,n.$

Here, one can notice that the right-hand sides of the fractional q-Langevin equations in the system (1) involve the usual as well as q-integral-type nonlinearities. These equations correspond to different combinations of nonlinearities, such as ordinary nonlinearities, $f_1(t,x\left(t\right),y\left(t\right))$ and $f_2(t,x\left(t\right),y\left(t\right))$ for ${\beta }_1=0={\beta }_2$, purely q-integral-type nonlinearities, $I_q^{{\xi }_1}{g}_1(t,x\left(t\right),y\left(t\right))$ and $I_q^{{\xi }_2}{g}_2(t,x\left(t\right),y\left(t\right))$ for ${\alpha }_1=0={\alpha }_2,$ and so on.

The paper is organized as follows. In Section 2, we recall some general concepts and results on q-calculus and fractional calculus. We then solve a linear variant of the given problem that provides a platform to define the solution for the problem at hand. Section 3 is devoted to the main existence results, which are established with the aid of some classical fixed-point theorems. The paper concludes with an illustrative example.

2. Preliminaries on Fractional q-Calculus

Here, we recall some basic definitions and known results on fractional q-calculus.

Definition 1.

Let $\beta \ge 0,0<q<1,$ and f be a function defined on $[0,1].$ The fractional q-integral of the Riemann–Liouville type is $\left(I_q^{0}f\right)\left(t\right)=f\left(t\right)$ and

$$\left(I_q^{\beta }f\right)\left(t\right)={\int }_0^t\frac{{(t-qs)}^{(\beta -1)}}{{\Gamma }_q\left(\beta \right)}f\left(s\right)d_q\left(s\right),\beta >0,t\in [0,1],$$

where

$${\Gamma }_q\left(\beta \right)=\frac{{(1-q)}^{(\beta -1)}}{{(1-q)}^{\beta -1}},0<q<1$$

and satisfies the relation:

${\Gamma }_q(\beta +1)={\left[\beta \right]}_q{\Gamma }_q\left(\beta \right),$with

$${\left[\beta \right]}_q=\frac{q^{\beta }-1}{q-1},{(1-q)}^{\left(0\right)}=1,{(1-q)}^{\left(n\right)}=\prod _{k=0}^{n-1}(1-q^{k+1}),n\in \mathbb{N}.$$

More generally, if $\alpha \in \mathbb{R}$, then

$${(1-q)}^{\left(\alpha \right)}=\prod _{i=0}^{\infty }\frac{(1-q^{i+1})}{(1-q^{1+\alpha +i})}.$$

For $0<q<1,$ we define the q-derivative of a real valued function f as

$$D_{q}f\left(t\right)=\frac{f\left(t\right)-f\left(qt\right)}{(1-q)t},t\ne 0,D_{q}f\left(0\right)=\underset{n\to \infty }{lim}\frac{f\left(s{q}^n\right)-f\left(0\right)}{s{q}^n},s\ne 0.$$

For more details, see [22].

Definition 2

([45]). The fractional q-derivative of the Riemann–Liouville type of order $\beta \ge 0$ is defined by $\left(D_q^{0}f\right)\left(t\right)=f\left(t\right)$ and

$$\left(D_q^{\beta }f\right)\left(t\right)=\left(D_q^{\left[\beta \right]}{I}_q^{\left[\beta \right]-\beta }f\right)\left(t\right),\beta >0,$$

where $\left[\beta \right]$ is the smallest integer greater than or equal to $\beta .$

Definition 3

([45]). The fractional q-derivative of the Caputo type of order $\beta \ge 0$ is defined by

$${(}^c{D}_q^{\beta }f)\left(t\right)=\left(I_q^{\left[\beta \right]-\beta }{D}_q^{\left[\beta \right]}f\right)\left(t\right),\beta >0,$$

where $\left[\beta \right]$ is the smallest integer greater than or equal to $\beta .$

Definition 4.

(q-Beta function) For any $x,y>0$,

$$B_q(x,y)={\int }_0^1{t}^{(x-1)}{(1-qt)}^{(y-1)}{d}_{q}t$$

is called the q-beta function.

Recall that

$$B_q(x,y)=\frac{{\Gamma }_q\left(x\right){\Gamma }_q\left(y\right)}{{\Gamma }_q(x+y)}.$$

(3)

Lemma 1

([45]). Let $\beta ,\gamma \ge 0$ and let f be a function defined on $[0,1].$ Then

(i) $\left(I_q^{\gamma }{I}_q^{\beta }f\right)\left(t\right)=\left(I_q^{\beta +\gamma }f\right)\left(t\right),$

(ii) $\left(D_q^{\beta }{I}_q^{\beta }f\right)\left(t\right)=f\left(t\right).$

Lemma 2

([45]). Let $\beta >0.$ Then the following equality holds:

$$\left(I_q^{\beta }{}^c{D}_q^{\beta }f\right)\left(t\right)=f\left(t\right)-\sum _{k=0}^{\left[\beta \right]-1}\frac{t^k}{{\Gamma }_q(k+1)}\left(D_q^{k}f\right)\left(0\right).$$

Lemma 3

([25]). Let $\beta \ge 0$ and $n\in \mathbb{N}.$ Then the following equality holds:

$$\left(I_q^{\beta }{D}_q^{n}f\right)\left(t\right)=D_q^n{I}_q^{\beta }f\left(t\right)-\sum _{k=0}^{\left[\beta \right]-1}\frac{t^{\beta -n+k}}{{\Gamma }_q(\beta -n+k)}\left(D_q^{k}f\right)\left(0\right).$$

Lemma 4

([46]). For $\beta \in {\mathbb{R}}^{+},$$\lambda \in (-1,\infty ),$ the following is valid

$$I_q^{\beta }\left({(x-a)}^{\left(\lambda \right)}\right)=\frac{{\Gamma }_q(\lambda +1)}{{\Gamma }_q(\beta +\lambda +1)}{(x-a)}^{(\beta +\lambda )},0<a<x<b.$$

In particular, for $\lambda =0,a=0,$ using q-integration by parts, we have

$$\begin{array}{ccc}\hfill \left(I_q^{\beta }1\right)\left(x\right)& =& \frac{1}{{\Gamma }_q\left(\beta \right)}{\int }_0^x{(x-qt)}^{(\beta -1)}{d}_{q}t=-\frac{1}{{\Gamma }_q\left(\beta \right)}{\int }_0^x\frac{D_q\left({(x-t)}^{\left(\beta \right)}\right)}{{\left[\beta \right]}_q}{d}_{q}t\hfill \\ & =& -\frac{1}{{\Gamma }_q(\beta +1)}{\int }_0^x{D}_q\left({(x-t)}^{\left(\beta \right)}\right)d_{q}t=\frac{1}{{\Gamma }_q(\beta +1)}{x}^{\left(\beta \right)}.\hfill \end{array}$$

In order to define the solution for the problem (1) and (2), we need the following lemma.

Lemma 5.

Let $\Lambda \ne 0$ and $h_1,h_2\in C([0,1],\mathbb{R}).$ Then the unique solution of the following linear system of equations:

$$\left\{\begin{array}{c}{\displaystyle {}^c{D}_q^{p_1}{(}^c{D}_q^{p_2}+{\lambda }_1)x\left(t\right)=h_1\left(t\right),0\le t\le 1,}\\ {\displaystyle {}^c{D}_q^{r_1}{(}^c{D}_q^{r_2}+{\lambda }_2)y\left(t\right)=h_2\left(t\right),0\le t\le 1,}\end{array}\right.$$

(4)

subject to the boundary conditions (2) is given by

$$\begin{array}{ccc}x\left(t\right)& =& {\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\hfill \\ & -& {\delta }_1\left(\frac{{\rho }_1{t}^{p_2}+{\rho }_5}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\right)\hfill \\ & -& {\delta }_2\left(\frac{{\rho }_2{t}^{p_2}+{\rho }_6}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\right)\hfill \\ & -& {\delta }_3\left(\frac{{\rho }_3{t}^{p_2}+{\rho }_7}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_1\left(\frac{{\rho }_3{t}^{p_2}+{\rho }_7}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_2\left(\frac{{\rho }_3{t}^{p_2}+{\rho }_7}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u\right)\hfill \\ & -& {\delta }_4\left(\frac{{\rho }_4{t}^{p_2}+{\rho }_8}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_3\left(\frac{{\rho }_4{t}^{p_2}+{\rho }_8}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_4\left(\frac{{\rho }_4{t}^{p_2}+{\rho }_8}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u\right),\hfill \end{array}$$

(5)

and

$$\begin{array}{ccc}y\left(t\right)& =& {\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\hfill \\ & -& {\delta }_1\left(\frac{{\rho }_9{t}^{r_2}+{\rho }_{13}}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\right)\hfill \\ & -& {\delta }_2\left(\frac{{\rho }_{10}{t}^{r_2}+{\rho }_{14}}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\right)\hfill \\ & -& {\delta }_3\left(\frac{{\rho }_{11}{t}^{r_2}+{\rho }_{15}}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_1\left(\frac{{\rho }_{11}{t}^{r_2}+{\rho }_{15}}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_2\left(\frac{{\rho }_{11}{t}^{r_2}+{\rho }_{15}}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u\right)\hfill \\ & -& {\delta }_4\left(\frac{{\rho }_{12}{t}^{r_2}+{\rho }_{16}}{\Lambda }\right)\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_3\left(\frac{{\rho }_{12}{t}^{r_2}+{\rho }_{16}}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\right)\hfill \\ & +& {\sigma }_4\left(\frac{{\rho }_{12}{t}^{r_2}+{\rho }_{16}}{\Lambda }\right)\left({\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u\right),\hfill \end{array}$$

(6)

where

$$\begin{array}{ccc}\Lambda & =& {\displaystyle ({\delta }_1{\delta }_2-{\mu }_1{\mu }_3)({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-{\sigma }_1({\delta }_1{\delta }_6+{\mu }_2{\mu }_3{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)}\hfill \\ & & +({\mu }_1{\mu }_3{\delta }_7{\delta }_8-{\mu }_2{\delta }_2{\delta }_7{\sigma }_3-{\mu }_2{\mu }_4{\sigma }_1{\sigma }_3-{\delta }_1{\delta }_2{\delta }_7{\delta }_8-{\mu }_4{\delta }_1{\delta }_8{\sigma }_1){\left[p_2\right]}_q{\left[r_2\right]}_q\hfill \\ & & -{\sigma }_3({\delta }_2{\delta }_5+{\mu }_1{\mu }_4{\left[r_2\right]}_q)({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)-{\mu }_3{\delta }_5{\delta }_8{\sigma }_1{\left[p_2\right]}_q-{\mu }_1{\delta }_6{\delta }_7{\sigma }_3{\left[r_2\right]}_q+{\delta }_5{\delta }_6{\sigma }_1{\sigma }_3,\hfill \end{array}$$

(7)

$$\left\{\begin{array}{c}{\displaystyle {\delta }_1=\sum _{j=1}^n{a}_j,{\delta }_2=\sum _{j=1}^n{b}_j,{\delta }_3=\sum _{j=1}^n{k}_j,{\delta }_4=\sum _{j=1}^n{m}_j,}\hfill \\ {\displaystyle {\delta }_5=\sum _{j=1}^n{a}_j{\eta }_j^{r_2},{\delta }_6=\sum _{j=1}^n{b}_j{\eta }_j^{p_2},{\delta }_7=\sum _{j=1}^n{k}_j{\eta }_j^{r_2-1},{\delta }_8=\sum _{j=1}^n{m}_j{\eta }_j^{p_2-1},}\hfill \\ {\displaystyle {\rho }_1=-{\mu }_3{\sigma }_1({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-({\delta }_2{\delta }_7{\sigma }_3+{\mu }_4{\sigma }_1{\sigma }_3){\left[r_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_2=-{\delta }_1{\sigma }_1({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-{\mu }_1{\delta }_7{\sigma }_3{\left[r_2\right]}_q+{\delta }_5{\sigma }_1{\sigma }_3,}\hfill \\ {\displaystyle {\rho }_3=({\mu }_1{\mu }_3-{\delta }_1{\delta }_2)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)+{\mu }_1{\mu }_4{\sigma }_3{\left[r_2\right]}_q+{\delta }_2{\delta }_5{\sigma }_3,}\hfill \\ {\displaystyle {\rho }_4=({\mu }_1{\mu }_3{\delta }_7-{\delta }_1{\delta }_2{\delta }_7-{\mu }_4{\delta }_1{\sigma }_1){\left[r_2\right]}_q-{\mu }_3{\delta }_5{\sigma }_1,}\hfill \\ {\displaystyle {\rho }_5={\mu }_3({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)+{\mu }_4{\sigma }_3{\left[r_2\right]}_q({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)-{\mu }_3{\delta }_7{\delta }_8{\left[p_2\right]}_q{\left[r_2\right]}_q+{\delta }_6{\delta }_7{\sigma }_3{\left[r_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_6={\delta }_1({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-{\delta }_5{\sigma }_3({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)-({\delta }_1{\delta }_7{\delta }_8+{\mu }_2{\delta }_7{\sigma }_3){\left[p_2\right]}_q{\left[r_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_7=({\delta }_1{\delta }_6+{\mu }_2{\mu }_3{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)+({\mu }_4{\delta }_1{\delta }_8+{\mu }_2{\mu }_4{\sigma }_3){\left[p_2\right]}_q{\left[r_2\right]}_q+{\mu }_3{\delta }_5{\delta }_8{\left[p_2\right]}_q-{\delta }_5{\delta }_6{\sigma }_3,}\hfill \\ {\displaystyle {\rho }_8=({\mu }_3{\delta }_5+{\mu }_4{\delta }_1{\left[r_2\right]}_q)({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)+{\mu }_2{\mu }_3{\delta }_7{\left[p_2\right]}_q{\left[r_2\right]}_q+{\delta }_1{\delta }_6{\delta }_7{\left[r_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_9=-{\delta }_2{\sigma }_3({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)-{\mu }_3{\delta }_8{\sigma }_1{\left[p_2\right]}_q+{\delta }_6{\sigma }_1{\sigma }_3,}\hfill \\ {\displaystyle {\rho }_{10}=-{\mu }_1{\sigma }_3({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)-({\mu }_2{\sigma }_1{\sigma }_3+{\delta }_1{\delta }_8{\sigma }_1){\left[p_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_{11}=({\mu }_1{\mu }_3{\delta }_8-{\delta }_1{\delta }_2{\delta }_8-{\mu }_2{\delta }_2{\sigma }_3){\left[p_2\right]}_q-{\mu }_1{\delta }_6{\sigma }_3,}\hfill \\ {\displaystyle {\rho }_{12}=({\mu }_1{\mu }_3-{\delta }_1{\delta }_2)({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)+{\mu }_2{\mu }_3{\sigma }_1{\left[p_2\right]}_q+{\delta }_1{\delta }_6{\sigma }_1,}\hfill \\ {\displaystyle {\rho }_{13}={\delta }_2({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-{\delta }_6{\sigma }_1({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-({\delta }_2{\delta }_7{\delta }_8+{\mu }_4{\delta }_8{\sigma }_1){\left[p_2\right]}_q{\left[r_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_{14}={\mu }_1({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)+{\mu }_2{\sigma }_1{\left[p_2\right]}_q({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)-{\mu }_1{\delta }_7{\delta }_8{\left[p_2\right]}_q{\left[r_2\right]}_q+{\delta }_5{\delta }_8{\sigma }_1{\left[p_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_{15}=({\mu }_1{\delta }_6+{\mu }_2{\delta }_2{\left[p_2\right]}_q)({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)+{\mu }_1{\mu }_4{\delta }_8{\left[p_2\right]}_q{\left[r_2\right]}_q+{\delta }_2{\delta }_5{\delta }_8{\left[p_2\right]}_q,}\hfill \\ {\displaystyle {\rho }_{16}=({\delta }_2{\delta }_5+{\mu }_1{\mu }_4{\left[r_2\right]}_q)({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)+({\mu }_2{\delta }_2{\delta }_7+{\mu }_2{\mu }_4{\sigma }_1){\left[p_2\right]}_q{\left[r_2\right]}_q+{\mu }_1{\delta }_6{\delta }_7{\left[r_2\right]}_q-{\delta }_5{\delta }_6{\sigma }_1.}\hfill \end{array}\right.$$

(8)

Proof. 

Applying the q-integral operators $I_q^{p_1}$ and $I_q^{r_1}$, respectively, on the first and second equations of (4), we obtain

$$\begin{array}{c}\hfill {(}^c{D}_q^{p_2}+{\lambda }_1)x\left(t\right)=I_q^{p_1}{h}_1\left(t\right)-c_0,{(}^c{D}_q^{r_2}+{\lambda }_2)y\left(t\right)=I_q^{r_1}{h}_2\left(t\right)-d_0,\end{array}$$

where $c_0$ and $d_0$ are arbitrary real constants. Now, applying the q-integral operators $I_q^{p_2}$ and $I_q^{r_2}$, respectively, to both sides of the above equations, we obtain

$$\begin{array}{c}{\displaystyle x\left(t\right)={\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u-c_0\frac{t^{p_2}}{{\Gamma }_q(p_2+1)}-c_1,}\hfill \end{array}$$

(9)

$$\begin{array}{c}{\displaystyle y\left(t\right)={\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u-d_0\frac{t^{r_2}}{{\Gamma }_q(r_2+1)}-d_1,}\hfill \end{array}$$

(10)

where $c_1,d_1\in \mathbb{R}$ are arbitrary constants. By using the conditions (2), we obtain a system of equations in the unknown constants $c_0,c_1,d_0$ and $d_1$ given by

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mu }_2{\left[p_2\right]}_q}{{\Gamma }_q(p_2+1)}{c}_0-{\mu }_1{c}_1+\frac{{\delta }_5}{{\Gamma }_q(r_2+1)}{d}_0+{\delta }_1{d}_1=F_1,}\\ {\displaystyle \frac{{\delta }_6}{{\Gamma }_q(p_2+1)}{c}_0+{\delta }_2{c}_1+\frac{{\mu }_4{\left[r_2\right]}_q}{{\Gamma }_q(r_2+1)}{d}_0-{\mu }_3{d}_1=F_2,}\\ {\displaystyle -\frac{({\sigma }_1+{\sigma }_2{\left[p_2\right]}_q)}{{\Gamma }_q(p_2+1)}{c}_0-{\sigma }_1{c}_1+\frac{{\delta }_7{\left[r_2\right]}_q}{{\Gamma }_q(r_2+1)}{d}_0=F_3,}\\ {\displaystyle \frac{{\delta }_8{\left[p_2\right]}_q}{{\Gamma }_q(p_2+1)}{c}_0-\frac{({\sigma }_3+{\sigma }_4{\left[r_2\right]}_q)}{{\Gamma }_q(r_2+1)}{d}_0-{\sigma }_3{d}_1=F_4,}\end{array}\right.$$

(11)

where ${\delta }_1,{\delta }_2,{\delta }_5,{\delta }_6,{\delta }_7,{\delta }_8$ are given in (8), and

$$\begin{array}{ccc}\hfill F_1& =& {\delta }_1\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\right),\hfill \\ \hfill F_2& =& {\delta }_2\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\right),\hfill \\ \hfill F_3& =& {\delta }_3\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u\right)\hfill \\ & & -{\sigma }_1\left({\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\right)\hfill \\ & & -{\sigma }_2\left({\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u\right),\hfill \\ \hfill F_4& =& {\delta }_4\left({\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{h}_1\left(u\right)d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u\right)\hfill \\ & & -{\sigma }_3\left({\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\right)\hfill \\ & & -{\sigma }_4\left({\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{h}_2\left(u\right)d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u\right).\hfill \end{array}$$

Solving the system (11) for $c_0,c_1,d_0$ and $d_1$, we find that

$$\begin{array}{ccc}\hfill c_0& =& \frac{{\Gamma }_q(p_2+1)}{\Lambda }\left({\rho }_1{F}_1+{\rho }_2{F}_2+{\rho }_3{F}_3+{\rho }_4{F}_4\right),c_1=\frac{1}{\Lambda }\left({\rho }_5{F}_1+{\rho }_6{F}_2+{\rho }_7{F}_3+{\rho }_8{F}_4\right),\hfill \\ \hfill d_0& =& \frac{{\Gamma }_q(r_2+1)}{\Lambda }\left({\rho }_9{F}_1+{\rho }_{10}{F}_2+{\rho }_{11}{F}_3+{\rho }_{12}{F}_4\right),d_1=\frac{1}{\Lambda }\left({\rho }_{13}{F}_1+{\rho }_{14}{F}_2+{\rho }_{15}{F}_3+{\rho }_{16}{F}_4\right),\hfill \end{array}$$

where $\Lambda$ is given by (7). Substituting the values of $c_0,c_1,d_0$ and $d_1$ in (9) and (10) yields the solution (5) and (6). By direct computation, one can obtain the converse of the lemma. This completes the proof.  □

Let $\mathcal{C}=\{x|x\in C\left(\right[0,1],\mathbb{R})\}$ be the space equipped with the norm $\parallel x\parallel ={sup}_{t\in [0,1]}|x\left(t\right)|.$ Obviously, $(\mathcal{C},\parallel .\parallel )$ is a Banach space. Then, the product space $(\mathcal{C}\times \mathcal{C},\parallel .\parallel )$ is also a Banach space with the norm $\parallel (x,y)\parallel =\parallel x\parallel +\parallel y\parallel$ for $(x,y)\in \mathcal{C}\times \mathcal{C}.$

In view of Lemma 5, we define an operator $\mathcal{G}:\mathcal{C}\times \mathcal{C}\to \mathcal{C}\times \mathcal{C}$ by

$$\mathcal{G}(x,y)\left(t\right)=\left(\begin{array}{c}{\mathcal{G}}_1(x,y)\left(t\right)\\ {\mathcal{G}}_2(x,y)\left(t\right)\end{array}\right),$$

(12)

where

$$\begin{array}{ccc}{\mathcal{G}}_1(x,y)\left(t\right)& =& {\alpha }_1{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+\xi )}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u\hfill \\ & -& {\delta }_1\left(\frac{{\rho }_1{t}^{p_2}+{\rho }_5}{\Lambda }\right)({\alpha }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u)\hfill \\ & -& {\delta }_2\left(\frac{{\rho }_2{t}^{p_2}+{\rho }_6}{\Lambda }\right)({\alpha }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u)\hfill \\ & -& {\delta }_3\left(\frac{{\rho }_3{t}^{p_2}+{\rho }_7}{\Lambda }\right)({\alpha }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_1\left(\frac{{\rho }_3{t}^{p_2}+{\rho }_7}{\Lambda }\right)({\alpha }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_2\left(\frac{{\rho }_3{t}^{p_2}+{\rho }_7}{\Lambda }\right)({\alpha }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u)\hfill \\ & -& {\delta }_4\left(\frac{{\rho }_4{t}^{p_2}+{\rho }_8}{\Lambda }\right)({\alpha }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_3\left(\frac{{\rho }_4{t}^{p_2}+{\rho }_8}{\Lambda }\right)({\alpha }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_4\left(\frac{{\rho }_4{t}^{p_2}+{\rho }_8}{\Lambda }\right)({\alpha }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u),\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{G}}_2(x,y)\left(t\right)& =& {\alpha }_2{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u\hfill \\ & -& {\delta }_1\left(\frac{{\rho }_9{t}^{r_2}+{\rho }_{13}}{\Lambda }\right)({\alpha }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u)\hfill \\ & -& {\delta }_2\left(\frac{{\rho }_{10}{t}^{r_2}+{\rho }_{14}}{\Lambda }\right)({\alpha }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u)\hfill \\ & -& {\delta }_3\left(\frac{{\rho }_{11}{t}^{r_2}+{\rho }_{15}}{\Lambda }\right)({\alpha }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_1\left(\frac{{\rho }_{11}{t}^{r_2}+{\rho }_{15}}{\Lambda }\right)({\alpha }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}x\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_2\left(\frac{{\rho }_{11}{t}^{r_2}+{\rho }_{15}}{\Lambda }\right)({\alpha }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u)\hfill \\ & -& {\delta }_4\left(\frac{{\rho }_{12}{t}^{r_2}+{\rho }_{16}}{\Lambda }\right)({\alpha }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{f}_1(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{g}_1(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_1{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}x\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_3\left(\frac{{\rho }_{12}{t}^{r_2}+{\rho }_{16}}{\Lambda }\right)({\alpha }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}y\left(u\right)d_{q}u)\hfill \\ & +& {\sigma }_4\left(\frac{{\rho }_{12}{t}^{r_2}+{\rho }_{16}}{\Lambda }\right)({\alpha }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{f}_2(u,x\left(u\right),y\left(u\right))d_{q}u\hfill \\ & +& {\beta }_2{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{g}_2(u,x\left(u\right),y\left(u\right))d_{q}u-{\lambda }_2{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}y\left(u\right)d_{q}u).\hfill \end{array}$$

3. Existence and Uniqueness Results

In the sequel, we set the notation

$$\left\{\begin{array}{c}{\Psi }_1=\frac{|{\alpha }_1|}{{\Gamma }_q(p_1+p_2+1)}+\frac{|{\alpha }_1|({\gamma }_1+{\gamma }_2)}{|\Lambda |},{\Psi }_2=\frac{|{\alpha }_2|({\gamma }_3+{\gamma }_4)}{|\Lambda |},{\Psi }_3=\frac{|{\alpha }_1|({\gamma }_{13}+{\gamma }_{14})}{|\Lambda |},\hfill \\ {\Psi }_4=\frac{|{\alpha }_2|}{{\Gamma }_q(r_1+r_2+1)}+\frac{|{\alpha }_2|({\gamma }_{15}+{\gamma }_{16})}{|\Lambda |},{\Phi }_1=\frac{|{\beta }_1|}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)}+\frac{|{\beta }_1|({\gamma }_5+{\gamma }_6)}{|\Lambda |},\hfill \\ {\Phi }_2=\frac{|{\beta }_2|({\gamma }_7+{\gamma }_8)}{|\Lambda |},{\Phi }_3=\frac{|{\beta }_1|({\gamma }_{17}+{\gamma }_{18})}{|\Lambda |},{\Phi }_4=\frac{|{\beta }_2|}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)}+\frac{|{\beta }_2|({\gamma }_{19}+{\gamma }_{20})}{|\Lambda |},\hfill \\ {\Theta }_1=\frac{|{\lambda }_1|}{{\Gamma }_q(p_2+1)}+\frac{|{\lambda }_1|({\gamma }_9+{\gamma }_{10})}{|\Lambda |}+\frac{|{\lambda }_2|({\gamma }_{11}+{\gamma }_{12})}{|\Lambda |},\hfill \\ {\Theta }_2=\frac{|{\lambda }_1|({\gamma }_{21}+{\gamma }_{22})}{|\Lambda |}+\frac{|{\lambda }_2|}{{\Gamma }_q(r_2+1)}+\frac{|{\lambda }_2|({\gamma }_{23}+{\gamma }_{24})}{|\Lambda |},\hfill \end{array}\right.$$

(13)

$$\left\{\begin{array}{c}{\displaystyle {\gamma }_1=\frac{|{\delta }_2||{\rho }_2+{\rho }_6|{\eta }_j^{p_1+p_2}+|{\sigma }_1||{\rho }_3+{\rho }_7|}{{\Gamma }_q(p_1+p_2+1)},{\gamma }_2=\frac{|{\delta }_4||{\rho }_4+{\rho }_8|{\eta }_j^{p_1+p_2-1}+|{\sigma }_2||{\rho }_3+{\rho }_7|}{{\Gamma }_q(p_1+p_2)},}\hfill \\ {\displaystyle {\gamma }_3=\frac{|{\delta }_1||{\rho }_1+{\rho }_5|{\eta }_j^{r_1+r_2}+|{\sigma }_3||{\rho }_4+{\rho }_8|}{{\Gamma }_q(r_1+r_2+1)},{\gamma }_4=\frac{|{\delta }_3||{\rho }_3+{\rho }_7|{\eta }_j^{r_1+r_2-1}+|{\sigma }_4||{\rho }_4+{\rho }_8|}{{\Gamma }_q(r_1+r_2)},}\hfill \\ {\displaystyle {\gamma }_5=\frac{|{\delta }_2||{\rho }_2+{\rho }_6|{\eta }_j^{p_1+p_2+{\xi }_1}+|{\sigma }_1||{\rho }_3+{\rho }_7|}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)},{\gamma }_6=\frac{|{\delta }_4||{\rho }_4+{\rho }_8|{\eta }_j^{p_1+p_2+{\xi }_1-1}+|{\sigma }_2||{\rho }_3+{\rho }_7|}{{\Gamma }_q(p_1+p_2+{\xi }_1)},}\hfill \\ {\displaystyle {\gamma }_7=\frac{|{\delta }_1||{\rho }_1+{\rho }_5|{\eta }_j^{r_1+r_2+{\xi }_2}+|{\sigma }_3||{\rho }_4+{\rho }_8|}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)},{\gamma }_8=\frac{|{\delta }_3||{\rho }_3+{\rho }_7|{\eta }_j^{r_1+r_2+{\xi }_2-1}+|{\sigma }_4||{\rho }_4+{\rho }_8|}{{\Gamma }_q(r_1+r_2+{\xi }_2)},}\hfill \\ {\displaystyle {\gamma }_9=\frac{|{\delta }_2||{\rho }_2+{\rho }_6|{\eta }_j^{p_2}+|{\sigma }_1||{\rho }_3+{\rho }_7|}{{\Gamma }_q(p_2+1)},{\gamma }_{10}=\frac{|{\delta }_4||{\rho }_4+{\rho }_8|{\eta }_j^{p_2-1}+|{\sigma }_2||{\rho }_3+{\rho }_7|}{{\Gamma }_q\left(p_2\right)},}\hfill \\ {\displaystyle {\gamma }_{11}=\frac{|{\delta }_1||{\rho }_1+{\rho }_5|{\eta }_j^{r_2}+|{\sigma }_3||{\rho }_4+{\rho }_8|}{{\Gamma }_q(r_2+1)},{\gamma }_{12}=\frac{|{\delta }_3||{\rho }_3+{\rho }_7|{\eta }_j^{r_2-1}+|{\sigma }_4||{\rho }_4+{\rho }_8|}{{\Gamma }_q\left(r_2\right)},}\hfill \\ {\displaystyle {\gamma }_{13}=\frac{|{\delta }_2||{\rho }_{10}+{\rho }_{14}|{\eta }_j^{p_1+p_2}+|{\sigma }_1||{\rho }_{11}+{\rho }_{15}|}{{\Gamma }_q(p_1+p_2+1)},{\gamma }_{14}=\frac{|{\delta }_4||{\rho }_{12}+{\rho }_{16}|{\eta }_j^{p_1+p_2-1}+|{\sigma }_2||{\rho }_{11}+{\rho }_{15}|}{{\Gamma }_q(p_1+p_2)},}\hfill \\ {\displaystyle {\gamma }_{15}=\frac{|{\delta }_1||{\rho }_9+{\rho }_{13}|{\eta }_j^{r_1+r_2}+|{\sigma }_3||{\rho }_{12}+{\rho }_{16}|}{{\Gamma }_q(r_1+r_2+1)},{\gamma }_{16}=\frac{|{\delta }_3||{\rho }_{11}+{\rho }_{15}|{\eta }_j^{r_1+r_2-1}+|{\sigma }_4||{\rho }_{12}+{\rho }_{16}|}{{\Gamma }_q(r_1+r_2)},}\hfill \\ {\displaystyle {\gamma }_{17}=\frac{|{\delta }_2||{\rho }_{10}+{\rho }_{14}|{\eta }_j^{p_1+p_2+{\xi }_1}+|{\sigma }_1||{\rho }_{11}+{\rho }_{15}|}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)},{\gamma }_{18}=\frac{|{\delta }_4||{\rho }_{12}+{\rho }_{16}|{\eta }_j^{p_1+p_2+{\xi }_1-1}+|{\sigma }_2||{\rho }_{11}+{\rho }_{15}|}{{\Gamma }_q(p_1+p_2+{\xi }_1)},}\hfill \\ {\displaystyle {\gamma }_{19}=\frac{|{\delta }_1||{\rho }_9+{\rho }_{13}|{\eta }_j^{r_1+r_2+{\xi }_2}+|{\sigma }_3||{\rho }_{12}+{\rho }_{16}|}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)},{\gamma }_{20}=\frac{|{\delta }_3||{\rho }_{11}+{\rho }_{15}|{\eta }_j^{r_1+r_2+{\xi }_2-1}+|{\sigma }_4||{\rho }_{12}+{\rho }_{16}|}{{\Gamma }_q(r_1+r_2+{\xi }_2)},}\hfill \\ {\displaystyle {\gamma }_{21}=\frac{|{\delta }_2||{\rho }_{10}+{\rho }_{14}|{\eta }_j^{p_2}+|{\sigma }_1||{\rho }_{11}+{\rho }_{15}|}{{\Gamma }_q(p_2+1)},{\gamma }_{22}=\frac{|{\delta }_4||{\rho }_{12}+{\rho }_{16}|{\eta }_j^{p_2-1}+|{\sigma }_2||{\rho }_{11}+{\rho }_{15}|}{{\Gamma }_q\left(p_2\right)},}\hfill \\ {\displaystyle {\gamma }_{23}=\frac{|{\delta }_1||{\rho }_9+{\rho }_{13}|{\eta }_j^{r_2}+|{\sigma }_3||{\rho }_{12}+{\rho }_{16}|}{{\Gamma }_q(r_2+1)},{\gamma }_{24}=\frac{|{\delta }_3||{\rho }_{11}+{\rho }_{15}|{\eta }_j^{r_2-1}+|{\sigma }_4||{\rho }_{12}+{\rho }_{16}|}{{\Gamma }_q\left(r_2\right)}.}\hfill \end{array}\right.$$

(14)

In the following theorem, we prove the existence of a unique solution to the system (1) and (2) by applying the Banach contraction mapping principle [47].

Theorem 1.

Let $f_1,f_2:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $g_1,g_2:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be continuous functions satisfying the following conditions:

(A₁)

There exist positive constants ${\iota }_1,{\iota }_2$ such that for each $t\in [0,1]$ and $x_i,y_i\in \mathbb{R},i=1,2$,

$$|f_1(t,x_1,y_1)-f_1(t,x_2,y_2)|\le {\iota }_1(|x_1-x_2|+|y_1-y_2|),$$

$$|f_2(t,x_1,y_1)-f_2(t,x_2,y_2)|\le {\iota }_2(|x_1-x_2|+|y_1-y_2|).$$

(A₂)

There exist positive constants ${\kappa }_1,{\kappa }_2$ such that for each $t\in [0,1]$ and $x_i,y_i\in \mathbb{R},i=1,2$,

$$|g_1(t,x_1,y_1)-g_1(t,x_2,y_2)|\le {\kappa }_1(|x_1-x_2|+|y_1-y_2|),$$

$$|g_2(t,x_1,y_1)-g_2(t,x_2,y_2)|\le {\kappa }_2(|x_1-x_2|+|y_1-y_2|).$$

Then the system (1) and (2) has a unique solution on $[0,1]$, provided that

$$\begin{array}{c}\hfill \mathrm{Y}=({\Psi }_1+{\Psi }_3){\iota }_1+({\Psi }_2+{\Psi }_4){\iota }_2+({\Phi }_1+{\Phi }_3){\kappa }_1+({\Phi }_2+{\Phi }_4){\kappa }_2+{\Theta }_1+{\Theta }_2<1.\end{array}$$

(15)

where ${\Psi }_1,{\Psi }_2,{\Psi }_3,{\Psi }_4,{\Phi }_1,{\Phi }_2,{\Phi }_3,{\Phi }_4,{\Theta }_1,{\Theta }_2$ are given in (13).

Proof. 

Let $N_1,N_2,M_1,M_2$ be finite numbers such that

$$\begin{array}{ccc}& & N_1=\underset{t\in [0,1]}{sup}|f_1(t,0,0)|,N_2=\underset{t\in [0,1]}{sup}|f_2(t,0,0)|,\hfill \\ & & M_1=\underset{t\in [0,1]}{sup}|g_1(t,0,0)|,M_2=\underset{t\in [0,1]}{sup}|g_2(t,0,0)|.\hfill \end{array}$$

(16)

Now we show that $\mathcal{G}{B}_{\varrho }\subset B_{\varrho },$ where $B_{\varrho }=\{(x,y)\in \mathcal{G}\times \mathcal{G}:\parallel (x,y)\parallel \le \varrho \}$ with

$$\varrho \ge \frac{({\Psi }_1+{\Psi }_3)N_1+({\Psi }_2+{\Psi }_4)N_2+({\Phi }_1+{\Phi }_3)M_1+({\Phi }_2+{\Phi }_4)M_2}{1-\mathrm{Y}},$$

where $\mathrm{Y}$ is given in (15). For any $(x,y)\in B_{\varrho },t\in [0,1],$ using $\left({\mathbf{A}}_1\right)$, we have

$$\begin{array}{ccc}\hfill |f_1(t,x\left(t\right),y\left(t\right))|& & \le |f_1(t,x\left(t\right),y\left(t\right))-f_1(t,0,0)|+|f_1(t,0,0)|\hfill \\ & & \le {\iota }_1(|x\left(t\right)|+|y\left(t\right)|)+|f_1(t,0,0)|\hfill \\ & & \le {\iota }_1(\parallel x\parallel +\parallel y\parallel )+N_1\hfill \\ & & \le {\iota }_1\varrho +N_1.\hfill \end{array}$$

Similarly, we can find that

$$|f_2(t,x\left(t\right),y\left(t\right))|\le {\iota }_2\varrho +N_2,|g_1(t,x\left(t\right),y\left(t\right))|\le {\kappa }_1\varrho +M_1,|g_2(t,x\left(t\right),y\left(t\right))|\le {\kappa }_2\varrho +M_2.$$

Then we have

$$\begin{array}{ccc}& & \parallel {\mathcal{G}}_1(x,y)\parallel \hfill \\ & & \le \underset{t\in [0,1]}{sup}\{|{\alpha }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+\xi )}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u\hfill \\ & & +\frac{|{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u\left)\right\}\hfill \\ & & \le ({\iota }_{1}r+N_1)\underset{t\in [0,1]}{sup}\{|{\alpha }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u+\frac{|{\alpha }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\}\hfill \\ & & +({\kappa }_{1}r+M_1)\underset{t\in [0,1]}{sup}\{|{\beta }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u\}+({\iota }_{2}r+N_2)\underset{t\in [0,1]}{sup}\{\frac{|{\alpha }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u\}\hfill \\ & & +({\kappa }_{2}r+M_2)\underset{t\in [0,1]}{sup}\{\frac{|{\beta }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u\}+\varrho \underset{t\in [0,1]}{sup}\{|{\lambda }_1|{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\lambda }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u+\frac{|{\lambda }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u+\frac{|{\lambda }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\lambda }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\}\hfill \\ & & \le ({\iota }_1\varrho +N_1){\Psi }_1+({\iota }_2\varrho +N_2){\Psi }_2+({\kappa }_1\varrho +M_1){\Phi }_1+({\kappa }_2\varrho +M_2){\Phi }_2+\varrho {\Theta }_1,\hfill \end{array}$$

where ${\Psi }_1,{\Psi }_2,{\Phi }_1,{\Phi }_2,{\Theta }_1$ are given in (13).

Furthermore, we obtain

$$\begin{array}{ccc}& & \parallel {\mathcal{G}}_2(x,y)\parallel \hfill \\ & & \le \underset{t\in [0,1]}{sup}\{|{\alpha }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u\hfill \\ & & +\frac{|{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u\left)\right\}\hfill \\ & & \le ({\iota }_{1}r+N_1)\underset{t\in [0,1]}{sup}\{\frac{|{\alpha }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\}+({\kappa }_{1}r+M_1)\underset{t\in [0,1]}{sup}\{\frac{|{\beta }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u\}+({\iota }_{2}r+N_2)\underset{t\in [0,1]}{sup}\{|{\alpha }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u\}\hfill \\ & & +({\kappa }_{2}r+M_2)\underset{t\in [0,1]}{sup}\{|{\beta }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}\hfill \\ & & \times {\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u\}+\varrho \underset{t\in [0,1]}{sup}\{|{\lambda }_2|{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\lambda }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u+\frac{|{\lambda }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u+\frac{|{\lambda }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\lambda }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\}\hfill \\ & & \le ({\iota }_1\varrho +N_1){\Psi }_3+({\iota }_2\varrho +N_2){\Psi }_4+({\kappa }_1\varrho +M_1){\Phi }_3+({\kappa }_2\varrho +M_2){\Phi }_4+\varrho {\Theta }_2,\hfill \end{array}$$

where ${\Psi }_3,{\Psi }_4,{\Phi }_3,{\Phi }_4,{\Theta }_2$ are given in (13).

From the foregoing inequalities, it follows that

$$\parallel \mathcal{G}(x,y)\parallel \le \mathrm{Y}\varrho +({\Psi }_1+{\Psi }_3)N_1+({\Psi }_2+{\Psi }_4)N_2+({\Phi }_1+{\Phi }_3)M_1+({\Phi }_2+{\Phi }_4)M_2,$$

which implies that $\mathcal{G}{B}_{\varrho }\subset B_{\varrho }$. Next we show that the operator $\mathcal{G}$ is a contraction. Using conditions $\left({\mathbf{A}}_1\right)$ and $\left({\mathbf{A}}_2\right)$, for any $(x_1,y_1),(x_2,y_2)\in \mathcal{C}\times \mathcal{C},t\in [0,1],$ we obtain

$$\begin{array}{ccc}& & \parallel {\mathcal{G}}_1(x_1,y_1)-{\mathcal{G}}_1(x_2,y_2)\parallel \hfill \\ & & \le \underset{t\in [0,1]}{sup}\{|{\alpha }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u+\frac{|{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}\hfill \\ & & \times |f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}\hfill \\ & & \times |g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)+\frac{|{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}\hfill \\ & & \times |f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}\hfill \\ & & \times |g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)+\frac{|{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}\hfill \\ & & \times |f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}\hfill \\ & & \times |g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)+\frac{|{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}\hfill \\ & & \times |f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}\hfill \\ & & \times |g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u\left)\right\}\hfill \\ & & \le {\iota }_1(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\underset{t\in [0,1]}{sup}\{|{\alpha }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u+\frac{|{\alpha }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\}\hfill \\ & & +{\kappa }_1(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\underset{t\in [0,1]}{sup}\{|{\beta }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u\}\hfill \\ & & +{\iota }_2(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\underset{t\in [0,1]}{sup}\{\frac{|{\alpha }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u\}+{\kappa }_2(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\hfill \\ & & \times \underset{t\in [0,1]}{sup}\{\frac{|{\beta }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u\}+(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\hfill \\ & & \times \underset{t\in [0,1]}{sup}\{|{\lambda }_1|{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u+\frac{|{\lambda }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u+\frac{|{\lambda }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u+\frac{|{\lambda }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u+\frac{|{\lambda }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\}\hfill \\ & & \le ({\iota }_1{\Psi }_1+{\iota }_2{\Psi }_2+{\kappa }_1{\Phi }_1+{\kappa }_2{\Phi }_2+{\Theta }_1)\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right),\hfill \end{array}$$

where ${\Psi }_1,{\Psi }_2,{\Phi }_1,{\Phi }_2,{\Theta }_1$ are given in (13).

Similarly, one can obtain

$$\begin{array}{ccc}& & \parallel {\mathcal{G}}_2(x_1,y_1)-{\mathcal{G}}_2(x_2,y_2)\parallel \hfill \\ & & \le \underset{t\in [0,1]}{sup}\{|{\alpha }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_2|{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u+\frac{|{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}\hfill \\ & & \times |f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}\hfill \\ & & \times |g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)+\frac{|{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}\hfill \\ & & \times |f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}\hfill \\ & & \times |g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)+\frac{|{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}\hfill \\ & & \times |f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}\hfill \\ & & \times |g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x_1\left(u\right),y_1\left(u\right))-f_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x_1\left(u\right),y_1\left(u\right))-g_1(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x_1\left(u\right)-x_2\left(u\right)|d_{q}u)+\frac{|{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}\hfill \\ & & \times |f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}\hfill \\ & & \times |g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x_1\left(u\right),y_1\left(u\right))-f_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x_1\left(u\right),y_1\left(u\right))-g_2(u,x_2\left(u\right),y_2\left(u\right))|d_{q}u\hfill \\ & & +|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y_1\left(u\right)-y_2\left(u\right)|d_{q}u\left)\right\}\hfill \\ & & \le {\iota }_1(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\underset{t\in [0,1]}{sup}\{\frac{|{\alpha }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\}+{\kappa }_1(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\hfill \\ & & \times \underset{t\in [0,1]}{sup}\{\frac{|{\beta }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}\hfill \\ & & \times {\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u\}+{\iota }_2(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\hfill \\ & & \times \underset{t\in [0,1]}{sup}\{|{\alpha }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u\}+{\kappa }_2(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\hfill \\ & & \times \underset{t\in [0,1]}{sup}\{|{\beta }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}\hfill \\ & & \times {\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u\}\hfill \\ & & +(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel )\underset{t\in [0,1]}{sup}\{|{\lambda }_2|{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\lambda }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u+\frac{|{\lambda }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u+\frac{|{\lambda }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\lambda }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\lambda }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\}\hfill \\ & & \le ({\iota }_1{\Psi }_3+{\iota }_2{\Psi }_4+{\kappa }_1{\Phi }_3+{\kappa }_2{\Phi }_4+{\Theta }_2)\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right),\hfill \end{array}$$

where ${\Psi }_3,{\Psi }_4,{\Phi }_3,{\Phi }_4,{\Theta }_2$ are given in (13). Consequently, we obtain

$$\parallel \mathcal{G}(x_1,y_1)-\mathcal{G}(x_2,y_2)\parallel \le \mathrm{Y}\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right).$$

As $\mathrm{Y}<1$ by (15), therefore $\mathcal{G}$ is a contraction. Hence, we deduce by the conclusion of the Banach contraction mapping principle that the operator $\mathcal{G}$ has a unique fixed point, which is indeed the unique solution of the problem (1) and (2). The proof is completed.  □

Next, we present an existence result for the problem (1) and (2) which is proved by means of the Leray–Schauder nonlinear alternative [48].

Theorem 2.

Assume that

$\left({\mathbf{A}}_3\right)$

$f_1,f_2,g_1,g_2:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions and that there exist real constants ${\tau }_i,{\tilde{\tau }}_i,{ϵ}_i,{\widetilde{ϵ}}_i\ge 0,(i=1,2)$ and ${\tau }_0,{\tilde{\tau }}_0,{ϵ}_0,{\widetilde{ϵ}}_0>0$ such that, $\forall x,y\in \mathbb{R},$

$$|f_1(t,x,y)|\le {\tau }_0+{\tau }_1|x|+{\tau }_2|y|,|f_2(t,x,y)|\le {\tilde{\tau }}_0+{\tilde{\tau }}_1|x|+{\tilde{\tau }}_2|y|,$$

$$|g_1(t,x,y)|\le {ϵ}_0+{ϵ}_1|x|+{ϵ}_2|y|,|g_2(t,x,y)|\le {\widetilde{ϵ}}_0+{\widetilde{ϵ}}_1|x|+{\widetilde{ϵ}}_2|y|,$$

Then the system (1) and (2) has at least one solution on $[0,1]$ provided that

$$({\Psi }_1+{\Psi }_3){\tau }_1+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_1+({\Phi }_1+{\Phi }_3){ϵ}_1+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_1+{\upsilon }_1<1,$$

where ${\Psi }_i,{\Phi }_i,i=1,2,3,4$ are given by (13) and

$${\upsilon }_1=|{\lambda }_1|\left(\frac{1}{{\Gamma }_q(p_2+1)}+\frac{|{\gamma }_9+{\gamma }_{10}|}{|\Lambda |}+\frac{|{\gamma }_{21}+{\gamma }_{22}|}{|\Lambda |}\right),$$

Proof. 

In the first step, it will be shown that the operator $\mathcal{G}:\mathcal{C}\times \mathcal{C}\to \mathcal{C}\times \mathcal{C}$ is completely continuous. Notice that the operator $\mathcal{G}$ is continuous in view of the continuity of the functions $f_1,f_2,g_1,g_2.$ Let $\mathcal{Y}\subset \mathcal{C}\times \mathcal{C}$ be bounded. Then, for all $(x,y)\in \mathcal{Y},$ there exist constants ${\hslash }_1,{\hslash }_2,{\varpi }_1,{\varpi }_2$ such that $|f_1(t,x\left(t\right),y\left(t\right))|\le {\hslash }_1,|f_2(t,x\left(t\right),y\left(t\right))|\le {\hslash }_2,|g_1(t,x\left(t\right),y\left(t\right))|\le {\varpi }_1,|g_2(t,x\left(t\right),y\left(t\right))|\le {\varpi }_2.$ Let $(x,y)\in \mathcal{Y}.$ Then there exists $\phi$ such that $\parallel (x,y)\parallel =\parallel x\parallel +\parallel y\parallel \le \phi$, and for any $(x,y)\in \mathcal{Y},$ we have

$$\begin{array}{ccc}& & \parallel {\mathcal{G}}_1(x,y)\parallel \hfill \\ & & \le \underset{t\in [0,1]}{sup}\{|{\alpha }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+\xi )}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u\hfill \\ & & +\frac{|{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u\left)\right\}\hfill \\ & & \le {\hslash }_1{\Psi }_1+{\hslash }_2{\Psi }_2+{\varpi }_1{\Phi }_1+{\varpi }_2{\Phi }_2+\phi {\Theta }_1\hfill \end{array}$$

where ${\Psi }_1,{\Psi }_2,{\Phi }_1,{\Phi }_2,{\Theta }_1$ are given in (13). Similarly, we can find that

$$\begin{array}{ccc}& & \parallel {\mathcal{G}}_2(x,y)\parallel \hfill \\ & & \le \underset{t\in [0,1]}{sup}\{|{\alpha }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u\hfill \\ & & +\frac{|{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}|f_1(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}|g_1(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_1|{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}|x\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}|y\left(u\right)|d_{q}u)\hfill \\ & & +\frac{|{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}(|{\alpha }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}|f_2(u,x\left(u\right),y\left(u\right))|d_{q}u\hfill \\ & & +|{\beta }_2|{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}|g_2(u,x\left(u\right),y\left(u\right))|d_{q}u+|{\lambda }_2|{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}|y\left(u\right)|d_{q}u\left)\right\}\hfill \\ & & \le {\hslash }_1{\Psi }_3+{\hslash }_2{\Psi }_4+{\varpi }_1{\Phi }_3+{\varpi }_2{\Phi }_4+\phi {\Theta }_2,\hfill \end{array}$$

where ${\Psi }_3,{\Psi }_4,{\Phi }_3,{\Phi }_4,{\Theta }_2$ are given in (13).

Consequently, we obtain

$$\parallel \mathcal{G}(x,y)\parallel \le ({\Psi }_1+{\Psi }_3){\hslash }_1+({\Psi }_2+{\Psi }_4){\hslash }_2+({\Phi }_1+{\Phi }_3){\varpi }_1+({\Phi }_2+{\Phi }_4){\varpi }_2+({\Theta }_1+{\Theta }_2)\phi .$$

Therefore, the operator $\mathcal{G}$ is uniformly bounded. Next, we show that the operator $\mathcal{G}$ is equicontinuous. Let $t_1,t_2\in [0,1]$ with $t_1<t_2$. Then we have

$$\begin{array}{ccc}& & |{\mathcal{G}}_1(x\left(t_2\right),y\left(t_2\right))-{\mathcal{G}}_1(x\left(t_1\right),y\left(t_1\right))|\hfill \\ & & \le \frac{|{\alpha }_1|{\hslash }_1}{{\Gamma }_q(p_1+p_2)}|{\int }_0^{t_1}[{(t_2-qu)}^{(p_1+p_2-1)}-{(t_1-qu)}^{(p_1+p_2-1)}]d_{q}u+{\int }_{t_1}^{t_2}{(t_2-qu)}^{(p_1+p_2-1)}{d}_{q}u|\hfill \\ & & +\frac{|{\beta }_1|{\varpi }_1}{{\Gamma }_q(p_1+p_2+{\xi }_1)}|{\int }_0^{t_1}[{(t_2-qu)}^{(p_1+p_2+{\xi }_1-1)}-{(t_1-qu)}^{(p_1+p_2+{\xi }_1-1)}]d_{q}u\hfill \\ & & +{\int }_{t_1}^{t_2}{(t_2-qu)}^{(p_1+p_2+{\xi }_1-1)}{d}_{q}u|+\frac{|{\lambda }_1|\phi }{{\Gamma }_q\left(p_2\right)}|{\int }_0^{t_1}[{(t_2-qu)}^{(p_2-1)}-{(t_1-qu)}^{(p_2-1)}]d_{q}u\hfill \\ & & +{\int }_{t_1}^{t_2}{(t_2-qu)}^{(p_2-1)}{d}_{q}u|+\frac{|t_2^{p_2}-t_1^{p_2}|}{|\Lambda |}[|{\delta }_1||{\rho }_1|(\frac{|{\alpha }_2|{\eta }_j^{r_1+r_2}{\hslash }_2}{{\Gamma }_q(r_1+r_2+1)}+\frac{|{\beta }_2|{\eta }_j^{r_1+r_2+{\xi }_2}{\varpi }_2}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)}\hfill \\ & & +\frac{|{\lambda }_2|{\eta }_j^{r_2}\phi }{{\Gamma }_q(r_2+1)})+|{\delta }_2||{\rho }_2|\left(\frac{|{\alpha }_1|{\eta }_j^{p_1+p_2}{\hslash }_1}{{\Gamma }_q(p_1+p_2+1)}+\frac{|{\beta }_1|{\eta }_j^{p_1+p_2+{\xi }_1}{\varpi }_1}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)}+\frac{|{\lambda }_1|{\eta }_j^{p_2}\phi }{{\Gamma }_q(p_2+1)}\right)\hfill \\ & & +|{\delta }_3||{\rho }_3|\left(\frac{|{\alpha }_2|{\eta }_j^{r_1+r_2-1}{\hslash }_2}{{\Gamma }_q(r_1+r_2)}+\frac{|{\beta }_2|{\eta }_j^{r_1+r_2+{\xi }_2-1}{\varpi }_2}{{\Gamma }_q(r_1+r_2+{\xi }_2)}+\frac{|{\lambda }_2|{\eta }_j^{r_2-1}\phi }{{\Gamma }_q\left(r_2\right)}\right)+|{\sigma }_1||{\rho }_3|(\frac{|{\alpha }_1|{\hslash }_1}{{\Gamma }_q(p_1+p_2+1)}\hfill \\ & & +\frac{|{\beta }_1|{\varpi }_1}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)}+\frac{|{\lambda }_1|\phi }{{\Gamma }_q(p_2+1)})+|{\sigma }_2||{\rho }_3|\left(\frac{|{\alpha }_1|{\hslash }_1}{{\Gamma }_q(p_1+p_2)}+\frac{|{\beta }_1|{\varpi }_1}{{\Gamma }_q(p_1+p_2+{\xi }_1)}+\frac{|{\lambda }_1|\phi }{{\Gamma }_q\left(p_2\right)}\right)\hfill \\ & & +|{\delta }_4||{\rho }_4|\left(\frac{|{\alpha }_1|{\eta }_j^{p_1+p_2-1}{\hslash }_1}{{\Gamma }_q(p_1+p_2)}+\frac{|{\beta }_1|{\eta }_j^{p_1+p_2+{\xi }_1-1}{\varpi }_1}{{\Gamma }_q(p_1+p_2+{\xi }_1)}+\frac{|{\lambda }_1|{\eta }_j^{p_2-1}\phi }{{\Gamma }_q\left(p_2\right)}\right)+|{\sigma }_3||{\rho }_4|(\frac{|{\alpha }_2|{\hslash }_2}{{\Gamma }_q(r_1+r_2+1)}\hfill \\ & & +\frac{|{\beta }_2|{\varpi }_2}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)}+\frac{|{\lambda }_2|\phi }{{\Gamma }_q(r_2+1)})+|{\sigma }_4||{\rho }_4|\left(\frac{|{\alpha }_2|{\hslash }_2}{{\Gamma }_q(r_1+r_2)}+\frac{|{\beta }_2|{\varpi }_2}{{\Gamma }_q(r_1+r_2+{\xi }_2)}+\frac{|{\lambda }_2|\phi }{{\Gamma }_q\left(r_2\right)}\right)\hfill \\ & & \le \frac{|{\alpha }_1|{\hslash }_1}{{\Gamma }_q(p_1+p_2+1)}\left[2{(t_2-t_1)}^{p_1+p_2}+|t_2^{p_1+p_2}-t_1^{p_1+p_2}|\right]+\frac{|{\beta }_1|{\varpi }_1}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)}\hfill \\ & & \times \left[2{(t_2-t_1)}^{p_1+p_2+{\xi }_1}+|t_2^{p_1+p_2+{\xi }_1}-t_1^{p_1+p_2+{\xi }_1}|\right]+\frac{|{\lambda }_1|\phi }{{\Gamma }_q(p_2+1)}\left[2{(t_2-t_1)}^{p_2}+|t_2^{p_2}-t_1^{p_2}|\right]\hfill \\ & & +\frac{|t_2^{p_2}-t_1^{p_2}|}{|\Lambda |}[|{\alpha }_1|{\hslash }_1\left(\frac{|{\delta }_2||{\rho }_2|{\eta }_j^{p_1+p_2}+|{\sigma }_1||{\rho }_3|}{{\Gamma }_q(p_1+p_2+1)}+\frac{|{\delta }_4||{\rho }_4|{\eta }_j^{p_1+p_2-1}+|{\sigma }_2||{\rho }_3|}{{\Gamma }_q(p_1+p_2)}\right)\hfill \\ & & +|{\alpha }_2|{\hslash }_2\left(\frac{|{\delta }_1||{\rho }_1|{\eta }_j^{r_1+r_2}+|{\sigma }_3||{\rho }_4|}{{\Gamma }_q(r_1+r_2+1)}+\frac{|{\delta }_3||{\rho }_3|{\eta }_j^{r_1+r_2-1}+|{\sigma }_4||{\rho }_4|}{{\Gamma }_q(r_1+r_2)}\right)\hfill \\ & & +|{\beta }_1|{\varpi }_1\left(\frac{|{\delta }_2||{\rho }_2|{\eta }_j^{p_1+p_2+{\xi }_1}+|{\sigma }_1||{\rho }_3|}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)}+\frac{|{\delta }_4||{\rho }_4|{\eta }_j^{p_1+p_2+{\xi }_1-1}+|{\sigma }_2||{\rho }_3|}{{\Gamma }_q(p_1+p_2+{\xi }_1)}\right)\hfill \\ & & +|{\beta }_2|{\varpi }_2\left(\frac{|{\delta }_1||{\rho }_1|{\eta }_j^{r_1+r_2+{\xi }_2}+|{\sigma }_3||{\rho }_4|}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)}+\frac{|{\delta }_3||{\rho }_3|{\eta }_j^{r_1+r_2+{\xi }_2-1}+|{\sigma }_4||{\rho }_4|}{{\Gamma }_q(r_1+r_2+{\xi }_2)}\right)\hfill \\ & & +\phi (|{\lambda }_1|\left(\frac{|{\delta }_2||{\rho }_2|{\eta }_j^{p_2}+|{\sigma }_1||{\rho }_3|}{{\Gamma }_q(p_2+1)}+\frac{|{\delta }_4||{\rho }_4|{\eta }_j^{p_2-1}+|{\sigma }_2||{\rho }_3|}{{\Gamma }_q\left(p_2\right)}\right)+|{\lambda }_2|(\frac{|{\delta }_1||{\rho }_1|{\eta }_j^{r_2}+|{\sigma }_3||{\rho }_4|}{{\Gamma }_q(r_2+1)}\hfill \\ & & +\frac{|{\delta }_3||{\rho }_3|{\eta }_j^{r_2-1}+|{\sigma }_4||{\rho }_4|}{{\Gamma }_q\left(r_2\right)}\left)\right)],\hfill \end{array}$$

which tends to zero as $t_2-t_1\to 0$ independent of $(x,y).$ Analogously, we can obtain

$$\begin{array}{ccc}& & |{\mathcal{G}}_2(x\left(t_2\right),y\left(t_2\right))-{\mathcal{G}}_2(x\left(t_1\right),y\left(t_1\right))|\hfill \\ & & \le \frac{|{\alpha }_2|{\hslash }_2}{{\Gamma }_q(r_1+r_2+1)}\left[2{(t_2-t_1)}^{r_1+r_2}+|t_2^{r_1+r_2}-t_1^{r_1+r_2}|\right]+\frac{|{\beta }_2|{\varpi }_2}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)}\hfill \\ & & \times \left[2{(t_2-t_1)}^{r_1+r_2+{\xi }_2}+|t_2^{r_1+r_2+{\xi }_2}-t_1^{r_1+r_2+{\xi }_2}|\right]+\frac{|{\lambda }_2|\phi }{{\Gamma }_q(r_2+1)}\left[2{(t_2-t_1)}^{r_2}+|t_2^{r_2}-t_1^{r_2}|\right]\hfill \\ & & +\frac{|t_2^{r_2}-t_1^{r_2}|}{|\Lambda |}[|{\alpha }_1|{\hslash }_1\left(\frac{|{\delta }_2||{\rho }_{10}|{\eta }_j^{p_1+p_2}+|{\sigma }_1||{\rho }_{11}|}{{\Gamma }_q(p_1+p_2+1)}+\frac{|{\delta }_4||{\rho }_{12}|{\eta }_j^{p_1+p_2-1}+|{\sigma }_2||{\rho }_{11}|}{{\Gamma }_q(p_1+p_2)}\right)\hfill \\ & & +|{\alpha }_2|{\hslash }_2\left(\frac{|{\delta }_1||{\rho }_9|{\eta }_j^{r_1+r_2}+|{\sigma }_3||{\rho }_{12}|}{{\Gamma }_q(r_1+r_2+1)}+\frac{|{\delta }_3||{\rho }_{11}|{\eta }_j^{r_1+r_2-1}+|{\sigma }_4||{\rho }_{12}|}{{\Gamma }_q(r_1+r_2)}\right)\hfill \\ & & +|{\beta }_1|{\varpi }_1\left(\frac{|{\delta }_2||{\rho }_{10}|{\eta }_j^{p_1+p_2+{\xi }_1}+|{\sigma }_1||{\rho }_{11}|}{{\Gamma }_q(p_1+p_2+{\xi }_1+1)}+\frac{|{\delta }_4||{\rho }_{12}|{\eta }_j^{p_1+p_2+{\xi }_1-1}+|{\sigma }_2||{\rho }_{11}|}{{\Gamma }_q(p_1+p_2+{\xi }_1)}\right)\hfill \\ & & +|{\beta }_2|{\varpi }_2\left(\frac{|{\delta }_1||{\rho }_9|{\eta }_j^{r_1+r_2+{\xi }_2}+|{\sigma }_3||{\rho }_{12}|}{{\Gamma }_q(r_1+r_2+{\xi }_2+1)}+\frac{|{\delta }_3||{\rho }_{11}|{\eta }_j^{r_1+r_2+{\xi }_2-1}+|{\sigma }_4||{\rho }_{12}|}{{\Gamma }_q(r_1+r_2+{\xi }_2)}\right)\hfill \\ & & +\phi (|{\lambda }_1|(\frac{|{\delta }_2||{\rho }_{10}|{\eta }_j^{p_2}+|{\sigma }_1||{\rho }_{11}|}{{\Gamma }_q(p_2+1)}+\frac{|{\delta }_4||{\rho }_{12}|{\eta }_j^{p_2-1}+|{\sigma }_2||{\rho }_{11}|}{{\Gamma }_q(p_2)})+|{\lambda }_2|(\frac{|{\delta }_1||{\rho }_9|{\eta }_j^{r_2}+|{\sigma }_3||{\rho }_{12}|}{{\Gamma }_q(r_2+1)}\hfill \\ & & +\frac{|{\delta }_3||{\rho }_{11}|{\eta }_j^{r_2-1}+|{\sigma }_4||{\rho }_{12}|}{{\Gamma }_q\left(r_2\right)}))].\hfill \end{array}$$

Note that the right-hand side of the above inequality tends to zero as $t_2-t_1\to 0$ independent of $(x,y).$ Thus the operator $\mathcal{G}(x,y)$ is equicontinuous. In view of the foregoing arguments, we deduce that the operator $\mathcal{G}(x,y)$ is completely continuous.

Finally, we show that $\Omega =\left\{\right(x,y)\in \mathcal{C}\times \mathcal{C}|(x,y)=\zeta \mathcal{G}(x,y),0<\zeta <1\}$ is bounded. Let $(x,y)\in \Omega ,$ with $(x,y)=\zeta \mathcal{G}(x,y)\left(t\right)$ and for any $t\in [0,1]$, we have

$$x\left(t\right)=\zeta {\mathcal{G}}_1(x,y)\left(t\right),y\left(t\right)=\zeta {\mathcal{G}}_2(x,y)\left(t\right).$$

In view of condition $\left({\mathbf{A}}_3\right),$ we can find that

$$\begin{array}{ccc}& & |x(t)|\hfill \\ & & \le ({\tau }_0+{\tau }_1|x|+{\tau }_2|y|\left)\right[|{\alpha }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ & & +\frac{|{\alpha }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u+\frac{|{\alpha }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u]\hfill \\ & & +({\tilde{\tau }}_0+{\tilde{\tau }}_1|x|+{\tilde{\tau }}_2|y|\left)\right[\frac{|{\alpha }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}\hfill \\ & & \times {\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u]+({ϵ}_0+{ϵ}_1|x|+{ϵ}_2|y|\left)\right[|{\beta }_1|{\int }_0^t\frac{{(t-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_1||{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_1||{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u]\hfill \\ & & +({\widetilde{ϵ}}_0+{\widetilde{ϵ}}_1|x|+{\widetilde{ϵ}}_2|y|\left)\right[\frac{|{\beta }_2||{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}\hfill \\ & & \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u\hfill \\ & & +\frac{|{\beta }_2||{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u]+|{\lambda }_1|[{\int }_0^t\frac{{(t-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\delta }_2||{\rho }_2{t}^{p_2}+{\rho }_6|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u+\frac{|{\sigma }_1||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ & & +\frac{|{\sigma }_2||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u+\frac{|{\delta }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u]|x|\hfill \\ & & +|{\lambda }_2|[\frac{|{\delta }_1||{\rho }_1{t}^{p_2}+{\rho }_5|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\delta }_3||{\rho }_3{t}^{p_2}+{\rho }_7|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\hfill \\ & & +\frac{|{\sigma }_3||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\sigma }_4||{\rho }_4{t}^{p_2}+{\rho }_8|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u]|y|,\hfill \end{array}$$

and

$$\begin{array}{c}|y(t)|\hfill \\ \le ({\tau }_0+{\tau }_1|x|+{\tau }_2|y|\left)\right[\frac{|{\alpha }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u\hfill \\ +\frac{|{\alpha }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-1)}}{{\Gamma }_q(p_1+p_2)}{d}_{q}u+\frac{|{\alpha }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u\hfill \\ +\frac{|{\alpha }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2-2)}}{{\Gamma }_q(p_1+p_2-1)}{d}_{q}u]\hfill \\ +({\tilde{\tau }}_0+{\tilde{\tau }}_1|x|+{\tilde{\tau }}_2|y|\left)\right[|{\alpha }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u+\frac{|{\alpha }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ +\frac{|{\alpha }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u+\frac{|{\alpha }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-1)}}{{\Gamma }_q(r_1+r_2)}{d}_{q}u\hfill \\ +\frac{|{\alpha }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2-2)}}{{\Gamma }_q(r_1+r_2-1)}{d}_{q}u]\hfill \\ +({ϵ}_0+{ϵ}_1|x|+{ϵ}_2|y|\left)\right[\frac{|{\beta }_1||{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u\hfill \\ +\frac{|{\beta }_1||{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-1)}}{{\Gamma }_q(p_1+p_2+{\xi }_1)}{d}_{q}u+\frac{|{\beta }_1||{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}\hfill \\ \times {\int }_0^1\frac{{(1-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u+\frac{|{\beta }_1||{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_1+p_2+{\xi }_1-2)}}{{\Gamma }_q(p_1+p_2+{\xi }_1-1)}{d}_{q}u]\hfill \\ +({\widetilde{ϵ}}_0+{\widetilde{ϵ}}_1|x|+{\widetilde{ϵ}}_2|y|\left)\right[|{\beta }_2|{\int }_0^t\frac{{(t-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}\hfill \\ \times {\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u\hfill \\ +\frac{|{\beta }_2||{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-1)}}{{\Gamma }_q(r_1+r_2+{\xi }_2)}{d}_{q}u+\frac{|{\beta }_2||{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}\hfill \\ \times {\int }_0^1\frac{{(1-qu)}^{(r_1+r_2+{\xi }_2-2)}}{{\Gamma }_q(r_1+r_2+{\xi }_2-1)}{d}_{q}u]+|{\lambda }_1|[\frac{|{\delta }_2||{\rho }_{10}{t}^{r_2}+{\rho }_{14}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u\hfill \\ +\frac{|{\sigma }_1||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-1)}}{{\Gamma }_q\left(p_2\right)}{d}_{q}u+\frac{|{\sigma }_2||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u\hfill \\ +\frac{|{\delta }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(p_2-2)}}{{\Gamma }_q(p_2-1)}{d}_{q}u]|x|+|{\lambda }_2|[{\int }_0^t\frac{{(t-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u\hfill \\ +\frac{|{\delta }_1||{\rho }_9{t}^{r_2}+{\rho }_{13}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\delta }_3||{\rho }_{11}{t}^{r_2}+{\rho }_{15}|}{|\Lambda |}{\int }_0^{{\eta }_j}\frac{{({\eta }_j-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u\hfill \\ +\frac{|{\sigma }_3||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-1)}}{{\Gamma }_q\left(r_2\right)}{d}_{q}u+\frac{|{\sigma }_4||{\rho }_{12}{t}^{r_2}+{\rho }_{16}|}{|\Lambda |}{\int }_0^1\frac{{(1-qu)}^{(r_2-2)}}{{\Gamma }_q(r_2-1)}{d}_{q}u]|y|.\hfill \end{array}$$

In consequence, we obtain

$$\begin{array}{ccc}\hfill \parallel x\parallel & & \le ({\tau }_0+{\tau }_1\parallel x\parallel +{\tau }_2\parallel y\parallel ){\Psi }_1+({\tilde{\tau }}_0+{\tilde{\tau }}_1\parallel x\parallel +{\tilde{\tau }}_2\parallel y\parallel ){\Psi }_2\hfill \\ & & +({ϵ}_0+{ϵ}_1\parallel x\parallel +{ϵ}_2\parallel y\parallel ){\Phi }_1+({\widetilde{ϵ}}_0+{\widetilde{ϵ}}_1\parallel x\parallel +{\widetilde{ϵ}}_2\parallel y\parallel ){\Phi }_2\hfill \\ & & +|{\lambda }_1|\left(\frac{1}{{\Gamma }_q(p_2+1)}+\frac{({\gamma }_9+{\gamma }_{10})}{|\Lambda |}\right)\parallel x\parallel +\frac{|{\lambda }_2|({\gamma }_{11}+{\gamma }_{12})}{|\Lambda |}\parallel y\parallel \hfill \\ & & ={\tau }_0{\Psi }_1+{\tilde{\tau }}_0{\Psi }_2+{ϵ}_0{\Phi }_1+{\widetilde{ϵ}}_0{\Phi }_2\hfill \\ & & +({\tau }_1{\Psi }_1+{\tilde{\tau }}_1{\Psi }_2+{ϵ}_1{\Phi }_1+{\widetilde{ϵ}}_1{\Phi }_2)\parallel x\parallel \hfill \\ & & +({\tau }_2{\Psi }_1+{\tilde{\tau }}_2{\Psi }_2+{ϵ}_2{\Phi }_1+{\widetilde{ϵ}}_2{\Phi }_2)\parallel y\parallel \hfill \\ & & +|{\lambda }_1|\left(\frac{1}{{\Gamma }_q(p_2+1)}+\frac{({\gamma }_9+{\gamma }_{10})}{|\Lambda |}\right)\parallel x\parallel +\frac{|{\lambda }_2|({\gamma }_{11}+{\gamma }_{12})}{|\Lambda |}\parallel y\parallel ,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel y\parallel & & \le ({\tau }_0+{\tau }_1\parallel x\parallel +{\tau }_2\parallel y\parallel ){\Psi }_3+({\tilde{\tau }}_0+{\tilde{\tau }}_1\parallel x\parallel +{\tilde{\tau }}_2\parallel y\parallel ){\Psi }_4\hfill \\ & & +({ϵ}_0+{ϵ}_1\parallel x\parallel +{ϵ}_2\parallel y\parallel ){\Phi }_3+({\widetilde{ϵ}}_0+{\widetilde{ϵ}}_1\parallel x\parallel +{\widetilde{ϵ}}_2\parallel y\parallel ){\Phi }_4\hfill \\ & & +\frac{|{\lambda }_1|({\gamma }_{21}+{\gamma }_{22})}{|\Lambda |}\parallel x\parallel +|{\lambda }_2|\left(\frac{1}{{\Gamma }_q(r_2+1)}+\frac{({\gamma }_{23}+{\gamma }_{24})}{|\Lambda |}\right)\parallel y\parallel \hfill \\ & & ={\tau }_0{\Psi }_3+{\tilde{\tau }}_0{\Psi }_4+{ϵ}_0{\Phi }_3+{\widetilde{ϵ}}_0{\Phi }_4\hfill \\ & & +({\tau }_1{\Psi }_3+{\tilde{\tau }}_1{\Psi }_4+{ϵ}_1{\Phi }_3+{\widetilde{ϵ}}_1{\Phi }_4)\parallel x\parallel \hfill \\ & & +({\tau }_2{\Psi }_3+{\tilde{\tau }}_2{\Psi }_4+{ϵ}_2{\Phi }_3+{\widetilde{ϵ}}_2{\Phi }_4)\parallel y\parallel \hfill \\ & & +\frac{|{\lambda }_1|({\gamma }_{21}+{\gamma }_{22})}{|\Lambda |}\parallel x\parallel +|{\lambda }_2|\left(\frac{1}{{\Gamma }_q(r_2+1)}+\frac{({\gamma }_{23}+{\gamma }_{24})}{|\Lambda |}\right)\parallel y\parallel ,\hfill \end{array}$$

which imply that

$$\begin{array}{ccc}\hfill \parallel x\parallel +\parallel y\parallel & & \le ({\Psi }_1+{\Psi }_3){\tau }_0+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_0+({\Phi }_1+{\Phi }_3){ϵ}_0+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_0\hfill \\ & & +\left[({\Psi }_1+{\Psi }_3){\tau }_1+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_1+({\Phi }_1+{\Phi }_3){ϵ}_1+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_1+{\upsilon }_1\right]\parallel x\parallel \hfill \\ & & +\left[({\Psi }_1+{\Psi }_3){\tau }_2+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_2+({\Phi }_1+{\Phi }_3){ϵ}_2+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_2+{\upsilon }_2\right]\parallel y\parallel .\hfill \end{array}$$

Thus we have

$$\begin{array}{ccc}\hfill \parallel (x,y)\parallel & & \le \frac{({\Psi }_1+{\Psi }_3){\tau }_0+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_0+({\Phi }_1+{\Phi }_3){ϵ}_0+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_0}{H_0},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill H_0& =& min\{1-[({\Psi }_1+{\Psi }_3){\tau }_1+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_1+({\Phi }_1+{\Phi }_3){ϵ}_1+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_1+{\upsilon }_1],\hfill \\ & & 1-[({\Psi }_1+{\Psi }_3){\tau }_2+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_2+({\Phi }_1+{\Phi }_3){ϵ}_2+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_2+{\upsilon }_2]\},\hfill \end{array}$$

which establishes that the set $\Omega$ is bounded. Thus, by Leray–Schauder nonlinear alternative [48], there exists a solution of the system (1)–(2) on $[0,1]$. The proof is complete.  □

4. Examples

I.

Illustration of Theorem 1

Example 1.Let us consider a nonlinear system of coupled fractional q-integro-difference equations:

$${\displaystyle \left\{\begin{array}{l}{\displaystyle {}^c{D}_{0.5}^{0.05}{(}^c{D}_{0.5}^{0.05}+0.02)x\left(t\right)=0.09f_1(t,x\left(t\right),y\left(t\right))+0.03I_{0.5}^{0.25}{g}_1(t,x\left(t\right),y\left(t\right)),0\le t\le 1,}\\ {\displaystyle {}^c{D}_{0.5}^{0.35}{(}^c{D}_{0.5}^{0.35}+0.06)y\left(t\right)=0.08f_2(t,x\left(t\right),y\left(t\right))+0.07I_{0.5}^{0.25}{g}_2(t,x\left(t\right),y\left(t\right)),0\le t\le 1,}\end{array}\right.}$$

(17)

supplemented with four-point coupled boundary conditions

$$\left\{\begin{array}{c}{\displaystyle 0.4x\left(0\right)-0.2\left(t^{(1-0.05)}{D}_{q}x\left(t\right)\right){|}_{t=0}=\sum _{j=1}^2{a}_{j}y\left({\eta }_j\right),}\\ {\displaystyle 0.4y\left(0\right)-0.2\left(t^{(1-0.35)}{D}_{q}y\left(t\right)\right){|}_{t=0}=\sum _{j=1}^2{b}_{j}x\left({\eta }_j\right),}\\ {\displaystyle 0.1x\left(1\right)+0.2D_{q}x\left(1\right)=\sum _{j=1}^2{k}_j{D}_{q}y\left({\eta }_j\right),}\\ {\displaystyle 0.1y\left(1\right)+0.2D_{q}y\left(1\right)=\sum _{j=1}^2{m}_j{D}_{q}x\left({\eta }_j\right),}\end{array}\right.$$

(18)

where $p_1=p_2=0.05$, $q=0.5$, $r_1=r_2=0.35$, ${\alpha }_1=0.09$, ${\alpha }_2=0.08$, ${\beta }_1=0.03$, ${\beta }_2=0.07$, ${\xi }_1={\xi }_2=0.25$, ${\lambda }_1=0.02,{\lambda }_2=0.06$, ${\mu }_1={\mu }_3=0.4$, ${\mu }_2={\mu }_4=0.2$, ${\sigma }_1={\sigma }_3=0.1,{\sigma }_2={\sigma }_4=0.2$, $a_1=0.35,a_2=0.3,$$b_1=0.2,b_2=0.25,$$k_1=0.7,k_2=0.1,$$m_1=0.6,m_2=0.8$, ${\eta }_1=0.45,{\eta }_2=0.65$, $t\in [0,1]$ and

$$\begin{array}{ccc}\hfill f_1(t,x\left(t\right),y\left(t\right))& =& \frac{1}{196}\frac{|x(t)|}{1+|x(t)|}+\frac{arctany\left(t\right)}{{(t^2+14)}^2}-10t,\hfill \\ \hfill f_2(t,x\left(t\right),y\left(t\right))& =& \frac{1}{\sqrt{225}+105+t^3}(x\left(t\right)+cosy\left(t\right)),\hfill \\ \hfill g_1(t,x\left(t\right),y\left(t\right))& =& \frac{1}{16\sqrt{t^6+81}}(sinx\left(t\right)+arctany\left(t\right)-cos2t),\hfill \\ \hfill g_2(t,x\left(t\right),y\left(t\right))& =& \frac{1}{20\sqrt{t+49}}(sinx\left(t\right)+\frac{|y(t)|}{1+|y(t)|})+16e^{-t}.\hfill \end{array}$$

(19)

Then ${\iota }_1=1/196,{\iota }_2=1/120,{\kappa }_1=1/144,{\kappa }_2=1/140$ as

$$|f_1(t,x_1\left(t\right),y_1\left(t\right))-f_1(t,x_2\left(t\right),y_2\left(t\right))|=\frac{1}{196}(|x_1-x_2|+|y_1-y_2|),$$

$$|f_2(t,x_1\left(t\right),y_1\left(t\right))-f_2(t,x_2\left(t\right),y_2\left(t\right))|=\frac{1}{120}(|x_1-x_2|+|y_1-y_2|),$$

$$|g_1(t,x_1\left(t\right),y_1\left(t\right))-g_1(t,x_2\left(t\right),y_2\left(t\right))|=\frac{1}{144}(|x_1-x_2|+|y_1-y_2|),$$

$$|g_2(t,x_1\left(t\right),y_1\left(t\right))-g_2(t,x_2\left(t\right),y_2\left(t\right))|=\frac{1}{140}(|x_1-x_2|+|y_1-y_2|).$$

Using the given data, it is found that ${\Psi }_1=0.393067,{\Psi }_2=0.476841,{\Psi }_3=0.356139,{\Psi }_4=0.451896,$${\Phi }_1=0.248188,{\Phi }_2=0.414528,{\Phi }_3=0.271996,{\Phi }_4=0.383275,{\Theta }_1=0.359396,{\Theta }_2=0.363914,$ and $\mathrm{Y}\approx 0.744182<1$. Clearly the hypothesis of Theorem 1 holds true. So, by the conclusion of Theorem 1, the system (17) and (18) has a unique solution on $[0,1]$.

II.

Illustration of Theorem 2

Example 2.Let us consider the system (17) and (18) with nonlinearities:

$$\begin{array}{ccc}\hfill f_1(t,x\left(t\right),y\left(t\right))& =& \frac{1}{(60+t)}+\frac{1}{263}x\left(t\right)sin\left(t\right)+\frac{1}{170}arctany\left(t\right),\hfill \\ \hfill f_2(t,x\left(t\right),y\left(t\right))& =& \frac{1}{5\sqrt{1600t}}+\frac{1}{2(t^3+9)}cosx\left(t\right)+\frac{1}{2{(11+t^5)}^{2}y\left(t\right)},\hfill \\ \hfill g_1(t,x\left(t\right),y\left(t\right))& =& \frac{1}{122\sqrt{t}}+\frac{2}{139}x\left(t\right)+\frac{1}{12\sqrt{256+t}}cosy\left(t\right),\hfill \\ \hfill g_2(t,x\left(t\right),y\left(t\right))& =& \frac{5sint}{(t^3+126)}+\frac{1}{280}x\left(t\right)+\frac{2}{153}y\left(t\right).\hfill \end{array}$$

(20)

Notice that the condition $\left({\mathbf{A}}_3\right)$ holds true as

$$\begin{array}{ccc}\hfill |f_1(t,x\left(t\right),y\left(t\right))|& \le & \frac{1}{60}+\frac{1}{263}|x\left(t\right)|+\frac{1}{170}|y\left(t\right)|,|f_2(t,x\left(t\right),y\left(t\right))|\le \frac{1}{20}+\frac{1}{162}|x\left(t\right)|+\frac{1}{242}|y\left(t\right)|,\hfill \\ \hfill |g_1(t,x\left(t\right),y\left(t\right))|& \le & \frac{1}{122}+\frac{2}{139}|x\left(t\right)|+\frac{1}{192}|y\left(t\right)|,|g_2(t,x\left(t\right),y\left(t\right))|\le \frac{5}{126}+\frac{1}{280}|x\left(t\right)|+\frac{2}{153}|y\left(t\right)|,\hfill \end{array}$$

with ${\tau }_0=1/60,{\tau }_1=1/263,{\tau }_2=1/170,{\tilde{\tau }}_0=1/200,{\tilde{\tau }}_1=1/162,{\tilde{\tau }}_2=1/242,{ϵ}_0=1/122,{ϵ}_1=2/139,{ϵ}_2=1/192,{\widetilde{ϵ}}_0=5/126,{\widetilde{ϵ}}_1=1/280,{\widetilde{ϵ}}_2=2/153.$ Moreover,

$$({\Psi }_1+{\Psi }_3){\tau }_1+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_1+({\Phi }_1+{\Phi }_3){ϵ}_1+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_1+{\upsilon }_1\approx 0.138309<1,$$

$$({\Psi }_1+{\Psi }_3){\tau }_2+({\Psi }_2+{\Psi }_4){\tilde{\tau }}_2+({\Phi }_1+{\Phi }_3){ϵ}_2+({\Phi }_2+{\Phi }_4){\widetilde{ϵ}}_2+{\upsilon }_2\approx 0.625299<1.$$

Thus, all the assumptions of Theorem 2 are satisfied. Therefore, the conclusion of Theorem 2 applies and hence the system (17) and (18) with the nonlinearities (20) has at least one solution on $[0,1].$

5. Conclusions

We have studied a new class of nonlocal multipoint boundary value problems of Langevin-type nonlinear coupled q-fractional integro-difference equations. First of all, the given problem was converted into an equivalent fixed-point problem. Then, we proved an existence and uniqueness result for the problem at hand by applying the Banach contraction mapping principle. In our second result, we presented the criteria ensuring the existence of a solution for the given problem. We also demonstrated the application of the obtained results by solving some particular problems. We emphasize that our results are new and contribute significantly to the literature on nonlocal multipoint boundary value problems of nonlinear coupled q-fractional integro-difference equations. It is imperative to note that our results correspond to the non-coupled separated boundary conditions for all $a_j=0,b_j=0,k_j=0,m_j=0,j=1,\cdots ,n,$ which are indeed new in the given configuration.