Solvability Criterion for Fractional q-Integro-Difference System with Riemann-Stieltjes Integrals Conditions

Abstract

Due to the great application potential of fractional q-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional q-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional q-integro-difference system with the nonlocal boundary conditions involving diverse fractional q-derivatives and Riemann-Stieltjes q-integrals. We acquire the existence results of solutions for the systems by applying Schauder fixed point theorem, Krasnoselskii’s fixed point theorem, Schaefer’s fixed point theorem and nonlinear alternative for single-valued maps, and a uniqueness result is obtained through the Banach contraction mapping principle. Finally, we give some examples to illustrate the main results.

1. Introduction

In the early twentieth century, Jackson [1] proposed a new mathematical direction of q-calculus, and it plays an indispensable role in the fields of nuclear, conformal quantum mechanics and dynamics. In the 1960s, Agarwal [2] and Al-Salam [3] put forward a novel concept of fractional q-calculus, its relevant application and development can be seen in the literature [4,5,6]. Compared with classical q-calculus, fractional q-calculus can more accurately describe some phenomena in nature, and many practical problems can be abstracted into fractional q-difference equations or a system of fractional q-difference equations by mathematical modeling. In recent years, abundant theoretical achievements have been made in the research of boundary value problems (BVPs) for fractional q-difference equations, according to the literature [7,8,9,10,11,12,13,14,15,16] and the references therein.

Riemann-Stieltjes integral is a generalization of Riemann integral. As well as we known, the classical Riemann-Stieltjes integral can be widely applied in several areas of analysis, such as probability theory, stochastic processes, physics, econometrics, biometrics and informetrics and so on. BVPs with Riemann-Stieltjes integral boundary condition (BC) have been considered as both multi-point and integral type BCs are treated in a single framework. In recent years, some interesting results about the existence of solutions for nonlinear fractional differential equations with the Riemann-Stieltjes integral BC have been researched, see [17,18] and the references therein.

Nowadays, the system of nonlinear fractional differential equations has important applications in engineering, economy and other fields. This is mainly because the effect of using fractional calculus to solve problems is more practical and efficient than that of classical calculus. Over the years, the BVPs for a system of fractional differential equations have developed rapidly, and numerous mature conclusions have been obtained, which can be referred to the literature [19,20,21,22,23,24,25].

In [24], Tudorache, A. and Luca, R. applied the Guo-Krasnoselskii fixed point theorem to study the existence of solutions for a system of fractional differential equations with p-Laplacian operators

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_{0^{+}}^{{\alpha }_1}({\phi }_{{\varrho }_1}(D_{0_{+}}^{{\beta }_1}x(t)))+\lambda f(t,x(t),y(t))=0,t\in (0,1),\\ D_{0^{+}}^{{\alpha }_2}({\phi }_{{\varrho }_2}(D_{0_{+}}^{{\beta }_2}y(t)))+\mu g(t,x(t),y(t))=0,t\in (0,1),\end{array}\right.\end{array}$$

with the nonlocal BCs

$$\begin{array}{l}\left\{\begin{array}{l}{x}^{(j)}(0)=0,j=0,\cdots ,n-2;D_{0^{+}}^{{\beta }_1}x(0)=0,\\ D_{0^{+}}^{{\gamma }_0}x(1)=\sum _{i=1}^p{\int }_0^1{D}_{0^{+}}^{{\gamma }_i}y(t)d{H}_{i}t,\\ y^{(j)}(0)=0,j=0,\cdots ,m-2;D_{0^{+}}^{{\beta }_2}y(0)=0,\\ D_{0^{+}}^{{\delta }_0}y(1)=\sum _{i=1}^q{\int }_0^1{D}_{0^{+}}^{{\delta }_i}x(t)d{K}_{i}t.\end{array}\right.\end{array}$$

In [25], Luca, R. considered the existence of solutions of the nonlinear system of fractional differential equations by using a variety of fixed point theorems

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_{0^{+}}^{\alpha }x(t)+f(t,x(t),y(t),I_{0^{+}}^{{\theta }_1}x(t),I_{0^{+}}^{{\sigma }_1}y(t))=0,t\in (0,1),\\ D_{0^{+}}^{\beta }y(t)+g(t,x(t),y(t),I_{0^{+}}^{{\theta }_2}x(t),I_{0^{+}}^{{\sigma }_2}y(t))=0,t\in (0,1),\end{array}\right.\end{array}$$

with the nonlocal BCs

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x(0)=x^{{}^{\prime }}(0)=\cdots =x^{(n-2)}(0)=0,D_{0^{+}}^{{\gamma }_0}x(1)=\sum _{i=1}^p{\int }_0^1{D}_{0^{+}}^{{\gamma }_i}y(t)d{H}_i(t),\\ y(0)=y^{{}^{\prime }}(0)=\cdots =y^{(m-2)}(0)=0,D_{0^{+}}^{{\delta }_0}y(1)=\sum _{i=1}^q{\int }_0^1{D}_{0^{+}}^{{\delta }_i}x(t)d{K}_i(t).\end{array}\right.\end{array}$$

Despite quite a number of contributions dealing with the solvability for the system of classical fractional difference equations. However, as the generalization of the above system, limited work has been done in the nonlinear system of fractional q-difference equations. In particular, there is little research on the existence and uniqueness of solutions for the system of fractional q-difference equations with Riemann-Stieltjes integral BC. To fill this gap, we investigate the system of nonlinear fractional q-difference equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(D_q^{\alpha }u)(t)+P(t,u(t),v(t),I_q^{{\omega }_1}u(t),I_q^{{\delta }_1}v(t))=0,\\ (D_q^{\beta }v)(t)+Q(t,u(t),v(t),I_q^{{\omega }_2}u(t),I_q^{{\delta }_2}v(t))=0,\end{array}\right.\end{array}$$

(1)

with the nonlocal BCs

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u(0)=D_{q}u(0)=\cdots =D_q^{n-2}u(0)=0,D_q^{{\zeta }_0}u(1)={\int }_0^1{D}_q^{\zeta }v(t)d_{q}H(t),\\ v(0)=D_{q}v(0)=\cdots =D_q^{m-2}v(0)=0,D_q^{{\xi }_0}v(1)={\int }_0^1{D}_q^{\xi }u(t)d_{q}K(t),\end{array}\right.\end{array}$$

(2)

where $t\in (0,1),0<q<1,\alpha \in (n-1,n],\beta \in (z-1,z],n,z\in \mathbb{N},n\ge 2$ and $z\ge 2$, ${\omega }_1,{\omega }_2,{\delta }_1,{\delta }_2>0,0\le \zeta <\beta -1,0\le \xi <\alpha -1,{\zeta }_0\in [0,\alpha -1),$ ${\xi }_0\in [0,\beta -1),$ $D_q^i$ denotes the Riemann-Liouville q-derivative of order i ($i=\alpha ,\beta ,{\zeta }_0,\zeta ,{\xi }_0,\xi$), $I_q^{\varpi }$ is the Riemann-Liouville q-integral of order $\varpi$ ($\varpi ={\omega }_1,{\omega }_2,{\delta }_1,{\delta }_2$), P and Q are nonlinear functions. The BCs include Riemann-Stieltjes integrals, where $H(t),K(t)$ are the bounded variation functions. In the case where $H(t)=K(t)=t$, the Riemann–Stieltjes integrals in (2) reduce to the classical q-integral.

The present paper is bulit up as follows. The second part offers the necessary definitions, lemmas and theorems needed in the following. The third part obtains the important conclusions by applying various fixed point theorems, including nine theorems or corollaries. In the final part, four examples are provided to verify our main results.

2. Preliminaries

In this section, we present some definitions, lemmas and theorems.

Definition 1

([11]). Let $\beta \ge 0$ and f be a function defined on $[0,1]$. The fractional q-integral of the Riemann-Liouville type is

$$\begin{array}{c}\hfill (I_q^{\beta }f)(s)=\frac{1}{{\Gamma }_q(\beta )}{\int }_0^s{(s-qt)}^{(\beta -1)}f(t)d_{q}t,\beta >0,s\in [0,1].\end{array}$$

Obviously, $(I_q^{\beta }f)(s)=(I_{q}f)(s)$, when $\beta =1$.

Definition 2

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

$$\begin{array}{c}\hfill (D_q^{\beta }f)(s)=(D_q^l{I}_q^{l-\beta }f)(s),\beta >0,s\in [0,1],\end{array}$$

where l is the smallest integer greater than or equal to β.

Lemma 1

([11]). Let $\alpha ,\beta \ge 0$ and f be a function defined on $[0,1]$. Then, the following formulas hold:

1.

$(I_q^{\beta }{I}_q^{\alpha }f)(x)=(I_q^{\alpha +\beta }f)(x),$

2.

$(D_q^{\alpha }{I}_q^{\alpha }f)(x)=f(x).$

Lemma 2

([11]). Let $\alpha >0$ and p be a positive integer. Then, the following equality holds:

$$\begin{array}{c}\hfill (I_q^{\alpha }{D}_q^{p}f)(x)=(D_q^p{I}_q^{\alpha }f)(x)-\sum _{k=0}^{p-1}\frac{x^{\alpha -p+k}}{{\Gamma }_q(\alpha +k-p+1)}(D_q^{k}f)(0).\end{array}$$

Lemma 3.

If $x\in C[0,1]$, then for $\kappa >0$, we get

$$\begin{array}{c}\hfill |I_q^{\kappa }x(t)|\le \frac{\Vert x\Vert }{{\Gamma }_q(\kappa )},\end{array}$$

where $\Vert x\Vert ={sup}_{t\in [0,1]}\left|x(t)\right|$.

Proof. 

According to Definition 1, this lemma clearly holds. □

Definition 3

([15]). The function $f:I\times {\mathbb{R}}^4\to \mathbb{R}$ is called an S-Carathéodory function if and only if

 (i)

for each $(u,v,x,y)\in {\mathbb{R}}^4,t\to f(t,u,v,x,y)$ is measurable on I;

 (ii)

for a.e. $t\in I,(u,v,x,y)\to f(t,u,v,x,y)$ is continuous on ${\mathbb{R}}^4$;

 (iii)

for each $r>0$, there exists ${\psi }_r(t)\in L^1(I,{\mathbb{R}}^{+})$ with $t{\psi }_r(t)\in L^1(I,{\mathbb{R}}^{+})$ on I such that $max\left\{\right|u|,|v|,|x|,|y\left|\right\}\le r$ implies $\left|f(t,u,v,x,y)\right|\le {\psi }_r(t)$, for a.e.I, where $L^1(I,{\mathbb{R}}^{+})=\{u\in X:{\int }_0^{1}u(t)d_{q}t\mathrm{exists}\}$, and normed ${\Vert u\Vert }_{L^1}={\int }_0^1\left|u(t)\right|d_{q}t$ for all $u\in L^1(I,{\mathbb{R}}^{+})$.

Theorem 1

([26]). (Schauder fixed point theorem) Let D be a bounded closed convex set in E (D does not necessarily have an interior point), and $A:D\to D$ is completely continuous, then A must have a fixed point in D.

Theorem 2

([12]). (Krasnoselskii’s fixed point theorem) Let K be a closed convex and nonempty subset of a Banach space X. Let $T,S$ be the operators such that

 (i)

$Tu+Sv\in K$ whenever $u,v\in K$;

 (ii)

T is compact and continuous;

 (iii)

S is a contraction mapping.

Then, there exists $z\in K$ such that $z=Tz+Sz.$

Theorem 3

([16]). (Schaefer’s fixed point theorem) Let T be a continuous and compact mapping of a Banach space X into itself, such that the set $E=\left\{x\right|x\in X:x=\lambda Tx,0\le \lambda \le 1\}$ is bounded. Then T has a fixed point.

Theorem 4

([15]). (Nonlinear alternative for single-valued maps) Let E be a Banach space, let C be a closed and convex subset of E, and let U be an open subset of C and $0\in U$. Suppose that $F:\overline{U}\to C$ is a continuous, compact (that is, $F(\overline{U})$ is a relatively compact subset of C) map. Then either

 (i)

F has a fixed point in $\overline{U}$, or

 (ii)

there is a $u\in \partial U$ (the boundary of U in C) and $\lambda \in (0,1)$ with $u=\lambda Fu$.

Throughout this paper, we adopt the following assumptions:

$({\mathcal{H}}_1)$ The functions $P,Q\in C([0,1]\times {\mathbb{R}}^4,\mathbb{R})$ and for $x_i,y_i\in \mathbb{R}$, there exist $L_i(t),l_i(t)\in C([0,1],[0,+\infty )),i=1,2,3,4$, such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)-P(t,y_1,y_2,y_3,y_4)|\le \sum _{i=1}^4{L}_i(t)|x_i-y_i|,\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)-Q(t,y_1,y_2,y_3,y_4)|\le \sum _{i=1}^4{l}_i(t)|x_i-y_i|.\hfill \end{array}$$

$({\mathcal{H}}_1^{\prime })$ The functions $P,Q\in C([0,1]\times {\mathbb{R}}^4,\mathbb{R})$ and for $x_i,y_i\in \mathbb{R}$, there exist real constants $L_i,l_i>0,i=1,2,3,4,$ such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)-P(t,y_1,y_2,y_3,y_4)|\le \sum _{i=1}^4{L}_i|x_i-y_i|,\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)-Q(t,y_1,y_2,y_3,y_4)|\le \sum _{i=1}^4{l}_i|x_i-y_i|.\hfill \end{array}$$

$({\mathcal{H}}_1^{″})$ The functions $P,Q\in C([0,1]\times {\mathbb{R}}^4,\mathbb{R})$ and for $x_i,y_i\in \mathbb{R}$, there exist real functions ${\rho }_i(t),{\varrho }_i(t)\in C([0,1],{\mathbb{R}}^{+}),i=1,2,3,4$, such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)-P(t,y_1,y_2,y_3,y_4)|\le \sum _{i=1}^4{\rho }_i(t)|x_i-y_i|,\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)-Q(t,y_1,y_2,y_3,y_4)|\le \sum _{i=1}^4{\varrho }_i(t)|x_i-y_i|.\hfill \end{array}$$

$({\mathcal{H}}_2)$ The functions $P,Q\in C([0,1]\times {\mathbb{R}}^4,\mathbb{R})$, and for $x_i\in \mathbb{R}$, there exist functions $c_i(t),d_i(t)\in C([0,1],{\mathbb{R}}^{+}),$ and $h_i,m_i\in (0,1),i=1,2,3,4$, such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le c_0(t)+\sum _{i=1}^4{c}_i(t){\left|x_i\right|}^{h_i},\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le d_0(t)+\sum _{i=1}^4{d}_i(t){\left|x_i\right|}^{m_i}.\hfill \end{array}$$

$({\mathcal{H}}_3)$ The functions $P,Q\in C([0,1]\times {\mathbb{R}}^4,\mathbb{R})$, and for $x_i\in \mathbb{R},i=1,2,3,4$, there exist functions ${\sigma }_1(t),{\sigma }_2(t)\in C([0,1],{\mathbb{R}}^{+})$ such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le {\sigma }_1(t),\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le {\sigma }_2(t).\hfill \end{array}$$

$({\mathcal{H}}_4)$ The functions $P,Q:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ and for a.e. $t\in [0,1],x_i\in \mathbb{R}$, there exist $r_1(t),r_2(t),L_i(t),l_i(t)\in C([0,1],{\mathbb{R}}^{+}),i=1,2,3,4,$ such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le \sum _{i=1}^4{L}_i(t)\left|x_i\right|+r_1(t),\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le \sum _{i=1}^4{l}_i(t)\left|x_i\right|+r_2(t).\hfill \end{array}$$

$({\mathcal{H}}_4^{\prime })$ The functions $P,Q:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ and for a.e. $t\in [0,1],x_i\in \mathbb{R}$, there exist non-negative real numbers $L_i,l_i(i=1,2,3,4)$, and $r_1,r_2$, where at least one of $r_1$ and $r_2$ is positive, such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le \sum _{i=1}^4{L}_i\left|x_i\right|+r_1,\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le \sum _{i=1}^4{l}_i\left|x_i\right|+r_2.\hfill \end{array}$$

$({\mathcal{H}}_5)$ The functions $P,Q:[0,1]\times {\mathbb{R}}^4\to \mathbb{R}$ and for a.e. $t\in [0,1],x_i\in \mathbb{R}$, there exist functions $p_i(t),q_i(t)\in C([0,1],{\mathbb{R}}^{+})$, where $p_i(t),q_i(t)$ have at least one non-zero function, and there exist nondecreasing functions ${\phi }_i,{\eta }_i\in C([0,\infty ),{\mathbb{R}}^{+}),i=1,2,3,4$, such that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le \sum _{i=1}^4{p}_i(t){\phi }_i(|x_i|)+p_0(t),\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le \sum _{i=1}^4{q}_i(t){\eta }_i(|x_i|)+q_0(t).\hfill \end{array}$$

For convenience, we denote

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_1=& 1+\frac{1}{{\Gamma }_q({\omega }_1)},C_2=1+\frac{1}{{\Gamma }_q({\delta }_1)},C_3=max\{C_1,C_2\},\hfill \\ \hfill C_4=& 1+\frac{1}{{\Gamma }_q({\omega }_2)},C_5=1+\frac{1}{{\Gamma }_q({\delta }_2)},C_6=max\{C_4,C_5\},\hfill \\ \hfill C_7=& \frac{1}{{\Gamma }_q(\alpha )}+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}\hfill \\ & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|·\left|{\int }_0^1{d}_{q}K(s)\right|,\hfill \\ \hfill C_8=& \frac{1}{{\Gamma }_q(\beta )}+\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}\hfill \\ & +\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{s}^{\alpha -\xi -1}{d}_{q}K(s)\right|·\left|{\int }_0^1{d}_{q}H(s)\right|,\hfill \\ \hfill C_9=& \frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\alpha -\xi )}\left|{\int }_0^1{s}^{\alpha -\xi -1}{d}_{q}K(s)\right|\hfill \\ & +\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\alpha -\xi )}\left|{\int }_0^1{d}_{q}K(s)\right|,\hfill \\ \hfill C_{10}=& \frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|\hfill \\ & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|,\hfill \\ \hfill C_{11}=& C_7-\frac{1}{{\Gamma }_q(\alpha )},C_{12}=C_8-\frac{1}{{\Gamma }_q(\beta )}.\hfill \\ \hfill {\Omega }_1=& \frac{{\Gamma }_q(\beta )}{{\Gamma }_q(\beta -\zeta )}{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s),{\Omega }_2=\frac{{\Gamma }_q(\alpha )}{{\Gamma }_q(\alpha -\xi )}{\int }_0^1{s}^{\alpha -\xi -1}{d}_{q}K(s),\hfill \\ \hfill \Omega =& \frac{{\Gamma }_q(\alpha ){\Gamma }_q(\beta )}{{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}-{\Omega }_1{\Omega }_2.\hfill \end{array}\end{array}$$

(3)

3. Criterion of Uniqueness and Existence

In this section, we show some existence and uniqueness results for the Systems (1)–(2).

Lemma 4.

Let $h,k\in C(0,1)\cap L^1(0,1)$ and $\Omega \ne 0$, then the system of fractional q-difference equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_q^{\alpha }u(t)+h(t)=0,t\in (0,1),\\ D_q^{\beta }v(t)+k(t)=0,t\in (0,1),\end{array}\right.\end{array}$$

(4)

with the coupled BCs (2) has a unique solution $(u(t),v(t))$, namely

$$\begin{array}{cc}\hfill u(t)=& -\frac{1}{{\Gamma }_q(\alpha )}{\int }_0^t{(t-qs)}^{(\alpha -1)}h(s)d_{q}s+\frac{t^{\alpha -1}}{\Omega }[\frac{{\Omega }_1}{{\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}k(s)d_{q}s\hfill \\ \hfill & -\frac{{\Omega }_1}{{\Gamma }_q(\alpha -\xi )}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}h(\tau )d_q\tau \right]d_{q}K(s)\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}h(s)d_{q}s\hfill \\ \hfill & -\frac{{\Gamma }_q(\beta )}{{\Gamma }_q(\beta -\zeta ){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}k(\tau )d_q\tau \right]d_{q}H(s)],\hfill \\ \hfill v(t)=& -\frac{1}{{\Gamma }_q(\beta )}{\int }_0^t{(t-qs)}^{(\beta -1)}k(s)d_{q}s+\frac{t^{\beta -1}}{\Omega }[\frac{{\Omega }_2}{{\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}h(s)d_{q}s\hfill \\ \hfill & -\frac{{\Omega }_2}{{\Gamma }_q(\beta -\zeta )}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}k(\tau )d_q\tau \right]d_{q}H(s)\hfill \\ \hfill & +\frac{{\Gamma }_q(\alpha )}{{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}k(s)d_{q}s\hfill \\ \hfill & -\frac{{\Gamma }_q(\alpha )}{{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}h(\tau )d_q\tau \right]d_{q}K(s)],t\in [0,1].\hfill \end{array}$$

Proof. 

The proof is similar to the Lemma 2.1 in [24]. □

Let $U=C[0,1]$ and $V=U\times U$ be the Banach spaces with the norms $\Vert u\Vert ={sup}_{t\in [0,1]}\left|u(t)\right|$ and ${\Vert (u,v)\Vert }_V=\Vert u\Vert +\Vert v\Vert$, respectively. Nowdays, we introduce the operator $\mathcal{T}:V\to V$, where $\mathcal{T}(x,y)=({\mathcal{T}}_1(x,y),{\mathcal{T}}_2(x,y))$ for $(x,y)\in V$, and ${\mathcal{T}}_1,{\mathcal{T}}_2:V\to U$ are defined by

$$\begin{array}{cc}\hfill {\mathcal{T}}_1(u,v)(t)=& -\frac{1}{{\Gamma }_q(\alpha )}{\int }_0^t{(t-qs)}^{(\alpha -1)}{F}_{uv}(s)d_{q}s+\frac{{\Omega }_1{t}^{\alpha -1}}{\Omega {\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}\hfill \\ \hfill & ·G_{uv}(s)d_{q}s-\frac{{\Omega }_1{t}^{\alpha -1}}{\Omega {\Gamma }_q(\alpha -\xi )}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}{F}_{uv}(\tau )d_q\tau \right]d_{q}K(s)\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{\Omega {\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}{F}_{uv}(s)d_{q}s\hfill \\ \hfill & -\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{\Omega {\Gamma }_q(\beta -\zeta ){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}{G}_{uv}(\tau )d_q\tau \right]d_{q}H(s),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathcal{T}}_2(u,v)(t)=& -\frac{1}{{\Gamma }_q(\beta )}{\int }_0^t{(t-qs)}^{(\beta -1)}{G}_{uv}(s)d_{q}s+\frac{{\Omega }_2{t}^{\beta -1}}{\Omega {\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}\hfill \\ \hfill & ·F_{uv}(s)d_{q}s-\frac{{\Omega }_2{t}^{\beta -1}}{\Omega {\Gamma }_q(\beta -\zeta )}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}{G}_{uv}(\tau )d_q\tau \right]d_{q}H(s)\hfill \\ \hfill & +\frac{{\Gamma }_q(\alpha )t^{\beta -1}}{\Omega {\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}{G}_{uv}(s)d_{q}s\hfill \\ \hfill & -\frac{{\Gamma }_q(\alpha )t^{\beta -1}}{\Omega {\Gamma }_q(\alpha -\xi ){\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}{F}_{uv}(\tau )d_q\tau \right]d_{q}K(s),\hfill \end{array}$$

for $t\in [0,1]$ and $(u,v)\in V$, where

$$\begin{array}{c}\hfill F_{uv}(s)=P(s,u(s),v(s),I_q^{{\omega }_1}u(s),I_q^{{\delta }_1}v(s)),G_{uv}(s)=Q(s,u(s),v(s),I_q^{{\omega }_2}u(s),I_q^{{\delta }_2}v(s)).\end{array}$$

According to Lemma 4, it is easy to see that $(u(t),v(t))$ is a solution of the Systems (1)–(2) if and only if $(u(t),v(t))$ is a fixed point of operator $\mathcal{T}$.

At first, we prove the existence and uniqueness theorem of the Systems (1)–(2) by Banach contraction mapping principle.

Theorem 5.

Suppose that $({\mathcal{H}}_1)$ holds. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill \Lambda ={\Lambda }_1{C}_3(C_7+C_9)+{\Lambda }_2{C}_6(C_8+C_{10})<1,\end{array}$$

where ${\Lambda }_1={max}_{t\in [0,1]}\left\{{\sum }_{i=1}^4{L}_i(t)\right\},{\Lambda }_2={max}_{t\in [0,1]}\left\{{\sum }_{i=1}^4{l}_i(t)\right\}$. Then the Systems (1)–(2) has a unique solution.

Proof. 

Let $r>0$ such that

$$\begin{array}{c}\hfill r=\frac{C_0(C_7+C_9)+{\tilde{C}}_0(C_8+C_{10})}{1-{\Lambda }_1{C}_3(C_7+C_9)-{\Lambda }_2{C}_6(C_8+C_{10})},\end{array}$$

where $C_0={sup}_{t\in [0,1]}\left|P(t,0,0,0,0)\right|,{\tilde{C}}_0={sup}_{t\in [0,1]}\left|Q(t,0,0,0,0)\right|$.

We divide two steps to prove the theorem.

(i) Our first task is to show that $\mathcal{T}$ maps bounded sets into bounded sets in V.

Let $B_r=\left\{(u,v)\in {V,\Vert (u,v)\Vert }_V\le r\right\}$ be a bounded set in V and $(u,v)\in B_r$. Then we show that $\mathcal{T}(B_r)\subset B_r$. By $({\mathcal{H}}_1)$ and Lemma 3, we get

$$\begin{array}{cc}\hfill \left|F_{uv}(t)\right|\le & |P(t,u(t),v(t),I_q^{{\omega }_1}u(t),I_q^{{\delta }_1}v(t))-P(t,0,0,0,0)|+\left|P(t,0,0,0,0)\right|\hfill \\ \hfill \le & \left[L_1(t)\left|u(t)\right|+L_2(t)\left|v(t)\right|+L_3(t)|I_q^{{\omega }_1}u(t)|+L_4(t)\left|I_q^{{\delta }_1}v(t)\right|\right]+C_0\hfill \\ \hfill \le & {\Lambda }_1\left[\Vert u\Vert +\Vert v\Vert +\frac{\Vert u\Vert }{{\Gamma }_q({\omega }_1)}+\frac{\Vert v\Vert }{{\Gamma }_q({\delta }_1)}\right]+C_0\hfill \\ \hfill =& {\Lambda }_1(C_1\Vert u\Vert +C_2\Vert v\Vert )+C_0\hfill \\ \hfill \le & {\Lambda }_1{C}_3{\Vert (u,v)\Vert }_V+C_0\le {\Lambda }_1{C}_{3}r+C_0,\hfill \end{array}$$

similarly,

$$\begin{array}{c}\hfill \left|G_{uv}(t)\right|\le {\Lambda }_2{C}_{6}r+{\tilde{C}}_0.\end{array}$$

According to the expression of operators ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$, we obtain

$$\begin{array}{cc}\hfill |{\mathcal{T}}_1(u,v)(t)|\le & \frac{1}{{\Gamma }_q(\alpha )}{\int }_0^t{(t-qs)}^{(\alpha -1)}\left|F_{uv}(s)\right|d_{q}s\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}\left|G_{uv}(s)\right|d_{q}s\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\hfill \\ \hfill & ·\left|{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}\left|F_{uv}(\tau )\right|d_q\tau \right]d_{q}K(s)\right|·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}\left|F_{uv}(s)\right|d_{q}s\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}\left|G_{uv}(\tau )\right|d_q\tau \right]d_{q}H(s)\right|\hfill \\ \hfill \le & [\frac{1}{{\Gamma }_q(\alpha )}+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|·\left|{\int }_0^1{d}_{q}K(s)\right|\left]{\int }_0^1\right|F_{uv}(s)|d_{q}s\hfill \\ \hfill & +[\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}·|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|\left]{\int }_0^1\right|G_{uv}(s)|d_{q}s,\hfill \end{array}$$

thus, we have

$$\begin{array}{c}\hfill \Vert {\mathcal{T}}_1(u,v)\Vert \le C_7({\Lambda }_1{C}_{3}r+C_0)+C_{10}({\Lambda }_2{C}_{6}r+{\tilde{C}}_0),\end{array}$$

(5)

in like wise,

$$\begin{array}{cc}\hfill \Vert {\mathcal{T}}_2(u,v)\Vert \le & C_9({\Lambda }_1{C}_{3}r+C_0)+C_8({\Lambda }_2{C}_{6}r+{\tilde{C}}_0).\hfill \end{array}$$

(6)

Using (5) and (6), we obtain that for $\forall (u,v)\in B_r$,

$$\begin{array}{cc}\hfill {\Vert \mathcal{T}(u,v)\Vert }_V=& \Vert {\mathcal{T}}_1(u,v)\Vert +\Vert {\mathcal{T}}_2(u,v)\Vert \hfill \\ \hfill \le & ({\Lambda }_1{C}_{3}r+C_0)(C_7+C_9)+({\Lambda }_2{C}_{6}r+{\tilde{C}}_0)(C_8+C_{10})=r,\hfill \end{array}$$

that is $\mathcal{T}(B_r)\subset B_r$.

(ii) The next step is to prove that operator $\mathcal{T}$ is a contraction.

For $(u_i,v_i)\in B_r(i=1,2),t\in [0,1]$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |{\mathcal{T}}_1(u_1,v_1)(t)-{\mathcal{T}}_1(u_2,v_2)(t)|\hfill \\ \hfill \le & \frac{1}{{\Gamma }_q(\alpha )}{\int }_0^t{(t-qs)}^{(\alpha -1)}|F_{u_1{v}_1}(s)-F_{u_2{v}_2}(s)|d_{q}s\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}|G_{u_1{v}_1}(s)-G_{u_2{v}_2}(s)|d_{q}s\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}|F_{u_1{v}_1}(\tau )-F_{u_2{v}_2}(\tau )|d_q\tau \right]d_{q}K(s)\right|\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}\hfill \\ \hfill & ·|F_{u_1{v}_1}(s)-F_{u_2{v}_2}(s)|d_{q}s+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\hfill \\ \hfill & ·\left|{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}|G_{u_1{v}_1}(\tau )-G_{u_2{v}_2}(\tau )|d_q\tau \right]d_{q}H(s)\right|.\hfill \end{array}\end{array}$$

(7)

Since

$$\begin{array}{cc}\hfill |F_{u_1{v}_1}(s)-F_{u_2{v}_2}(s)|\le & \left[L_1(s)\right|u_1(s)-u_2(s)|+L_2(s)|v_1(s)-v_2(s)|\hfill \\ \hfill & +L_3(s)|I_q^{{\omega }_1}{u}_1(s)-I_q^{{\omega }_1}{u}_2(s)|+L_4(s)|I_q^{{\delta }_1}{v}_1(s)-I_q^{{\delta }_1}{v}_2(s)|]\hfill \\ \hfill \le & {\Lambda }_1(C_1\Vert u_1-u_2\Vert +C_2\Vert v_1-v_2\Vert )\hfill \\ \hfill \le & {\Lambda }_1{C}_3{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & |G_{u_1{v}_1}(s)-G_{u_2{v}_2}(s)|\le {\Lambda }_2{C}_6{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V.\hfill \end{array}$$

By (7), we have

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1(u_1,v_1)(t)-{\mathcal{T}}_1(u_2,v_2)(t)|\hfill \\ \hfill \le & \frac{1}{{\Gamma }_q(\alpha )}{\int }_0^1|F_{u_1{v}_1}(s)-F_{u_2{v}_2}(s)|d_{q}s+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1|F_{u_1{v}_1}(s)-F_{u_2{v}_2}(s)|d_{q}s\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}{\int }_0^1|G_{u_1{v}_1}(s)-G_{u_2{v}_2}(s)|d_{q}s·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1\left[{\int }_0^1|F_{u_1{v}_1}(\tau )-F_{u_2{v}_2}(\tau )|d_q\tau \right]d_{q}K(s)\right|\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\hfill \\ \hfill & ·\left|{\int }_0^1\left[{\int }_0^1|G_{u_1{v}_1}(\tau )-G_{u_2{v}_2}(\tau )|d_q(\tau )\right]d_{q}H(s)\right|\hfill \\ \hfill \le & {\Lambda }_1{C}_3{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V[\frac{1}{{\Gamma }_q(\alpha )}+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|]\hfill \\ \hfill & +{\Lambda }_2{C}_6{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V\left[\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\right|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|],\hfill \end{array}$$

hence, we deduce

$$\begin{array}{c}\hfill \Vert {\mathcal{T}}_1(u_1,v_1)-{\mathcal{T}}_1(u_2,v_2)\Vert \le ({\Lambda }_1{C}_3{C}_7+{\Lambda }_2{C}_6{C}_{10}){\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V.\end{array}$$

(8)

For the same way, we can obtain

$$\begin{array}{c}\hfill \Vert {\mathcal{T}}_2(u_1,v_1)-{\mathcal{T}}_2(u_2,v_2)\Vert \le ({\Lambda }_1{C}_3{C}_9+{\Lambda }_2{C}_6{C}_8){\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V.\end{array}$$

(9)

From (8) and (9), we have

$$\begin{array}{cc}\hfill & \Vert \mathcal{T}(u_1,v_1)-\mathcal{T}(u_2,v_2){\Vert }_V\hfill \\ \hfill =& \Vert {\mathcal{T}}_1(u_1,v_1)-{\mathcal{T}}_1(u_2,v_2)\Vert +\Vert {\mathcal{T}}_2(u_1,v_1)-{\mathcal{T}}_2(u_2,v_2)\Vert \hfill \\ \hfill \le & \left[{\Lambda }_1{C}_3(C_7+C_9)+{\Lambda }_2{C}_6(C_8+C_{10})\right]{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V\hfill \\ \hfill =& \Lambda \Vert (u_1,v_1)-(u_2,v_2){\Vert }_V.\hfill \end{array}$$

Due to $\Lambda <1$, it follows that $\Vert \mathcal{T}(u_1,v_1)-\mathcal{T}(u_2,v_2){\Vert }_V<{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V$, so operator $\mathcal{T}$ is a contraction. Hence, we obtain that the Systems (1)–(2) has a unique solution $(u,v)\in B_r$ by using Banach contraction mapping principle. The proof is completed. □

Corollary 1.

Suppose that $({\mathcal{H}}_1^{\prime })$ holds. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill {\Lambda }^{*}={\Lambda }_3{C}_3(C_7+C_9)+{\Lambda }_4{C}_6(C_8+C_{10})<1,\end{array}$$

where ${\Lambda }_3={\sum }_{i=1}^4{L}_i,{\Lambda }_4={\sum }_{i=1}^4{l}_i$. Then the Systems (1)–(2) has a unique solution.

Corollary 2.

Suppose that $({\mathcal{H}}_1^{\prime \prime })$ holds. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill \tilde{\Lambda }={\Lambda }_5{C}_3(C_7+C_9)+{\Lambda }_6{C}_6(C_8+C_{10})<1,\end{array}$$

where ${\Lambda }_5={sup}_{t\in [0,1]}\left\{{\sum }_{i=1}^4{\rho }_i(t)\right\},{\Lambda }_6={sup}_{t\in [0,1]}\left\{{\sum }_{i=1}^4{\varrho }_i(t)\right\}$. Then the Systems (1)–(2) has a unique solution.

Next, we apply several kinds of fixed point theorems to achieve the existence results of solutions for the Systems (1)–(2).

Theorem 6.

Suppose that $({\mathcal{H}}_2)$ and $\Omega \ne 0$ hold. Then the System (1)–(2) has at least one solution.

Proof. 

Let $B_R={\{(u,v)\in V,\Vert (u,v)\Vert }_V\le R\}$, and we denote

$$\begin{array}{cc}\hfill R_1=max& \left\{\right[\Vert c_0\Vert +\Vert c_1\Vert {({\mathcal{N}}_1)}^{h_1}+\Vert c_2\Vert {({\mathcal{N}}_2)}^{h_2}+\Vert c_3\Vert {\left(\frac{{\mathcal{N}}_1}{{\Gamma }_q({\omega }_1)}\right)}^{h_3}\hfill \\ \hfill & +\Vert c_4\Vert {\left(\frac{{\mathcal{N}}_2}{{\Gamma }_q({\delta }_1)}\right)}^{h_4}]C_7,[\Vert d_0\Vert +\Vert d_1\Vert {({\mathcal{N}}_1)}^{m_1}+\Vert d_2\Vert {({\mathcal{N}}_2)}^{m_2}\hfill \\ \hfill & +\Vert d_3\Vert {\left(\frac{{\mathcal{N}}_1}{{\Gamma }_q({\omega }_2)}\right)}^{m_3}+\Vert d_4\Vert {\left(\frac{{\mathcal{N}}_2}{{\Gamma }_q({\delta }_2)}\right)}^{m_4}\left]C_{10}\right\},\hfill \\ \hfill R_2=max& \left\{\right[\Vert c_0\Vert +\Vert c_1\Vert {({\mathcal{N}}_1)}^{h_1}+\Vert c_2\Vert {({\mathcal{N}}_2)}^{h_2}+\Vert c_3\Vert {\left(\frac{{\mathcal{N}}_1}{{\Gamma }_q({\omega }_1)}\right)}^{h_3}\hfill \\ \hfill & +\Vert c_4\Vert {\left(\frac{{\mathcal{N}}_2}{{\Gamma }_q({\delta }_1)}\right)}^{h_4}]C_9,[\Vert d_0\Vert +\Vert d_1\Vert {({\mathcal{N}}_1)}^{m_1}+\Vert d_2\Vert {({\mathcal{N}}_2)}^{m_2}\hfill \\ \hfill & +\Vert d_3\Vert {\left(\frac{{\mathcal{N}}_1}{{\Gamma }_q({\omega }_2)}\right)}^{m_3}+\Vert d_4\Vert {\left(\frac{{\mathcal{N}}_2}{{\Gamma }_q({\delta }_2)}\right)}^{m_4}\left]C_8\right\},\hfill \\ \hfill R=2max& \left\{R_1,R_2\right\}.\hfill \end{array}$$

where there exist ${\mathcal{N}}_1,{\mathcal{N}}_2\in \mathbb{R}$ such that $\left|u(t)\right|\le {\mathcal{N}}_1,\left|v(t)\right|\le {\mathcal{N}}_2$.

Firstly, we show that $\mathcal{T}$ maps bounded sets into bounded sets in V. For $(u,v)\in B_R$, we obtain

$$\begin{array}{cc}\hfill \Vert {\mathcal{T}}_1(u,v)\Vert \le & [\Vert c_0\Vert +\Vert c_1\Vert {({\mathcal{N}}_1)}^{h_1}+\Vert c_2\Vert {({\mathcal{N}}_2)}^{h_2}+\Vert c_3\Vert {\left(\frac{{\mathcal{N}}_1}{{\Gamma }_q({\omega }_1)}\right)}^{h_3}\hfill \\ \hfill & +\Vert c_4\Vert {\left(\frac{{\mathcal{N}}_2}{{\Gamma }_q({\delta }_1)}\right)}^{h_4}]C_7+[\Vert d_0\Vert +\Vert d_1\Vert {({\mathcal{N}}_1)}^{m_1}+\Vert d_2\Vert {({\mathcal{N}}_2)}^{m_2}\hfill \\ \hfill & +\Vert d_3\Vert {\left(\frac{{\mathcal{N}}_1}{{\Gamma }_q({\omega }_2)}\right)}^{m_3}+\Vert d_4\Vert {\left(\frac{{\mathcal{N}}_2}{{\Gamma }_q({\delta }_2)}\right)}^{m_4}]C_{10}\le 2R_1,\hfill \end{array}$$

similarly, $\Vert {\mathcal{T}}_2(u,v)\Vert \le 2R_2$, then

$$\begin{array}{c}\hfill {\Vert \mathcal{T}(u,v)\Vert }_V=\Vert {\mathcal{T}}_1(u,v)\Vert +\Vert {\mathcal{T}}_2(u,v)\Vert \le R,(u,v)\in B_R,\end{array}$$

as above, we obtain $\mathcal{T}(B_R)\subset B_R$.

Secondly, we prove that $\mathcal{T}$ maps bounded sets into equicontinuous sets of V. Let $\mathcal{N}=max\{{\mathcal{N}}_1,{\mathcal{N}}_2\}$, for simplicity of presentation, we denote that

$$\begin{array}{cc}\hfill & {\Psi }_{\mathcal{N}}=\underset{t\in [0,1]}{sup}\left\{\left|P(t,u,v,x,y)\right|,\left|u\right|\le \mathcal{N},\left|v\right|\le \mathcal{N},\left|x\right|\le \frac{\mathcal{N}}{{\Gamma }_q({\omega }_1)},\left|y\right|\le \frac{\mathcal{N}}{{\Gamma }_q({\delta }_1)}\right\},\hfill \\ \hfill & {\Theta }_{\mathcal{N}}=\underset{t\in [0,1]}{sup}\left\{\left|Q(t,u,v,x,y)\right|,\left|u\right|\le \mathcal{N},\left|v\right|\le \mathcal{N},\left|x\right|\le \frac{\mathcal{N}}{{\Gamma }_q({\omega }_2)},\left|y\right|\le \frac{\mathcal{N}}{{\Gamma }_q({\delta }_2)}\right\},\hfill \end{array}$$

then for $(u,v)\in B_R$ and $t_1,t_2\in [0,1]$ with $t_1<t_2$, we have

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_1(u,v)(t_2)-{\mathcal{T}}_1(u,v)(t_1)|\hfill \\ \hfill \le & \frac{{\Psi }_{\mathcal{N}}}{{\Gamma }_q(\alpha +1)}(t_2^{\alpha }-t_1^{\alpha })+{\Psi }_{\mathcal{N}}(t_2^{\alpha -1}-t_1^{\alpha -1})\left[\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\right|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)|\hfill \\ \hfill & ·\left|{\int }_0^1{d}_{q}K(s)\right|+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}]+{\Theta }_{\mathcal{N}}(t_2^{\alpha -1}-t_1^{\alpha -1})[\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|]\hfill \\ \hfill =& \frac{{\Psi }_{\mathcal{N}}}{{\Gamma }_q(\alpha +1)}(t_2^{\alpha }-t_1^{\alpha })+({\Psi }_{\mathcal{N}}{C}_{11}+{\Theta }_{\mathcal{N}}{C}_{10})(t_2^{\alpha -1}-t_1^{\alpha -1}).\hfill \end{array}$$

The same can be proved that

$$\begin{array}{c}\hfill |{\mathcal{T}}_2(u,v)(t_2)-{\mathcal{T}}_2(u,v)(t_1)|\le \frac{{\Theta }_{\mathcal{N}}}{{\Gamma }_q(\beta +1)}(t_2^{\beta }-t_1^{\beta })+({\Psi }_{\mathcal{N}}{C}_9+{\Theta }_{\mathcal{N}}{C}_{12})(t_2^{\beta -1}-t_1^{\beta -1}).\end{array}$$

Hence, we conclude

$$\begin{array}{c}\hfill |{\mathcal{T}}_1(u,v)(t_2)-{\mathcal{T}}_1(u,v)(t_1)|\to 0,|{\mathcal{T}}_2(u,v)(t_2)-{\mathcal{T}}_2(u,v)(t_1)|\to 0,\end{array}$$

as $t_2\to t_1,(u,v)\in B_R$. Thus, $\mathcal{T}(B_R)$ is equicontinuous. According to the Arzela-Ascoli theorem, it follows that the set $\mathcal{T}(B_R)$ is relatively compact. Therefore, $\mathcal{T}$ is compact on $B_R$. By Theorem 1, we get that the System (1)–(2) has at least one solution. The proof is completed. □

Theorem 7.

Suppose that $({\mathcal{H}}_1^{\prime })$ and $({\mathcal{H}}_3)$ hold. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill \overline{\Lambda }={\Lambda }_3{C}_3\frac{1}{{\Gamma }_q(\alpha )}+{\Lambda }_4{C}_6\frac{1}{{\Gamma }_q(\beta )}<1.\end{array}$$

Then the System (1)–(2) has at least one solution.

Proof. 

Take $r_0>0$ such that

$$\begin{array}{c}\hfill r_0\ge (C_7+C_9)\Vert {\sigma }_1\Vert +(C_8+C_{10})\Vert {\sigma }_2\Vert .\end{array}$$

Let $B_{r_0}={\{(u,v)\in V,\Vert (u,v)\Vert }_V\le r_0\}$, and let the operators be $\mathcal{X}=({\mathcal{X}}_1,{\mathcal{X}}_2):B_{r_0}\to V$ and $\mathcal{Y}=({\mathcal{Y}}_1,{\mathcal{Y}}_2):B_{r_0}\to V,$ where ${\mathcal{X}}_1,{\mathcal{X}}_2,{\mathcal{Y}}_1,{\mathcal{Y}}_2:B_{r_0}\to U$ are denoted by

$$\begin{array}{cc}\hfill {\mathcal{X}}_1(u,v)(t)=& -\frac{1}{{\Gamma }_q(\alpha )}{\int }_0^t{(t-qs)}^{(\alpha -1)}{F}_{uv}(s)d_{q}s,\hfill \\ \hfill {\mathcal{Y}}_1(u,v)(t)=& \frac{{\Omega }_1{t}^{\alpha -1}}{\Omega {\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}{G}_{uv}(s)d_{q}s\hfill \\ \hfill & -\frac{{\Omega }_1{t}^{\alpha -1}}{\Omega {\Gamma }_q(\alpha -\xi )}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}{F}_{uv}(\tau )d_q\tau \right]d_{q}K(s)\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{\Omega {\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}{F}_{uv}(s)d_{q}s\hfill \\ \hfill & -\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{\Omega {\Gamma }_q(\beta -\zeta ){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}{G}_{uv}(\tau )d_q\tau \right]d_{q}H(s),\hfill \\ \hfill {\mathcal{X}}_2(u,v)(t)=& -\frac{1}{{\Gamma }_q(\beta )}{\int }_0^t{(t-qs)}^{(\beta -1)}{G}_{uv}(s)d_{q}s,\hfill \\ \hfill {\mathcal{Y}}_2(u,v)(t)=& \frac{{\Omega }_2{t}^{\beta -1}}{\Omega {\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}{F}_{uv}(s)d_{q}s\hfill \\ \hfill & -\frac{{\Omega }_2{t}^{\beta -1}}{\Omega {\Gamma }_q(\beta -\zeta )}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}{G}_{uv}(\tau )d_q\tau \right]d_{q}H(s)\hfill \\ \hfill & +\frac{{\Gamma }_q(\alpha )t^{\beta -1}}{\Omega {\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}{G}_{uv}(s)d_{q}s\hfill \\ \hfill & -\frac{{\Gamma }_q(\alpha )t^{\beta -1}}{\Omega {\Gamma }_q(\alpha -\xi ){\Gamma }_q(\alpha -{\zeta }_0)}{\int }_0^1\left[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}{F}_{uv}(\tau )d_q\tau \right]d_{q}K(s),\hfill \end{array}$$

where $t\in [0,1],(u,v)\in B_{r_0}$. Thus, ${\mathcal{T}}_1={\mathcal{X}}_1+{\mathcal{Y}}_1,{\mathcal{T}}_2={\mathcal{X}}_2+{\mathcal{Y}}_2$ and $\mathcal{T}=\mathcal{X}+\mathcal{Y}$.

By $({\mathcal{H}}_3)$, we know that $\forall (u_1,v_1),(u_2,v_2)\in B_{r_0}$,

$$\begin{array}{cc}\hfill & \Vert \mathcal{X}(u_1,v_1)+\mathcal{Y}(u_2,v_2){\Vert }_V\hfill \\ \hfill \le & \Vert \mathcal{X}(u_1,v_1){\Vert }_V+{\Vert \mathcal{Y}(u_2,v_2)\Vert }_V\hfill \\ \hfill =& \Vert {\mathcal{X}}_1(u_1,v_1)\Vert +\Vert {\mathcal{X}}_2(u_1,v_1)\Vert +\Vert {\mathcal{Y}}_1(u_2,v_2)\Vert +\Vert {\mathcal{Y}}_2(u_2,v_2)\Vert \hfill \\ \hfill \le & \frac{1}{{\Gamma }_q(\alpha )}\Vert {\sigma }_1\Vert +\frac{1}{{\Gamma }_q(\beta )}\Vert {\sigma }_2\Vert +\Vert {\sigma }_1\Vert [\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}K(s)\right|\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}]+\Vert {\sigma }_2\Vert [\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\hfill \\ \hfill & ·\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|]+\Vert {\sigma }_1\Vert \hfill \\ \hfill & ·\left[\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\alpha -\xi )}\left|{\int }_0^1{s}^{\alpha -\xi -1}{d}_{q}K(s)\right|+\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\alpha -{\zeta }_0)}\left|{\int }_0^1{d}_{q}K(s)\right|\right]\hfill \\ \hfill & +\Vert {\sigma }_2\Vert \left[\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|+\frac{{\Gamma }_q(\alpha )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}\right]\hfill \\ \hfill =& (C_7+C_9)\Vert {\sigma }_1\Vert +(C_8+C_{10})\Vert {\sigma }_2\Vert \le r_0.\hfill \end{array}$$

For $\forall (u_1,v_1),(u_2,v_2)\in B_{r_0}$, and $\overline{\Lambda }<1$, we have

$$\begin{array}{cc}\hfill & \Vert \mathcal{X}(u_1,v_1)-\mathcal{X}(u_2,v_2){\Vert }_V\hfill \\ \hfill =& \Vert {\mathcal{X}}_1(u_1,v_1)-{\mathcal{X}}_1(u_2,v_2)\Vert +\Vert {\mathcal{X}}_2(u_1,v_1)-{\mathcal{X}}_2(u_2,v_2)\Vert \hfill \\ \hfill \le & ({\Lambda }_3{C}_3\frac{1}{{\Gamma }_q(\alpha )}+{\Lambda }_4{C}_6\frac{1}{{\Gamma }_q(\beta )})(\Vert u_1-u_2\Vert +\Vert v_1-v_2\Vert )\hfill \\ \hfill =& \overline{\Lambda }{\Vert (u_1,v_1)-(u_2,v_2)\Vert }_V\hfill \\ \hfill <& \Vert (u_1,v_1)-(u_2,v_2){\Vert }_V.\hfill \end{array}$$

Hence, the operator $\mathcal{X}$ is a contraction.

Owing to the continuity of P and Q, $\mathcal{Y}$ is continuous. Next, we need to verify that $\mathcal{Y}$ is a compact operator. Due to $\forall (u,v)\in B_{r_0}$,

$$\begin{array}{c}\hfill {\Vert \mathcal{Y}(u,v)\Vert }_V=\Vert {\mathcal{Y}}_1(u,v)\Vert +\Vert {\mathcal{Y}}_2(u,v)\Vert \le (C_9+C_{11})\Vert {\sigma }_1\Vert +(C_{10}+C_{12})\Vert {\sigma }_2\Vert ,\end{array}$$

we have derived that the functions from $\mathcal{Y}$ are uniformly bounded.

We can show the equicontinuous of the functions from $\mathcal{Y}(B_{r_0})$. We denote that

$$\begin{array}{cc}\hfill & {\Psi }_{r_0}=\underset{t\in [0,1]}{sup}\left\{\left|P(t,u,v,x,y)\right|,\left|u\right|\le r_0,\left|v\right|\le r_0,\left|x\right|\le \frac{r_0}{{\Gamma }_q({\omega }_1)},\left|y\right|\le \frac{r_0}{{\Gamma }_q({\delta }_1)}\right\},\hfill \\ \hfill & {\Theta }_{r_0}=\underset{t\in [0,1]}{sup}\left\{\left|Q(t,u,v,x,y)\right|,\left|u\right|\le r_0,\left|v\right|\le r_0,\left|x\right|\le \frac{r_0}{{\Gamma }_q({\omega }_2)},\left|y\right|\le \frac{r_0}{{\Gamma }_q({\delta }_2)}\right\},\hfill \end{array}$$

for $(u,v)\in B_{r_0}$ and $t_1,t_2\in [0,1]$ with $t_1<t_2$. An argument similar to the one used in the proof of Theorem 6 shows that

$$\begin{array}{c}\hfill |{\mathcal{Y}}_1(u,v)(t_2)-{\mathcal{Y}}_1(u,v)(t_1)|\to 0,|{\mathcal{Y}}_2(u,v)(t_2)-{\mathcal{Y}}_2(u,v)(t_1)|\to 0,\end{array}$$

as $t_2\to t_1,(u,v)\in B_{r_0}$. Therefore, $\mathcal{Y}(B_{r_0})$ is equicontinuous. Then, we can see that $\mathcal{Y}(B_{r_0})$ is relatively compact. Hence, $\mathcal{Y}$ is compact on $B_{r_0}$. Using Theorem 2, we know that the System (1)–(2) has at least one solution. The proof is completed. □

Remark 1.

Evidently, we prove that the operator $\mathcal{X}$ is a contraction, the operator $\mathcal{Y}$ is compact and continuous in Theorem 7. An alternative method of proof is to show that $\mathcal{X}$ is compact and continuous, $\mathcal{Y}$ is a contraction, that is Theorem 8.

Theorem 8.

Suppose that $({\mathcal{H}}_1^{\prime })$ and $({\mathcal{H}}_3)$ hold. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill \widehat{\Lambda }={\Lambda }_3{C}_3(C_9+C_{11})+{\Lambda }_4{C}_6(C_8+C_{10})<1.\end{array}$$

Then the Systems (1)–(2) has at least one solution.

Proof. 

On the basis of Remark 1, this theorem can be proved by the same method as employed in Theorem 7. □

Theorem 9.

Suppose that $P,Q$ are S-Carathéodory functions and $({\mathcal{H}}_4)$ hold. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill \Xi =max\{C_{13},C_{14}\}<1,\end{array}$$

where $C_{13}=(j_1+\frac{j_3}{{\Gamma }_q({\omega }_1)})(C_7+C_9)+(k_1+\frac{k_3}{{\Gamma }_q({\omega }_2)})(C_8+C_{10}),C_{14}=(j_2+\frac{j_4}{{\Gamma }_q({\delta }_1)})(C_7+C_9)+(k_2+\frac{k_4}{{\Gamma }_1({\delta }_2)})(C_8+C_{10})$, and there exist $A_1,A_2,j_i,k_i>0$ such that $|r_1(t)|\le A_1,$ $|r_2(t)|\le A_2$, $|L_i(t)|\le j_i$ and $|l_i(t)|\le k_i(i=1,2,3,4)$. Then, the System (1)–(2) has at least one solution.

Proof. 

The main point of Theorem 9 is to prove $\mathcal{T}$ is completely continuous. Firstly, for the continuity of functions P and Q, we obtain that the operator $\mathcal{T}$ is continuous. Secondly, we show that $\mathcal{T}$ is compact.

Let the set $\Phi \subset V$ be bounded. Then, there exist integrable functions $M_1(t)$ and $M_2(t)\in L^1([0,1],{\mathbb{R}}^{+})$ such that for $\forall t\in [0,1],(u,v)\in \Phi$, we have

$$\begin{array}{c}\hfill \left|P(t,u(t),v(t),I_q^{{\omega }_1}u(t),I_q^{{\delta }_1}v(t))\right|\le M_1(t),\\ \hfill \left|Q(t,u(t),v(t),I_q^{{\omega }_2}u(t),I_q^{{\delta }_2}v(t))\right|\le M_2(t).\end{array}$$

According to the Theorem 5, we get

$$\begin{array}{cc}\hfill \left|F_{uv}(t)\right|=& \left|P(t,u(t),v(t),I_q^{{\omega }_1}u(t),I_q^{{\delta }_1}v(t))\right|\le {\Vert M_1\Vert }_{L^1},\hfill \\ \hfill \left|G_{uv}(t)\right|=& \left|Q(t,u(t),v(t),I_q^{{\omega }_2}u(t),I_q^{{\delta }_2}v(t))\right|\le {\Vert M_2\Vert }_{L^1},\hfill \end{array}$$

where ${\Vert u\Vert }_{L^1}={\int }_0^1\left|u(t)\right|d_{q}t.$

Then

$$\begin{array}{cc}\hfill \Vert {\mathcal{T}}_1(u,v)\Vert \le & \Vert M_1{\Vert }_{L^1}[\frac{1}{{\Gamma }_q(\alpha )}+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|·\left|{\int }_0^1{d}_{q}K(s)\right|]\hfill \\ \hfill & +\Vert M_2{\Vert }_{L^1}\left[\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\right|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|]\hfill \\ \hfill =& \Vert M_1{\Vert }_{L^1}{C}_7+{\Vert M_2\Vert }_{L^1}{C}_{10},\hfill \end{array}$$

in a similar manner, we have

$$\begin{array}{cc}\hfill \Vert {\mathcal{T}}_2(u,v)\Vert \le & \Vert M_1{\Vert }_{L^1}{C}_9+{\Vert M_2\Vert }_{L^1}{C}_8,\hfill \end{array}$$

so $\forall (u,v)\in \Phi$,

$$\begin{array}{c}\hfill {\Vert \mathcal{T}(u,v)\Vert }_V=\Vert {\mathcal{T}}_1(u,v)\Vert +\Vert {\mathcal{T}}_2(u,v)\Vert \le \Vert M_1{\Vert }_{L^1}(C_7+C_9)+{\Vert M_2\Vert }_{L^1}(C_{10}+C_8),\end{array}$$

therefore, $\mathcal{T}(\Phi )$ is uniformly bounded.

Another step is to show that $\mathcal{T}(\Phi )$ is equicontinuous. Proceeding as in the proof of Theorem 6, we obtain $|{\mathcal{T}}_1(u,v)(t_2)-{\mathcal{T}}_1(u,v)(t_1)|\to 0$ and $|{\mathcal{T}}_2(u,v)(t_2)-{\mathcal{T}}_2(u,v)(t_1)|\to 0$, as $t_2\to t_1,(u,v)\in \Phi$. Thus, $\mathcal{T}(\Phi )$ is equicontinuous. At the same time, we can also obtain that $\mathcal{T}$ is completely continuous.

Finally, we illustrate that $\mathcal{S}=\{(u,v)\in V,(u,v)=\lambda \mathcal{T}(u,v),0\le \lambda \le 1\}$ is bounded. Let $(u,v)\in \mathcal{S}$, then $\forall t\in [0,1]$, we have $u(t)=\lambda {\mathcal{T}}_1(u,v)(t),v(t)=\lambda {\mathcal{T}}_2(u,v)(t)$. For simplicity, we denote that

$$\begin{array}{cc}\hfill & {\widehat{F}}_{uv}(s)=r_1(s)+L_1(s)\left|u(s)\right|+L_2(s)\left|v(s)\right|+L_3(s)\left|I_q^{{\omega }_1}u(s)\right|+L_4(s)\left|I_q^{{\delta }_1}v(s)\right|,\hfill \\ \hfill & {\widehat{G}}_{uv}(s)=r_2(s)+l_1(s)\left|u(s)\right|+l_2(s)\left|v(s)\right|+l_3(s)\left|I_q^{{\omega }_2}u(s)\right|+l_4(s)\left|I_q^{{\delta }_2}v(s)\right|,\hfill \end{array}$$

so,

$$\begin{array}{cc}\hfill & {\widehat{F}}_{uv}(s)\le A_1+j_1\left|u(s)\right|+j_2\left|v(s)\right|+j_3\left|I_q^{{\omega }_1}u(s)\right|+j_4\left|I_q^{{\delta }_1}v(s)\right|,\hfill \\ \hfill & {\widehat{G}}_{uv}(s)\le A_2+k_1\left|u(s)\right|+k_2\left|v(s)\right|+k_3\left|I_q^{{\omega }_2}u(s)\right|+k_4\left|I_q^{{\delta }_2}v(s)\right|,\hfill \end{array}$$

then

$$\begin{array}{cc}\hfill |u(t)|\le & \left|{\mathcal{T}}_1(u,v)(t)\right|\hfill \\ \hfill \le & \frac{1}{{\Gamma }_q(\alpha )}{\int }_0^t{(t-qs)}^{(\alpha -1)}{\widehat{F}}_{uv}(s)d_{q}s+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}{\int }_0^1{(1-qs)}^{(\alpha -{\zeta }_0-1)}\hfill \\ \hfill & ·{\widehat{F}}_{uv}(s)d_{q}s+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|{\int }_0^1{(1-qs)}^{(\beta -{\xi }_0-1)}\hfill \\ \hfill & ·{\widehat{G}}_{uv}(s)d_{q}s+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)\right|\left|{\int }_0^1\right[{\int }_0^s{(s-q\tau )}^{(\alpha -\xi -1)}\hfill \\ \hfill & ·{\widehat{F}}_{uv}(\tau )d_q\tau \left]d_{q}K(s)\right|+\frac{{\Gamma }_q(\beta )t^{\alpha -1}}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1\right[{\int }_0^s{(s-q\tau )}^{(\beta -\zeta -1)}\hfill \\ \hfill & ·{\widehat{G}}_{uv}(\tau )d_q\tau \left]d_{q}H(s)\right|,\hfill \end{array}$$

hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert u\Vert \le & {\int }_0^1\left|{\widehat{F}}_{uv}(s)\right|d_{q}s[\frac{1}{{\Gamma }_q(\alpha )}+\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\alpha -\xi ){\Gamma }_q(\beta -\zeta )}|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)|\hfill \\ \hfill & ·\left|{\int }_0^1{d}_{q}K(s)\right|+\frac{{\Gamma }_a(\beta )}{|\Omega |{\Gamma }_q(\alpha -{\zeta }_0){\Gamma }_q(\beta -{\xi }_0)}]+{\int }_0^1|{\widehat{G}}_{uv}(s)|d_{q}s\hfill \\ \hfill & ·[\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}·|{\int }_0^1{s}^{\beta -\zeta -1}{d}_{q}H(s)|\hfill \\ \hfill & +\frac{{\Gamma }_q(\beta )}{|\Omega |{\Gamma }_q(\beta -{\xi }_0){\Gamma }_q(\beta -\zeta )}\left|{\int }_0^1{d}_{q}H(s)\right|]\hfill \\ \hfill =& (A_1+j_1\Vert u\Vert +j_2\Vert v\Vert +\frac{j_3}{{\Gamma }_q({\omega }_1)}\Vert u\Vert +\frac{j_4}{{\Gamma }_q({\delta }_1)}\Vert v\Vert )C_7\hfill \\ \hfill & +(A_2+k_1\Vert u\Vert +k_2\Vert v\Vert +\frac{k_3}{{\Gamma }_q({\omega }_2)}\Vert u\Vert +\frac{k_4}{{\Gamma }_q({\delta }_2)}\Vert v\Vert )C_{10}.\hfill \end{array}\end{array}$$

(10)

Similarly,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Vert v\Vert \le & (A_1+j_1\Vert u\Vert +j_2\Vert v\Vert +\frac{j_3}{{\Gamma }_q({\omega }_1)}\Vert u\Vert +\frac{j_4}{{\Gamma }_q({\delta }_1)}\Vert v\Vert )C_9\hfill \\ \hfill & +(A_2+k_1\Vert u\Vert +k_2\Vert v\Vert +\frac{k_3}{{\Gamma }_q({\omega }_2)}\Vert u\Vert +\frac{k_4}{{\Gamma }_q({\delta }_2)}\Vert v\Vert )C_8,\hfill \end{array}\end{array}$$

(11)

by means of (11) and (12), we have

$$\begin{array}{cc}\hfill {\Vert (u,v)\Vert }_V=& \Vert u\Vert +\Vert v\Vert \hfill \\ \hfill \le & A_1(C_7+C_9)+A_2(C_8+C_{10})+C_{13}\Vert u\Vert +C_{14}\Vert v\Vert \hfill \\ \hfill \le & A_1(C_7+C_9)+A_2(C_8+C_{10})+\Xi {\Vert (u,v)\Vert }_V.\hfill \end{array}$$

Due to $\Xi <1$, we get

$$\begin{array}{c}\hfill {\Vert (u,v)\Vert }_V\le \left[A_1(C_7+C_9)+A_2(C_8+C_{10})\right]{(1-\Xi )}^{-1},(u,v)\in \mathcal{S},\end{array}$$

thus, $\mathcal{S}$ is bounded.

By Theorem 3, it is time to say that the Systems (1)–(2) has at least one solution. Hence, the statements in Theorem 9 are proved. □

Corollary 3.

Suppose that $P,Q$ are S-Carathéodory functions and $({\mathcal{H}}_4^{\prime })$ hold. If $\Omega \ne 0$, and

$$\begin{array}{c}\hfill \widehat{\Xi }=max\{C_{15},C_{16}\}<1,\end{array}$$

where $C_{15}=L_1(C_7+C_9)+\frac{L_3}{{\Gamma }_q({\omega }_1)}(C_7+C_9)+l_1(C_8+C_{10})+\frac{l_3}{{\Gamma }_q({\omega }_2)}(C_8+C_{10}),C_{16}=L_2(C_7+C_9)+\frac{L_4}{{\Gamma }_q({\delta }_1)}(C_7+C_9)+l_2(C_8+C_{10})+\frac{l_4}{{\Gamma }_q({\delta }_2)}(C_8+C_{10}).$ Then the Systems (1)–(2) has at least one solution.

Theorem 10.

Suppose that $P,Q$ are S-Carathéodory functions and $({\mathcal{H}}_5)$ hold. If $\Omega \ne 0$ and there exists $\Pi >0$ such that

$$\begin{array}{cc}\hfill & [\Vert p_0\Vert +\Vert p_1\Vert {\phi }_1(\Pi )+\Vert p_2\Vert {\phi }_2(\Pi )+\Vert p_3\Vert {\phi }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_1)}\right)\hfill \\ \hfill +& \Vert p_4\Vert {\phi }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_1)}\right)](C_7+C_9)+[\Vert q_0\Vert +\Vert q_1\Vert {\eta }_1(\Pi )\hfill \\ \hfill +& \Vert q_2\Vert {\eta }_2(\Pi )+\Vert q_3\Vert {\eta }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_2)}\right)+\Vert q_4\Vert {\eta }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_2)}\right)](C_8+C_{10})<\Pi .\hfill \end{array}$$

Then the Systems (1)–(2) has at least one solution.

Proof. 

Let $B_{\Pi }={\{(u,v)\in V,\Vert (u,v)\Vert }_V\le \Pi \}$. Firstly, we prove that $\mathcal{T}:B_{\Pi }\to B_{\Pi }$. For $(u,v)\in B_{\Pi }$ and $t\in [0,1]$, we have

$$\begin{array}{cc}\hfill \Vert {\mathcal{T}}_1(u,v)\Vert \le & C_7[\Vert p_0\Vert +\Vert p_1\Vert {\phi }_1(\Pi )+\Vert p_2\Vert {\phi }_2(\Pi )+\Vert p_3\Vert {\phi }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_1)}\right)\hfill \\ \hfill & +\Vert p_4\Vert {\phi }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_1)}\right)]+C_{10}[\Vert q_0\Vert +\Vert q_1\Vert {\eta }_1(\Pi )+\Vert q_2\Vert {\eta }_2(\Pi )\hfill \\ \hfill & +\Vert q_3\Vert {\eta }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_2)}\right)+\Vert q_4\Vert {\eta }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_2)}\right)],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \Vert {\mathcal{T}}_2(u,v)\Vert \le & C_9[\Vert p_0\Vert +\Vert p_1\Vert {\phi }_1(\Pi )+\Vert p_2\Vert {\phi }_2(\Pi )+\Vert p_3\Vert {\phi }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_1)}\right)\hfill \\ \hfill & +\Vert p_4\Vert {\phi }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_1)}\right)]+C_8[\Vert q_0\Vert +\Vert q_1\Vert {\eta }_1(\Pi )+\Vert q_2\Vert {\eta }_2(\Pi )\hfill \\ \hfill & +\Vert q_3\Vert {\eta }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_2)}\right)+\Vert q_4\Vert {\eta }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_2)}\right)].\hfill \end{array}$$

For $(u,v)\in B_{\Pi }$, we have

$$\begin{array}{cc}\hfill {\Vert \mathcal{T}(u,v)\Vert }_V=& \Vert {\mathcal{T}}_1(u,v)\Vert +\Vert {\mathcal{T}}_2(u,v)\Vert \hfill \\ \hfill \le & (C_7+C_9)[\Vert p_0\Vert +\Vert p_1\Vert {\phi }_1(\Pi )+\Vert p_2\Vert {\phi }_2(\Pi )+\Vert p_3\Vert {\phi }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_1)}\right)\hfill \\ \hfill & +\Vert p_4\Vert {\phi }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_1)}\right)]+(C_{10}+C_8)[\Vert q_0\Vert +\Vert q_1\Vert {\eta }_1(\Pi )+\Vert q_2\Vert {\eta }_2(\Pi )\hfill \\ \hfill & +\Vert q_3\Vert {\eta }_3\left(\frac{\Pi }{{\Gamma }_q({\omega }_2)}\right)+\Vert q_4\Vert {\eta }_4\left(\frac{\Pi }{{\Gamma }_q({\delta }_2)}\right)]<\Pi .\hfill \end{array}$$

Consequently, $\mathcal{T}(B_{\Pi })\subset B_{\Pi }$. At the same time, it is easy to see that $\mathcal{T}$ is completely continuous, which can be derived in the same way as employed in Theorem 6.

Furthermore, assume that there exists $(u,v)\in \partial B_{\Pi }$ such that $(u,v)=\lambda \mathcal{T}(u,v)$ for $\lambda \in (0,1)$, it is simple to get ${\Vert (u,v)\Vert }_V\le {\Vert \mathcal{T}(u,v)\Vert }_V<\Pi$, this leads to a contradiction for $(u,v)\in \partial B_{\Pi }$. Therefore, by applying Theorem 4, we deduce that $\mathcal{T}$ has a fixed point $(u,v)\in B_{\Pi }$, which is a solution of the Systems (1)–(2). The proof is completed. □

4. Application Examples

In this section, for the system with the different nonlinearity terms, some examples are appreciated to illustrate our main results.

We consider the following system of fractional q-difference equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(D_q^{\frac{3}{2}}u)(t)+P(t,u(t),v(t),I_q^{\frac{1}{4}}u(t),I_q^{\frac{4}{3}}v(t))=0,t\in (0,1),\\ (D_q^{\frac{5}{2}}v)(t)+Q(t,u(t),v(t),I_q^{\frac{9}{4}}u(t),I_q^{\frac{2}{3}}v(t))=0,t\in (0,1),\end{array}\right.\end{array}$$

(12)

with the nonlocal BCs

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u(0)=0,D_q^{\frac{1}{5}}u(1)={\int }_0^1{D}_q^{\frac{5}{4}}v(t)d_q(-t^2),\\ v(0)=D_{q}v(0)=0,D_q^{\frac{7}{5}}v(1)={\int }_0^1{D}_q^{\frac{1}{6}}u(t)d_{q}t,\end{array}\right.\end{array}$$

(13)

where $\alpha =\frac{3}{2},\beta =\frac{5}{2},{\omega }_1=\frac{1}{4},{\delta }_1=\frac{4}{3},{\omega }_2=\frac{9}{4},{\delta }_2=\frac{2}{3},{\zeta }_0=\frac{1}{5},{\xi }_0=\frac{7}{5},\zeta =\frac{5}{4},\xi =\frac{1}{6},$ $q=\frac{1}{2},H(t)=-t^2,K(t)=t.$

After a simple caculation, we obtain $\Omega =2.58954375\ne 0,C_1=1.34100597,$ $C_2=2.08201688,C_3=C_2,C_4=1.92455621,C_5=1.79251862,C_6=C_4,$ $C_7=2.29230629,C_8=1.75022309,C_9=0.7590784,C_{10}=1.42695841,C_{11}=1.20641206,$ $C_{12}=0.91031044.$

Example 1.

Consider the nonlinear terms of the system

$$\begin{array}{cc}\hfill & P(t,x_1,x_2,x_3,x_4)=e^t+\frac{t}{36}cos{x}_1-\frac{t}{54}sin{x}_2+\frac{1}{63+t}arctan{x}_3-\frac{x_4}{{(t+9)}^2},\hfill \\ \hfill & Q(t,x_1,x_2,x_3,x_4)=\frac{1}{\sqrt{5+t^2}}-\frac{t}{48}sin{x}_1+\frac{t}{64}cos{x}_2-\frac{1}{36+t}arctan{x}_3+\frac{x_4}{t^2+56},\hfill \end{array}$$

where $t\in [0,1],x_i\in \mathbb{R}(i=1,2,3,4).$ For $x_i,y_i\in \mathbb{R}(i=1,2,3,4),$ we obtain

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)-P(t,y_1,y_2,y_3,y_4)|\hfill \\ \hfill \le & \frac{t}{36}|x_1-y_1|+\frac{t}{54}|x_2-y_2|+\frac{1}{t+63}|x_3-y_3|+\frac{1}{{(t+9)}^2}|x_4-y_4|\le {\Lambda }_1\sum _{i=1}^4|x_i-y_i|,\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)-Q(t,y_1,y_2,y_3,y_4)|\hfill \\ \hfill \le & \frac{t}{48}|x_1-y_1|+\frac{t}{64}|x_2-y_2|+\frac{1}{t+36}|x_3-y_3|+\frac{1}{56+t^2}|x_4-y_4|\le {\Lambda }_2\sum _{i=1}^4|x_i-y_i|.\hfill \end{array}$$

It is obvious that $L_1(t)=\frac{t}{36}$,$L_2(t)=\frac{t}{54}$,$L_3(t)=\frac{1}{63+t}$,$L_4(t)=\frac{1}{{(t+9)}^2}$ and $l_1(t)=\frac{t}{48}$, $l_2(t)=\frac{t}{64}$,$l_3(t)=\frac{1}{36+t}$,$l_4(t)=\frac{1}{t^2+56}.$ By a simple computation, we obtain ${\Lambda }_1=0.07451499,{\Lambda }_2=0.08209325$ and $\Lambda =0.97536897<1$, respectively. By Theorem 5, the Systems (12)–(13) has a unique solution.

Example 2.

Consider the nonlinear terms of the system

$$\begin{array}{cc}\hfill P(t,x_1,x_2,x_3,x_4)=& {(3t+5)}^2+\frac{t}{30}|x_1{|}^{\frac{1}{3}}+\frac{t}{t^2+9}arctan|x_2{|}^{\frac{1}{2}}+\frac{1}{5+8t}sin{|x_3|}^{\frac{3}{4}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -\frac{1}{{(4t+9)}^2}{\left|x_4\right|}^{\frac{1}{5}},\hfill \\ \hfill Q(t,x_1,x_2,x_3,x_4)=& e^t+\frac{1}{46(t+1)}|x_1{|}^{\frac{3}{5}}+\frac{t}{37}|x_2{|}^{\frac{1}{6}}+\frac{1}{t+29}sin|x_3{|}^{\frac{2}{3}}-8tarctan{|x_4|}^{\frac{5}{6}},\hfill \end{array}$$

where $t\in [0,1],x_i\in \mathbb{R}(i=1,2,3,4).$ It is clear that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le {(3t+5)}^2+\frac{t}{30}|x_1{|}^{\frac{1}{3}}+\frac{t}{t^2+9}|x_2{|}^{\frac{1}{2}}+\frac{1}{8t+5}|x_3{|}^{\frac{3}{4}}+\frac{1}{{(4t+9)}^2}{|x_4|}^{\frac{1}{5}},\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le e^t+\frac{1}{46(t+1)}|x_1{|}^{\frac{3}{5}}+\frac{t}{37}|x_2{|}^{\frac{1}{6}}+\frac{1}{t+29}|x_3{|}^{\frac{2}{3}}+8t{|x_4|}^{\frac{5}{6}}.\hfill \end{array}$$

Therefore, the assumption $({\mathcal{H}}_2)$ is satisfied with $c_0(t)={(3t+5)}^2,c_1(t)=\frac{t}{30},$ $c_2(t)=\frac{t}{t^2+9},$ $c_3(t)=\frac{1}{8t+5},c_4(t)=\frac{1}{{(4t+9)}^2},d_0(t)=e^t,d_1(t)=\frac{1}{46(t+1)},d_2(t)=\frac{t}{37},$ $d_3(t)=\frac{1}{t+29},$ and $d_4(t)=8t$. By Theorem 6, the Systems (12)–(13) has at least one solution.

Example 3.

Consider the nonlinear terms of the system

$$\begin{array}{cc}\hfill & P(t,x_1,x_2,x_3,x_4)=e^t+\frac{t}{40}arctan{x}_1-\frac{1}{{(t+6)}^2}sin{x}_2+\frac{1}{4(t+9)}sin{x}_3-\frac{t}{32}cos{x}_4,\hfill \\ \hfill & Q(t,x_1,x_2,x_3,x_4)=\frac{5t}{6+t^2}-\frac{3t}{56}cos{x}_1+\frac{1}{t+28}sin{x}_2-\frac{t}{72}{sin}^2{x}_3+\frac{t}{18}arctan{x}_4,\hfill \end{array}$$

where $t\in [0,1],x_i\in \mathbb{R}(i=1,2,3,4).$ For $\forall t\in [0,1],x_i,y_i\in \mathbb{R}(i=1,2,3,4),$ We obtain

$$\begin{array}{cc}\hfill |P(t,x_1,x_2,x_3,x_4)-P(t,y_1,y_2,y_3,y_4)|\le & \frac{1}{40}|x_1-y_1|+\frac{1}{36}|x_2-y_2|+\frac{1}{36}|x_3-y_3|\hfill \\ \hfill & +\frac{1}{32}|x_4-y_4|,\hfill \\ \hfill |Q(t,x_1,x_2,x_3,x_4)-Q(t,y_1,y_2,y_3,y_4)|\le & \frac{3}{56}|x_1-y_1|+\frac{1}{28}|x_2-y_2|+\frac{1}{36}|x_3-y_3|\hfill \\ \hfill & +\frac{1}{18}|x_4-y_4|,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le e^t+\frac{\pi t}{80}+\frac{1}{{(t+6)}^2}+\frac{1}{4(t+9)}+\frac{t}{32},\hfill \\ \hfill & |Q(t,x_1,x_2,x_3,x_4)|\le \frac{5t}{6+t^2}+\frac{3t}{56}+\frac{1}{t+28}+\frac{t}{72}+\frac{\pi t}{36},\hfill \end{array}$$

It is obvious that $L_1=\frac{1}{40},L_2=\frac{1}{36},L_3=\frac{1}{36},L_4=\frac{1}{32},l_1=\frac{3}{56},l_2=\frac{1}{28},$ $l_3=\frac{1}{36},$ $l_4=\frac{1}{18}$. By a simple computation, we have ${\Lambda }_3=0.11180556,{\Lambda }_4=0.17261905$, and $\overline{\Lambda }=0.53180725<1$, respectively. Therefore, the assumptions $({\mathcal{H}}_1^{\prime })$, and $({\mathcal{H}}_3)$ are satisfied, by Theorem 7, the Systems (12)–(13) has at least one solution.

Example 4.

Consider the nonlinear terms of the system

$$\begin{array}{cc}\hfill & P(t,x_1,x_2,x_3,x_4)=\frac{t}{20}+\frac{6t}{35}sin{x}_1-\frac{1}{4t+9\sqrt{6}}sin2x_2,\hfill \\ \hfill & Q(t,x_1,x_2,x_3,x_4)=\frac{2t}{3}+\frac{5t}{56}sin{x}_1-\frac{3}{2(7\sqrt{5}+t)}sin2x_2,\hfill \end{array}$$

where $t\in [0,1],x_i\in \mathbb{R}(i=1,2,3,4).$ It is clear that

$$\begin{array}{cc}\hfill & |P(t,x_1,x_2,x_3,x_4)|\le \frac{1}{20}+\frac{6}{35}|x_1|+\frac{2}{9\sqrt{6}}|x_2|,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left|Q(t,x_1,x_2,x_3,x_4)\right|\le \frac{2}{3}+\frac{5}{56}|x_1|+\frac{3}{7\sqrt{5}}\left|x_2\right|.\hfill \end{array}$$

Hence, $L_1=\frac{6}{35},L_2=\frac{2}{9\sqrt{6}},L_3=L_4=0,r_1=\frac{1}{20},l_1=\frac{5}{56},l_2=\frac{3}{7\sqrt{5}},l_3=l_4=0,$ $r_2=\frac{2}{3}$. By a simple computation, we obtain $C_{15}=0.80677144,C_{16}=0.88577528$, and $\widehat{\Xi }=0.88577528<1$, respectively. By Corollary 3, the Systems (12)–(13) has at least one solution.

5. Discussion

The system of fractional q-difference equations plays an extremely crucial role in many fields, such as quantum mechanics, dynamical systems, black holes, mathematical physics equations and so on, see [2,3,5,6,27,28,29,30] and the references therein. In this article, we are concerned with the solvability of a system of fractional q-difference equations with Riemann-Stieltjes integrals conditions based on some classical fixed point theorems. We obtain the multiple existence and uniqueness conclusions for the Systems (1)–(2). As a matter of fact, in the limit $q\to 1^{-}$, the system studied in this paper reduces to the classical system of fractional differential equations. It follows that the results we have discussed are the generalization of the classical analysis, they can extend classical theory in order to expand the range of the possible applications. In the future, we will devote ourselves to finding new inspirations and outstanding methods to overcome the more complex practical problems associated with the system of fractional q-difference equations. Moreover, we will investigate numerical methods for this kind of system.