On a coupled system of fractional sum-difference equations with p-Laplacian operator

Abstract

In this paper, we propose a nonlocal fractional sum-difference boundary value problem for a coupled system of fractional sum-difference equations with p-Laplacian operator. The problem contains both Riemann–Liouville and Caputo fractional difference with five fractional differences and four fractional sums. The existence and uniqueness result of the problem is studied by using the Banach fixed point theorem.

1 Introduction

Discrete fractional calculus and fractional difference equations have been widely studied. Goodrich and Peterson gave some useful basic definitions and properties of fractional difference calculus in the book [1]. Discrete fractional calculus can be applied in queuing problems, economics, logistic map, and electrical networks, see [2–4]. The extension of discrete fractional calculus has helped to build up some of the basic theory in this area, see [5–32] and the references cited therein.

The boundary value problem for fractional differential equations and the system of equations with p-Laplacian operator were presented in [33–39] and [40–44], respectively. Particularly, the boundary value problem for fractional difference equations with p-Laplacian operator was presented in [45–47]. In addition, the existence results of systems of fractional boundary value problems were presented in [48–55].

We observe that the boundary value problem of a coupled system of nonlinear fractional difference equations with p-Laplacian operator has not been studied. This result is the motivation for this research. In this paper, we aim to study the coupled system of nonlinear fractional sum-difference equations with p-Laplacian operator

$$\begin{aligned}& \begin{aligned}[b] \Delta ^{\alpha _{1}}_{C}\phi _{p} \bigl[ \Delta ^{\beta _{1}}_{C} u_{1}(t) \bigr] ={}&F_{1} \bigl[t+\alpha _{1}+\beta _{1}-1,t+\alpha _{2}+\beta _{2}-1, \Delta ^{\gamma _{1}}u_{1}(t+\alpha _{1}+\beta _{1}-\gamma _{1}), \\ &\varPsi ^{\omega _{2}} u_{2}(t+\alpha _{2}+\beta _{2}+\omega _{2}-1), u_{2}(t+\alpha _{2}+\beta _{2}+\omega _{2}-1) \bigr], \end{aligned} \\& \begin{aligned} \Delta ^{\alpha _{2}}_{C}\phi _{p} \bigl[ \Delta ^{\beta _{2}}_{C} u_{2}(t) \bigr] ={}&F_{2} \bigl[t+\alpha _{2}+\beta _{2}-1,t+\alpha _{1}+\beta _{1}-1, \Delta ^{\gamma _{2}}u_{2}(t+\alpha _{2}+\beta _{2}-\gamma _{2}), \\ &\varPsi ^{\omega _{1}} u_{1}(t+\alpha _{1}+\beta _{1}+\omega _{1}-1), u_{1}(t+\alpha _{1}+\beta _{1}+\omega _{1}-1) \bigr] \end{aligned} \end{aligned}$$

(1.1)

with the nonlocal fractional sum and fractional difference boundary conditions

$$ \begin{aligned} &\Delta ^{\beta _{1}}_{C} u_{1}(\alpha _{1}-1)=0,\qquad u_{1}(T+\alpha _{1}+ \beta _{1})=\lambda _{2}\Delta ^{-\theta _{2}}g_{2}(\eta _{2}+\theta _{2})u_{2}( \eta _{2}+\theta _{2}), \\ &\Delta ^{\beta _{2}}_{C} u_{2}(\alpha _{2}-1)=0,\qquad u_{2}(T+\alpha _{2}+ \beta _{2})=\lambda _{1}\Delta ^{-\theta _{1}}g_{1}( \eta _{1}+\theta _{1})u_{1}( \eta _{1}+\theta _{1}), \end{aligned} $$

(1.2)

where $t\in \mathbb{N}_{0,T}:=\{0,1,\ldots,T\}$, $\alpha _{i},\beta _{i},\gamma _{i},\omega _{i},\theta _{i}\in (0,1)$, $\alpha _{i}+\beta _{i}\in (1,2]$, $\lambda _{i}>0$, $\eta _{i}\in { \mathbb{N}}_{\alpha _{i}+\beta _{i}-1,T+\alpha _{i}+\beta _{i}-1}$, $F_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}\times \mathbb{R}^{3}, \mathbb{R} )$, $g_{i}\in C (\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R}^{+} )$, and for $\varphi _{i} : \mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}\rightarrow [0,\infty )$,

$$\begin{aligned} \varPsi ^{\omega _{i}}u_{i}(t+\omega _{i}) :=&\bigl[ \Delta ^{-\omega _{i}} \varphi _{i} u_{i}\bigr](t+ \omega _{i}) \\ =&\frac{1}{\varGamma (\omega _{i})}\sum_{s=\alpha _{i}+\beta _{i}- \omega _{i}-2}^{t-\omega _{i}} \bigl(t-\sigma (s)\bigr)^{ \underline{\omega _{i}-1}} \varphi _{i}(t,s+\omega _{i})u_{i}(s+ \omega _{i}) \end{aligned}$$

for $i\in \{1,2\}$. For $p>1$, the p-Laplacian operator is defined as $\phi _{p}(x)=|x|^{p-2}x$, where $\phi _{p}$ is invertible and its inverse operator is $\phi _{q}$, where $q>1$ is a constant such that $\frac{1}{p}+\frac{1}{q}=1$.

Our plan is as follows. In Sect. 2, we recall some basic knowledge and convert (1.1)–(1.2) to an equivalent summation equation and find its solution. In Sect. 3, we prove existence and uniqueness of the solution of boundary value problem (1.1)–(1.2) by using the Banach fixed point theorem. Some examples to illustrate our result are presented in the last section.

2 Preliminaries

Notations, definitions, and lemmas which are used in the main results are given as follows.

Definition 2.1

The generalized falling function is defined by $t^{\underline{\alpha }}:= \frac{\varGamma (t+1)}{\varGamma (t+1-\alpha )}$ for any t and α for which the right-hand side is defined. If $t+1-\alpha$ is a pole of the gamma function and $t+1$ is not a pole, then $t^{\underline{\alpha }}=0$.

Lemma 2.1

([5])

Assume that the following factorial functions are well defined:

• (i) $(t-\mu ) t^{\underline{\mu }}=t^{\underline{\mu +1}}$, where$\mu \in \mathbb{R}$.

• (ii) If$t\leq r$, then$t^{\underline{\alpha }}\leq r^{\underline{\alpha }}$for any$\alpha >0$.

Definition 2.2

Let $\alpha >0$ and f be defined on $\mathbb{N}_{a}$, the α-order fractional sum of f is defined by

$$ \Delta ^{-\alpha }f(t):=\frac{1}{\varGamma (\alpha )}\sum _{s=a}^{t- \alpha }\bigl(t-\sigma (s) \bigr)^{\underline{\alpha -1}}f(s), $$

where $t\in \mathbb{N}_{a+\alpha }$ and $\sigma (s)=s+1$.

Definition 2.3

For $\alpha >0$ and f defined on $\mathbb{N}_{a}$, the α-order Riemann–Liouville fractional difference of f is defined by

$$ \Delta ^{\alpha }f(t) := \Delta ^{N}\Delta ^{-(N-\alpha )}f(t)= \frac{1}{\varGamma (-\alpha )}\sum_{s=a}^{t+\alpha } \bigl(t-\sigma (s)\bigr)^{ \underline{-\alpha -1}} f(s). $$

The α-order Caputo fractional difference of f is defined by

$$ \Delta ^{\alpha }_{C}f(t):=\Delta ^{-(N-\alpha )}\Delta ^{N}f(t)= \frac{1}{\varGamma (N-\alpha )}\sum_{s=a}^{t-(N-\alpha )} \bigl(t-\sigma (s)\bigr)^{ \underline{N-\alpha -1}}\Delta ^{N}f(s), $$

where $t\in \mathbb{N}_{a+N-\alpha }$ and $N \in \mathbb{N}$ is chosen so that $0\leq N-1<\alpha < N$. If $\alpha =N$, then $\Delta ^{\alpha }f(t)=\Delta ^{\alpha }_{C} f(t)=\Delta ^{N} f(t)$.

Lemma 2.2

([7])

Let$0\leq N-1<\alpha \leq N$. Then

$$ \Delta ^{-\alpha }\Delta ^{\alpha }_{C} y(t)=y(t)+C_{0}+C_{1}t+\cdots+C_{N-1}t^{ \underline{N-1}} $$

for some$C_{i}\in \mathbb{R}$, with$1\leq i\leq N$.

We provide some properties of the p-Laplacian operator as follows.

• (A1) If $1< p<2$, $xy>0$ and $|x|,|y|\geq m>0$, then

  $$ \bigl\vert \phi _{p}(x)-\phi _{p}(y) \bigr\vert \leq (p-1)m^{p-2} \vert x-y \vert ; $$

• (A2) If $p>2$, $xy>0$ and $|x|,|y|\leq M$, then

  $$ \bigl\vert \phi _{p}(x)-\phi _{p}(y) \bigr\vert \leq (p-1)M^{p-2} \vert x-y \vert . $$

Next, we find a solution of the linear variant of boundary value problem (1.1)–(1.2) as shown in the following lemma.

Lemma 2.3

For$i,j\in \{1,2\}$and$i\neq j$, let$\varLambda \neq 0$, $\alpha _{i},\beta _{i},\theta _{i}\in (0, 1)$, $\alpha _{i}+\beta _{i}\in (1,2]$, $\lambda _{i}>0$be given constants, $h_{i}\in C (\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R} )$and$g_{i}\in C (\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R}^{+} )$be given functions. Then the linear variant problem given by

$$\begin{aligned}& \Delta ^{\alpha _{i}}_{C}\phi _{p} \bigl[ \Delta ^{\beta _{i}}_{C} u_{i}(t) \bigr] =h_{i}(t+\alpha _{i}-1),\quad t\in \mathbb{N}_{0,T}, \end{aligned}$$

(2.1)

$$\begin{aligned}& \Delta ^{\beta _{i}}_{C} u_{i}(\alpha _{i}-1)=0, \end{aligned}$$

(2.2)

$$\begin{aligned}& u_{i}(T+\alpha _{i}+\beta _{i})= \lambda _{j}\Delta ^{-\theta _{j}}g_{j}( \eta _{j}+\theta _{j})u_{j}(\eta _{j}+\theta _{j}), \quad \eta _{j} \in \mathbb{N}_{\alpha _{j}+\beta _{j}-1,T+\alpha _{j}+\beta _{j}-1} \end{aligned}$$

(2.3)

has the unique solution$(u_{1},u_{2})$, where

$$\begin{aligned}& u_{1}(t_{1})= \frac{1}{\varGamma (\beta _{1})}\sum _{s=\alpha _{1}-1}^{t_{1}- \beta _{1}}\bigl(t_{1}-\sigma (s) \bigr)^{\underline{\beta _{1}-1}} \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi =0}^{s-\alpha _{1}}\bigl(s- \sigma (\xi ) \bigr)^{\underline{\alpha _{1}-1}} h_{1}(\xi +\alpha _{1}+ \beta _{1}-1) \Biggr] \\& \hphantom{u_{1}(t_{1})={}}{} +\frac{1}{\varLambda } \Biggl\{ \Biggl( \frac{\lambda _{1}}{\varGamma (\theta _{1})}\sum _{s=\alpha _{1}+\beta _{1}-2}^{ \eta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \Biggr){\mathcal{P}[h_{1},h_{2}]} +{ \mathcal{Q}[h_{1},h_{2}]} \Biggr\} , \end{aligned}$$

(2.4)

$$\begin{aligned}& u_{2}(t_{2})= \frac{1}{\varGamma (\beta _{2})}\sum _{s=\alpha _{2}-1}^{t_{2}- \beta _{2}}\bigl(t_{2}-\sigma (s) \bigr)^{\underline{\beta _{2}-1}} \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi =0}^{s-\alpha _{2}}\bigl(s- \sigma (\xi ) \bigr)^{\underline{\alpha _{2}-1}} h_{2}(\xi +\alpha _{2}+ \beta _{2}-1) \Biggr] \\& \hphantom{u_{2}(t_{2})={}}{} +\frac{1}{\varLambda } \Biggl\{ \Biggl( \frac{\lambda _{2}}{\varGamma (\theta _{2})}\sum _{s=\alpha _{2}+\beta _{2}-2}^{ \eta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{2}(s) \Biggr){\mathcal{Q}[h_{1},h_{2}]} +{ \mathcal{P}[h_{1},h_{2}]} \Biggr\} , \end{aligned}$$

(2.5)

where$t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$, the constantΛis defined by

$$\begin{aligned} \varLambda ={}&\frac{\lambda _{1}\lambda _{2} }{ \varGamma (\theta _{1}) \varGamma (\theta _{2}) } \sum _{s=\alpha _{1}+\beta _{1}-1}^{\eta _{1}} \bigl(\eta _{1}+\theta _{1}- \sigma (s)\bigr)^{\underline{\theta _{1}-1}} g_{1}(s) \\ &{}\times \sum_{s=\alpha _{2}+\beta _{2}-1}^{\eta _{2}} \bigl(\eta _{2}+\theta _{2}- \sigma (s)\bigr)^{\underline{\theta _{2}-1}} g_{2}(s) -1, \end{aligned}$$

(2.6)

and the functionals${\mathcal{P}} [h_{1},h_{2}]$, ${\mathcal{Q}}[h_{1},h_{2}]$are defined by

$$\begin{aligned}& {\mathcal{P}} [h_{1},h_{2}] \\& \quad = \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{T+\alpha _{1}} \bigl(T+ \alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{\underline{\beta _{1}-1}} \\& \qquad {}\times\phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi =0}^{s- \alpha _{1}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{1}-1}} h_{1}(\xi + \alpha _{1}+\beta _{1}-1) \Biggr] \\& \qquad {} -\frac{\lambda _{2}}{\varGamma (\beta _{2})\varGamma (\theta _{2})}\sum_{r= \alpha _{2}+\beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r- \beta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (r)\bigr)^{ \underline{\theta _{2}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} g_{2}(r) \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi =0}^{s- \alpha _{2}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{2}-1}} h_{2}(\xi + \alpha _{2}+\beta _{2}-1) \Biggr], \end{aligned}$$

(2.7)

$$\begin{aligned}& {\mathcal{Q}} [h_{1},h_{2}] \\& \quad = \frac{1}{\varGamma (\beta _{2})}\sum_{s=\alpha _{2}-1}^{T+\alpha _{2}} \bigl(T+ \alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{\underline{\beta _{2}-1}} \\& \qquad {}\times\phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi =0}^{s- \alpha _{2}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{2}-1}} h_{2}(\xi + \alpha _{2}+\beta _{2}-1) \Biggr] \\& \qquad {} -\frac{\lambda _{1}}{\varGamma (\beta _{1})\varGamma (\theta _{1})}\sum_{r= \alpha _{1}+\beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r- \beta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (r)\bigr)^{ \underline{\theta _{1}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} g_{1}(r) \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi =0}^{s- \alpha _{1}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{1}-1}} h_{1}(\xi + \alpha _{1}+\beta _{1}-1) \Biggr]. \end{aligned}$$

(2.8)

Proof

For $i,j\in \{1,2\}$ and $i\neq j$, taking the fractional sum of order $\alpha _{i}$ for (2.1), we have

$$ \phi _{p} \bigl[ \Delta _{C}^{\beta _{i}} u_{i}(t) \bigr] =C_{0i} + \frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t-\alpha _{i}} \bigl(t-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}+\beta _{i}-1) $$

(2.9)

for $t\in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}}$.

From boundary condition (2.2), it implies that

$$ C_{0i}=0. $$

Then from (2.9) we have

$$ \Delta _{C}^{\beta _{i}} u_{i}(t) = \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{i})} \sum_{s=0}^{t-\alpha _{i}}\bigl(t-\sigma (s) \bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}+\beta _{i}-1) \Biggr]. $$

(2.10)

Next, taking the fractional sum of order $\beta _{i}$ for (2.10), we have

$$\begin{aligned} u_{i}(t) ={}& C_{1i}+ \frac{1}{\varGamma (\beta _{i})}\sum_{s=\alpha _{i}-1}^{t- \beta _{i}} \bigl(t-\sigma (s)\bigr)^{\underline{\beta _{i}-1}} \\ &{}\times\phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{i})}\sum _{\xi =0}^{s- \alpha _{i}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{i}-1}} h_{i}(\xi + \alpha _{i}+\beta _{i}-1) \Biggr] \end{aligned}$$

(2.11)

for $t\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$.

Using the fractional sum of order $\theta _{i}$ for (2.11), we get

$$\begin{aligned} \Delta ^{-\theta _{i}}u(t) =& \frac{C_{1i}}{\varGamma (\theta _{i})} \sum_{s=\alpha _{i}+\beta _{i}-2}^{t-\theta _{i}} \bigl(t-\sigma (s)\bigr)^{ \underline{\theta _{i}-1}} \\ &{}+\frac{1}{\varGamma (\theta _{i})\varGamma (\beta _{i})}\sum_{r=\alpha _{i}+ \beta _{i}-1}^{t-\theta _{i}} \sum_{s=\alpha _{i}-1}^{t-\beta _{i}}\bigl(t- \sigma (r) \bigr)^{\underline{\theta _{i}-1}} \bigl(r-\sigma (s)\bigr)^{ \underline{\beta _{i}-1}} \\ &{}\times\phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{i})}\sum _{\xi =0}^{s- \alpha _{i}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{i}-1}} h_{i}(\xi + \alpha _{i}+\beta _{i}-1) \Biggr] \end{aligned}$$

(2.12)

for $t\in \mathbb{N}_{\alpha _{i}+\beta _{i}+\theta _{i}-3,T+\alpha _{i}+ \beta _{i}+\theta _{i}}$.

Using boundary condition (2.3) implies

$$\begin{aligned}& C_{11}-C_{12}\frac{\lambda _{2}}{\varGamma (\theta _{2})} \sum_{s= \alpha _{2}+\beta _{2}-2}^{\eta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (s)\bigr)^{ \underline{\theta _{2}-1}} g_{2}(s) \\& \quad = \frac{\lambda _{2}}{\varGamma (\theta _{2})\varGamma (\beta _{2})}\sum_{r= \alpha _{2}+\beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r- \beta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (r)\bigr)^{ \underline{\theta _{2}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} g_{2}(r) \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi =0}^{s- \alpha _{2}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{2}-1}} h_{2}(\xi + \alpha _{2}+\beta _{2}-1) \Biggr] \\& \qquad {} -\frac{1}{\varGamma (\beta _{1})} \sum_{s=\alpha _{1}-1}^{T+\alpha _{1}} \bigl(T+ \alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{\underline{\beta _{1}-1}} \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi =0}^{s-\alpha _{1}}\bigl(s- \sigma (\xi ) \bigr)^{\underline{\alpha _{1}-1}} h_{1}(\xi +\alpha _{1}+ \beta _{1}-1) \Biggr] \end{aligned}$$

(2.13)

and

$$\begin{aligned}& C_{21}-C_{22}\frac{\lambda _{1}}{\varGamma (\theta _{1})} \sum_{s= \alpha _{1}+\beta _{1}-2}^{\eta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \\& \quad = \frac{\lambda _{1}}{\varGamma (\theta _{1})\varGamma (\beta _{1})}\sum_{r= \alpha _{1}+\beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r- \beta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (r)\bigr)^{ \underline{\theta _{1}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} g_{1}(r) \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi =0}^{s- \alpha _{1}}\bigl(s-\sigma (\xi ) \bigr)^{\underline{\alpha _{1}-1}} h_{1}(\xi + \alpha _{1}+\beta _{1}-1) \Biggr] \\& \qquad {} -\frac{1}{\varGamma (\beta _{2})} \sum_{s=\alpha _{2}-1}^{T+\alpha _{2}} \bigl(T+ \alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{\underline{\beta _{2}-1}} \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi =0}^{s-\alpha _{2}}\bigl(s- \sigma (\xi ) \bigr)^{\underline{\alpha _{2}-1}} h_{2}(\xi +\alpha _{2}+ \beta _{2}-1) \Biggr]. \end{aligned}$$

(2.14)

$C_{11}$, $C_{12}$ can be represented by solving equations (2.13) and (2.14) as

$$ C_{11} = \frac{1}{\varLambda } \Biggl\{ \Biggl( \frac{\lambda _{1}}{\varGamma (\theta _{1})}\sum_{s=\alpha _{1}+\beta _{1}-2}^{ \eta _{1}} \bigl(\eta _{1}+\theta _{1}-\sigma (s) \bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \Biggr){\mathcal{P}[h_{1},h_{2}]} +{\mathcal{Q}[h_{1},h_{2}]} \Biggr\} $$

(2.15)

and

$$ C_{12} =\frac{1}{\varLambda } \Biggl\{ \Biggl( \frac{\lambda _{2}}{\varGamma (\theta _{2})}\sum_{s=\alpha _{2}+\beta _{2}-2}^{ \eta _{2}} \bigl(\eta _{2}+\theta _{2}-\sigma (s) \bigr)^{ \underline{\theta _{1}-1}} g_{2}(s) \Biggr){\mathcal{Q}[h_{1},h_{2}]} +{\mathcal{P}[h_{1},h_{2}]} \Biggr\} , $$

(2.16)

where Λ, ${\mathcal{P}(h_{1},h_{2})}$ and ${\mathcal{Q}(h_{1},h_{2})}$ are defined as (2.6)–(2.8), respectively.

After substituting $C_{11}$ and $C_{12}$ into (2.11), we obtain (2.4) and (2.5). □

3 Existence and uniqueness result

In this section, we study the existence and uniqueness result for problem (1.1)–(1.2). For each $i,j \in \{1,2\}$ and $i\neq j$, we let $E_{i}:C ( \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R} )$ be the Banach space for all functions on $\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$. Clearly, the product space $\mathcal{C}=E_{1}\times E_{2}$ is the Banach space. Define the spaces

$$ \mathcal{C}_{i}= \bigl\{ (u_{1},u_{2}) \in {\mathcal{C}} : \Delta ^{ \gamma _{i}}u_{i}(t_{i}-\gamma _{i}+1) \in E_{i} \bigr\} , \quad t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}, $$

with the norm

$$ \bigl\| (u_{1},u_{2})\bigr\| _{\mathcal{C}_{i}}=\max \bigl\lbrace \bigl\| \Delta ^{ \gamma _{i}}u_{i}\bigr\| ,\|u_{j}\| \bigr\rbrace , $$

where

$$ \bigl\| \Delta ^{\gamma _{i}}u_{i}\bigr\| =\max_{t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}} \bigl| \Delta ^{\gamma _{i}}u_{i}(t_{i}-\gamma _{i}+1) \bigr| \quad \mbox{and}\quad \|u_{j}\|=\max_{t_{j}\in \mathbb{N}_{\alpha _{j}+ \beta _{j}-2,T+\alpha _{j}+\beta _{j}}} \bigl|u_{j}(t_{j})\bigr|. $$

Obviously, the space $( {\mathcal{C}_{1}\cap \mathcal{C}_{2}},\|(u_{1},u_{2})\|_{ \mathcal{C}_{1}\cap \mathcal{C}_{2}} )$ is also the Banach space with the norm

$$ \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{C}_{1}\cap \mathcal{C}_{2}}=\max \bigl\{ \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{C}_{1}}, \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{ \mathcal{C}_{2}} \bigr\} . $$

Let ${\mathcal{U}}=\mathcal{C}_{1}\cap \mathcal{C}_{2}$. The operator $\mathcal{T}:{\mathcal{U}}\rightarrow {\mathcal{U}}$ is defined by

$$ \bigl(\mathcal{T}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) = \bigl( \bigl( \mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2}), \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr) $$

(3.1)

and

$$\begin{aligned}& \bigl(\mathcal{T}_{1} (u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{t_{1}-\beta _{1}} \bigl(t_{1}- \sigma (s)\bigr)^{\underline{\beta _{1}-1}} \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi =\alpha _{1}+\beta _{1}-1}^{s+ \beta _{1}-1}\bigl(s+\alpha _{1}+ \beta _{1}-1-\sigma (\xi )\bigr)^{ \underline{\alpha _{1}-1}} F^{*}_{1}\bigl[u(t_{2},\xi )\bigr] \Biggr] \\& \qquad {} +\frac{1}{\varLambda } \Biggl\{ \Biggl( \frac{\lambda _{1}}{\varGamma (\theta _{1})}\sum _{s=\alpha _{1}+\beta _{1}-2}^{ \eta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \Biggr){\mathcal{P}[F_{1},F_{2}]}(u_{1},u_{2}) \\& \qquad {}+{\mathcal{Q}[F_{1},F_{2}]}(u_{1},u_{2}) \Biggr\} , \end{aligned}$$

(3.2)

$$\begin{aligned}& \bigl(\mathcal{T}_{2} (u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{2})}\sum_{s=\alpha _{2}-1}^{t_{2}-\beta _{2}} \bigl(t_{2}- \sigma (s)\bigr)^{\underline{\beta _{2}-1}} \\& \qquad {}\times\phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi =\alpha _{2}+\beta _{2}-1}^{s+ \beta _{2}-1}\bigl(s+\alpha _{2}+ \beta _{2}-1-\sigma (\xi )\bigr)^{ \underline{\alpha _{2}-1}} F^{*}_{2}\bigl[u(t_{1},\xi )\bigr] \Biggr] \\& \qquad {} +\frac{1}{\varLambda } \Biggl\{ \Biggl( \frac{\lambda _{2}}{\varGamma (\theta _{2})}\sum _{s=\alpha _{2}+\beta _{2}-2}^{ \eta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{2}(s) \Biggr){\mathcal{Q}[F_{1},F_{2}]}(u_{1},u_{2}) \\& \qquad {}+{ \mathcal{P}[F_{1},F_{2}]}(u_{1},u_{2}) \Biggr\} , \end{aligned}$$

(3.3)

where $t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$, Λ is defined as (2.6), and the functionals ${\mathcal{P} [F_{1},F_{2}]}(u_{1},u_{2})$, ${\mathcal{Q}[F_{1},F_{2}]}(u_{1},u_{2})$ are defined by

$$\begin{aligned}& {\mathcal{P}}[F_{1},F_{2}](u_{1},u_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{T+\alpha _{1}} \bigl(T+ \alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{\underline{\beta _{1}-1}} \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi = \alpha _{1}+\beta _{1}-1}^{s+\beta _{1}-1}\bigl(s+\alpha _{1}+ \beta _{1}-1- \sigma (\xi )\bigr)^{\underline{\alpha _{1}-1}} F^{*}_{1}\bigl[u(t_{2},\xi )\bigr] \Biggr] \\& \qquad {} -\frac{\lambda _{2}}{\varGamma (\beta _{2})\varGamma (\theta _{2})}\sum_{r= \alpha _{2}+\beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r- \beta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (r)\bigr)^{ \underline{\theta _{2}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} g_{2}(r) \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi = \alpha _{2}+\beta _{2}-1}^{s+\beta _{2}-1}\bigl(s+\alpha _{2}+ \beta _{2}-1- \sigma (\xi )\bigr)^{\underline{\alpha _{2}-1}} F^{*}_{2}\bigl[u(t_{1},\xi )\bigr] \Biggr], \end{aligned}$$

(3.4)

$$\begin{aligned}& {\mathcal{Q}}[F_{1},F_{2}](u_{1},u_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{2})}\sum_{s=\alpha _{2}-1}^{T+\alpha _{2}} \bigl(T+ \alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{\underline{\beta _{2}-1}} \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{2})}\sum _{\xi = \alpha _{2}+\beta _{2}-1}^{s+\beta _{2}-1}\bigl(s+\alpha _{2}+ \beta _{2}-1- \sigma (\xi )\bigr)^{\underline{\alpha _{2}-1}} F^{*}_{2}\bigl[u(t_{1},\xi )\bigr] \Biggr] \\& \qquad {} -\frac{\lambda _{1}}{\varGamma (\beta _{1})\varGamma (\theta _{1})}\sum_{r= \alpha _{1}+\beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r- \beta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (r)\bigr)^{ \underline{\theta _{1}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} g_{1}(r) \\& \qquad {}\times \phi _{q} \Biggl[ \frac{1}{\varGamma (\alpha _{1})}\sum _{\xi = \alpha _{1}+\beta _{1}-1}^{s+\beta _{1}-1}\bigl(s+\alpha _{1}+ \beta _{1}-1- \sigma (\xi )\bigr)^{\underline{\alpha _{1}-1}} F^{*}_{1}\bigl[u(t_{2},\xi )\bigr] \Biggr], \end{aligned}$$

(3.5)

with

$$ F^{*}_{i}\bigl[u(t_{j}, \xi )\bigr] =F_{i} \bigl[t_{j},\xi ,\Delta ^{\gamma _{i}}u_{i}( \xi -\gamma _{i}+1),\varPsi ^{\omega _{j}} u_{j}(t_{j}+\omega _{j}), u_{j}(t_{j}) \bigr]. $$

(3.6)

For each $i,j \in \{1,2\}$ and $i\neq j$, we define the operators $(\mathcal{T}_{i}^{0}(u_{1},u_{2}))(t_{1},t_{2})$ and $(\mathcal{T}_{i}^{*}(u_{1},u_{2}))(t_{1}, t_{2})$ by

$$\begin{aligned}& \bigl(\mathcal{T}_{1}^{0}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) = \phi _{q} \Biggl[ \sum_{\xi =\alpha _{1}+\beta _{1}-1}^{t_{1}+\beta _{1}-1} \frac{(t_{1}+\alpha _{1}+\beta _{1}-1-\sigma (\xi ))^{\underline{\alpha _{1}-1}}}{ \varGamma (\alpha _{1}) } F^{*}_{1}\bigl[u(t_{2},\xi )\bigr] \Biggr], \end{aligned}$$

(3.7)

$$\begin{aligned}& \bigl(\mathcal{T}_{2}^{0}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) = \phi _{q} \Biggl[ \sum_{\xi =\alpha _{2}+\beta _{2}-1}^{t_{2}+\beta _{2}-1} \frac{(t_{2}+\alpha _{2}+\beta _{2}-1-\sigma (\xi ))^{\underline{\alpha _{2}-1}}}{ \varGamma (\alpha _{2}) } F^{*}_{2}\bigl[u(t_{1},\xi )\bigr] \Biggr], \end{aligned}$$

(3.8)

and

$$\begin{aligned}& \bigl(\mathcal{T}_{1}^{*}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{t_{1}-\beta _{1}} \bigl(t_{1}- \sigma (s)\bigr)^{\underline{\beta _{1}-1}} (u_{1},u_{2}) (t_{2},s) + \frac{1}{\varLambda } \Biggl\{ {\mathcal{Q}^{*}[F_{1},F_{2}]}(u_{1},u_{2}) \\& \qquad {} + \Biggl( \frac{\lambda _{1}}{\varGamma (\theta _{1})}\sum_{s=\alpha _{1}+ \beta _{1}-2}^{\eta _{1}} \bigl(\eta _{1}+\theta _{1}-\sigma (s) \bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \Biggr){\mathcal{P}^{*}[F_{1},F_{2}]}(u_{1},u_{2}) \Biggr\} , \end{aligned}$$

(3.9)

$$\begin{aligned}& \bigl(\mathcal{T}_{2}^{*}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{2})}\sum_{s=\alpha _{2}-1}^{t_{2}-\beta _{2}} \bigl(t_{2}- \sigma (s)\bigr)^{\underline{\beta _{2}-1}} (u_{1},u_{2}) (t_{1},s) + \frac{1}{\varLambda } \Biggl\{ {\mathcal{P}^{*}[F_{1},F_{2}]}(u_{1},u_{2}) \\& \qquad {} + \Biggl( \frac{\lambda _{2}}{\varGamma (\theta _{2})}\sum_{s=\alpha _{2}+ \beta _{2}-2}^{\eta _{2}} \bigl(\eta _{2}+\theta _{2}-\sigma (s) \bigr)^{ \underline{\theta _{1}-1}} g_{2}(s) \Biggr){\mathcal{Q}^{*}[F_{1},F_{2}]}(u_{1},u_{2}) \Biggr\} , \end{aligned}$$

(3.10)

where $t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$, and the functionals ${\mathcal{P}^{*} [F_{1},F_{2}]}(u_{1},u_{2})$, ${\mathcal{Q}^{*}[F_{1},F_{2}]}(u_{1},u_{2})$ are defined by

$$\begin{aligned}& {\mathcal{P}}^{*} [F_{1},F_{2}](u_{1},u_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{T+\alpha _{1}} \bigl(T+ \alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{\underline{\beta _{1}-1}} (u_{1},u_{2}) (t_{2},s) \\& \qquad {} -\frac{\lambda _{2}}{\varGamma (\beta _{2})\varGamma (\theta _{2})}\sum_{r= \alpha _{2}+\beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r- \beta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (r)\bigr)^{ \underline{\theta _{2}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} \\& \qquad {}\times g_{2}(r) (u_{1},u_{2}) (t_{1},s), \end{aligned}$$

(3.11)

$$\begin{aligned}& {\mathcal{Q}}^{*} [F_{1},F_{2}](u_{1},u_{2}) \\& \quad = \frac{1}{\varGamma (\beta _{2})}\sum_{s=\alpha _{2}-1}^{T+\alpha _{2}} \bigl(T+ \alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{\underline{\beta _{2}-1}} (u_{1},u_{2}) (t_{1},s) \\& \qquad {} -\frac{\lambda _{1}}{\varGamma (\beta _{1})\varGamma (\theta _{1})}\sum_{r= \alpha _{1}+\beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r- \beta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (r)\bigr)^{ \underline{\theta _{1}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} \\& \qquad {}\times g_{1}(r) (u_{1},u_{2}) (t_{2},s). \end{aligned}$$

(3.12)

Let $\mathcal{T}_{i}=\mathcal{T}_{i}^{*} \circ \mathcal{T}_{i}^{0}$, then $\mathcal{T}_{i}$ and $\mathcal{T}:\mathcal{U}\rightarrow \mathcal{U}$ are continuous and compact operators. Note that problem (1.1)–(1.2) has solutions if and only if the operator $\mathcal{T}$ has fixed points.

In the case $p>2$, we have $1< q<2$ due to $\frac{1}{p}+\frac{1}{q}=1$ and the following theorem is obtained.

Theorem 3.1

Let$p>2$for each$i,j\in \{1,2\}$, $i\neq j$, $F_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}} \times\mathbb{R}^{3}, \mathbb{R} )$, $\varphi _{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}},[0,\infty ) )$with$\varphi ^{o}_{i}=\max \{\varphi (t_{i}-1,s) \}$. In addition, suppose that:

• (H1) There exist constants$\chi _{i}>0$and$0<\delta <\frac{1}{2-q}$such that

  $$\begin{aligned} \chi _{i}\Delta _{C}^{\alpha } \bigl( t_{i}^{\underline{\alpha _{i}}} \bigr)^{\delta }\leq F_{i} [t_{i},t_{j},x,y,z ] \end{aligned}$$

  for any$( t_{i},t_{j},x,y,z )\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+ \alpha _{i}+\beta _{i}}\times \mathbb{N}_{\alpha _{j}+\beta _{j}-2,T+ \alpha _{j}+\beta _{j}}\times \mathbb{R}^{3}$.

• (H2) There exist constants$L_{i},M_{i},N_{i}>0$such that, for each$t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$and$u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\in \mathbb{R}$,

  $$\begin{aligned} &\bigl\vert F_{i} [t_{i},t_{j},u_{1},u_{2},u_{3} ]-F_{i} [t_{i},t_{j},v_{1},v_{2},v_{3} ] \bigr\vert \\ &\quad \leq L_{i} \vert u_{1}-v_{1} \vert +M_{j} \vert u_{2}-v_{2} \vert +N_{j} \vert u_{3}-v_{3} \vert . \end{aligned}$$

• (H3) $g_{i}< g_{i}(t_{i})< G_{i}$for each$t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}$.

Then problem (1.1)–(1.2) has a unique solution provided that

$$\begin{aligned} \varPhi :=&\max \biggl\{ \mathcal{K}_{1}\chi _{1}^{q-2}\varTheta _{1} + \mathcal{K}_{2}\chi _{2}^{q-2}\varOmega _{2}, \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \varOmega _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \varTheta _{2} , \\ & \mathcal{K}_{1}\chi _{1}^{q-2}\varOmega _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \varTheta _{2}, \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \varTheta _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \varOmega _{2} \biggr\} \\ < &1, \end{aligned}$$

(3.13)

where

$$\begin{aligned}& \mathcal{K}_{i}= \biggl[ L_{i}+N_{j}+M_{j} \varphi _{j}^{o} \frac{(T+\omega _{j}+2)^{\underline{\omega _{j}}}}{\varGamma (\omega _{j}+1)} \biggr] \frac{(q-1)}{\varGamma (\alpha _{i}+1) }, \end{aligned}$$

(3.14)

$$\begin{aligned}& \varOmega _{i}= \biggl[ 1 + \frac{\lambda _{i}G_{i} (\eta _{i}-\alpha _{i}-\beta _{i}+\theta _{i}+2)^{\underline{\theta _{i}}} }{ \vert \varLambda \vert \varGamma (\theta _{i}+1) } \biggr] \frac{1}{\varGamma (\beta _{i})}\sum_{s=\alpha _{i}-1}^{T+ \alpha _{i}} \bigl(T+\alpha _{i}+\beta _{i}-\sigma (s) \bigr)^{ \underline{\beta _{i}-1}} \bigl( s^{\underline{\alpha _{i}}} \bigr)^{\delta (q-2)+1} \\& \hphantom{\varOmega _{i}={}}{}+ \frac{\lambda _{i}G_{i} (\eta _{i}-\alpha _{i}-\beta _{i}+\theta _{i}+1)^{\underline{\theta _{i}}} }{ \vert \varLambda \vert \varGamma (\theta _{i}+1) } \cdot \frac{1}{\varGamma (\beta _{i})} \sum _{s=\alpha _{i}-1}^{\eta _{i}- \beta _{i}} \bigl(\eta _{i}-\sigma (s)\bigr)^{\underline{\beta _{i}-1} } \bigl( s^{\underline{\alpha _{i}}} \bigr)^{\delta (q-2)+1}, \end{aligned}$$

(3.15)

$$\begin{aligned}& \varTheta _{i}= \frac{1}{ \vert \varLambda \vert \varGamma (\beta _{i})}\sum _{s= \alpha _{i}-1}^{T+\alpha _{i}}\bigl(T+\alpha _{i}+ \beta _{i}-\sigma (s)\bigr)^{ \underline{\beta _{i}-1}} \bigl( s^{\underline{\alpha _{i}}} \bigr)^{\delta (q-2)+1} \\& \hphantom{\varTheta _{i}={}}{}+ \frac{\lambda _{1}\lambda _{2}G_{1}G_{2} }{ \vert \varLambda \vert } \cdot \frac{ (\eta _{j}-\alpha _{j}-\beta _{j}+\theta _{j}+2)^{\underline{\theta _{j}}} }{ \varGamma (\theta _{j}+1)} \cdot \frac{(\eta _{i}-\alpha _{i}-\beta _{i}+\theta _{i}+1)^{\underline{\theta _{i}}} }{ \varGamma (\theta _{i}+1) } \\& \hphantom{\varTheta _{i}={}}{}\times \frac{1}{\varGamma (\beta _{i})} \sum_{s=\alpha _{i}-1}^{\eta _{i}- \beta _{i}} \bigl(\eta _{i}-\sigma (s)\bigr)^{\underline{\beta _{i}-1} } \bigl( s^{\underline{\alpha _{i}}} \bigr)^{\delta (q-2)+1} . \end{aligned}$$

(3.16)

Proof

For each $i,j\in \{1,2\}$, $i\neq j$, by (H1) we have

$$ \chi _{i} \bigl( t_{i}^{\underline{\alpha _{i}}} \bigr)^{\delta } \leq \frac{1}{\varGamma (\alpha _{i})} \sum _{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \bigl(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi )\bigr)^{ \underline{\alpha _{i}-1}} F^{*}_{i}(t_{j},\xi ). $$

(3.17)

By (A1), (H2), and the definition of operator $\mathcal{T}_{i}^{0}$, for any $(u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}$, we have

$$\begin{aligned}& \bigl\vert \bigl( \mathcal{T}_{i}^{0}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) - \bigl( \mathcal{T}_{i}^{0}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad = \Biggl\vert \phi _{q} \Biggl[ \sum _{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \frac{(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}}}{ \varGamma (\alpha _{i}) } F^{*}_{i}\bigl[u(t_{j},\xi )\bigr] \Biggr] \\& \qquad {} - \phi _{q} \Biggl[ \sum_{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \frac{(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}}}{ \varGamma (\alpha _{i}) } F^{*}_{i}\bigl[v(t_{j}, \xi )\bigr] \Biggr] \Biggr\vert \\& \quad \leq (q-1) \bigl( \chi _{i} \bigl( t_{i}^{\underline{\alpha _{i}}} \bigr)^{\delta } \bigr)^{q-2} \sum _{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \frac{(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}}}{ \varGamma (\alpha _{i}) } \bigl\vert F^{*}_{i}\bigl[u(t_{j},\xi )\bigr]- F^{*}_{i}\bigl[v(t_{j},\xi )\bigr] \bigr\vert \\& \quad \leq (q-1) \chi _{i}^{q-2} \frac{ ( t_{i}^{\underline{\alpha _{i}}} )^{\delta (q-2)+1}}{\varGamma (\alpha _{i}+1)} \bigl[ L_{i} \bigl\Vert \Delta ^{\gamma _{i}}u_{i}- \Delta ^{\gamma _{i}}v_{i} \bigr\Vert +M_{j} \bigl\Vert \varPsi ^{\omega _{i}}u_{i}-\varPsi ^{\omega _{j}}v_{i} \bigr\Vert +N_{j} \Vert u_{j}-v_{j} \Vert \bigr] \\& \quad \leq (q-1) \chi _{i}^{q-2} \frac{ ( t_{i}^{\underline{\alpha _{i}}} )^{\delta (q-2)+1}}{\varGamma (\alpha _{i}+1)} \biggl[ L_{i} \bigl\Vert \Delta ^{\gamma _{i}}u_{i}- \Delta ^{\gamma _{i}}v_{i} \bigr\Vert \\& \qquad {}+ \biggl( N_{j}+M_{j} \varphi _{j}^{o} \frac{(T+\omega _{j}+2)^{\underline{\omega _{j}}}}{\varGamma (\omega _{j}+1)} \biggr) \Vert u_{j}-v_{j} \Vert \biggr] \\& \quad \leq (q-1) \chi _{i}^{q-2} \frac{ ( t_{i}^{\underline{\alpha _{i}}} )^{\delta (q-2)+1}}{\varGamma (\alpha _{i}+1)} \biggl[ L_{i}+N_{j}+M_{j}\varphi _{j}^{o} \frac{(T+\omega _{j}+2)^{\underline{\omega _{j}}}}{\varGamma (\omega _{j}+1)} \biggr] \\& \qquad {}\times \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{i}}. \end{aligned}$$

(3.18)

Using (3.18) and (H3), we have

$$\begin{aligned}& \bigl\vert {\mathcal{P}}^{*} [F_{1},F_{2}](u_{1},u_{2}) - {\mathcal{P}}^{*}[F_{1},F_{2}](v_{1},v_{2}) \bigr\vert \\& \quad \leq \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{T+ \alpha _{1}} \bigl(T+\alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{ \underline{\beta _{1}-1}} \bigl\vert \bigl( \mathcal{T}_{i}^{0}(u_{1},u_{2}) \bigr) (s,t_{2}) - \bigl( \mathcal{T}_{i}^{0}(v_{1},v_{2}) \bigr) (s,t_{2}) \bigr\vert \\& \qquad {}- \frac{\lambda _{2}}{\varGamma (\beta _{2})\varGamma (\theta _{2})}\sum_{r= \alpha _{2}+\beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r- \beta _{2}}\bigl(\eta _{2}+\theta _{2}-\sigma (r)\bigr)^{ \underline{\theta _{2}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} g_{2}(r) \\& \qquad {}\times\bigl\vert \bigl( \mathcal{T}_{i}^{0}(u_{1},u_{2}) \bigr) (t_{1},s) - \bigl( \mathcal{T}_{i}^{0}(v_{1},v_{2}) \bigr) (t_{1},s) \bigr\vert \\& \quad \leq \frac{ \chi _{1}^{q-2} \mathcal{K}_{1} }{\varGamma (\beta _{1})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{1}} \sum_{s=\alpha _{1}-1}^{T+ \alpha _{1}} \bigl(T+\alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{ \underline{\beta _{1}-1}} \bigl( s^{\underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1} \\& \qquad {}- \frac{\lambda _{2}G_{2} \chi _{2}^{q-2} \mathcal{K}_{2} }{\varGamma (\beta _{2})\varGamma (\theta _{2})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{2}} \sum_{r=\alpha _{2}+ \beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r-\beta _{2}}\bigl(\eta _{2}+ \theta _{2}-\sigma (r)\bigr)^{\underline{\theta _{2}-1}} \\& \qquad {}\times \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} \bigl( s^{ \underline{\alpha _{2}}} \bigr)^{\delta (q-2)+1} , \end{aligned}$$

(3.19)

and

$$\begin{aligned}& \bigl\vert {\mathcal{Q}}^{*} [F_{1},F_{2}](u_{1},u_{2}) - {\mathcal{Q}}^{*}[F_{1},F_{2}](v_{1},v_{2}) \bigr\vert \\& \quad = \frac{ \chi _{2}^{q-2}\mathcal{K}_{2} }{\varGamma (\beta _{2})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{2}} \sum_{s=\alpha _{2}-1}^{T+\alpha _{2}} \bigl(T+ \alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{\underline{\beta _{2}-1}} \bigl( s^{\underline{\alpha _{2}}} \bigr)^{\delta (q-2)+1} \\& \qquad {}- \frac{\lambda _{1}G_{1} \chi _{1}^{q-2}\mathcal{K}_{1} }{\varGamma (\beta _{1})\varGamma (\theta _{1})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{1}} \sum_{r=\alpha _{1}+ \beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r-\beta _{1}}\bigl(\eta _{1}+ \theta _{1}-\sigma (r)\bigr)^{\underline{\theta _{1}-1}} \\& \qquad {}\times \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} \bigl( s^{ \underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1}. \end{aligned}$$

(3.20)

From (3.19)–(3.20), it implies that

$$\begin{aligned}& \bigl\vert \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \bigl( \mathcal{T}_{1}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad = \bigl\vert \bigl( \mathcal{T}_{1}^{*} \bigl( \mathcal{T}_{1}^{o} (u_{1},u_{2}) \bigr) \bigr) (t_{1},t_{2}) - \bigl( \mathcal{T}_{1}^{*} \bigl( \mathcal{T}_{1}^{o} (v_{1},v_{2}) \bigr) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \frac{1}{\varGamma (\beta _{1})}\sum_{s=\alpha _{1}-1}^{t_{1}- \beta _{1}} \bigl(t_{1}-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} \bigl\vert \bigl( \mathcal{T}_{1}^{0}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) - \bigl( \mathcal{T}_{1}^{0}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \qquad {}+ \frac{1}{ \vert \varLambda \vert } \Biggl\{ \Biggl( \frac{\lambda _{1}}{\varGamma (\theta _{1})}\sum _{s=\alpha _{1}+\beta _{1}-2}^{ \eta _{1}}\bigl(\eta _{1}+\theta _{1}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \Biggr) \\& \qquad {}\times \bigl\vert {\mathcal{P}}^{*}[F_{1},F_{2}](u_{1},u_{2}) - {\mathcal{P}}^{*}[F_{1},F_{2}](v_{1},v_{2}) \bigr\vert \\& \qquad {}+ \bigl\vert {\mathcal{Q}}^{*}[F_{1},F_{2}](u_{1},u_{2}) - { \mathcal{Q}}^{*}[F_{1},F_{2}](v_{1},v_{2}) \bigr\vert \Biggr\} \\& \quad \leq \frac{\chi _{1}^{q-2} \mathcal{K}_{1} \Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \Vert _{\mathcal{C}_{1}}}{\varGamma (\beta _{1})} \sum_{s=\alpha _{1}-1}^{t_{1}-\beta _{1}} \bigl(t_{1}-\sigma (s)\bigr)^{ \underline{\beta _{1}-1}} \bigl( s^{\underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1} \\& \qquad {}+ \frac{1}{ \vert \varLambda \vert } \Biggl\{ \frac{\lambda _{1}G_{1} (\eta _{1}-\alpha _{1}-\beta _{1}+\theta _{1}+2)^{\underline{\theta _{1}}} }{ \vert \varLambda \vert \varGamma (\theta _{1}+1) } \\& \qquad {}\times \Biggl[ \frac{ \chi _{1}^{q-2} \mathcal{K}_{1} }{\varGamma (\beta _{1})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{1}} \sum_{s=\alpha _{1}-1}^{T+ \alpha _{1}} \bigl(T+\alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{ \underline{\beta _{1}-1}} \bigl( s^{\underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1} \\& \qquad {}+ \frac{\lambda _{2}G_{2} \chi _{2}^{q-2} \mathcal{K}_{2} }{\varGamma (\beta _{2})\varGamma (\theta _{2})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{2}} \sum_{r=\alpha _{2}+ \beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r-\beta _{2}}\bigl(\eta _{2}+ \theta _{2}-\sigma (r)\bigr)^{\underline{\theta _{2}-1}} \\& \qquad {}\times \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} \bigl( s^{ \underline{\alpha _{2}}} \bigr)^{\delta (q-2)+1} \Biggr] \\& \qquad {}+ \Biggl[\frac{ \chi _{2}^{q-2}\mathcal{K}_{2} }{\varGamma (\beta _{2})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{2}} \sum_{s=\alpha _{2}-1}^{T+ \alpha _{2}} \bigl(T+\alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{ \underline{\beta _{2}-1}} \bigl( s^{\underline{\alpha _{2}}} \bigr)^{\delta (q-2)+1} \\& \qquad{} + \frac{\lambda _{1}G_{1} \chi _{1}^{q-2}\mathcal{K}_{1} }{\varGamma (\beta _{1})\varGamma (\theta _{1})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{1}} \sum_{r=\alpha _{1}+ \beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r-\beta _{1}}\bigl(\eta _{1}+ \theta _{1}-\sigma (r)\bigr)^{\underline{\theta _{1}-1}} \\& \qquad {}\times \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} \bigl( s^{ \underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1} \Biggr] \Biggr\} \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \bigl\{ \mathcal{K}_{1}\chi _{1}^{q-2}\varOmega _{1}+ \mathcal{K}_{2}\chi _{2}^{q-2} \varTheta _{2} \bigr\} . \end{aligned}$$

(3.21)

Similarly, we can find that

$$\begin{aligned}& \bigl\vert \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr)(t_{1},t_{2})- \bigl( \mathcal{T}_{2}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \bigl\{ \mathcal{K}_{1}\chi _{1}^{q-2}\varTheta _{1}+ \mathcal{K}_{2}\chi _{2}^{q-2} \varOmega _{2} \bigr\} . \end{aligned}$$

(3.22)

Next, taking the fractional difference of order $\gamma _{1}$, $\gamma _{2}$ for (3.2) and (3.3), respectively, we obtain

$$\begin{aligned}& \Delta ^{\gamma _{1}} \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\& \quad = \Delta ^{\gamma _{1}}\bigl( \mathcal{T}_{1}^{*}\bigl(\mathcal{T}_{1}^{o}(u_{1},u_{2}) \bigr)\bigr) (t_{1},t_{2}) \\& \quad = \frac{1}{ \varGamma (-\gamma _{1})\varGamma (\beta _{1}) } \sum_{r= \alpha _{1}+\beta _{1}-1}^{t_{1}+\gamma _{1}} \sum_{s=\alpha _{1}-1}^{r- \beta _{1}} \bigl(t_{1}- \sigma (r)\bigr)^{\underline{-\gamma _{1}-1}} \bigl(r- \sigma (s)\bigr)^{\underline{\beta _{1}-1}} \bigl(\mathcal{T}_{1}^{o}(u_{1},u_{2}) \bigr) (t_{2},s) \\& \qquad {} +\frac{1}{\varLambda } \Biggl\{ \frac{\lambda _{1}}{\varGamma (\theta _{1})\varGamma (-\gamma _{1}) }\sum _{r= \alpha _{1}+\beta _{1}-1}^{t_{1}+\gamma _{1}} \sum_{s=\alpha _{1}+ \beta _{1}-2}^{\eta _{1}} \bigl(t_{1}-\sigma (r)\bigr)^{ \underline{-\gamma _{1}-1}} \bigl(\eta _{1}+\theta _{1}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{1}(s) \\& \qquad {}\times {\mathcal{P}^{*}[F_{1},F_{2}]}(u_{1},u_{2})+ \frac{1}{ \varGamma (-\gamma _{1}) } \sum_{s=\alpha _{1}+\beta _{1}-1}^{t_{1}+ \gamma _{1}} \bigl(t_{1}-\sigma (s)\bigr)^{\underline{-\gamma _{1}-1}} { \mathcal{Q}^{*}[F_{1},F_{2}]}(u_{1},u_{2}) \Biggr\} \end{aligned}$$

(3.23)

and

$$\begin{aligned}& \Delta ^{\gamma _{2}} \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\& \quad = \Delta ^{\gamma _{2}}\bigl( \mathcal{T}_{2}^{*}\bigl(\mathcal{T}_{2}^{o}(u_{1},u_{2}) \bigr)\bigr) (t_{1},t_{2}) \\& \quad = \frac{1}{\varGamma (-\gamma _{2}) \varGamma (\beta _{2}) } \sum_{r= \alpha _{2}+\beta _{2}-1}^{t_{2}+\gamma _{2}} \sum_{s=\alpha _{2}-1}^{r- \beta _{2}} \bigl(t_{2}- \sigma (r)\bigr)^{\underline{-\gamma _{2}-1}} \bigl(r- \sigma (s)\bigr)^{\underline{\beta _{2}-1}} \bigl(\mathcal{T}_{2}^{o}(u_{1},u_{2}) \bigr) (t_{1},s) \\& \qquad {} +\frac{1}{\varLambda } \Biggl\{ \frac{\lambda _{2}}{\varGamma (-\gamma _{2})\varGamma (\theta _{2}) } \sum _{r= \alpha _{2}+\beta _{2}-1}^{t_{2}+\gamma _{2}}\sum_{s=\alpha _{2}+ \beta _{2}-2}^{\eta _{2}} \bigl(t_{2}-\sigma (r)\bigr)^{ \underline{-\gamma _{2}-1}} \bigl(\eta _{2}+\theta _{2}-\sigma (s)\bigr)^{ \underline{\theta _{1}-1}} g_{2}(s) \\& \qquad {} \times{\mathcal{Q}^{*}[F_{1},F_{2}]}(u_{1},u_{2})+ \frac{1}{\varGamma (-\gamma _{2}) } \sum_{s=\alpha _{2}+\beta _{2}-1}^{t_{2}+ \gamma _{2}} \bigl(t_{2}-\sigma (s)\bigr)^{\underline{-\gamma _{2}-1}} { \mathcal{P}^{*}[F_{1},F_{2}]}(u_{1},u_{2}) \Biggr\} , \end{aligned}$$

(3.24)

where $t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-\gamma _{i}+1,T+\alpha _{i}+ \beta _{i}-\gamma _{i}}$. Therefore,

$$\begin{aligned}& \bigl\vert \Delta ^{\gamma _{1}} \bigl( \mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \Delta ^{\gamma _{1}} \bigl( \mathcal{T}_{1}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \frac{1 }{ \varGamma (-\gamma _{1})\varGamma (\beta _{1}) } \sum_{r= \alpha _{1}+\beta _{1}-1}^{t_{1}+\gamma _{1}} \sum_{s=\alpha _{1}-1}^{r- \beta _{1}} \bigl(t_{1}- \sigma (r)\bigr)^{\underline{-\gamma _{1}-1}} \bigl(r- \sigma (s)\bigr)^{\underline{\beta _{1}-1}} \\& \qquad {}\times \bigl\vert \bigl( \mathcal{T}_{1}^{*} \bigl( \mathcal{T}_{1}^{o} (u_{1},u_{2}) \bigr) \bigr) (t_{1},t_{2}) - \bigl( \mathcal{T}_{1}^{*} \bigl( \mathcal{T}_{1}^{o} (v_{1},v_{2}) \bigr) \bigr) (t_{1},t_{2}) \bigr\vert \\& \qquad {}+ \frac{1}{\varLambda } \Biggl\{ \frac{\lambda _{1}G_{1}}{\varGamma (\theta _{1})\varGamma (-\gamma _{1}) } \sum _{x=\alpha _{1}+\beta _{1}-1}^{t_{1}+\gamma _{1}} \sum_{y= \alpha _{1}+\beta _{1}-2}^{\eta _{1}} \bigl(t_{1}-\sigma (x)\bigr)^{ \underline{-\gamma _{1}-1}} \bigl(\eta _{1}+\theta _{1}-\sigma (y)\bigr)^{ \underline{\theta _{1}-1}} \\& \qquad {}\times \bigl\vert {\mathcal{P}}^{*}[F_{1},F_{2}](u_{1},u_{2}) - {\mathcal{P}}^{*}[F_{1},F_{2}](v_{1},v_{2}) \bigr\vert \\& \qquad {}+ \frac{1}{ \varGamma (-\gamma _{1}) } \sum_{x=\alpha _{1}+\beta _{1}-1}^{t_{1}+ \gamma _{1}} \bigl(t_{1}-\sigma (x)\bigr)^{\underline{-\gamma _{1}-1}} \bigl\vert { \mathcal{Q}}^{*}[F_{1},F_{2}](u_{1},u_{2}) - {\mathcal{Q}}^{*}[F_{1},F_{2}](v_{1},v_{2}) \bigr\vert \Biggr\} \\& \quad \leq \frac{\chi _{1}^{q-2} \mathcal{K}_{1} \Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \Vert _{\mathcal{C}_{1}} }{ \varGamma (-\gamma _{1})\varGamma (\beta _{1}) } \cdot \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \sum _{s=\alpha _{1}-1}^{T+\alpha _{1}} \bigl(T+\alpha _{1}+ \beta _{1}- \sigma (s)\bigr)^{\underline{\beta _{1}-1}} \bigl( s^{ \underline{\alpha _{1}}} \bigr)^{ \delta (q-2)+1} \\& \qquad {}+ \frac{1}{\varLambda } \Biggl\{ \lambda _{1}G_{1} \cdot \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \cdot \frac{(\eta _{1}-\alpha _{1}-\beta _{1}+\theta _{1}+2)^{\underline{\theta _{1}}} }{ \varGamma (\theta _{1}+1) } \\& \qquad {}\times \Biggl[ \frac{ \chi _{1}^{q-2} \mathcal{K}_{1} }{\varGamma (\beta _{1})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{1}} \sum_{s=\alpha _{1}-1}^{T+ \alpha _{1}} \bigl(T+\alpha _{1}+\beta _{1}-\sigma (s) \bigr)^{ \underline{\beta _{1}-1}} \bigl( s^{\underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1} \\& \qquad {}+ \frac{\lambda _{2}G_{2} \chi _{2}^{q-2} \mathcal{K}_{2} }{\varGamma (\beta _{2})\varGamma (\theta _{2})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{2}} \sum_{r=\alpha _{2}+ \beta _{2}-1}^{\eta _{2}} \sum_{s=\alpha _{2}-1}^{r-\beta _{2}}\bigl(\eta _{2}+ \theta _{2}-\sigma (r)\bigr)^{\underline{\theta _{2}-1}} \\& \qquad {}\times \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{2}-1}} \bigl( s^{ \underline{\alpha _{2}}} \bigr)^{\delta (q-2)+1} \Biggr] + \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \\& \qquad {}\times \Biggl[\frac{ \chi _{2}^{q-2}\mathcal{K}_{2} }{\varGamma (\beta _{2})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{2}} \sum_{s=\alpha _{2}-1}^{T+ \alpha _{2}} \bigl(T+\alpha _{2}+\beta _{2}-\sigma (s) \bigr)^{ \underline{\beta _{2}-1}} \bigl( s^{\underline{\alpha _{2}}} \bigr)^{\delta (q-2)+1} \\& \qquad {}+ \frac{\lambda _{1}G_{1} \chi _{1}^{q-2}\mathcal{K}_{1} }{\varGamma (\beta _{1})\varGamma (\theta _{1})} \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{1}} \sum_{r=\alpha _{1}+ \beta _{1}-1}^{\eta _{1}} \sum_{s=\alpha _{1}-1}^{r-\beta _{1}}\bigl(\eta _{1}+ \theta _{1}-\sigma (r)\bigr)^{\underline{\theta _{1}-1}} \\& \qquad {}\times \bigl(r-\sigma (s)\bigr)^{\underline{\beta _{1}-1}} \bigl( s^{ \underline{\alpha _{1}}} \bigr)^{\delta (q-2)+1} \Biggr] \Biggr\} \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \biggl\{ \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \varOmega _{1}+\mathcal{K}_{2}\chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \varTheta _{2} \biggr\} . \end{aligned}$$

(3.25)

Similarly, we obtain

$$\begin{aligned}& \bigl\vert \Delta ^{\gamma _{2}} \bigl( \mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \Delta ^{\gamma _{2}} \bigl( \mathcal{T}_{2}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \biggl\{ \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \varTheta _{1}+\mathcal{K}_{2}\chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \varOmega _{2} \biggr\} . \end{aligned}$$

(3.26)

From (3.22) and (3.25), we find that

$$\begin{aligned}& \bigl\Vert \bigl(\mathcal{T}(u_{1},u_{2}) \bigr)- \bigl(\mathcal{T}(v_{1},v_{2}) \bigr) \bigr\Vert _{\mathcal{C}_{1}} \\& \quad < \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \\& \qquad {}\times \max \biggl\{ \mathcal{K}_{1}\chi _{1}^{q-2} \varTheta _{1} + \mathcal{K}_{2} \chi _{2}^{q-2}\varOmega _{2}, \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \varOmega _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \varTheta _{2} \biggr\} . \end{aligned}$$

(3.27)

In addition, by (3.21) and (3.26), we find that

$$\begin{aligned}& \bigl\Vert \bigl(\mathcal{T}(u_{1},u_{2}) \bigr)- \bigl(\mathcal{T}(v_{1},v_{2}) \bigr) \bigr\Vert _{\mathcal{C}_{2}} \\& \quad < \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \\& \qquad {}\times \max \biggl\{ \mathcal{K}_{1}\chi _{1}^{q-2} \varOmega _{1} + \mathcal{K}_{2} \chi _{2}^{q-2}\varTheta _{2}, \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \varTheta _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \varOmega _{2} \biggr\} . \end{aligned}$$

(3.28)

Hence, from (3.27) and (3.28), we can conclude that

$$\begin{aligned}& \bigl\Vert \bigl(\mathcal{T}(u_{1},u_{2}) \bigr)- \bigl(\mathcal{T}(v_{1},v_{2}) \bigr) \bigr\Vert _{\mathcal{U}} \\& \quad < \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \\& \qquad {}\times \max \biggl\{ \mathcal{K}_{1}\chi _{1}^{q-2} \varTheta _{1} + \mathcal{K}_{2} \chi _{2}^{q-2}\varOmega _{2}, \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \varOmega _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \varTheta _{2} , \\& \qquad \mathcal{K}_{1}\chi _{1}^{q-2}\varOmega _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \varTheta _{2}, \mathcal{K}_{1}\chi _{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \varTheta _{1} + \mathcal{K}_{2} \chi _{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \varOmega _{2} \biggr\} \\& \quad = \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \varPhi . \end{aligned}$$

(3.29)

By (3.13), $\mathcal{T}$ is a contraction mapping. Hence, by the Banach fixed point theorem, we get that $\mathcal{T}$ has a fixed point, which is a unique solution of problem (1.1)–(1.2). □

In the same manner as Theorem 3.1, we can obtain the following theorem.

Theorem 3.2

Let$p>2$, (H2)–(H3) hold, and the following condition hold:

• (H4) There exist constants$\chi _{i}>0$and$0<\delta <\frac{1}{2-q}$such that

  $$ F_{i} [t_{i},t_{j},x,y,z ] \leq -\chi _{i}\Delta _{C}^{\alpha } \bigl( t_{i}^{\underline{\alpha _{i}}} \bigr)^{\delta} $$

  for any$( t_{i},t_{j},x,y,z )\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+ \alpha _{i}+\beta _{i}}\times \mathbb{N}_{\alpha _{j}+\beta _{j}-2,T+ \alpha _{j}+\beta _{j}}\times \mathbb{R}^{3}$.

Then problem (1.1)–(1.2) has a unique solution.

In the case $1< p<2$ and $q>2$ since $\frac{1}{p}+\frac{1}{q}=1$, we obtain the following theorem.

Theorem 3.3

Let$1< p<2$and (H2)–(H3) hold. For each$i,j\in \{1,2\}$, $i\neq j$, $F_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}\times \mathbb{R}^{3}, \mathbb{R} )$, $\varphi _{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}, [0,\infty ) )$with$\varphi ^{o}_{i}=\max \{\varphi (t_{i}-1,s) \}$. Suppose that the following assumption holds:

• (H5) There exists a nonnegative function$k_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}, [0,\infty ) )$and$\mathcal{M}_{i}:=\frac{1}{\varGamma (\alpha _{i})} \sum_{ \xi =\alpha _{i}+\beta _{i}-1}^{T+\alpha _{i}+\beta _{i}-1}(T+2 \alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}} k_{i}(T+ \alpha _{j}+\beta _{j},\xi )>0$such that

  $$ F_{i}[ t_{i},t_{j},x,y,z]\leq k_{i}(t_{i},t_{j}) $$

  for any$( t_{i},t_{j},x,y,z )\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+ \alpha _{i}+\beta _{i}}\times \mathbb{N}_{\alpha _{j}+\beta _{j}-2,T+ \alpha _{j}+\beta _{j}}\times \mathbb{R}^{3}$.

Then problem (1.1)–(1.2) has a unique solution provided that

$$\begin{aligned} \varUpsilon :=&\max \biggl\{ \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \bar{\varTheta }_{1} + \mathcal{K}_{2}\mathcal{M}_{2}^{q-2}\bar{ \varOmega }_{2}, \mathcal{K}_{1}\mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \bar{\varOmega }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \bar{\varTheta }_{2} , \\ & \mathcal{K}_{1}\mathcal{M}_{1}^{q-2} \bar{\varOmega }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{\varTheta }_{2}, \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \bar{\varTheta }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \bar{\varOmega }_{2} \biggr\} \\ < &1, \end{aligned}$$

(3.30)

where$\mathcal{K}_{i}$is defined as (3.14), and

$$\begin{aligned}& \begin{aligned}[b] \bar{\varOmega }_{i}={}& \biggl[ 1 + \frac{\lambda _{i}G_{i} (\eta _{i}-\alpha _{i} -\beta _{i}+\theta _{i}+2)^{\underline{\theta _{i}}} }{ \vert \varLambda \vert \varGamma (\theta _{i}+1) } \biggr] \frac{1}{\varGamma (\beta _{i})}\sum_{s=\alpha _{i}-1}^{T+ \alpha _{i}} \bigl(T+\alpha _{i}+\beta _{i}-\sigma (s) \bigr)^{ \underline{\beta _{i}-1}} s^{\underline{\alpha _{i}}} \\ &{}+\frac{\lambda _{i}G_{i} (\eta _{i}-\alpha _{i}-\beta _{i}+\theta _{i}+1)^{\underline{\theta _{i}}} }{ \vert \varLambda \vert \varGamma (\theta _{i}+1) } \cdot \frac{1}{\varGamma (\beta _{i})} \sum _{s=\alpha _{i}-1}^{\eta _{i}- \beta _{i}} \bigl(\eta _{i}-\sigma (s)\bigr)^{\underline{\beta _{i}-1} } s^{ \underline{\alpha _{i}}}, \end{aligned} \end{aligned}$$

(3.31)

$$\begin{aligned}& \begin{aligned}[b] \bar{\varTheta }_{i}={}& \frac{1}{ \vert \varLambda \vert \varGamma (\beta _{i})}\sum _{s= \alpha _{i}-1}^{T+\alpha _{i}}\bigl(T+\alpha _{i}+ \beta _{i}-\sigma (s)\bigr)^{ \underline{\beta _{i}-1}} s^{\underline{\alpha _{i}}} \\ &{}+\frac{\lambda _{1}\lambda _{2}G_{1}G_{2} }{ \vert \varLambda \vert } \cdot \frac{ (\eta _{j}-\alpha _{j}-\beta _{j}+\theta _{j}+2)^{\underline{\theta _{j}}} }{ \varGamma (\theta _{j}+1)} \cdot \frac{(\eta _{i} -\alpha _{i}-\beta _{i}+\theta _{i}+1)^{\underline{\theta _{i}}} }{ \varGamma (\theta _{i}+1) } \\ &{}\times \frac{1}{\varGamma (\beta _{i})} \sum_{s=\alpha _{i}-1}^{\eta _{i}- \beta _{i}} \bigl(\eta _{i}-\sigma (s)\bigr)^{\underline{\beta _{i}-1} } s^{ \underline{\alpha _{i}}}. \end{aligned} \end{aligned}$$

(3.32)

Proof

For each $i,j\in \{1,2\}$, $i\neq j$, by (H5) we have

$$\begin{aligned}& \Biggl\vert \frac{1}{\varGamma (\alpha _{i})} \sum _{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \bigl(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi )\bigr)^{ \underline{\alpha _{i}-1}} F^{*}_{i}\bigl[u(t_{j},\xi )\bigr] \Biggr\vert \\& \quad \leq \frac{1}{\varGamma (\alpha _{i})} \sum_{\xi =\alpha _{i}+\beta _{i}-1}^{T+ \alpha _{i}+\beta _{i}-1} \bigl(T+2\alpha _{i}+\beta _{i}-1-\sigma (\xi ) \bigr)^{ \underline{\alpha _{i}-1}} k_{i}(t_{j},\xi ) \\& \quad \leq \mathcal{M}_{i}. \end{aligned}$$

(3.33)

By (A2), (H2), and the definition of operator $\mathcal{T}_{i}^{0}$, for any $(u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}$, we have

$$\begin{aligned}& \bigl\vert \bigl( \mathcal{T}_{i}^{0}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) - \bigl( \mathcal{T}_{i}^{0}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad = \Biggl\vert \phi _{q} \Biggl[ \sum _{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \frac{(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}}}{ \varGamma (\alpha _{i}) } F^{*}_{i}\bigl[u(t_{j},\xi )\bigr] \Biggr] \\& \qquad {} - \phi _{q} \Biggl[ \sum_{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \frac{(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}}}{ \varGamma (\alpha _{i}) } F^{*}_{i}\bigl[v(t_{j}, \xi )\bigr] \Biggr] \Biggr\vert \\& \quad \leq (q-1) \mathcal{M}_{i}^{q-2} \sum _{\xi =\alpha _{i}+\beta _{i}-1}^{t_{i}+ \beta _{i}-1} \frac{(t_{i}+\alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}}}{ \varGamma (\alpha _{i}) } \bigl\vert F^{*}_{i}\bigl[u(t_{j},\xi )\bigr]- F^{*}_{i}\bigl[v(t_{j},\xi )\bigr] \bigr\vert \\& \quad \leq (q-1) \mathcal{M}_{i}^{q-2} \frac{t_{i}^{\underline{\alpha _{i}}} }{\varGamma (\alpha _{i}+1)} \bigl[ L_{i} \bigl\Vert \Delta ^{\gamma _{i}}u_{i}-\Delta ^{\gamma _{i}}v_{i} \bigr\Vert +M_{j} \bigl\Vert \varPsi ^{\omega _{i}}u_{i}- \varPsi ^{\omega _{j}}v_{i} \bigr\Vert +N_{j} \Vert u_{j}-v_{j} \Vert \bigr] \\& \quad \leq (q-1) \mathcal{M}_{i}^{q-2} \frac{t_{i}^{\underline{\alpha _{i}}} }{\varGamma (\alpha _{i}+1)} \biggl[ L_{i} \bigl\Vert \Delta ^{\gamma _{i}}u_{i}- \Delta ^{\gamma _{i}}v_{i} \bigr\Vert + \biggl( N_{j}+M_{j} \varphi _{j}^{o} \frac{(T+\omega _{j}+2)^{\underline{\omega _{j}}}}{\varGamma (\omega _{j}+1)} \biggr) \Vert u_{j}-v_{j} \Vert \biggr] \\& \quad \leq (q-1) \mathcal{M}_{i}^{q-2} \frac{t_{i}^{\underline{\alpha _{i}}} }{\varGamma (\alpha _{i}+1)} \biggl[ L_{i}+N_{j}+M_{j}\varphi _{j}^{o} \frac{(T+\omega _{j}+2)^{\underline{\omega _{j}}}}{\varGamma (\omega _{j}+1)} \biggr] \\& \qquad {}\times\bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{C}_{i}}. \end{aligned}$$

(3.34)

Then, by (3.19) and (3.20), we have

$$\begin{aligned}& \bigl\vert \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \bigl( \mathcal{T}_{1}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \max \bigl\{ \mathcal{K}_{1} \mathcal{M}_{1}^{q-2}\bar{\varOmega }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{ \varTheta }_{2} \bigr\} , \end{aligned}$$

(3.35)

$$\begin{aligned}& \bigl\vert \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \bigl( \mathcal{T}_{2}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \max \bigl\{ \mathcal{K}_{1} \mathcal{M}_{1}^{q-2}\bar{\varTheta }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{ \varOmega }_{2} \bigr\} . \end{aligned}$$

(3.36)

Similarly as in Theorem 3.1, we obtain

$$\begin{aligned}& \bigl\vert \Delta ^{\gamma _{1}} \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \Delta ^{\gamma _{2}1} \bigl(\mathcal{T}_{1}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \biggl\{ \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varOmega }_{1}+\mathcal{K}_{2}\mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varTheta }_{2} \biggr\} , \end{aligned}$$

(3.37)

$$\begin{aligned}& \bigl\vert \Delta ^{\gamma _{2}} \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \Delta ^{\gamma _{2}} \bigl( \mathcal{T}_{2}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \bigl\Vert ( u_{1}-v_{1},u_{2}-v_{2} ) \bigr\Vert _{\mathcal{U}} \biggl\{ \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varTheta }_{1}+\mathcal{K}_{2}\mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varOmega }_{2} \biggr\} . \end{aligned}$$

(3.38)

By (3.36) and (3.37), we have

$$\begin{aligned}& \bigl\Vert \bigl(\mathcal{T}(u_{1},u_{2}) \bigr)- \bigl(\mathcal{T}(v_{1},v_{2}) \bigr) \bigr\Vert _{\mathcal{C}_{1}} \\& \quad < \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \\& \qquad {}\times \max \biggl\{ \mathcal{K}_{1}\mathcal{M}_{1}^{q-2} \bar{\varTheta }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{\varOmega }_{2}, \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varOmega }_{1} \\& \qquad {}+\mathcal{K}_{2} \mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varTheta }_{2} \biggr\} . \end{aligned}$$

(3.39)

By (3.35) and (3.38), we have

$$\begin{aligned}& \bigl\Vert \bigl(\mathcal{T}(u_{1},u_{2}) \bigr)- \bigl(\mathcal{T}(v_{1},v_{2}) \bigr) \bigr\Vert _{\mathcal{C}_{2}} \\& \quad < \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \\& \qquad {}\times \max \biggl\{ \mathcal{K}_{1}\mathcal{M}_{1}^{q-2} \bar{\varOmega }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{\varTheta }_{2} , \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varTheta }_{1} \\& \qquad {}+\mathcal{K}_{2} \mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}}}}{\varGamma (1-\gamma _{1})} \bar{\varOmega }_{2} \biggr\} . \end{aligned}$$

(3.40)

Therefore, by (3.39) and (3.40), we can conclude that

$$\begin{aligned}& \bigl\Vert \bigl(\mathcal{T}(u_{1},u_{2}) \bigr)- \bigl(\mathcal{T}(v_{1},v_{2}) \bigr) \bigr\Vert _{\mathcal{U}} \\& \quad < \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \\& \qquad {}\times \max \biggl\{ \mathcal{K}_{1}\mathcal{M}_{1}^{q-2} \bar{\varTheta }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{\varOmega }_{2}, \\& \qquad \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \bar{\varOmega }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{1}} }}{\varGamma (1-\gamma _{1})} \bar{\varTheta }_{2} , \\& \qquad \mathcal{K}_{1}\mathcal{M}_{1}^{q-2} \bar{\varOmega }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2}\bar{\varTheta }_{2}, \mathcal{K}_{1} \mathcal{M}_{1}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \bar{\varTheta }_{1} + \mathcal{K}_{2} \mathcal{M}_{2}^{q-2} \frac{(T+2)^{\underline{-\gamma _{2}} }}{\varGamma (1-\gamma _{2})} \bar{\varOmega }_{2} \biggr\} \\& \quad = \bigl\Vert (u_{1}-v_{1},u_{2}-v_{2}) \bigr\Vert _{\mathcal{U}} \varUpsilon . \end{aligned}$$

(3.41)

By (3.30), we can conclude that $\mathcal{T}$ is a contraction mapping. Hence, by the Banach fixed point theorem, $\mathcal{T}$ has a fixed point, which is a unique solution of problem (1.1)–(1.2). □

4 Some examples

In this section, we consider some examples to illustrate our main result.

Example 4.1

Consider the following fractional sum boundary value problem:

$$ \begin{aligned} &\Delta ^{\frac{1}{2}}_{C}\bigl[\phi _{\frac{5}{2}}\bigl(\Delta ^{\frac{2}{3}}_{C}u_{1} \bigr)\bigr](t) \\ &\quad = F_{1} \biggl[t+\frac{1}{6},t+ \frac{1}{8},\Delta ^{\frac{1}{3}}u_{1} \biggl(t+ \frac{5}{6} \biggr) ,\varPsi ^{\frac{1}{4}}u_{2} \biggl(t+ \frac{3}{8} \biggr) , u_{2} \biggl( t+ \frac{1}{8} \biggr) \biggr], \\ &\Delta ^{\frac{3}{4}}_{C}\bigl[\phi _{\frac{5}{2}}\bigl( \Delta ^{\frac{3}{8}}_{C}u_{2}\bigr)\bigr](t) \\ &\quad = F_{2} \biggl[t+\frac{1}{6},t+\frac{1}{8}, \Delta ^{\frac{2}{3}}u_{2} \biggl(t+\frac{11}{24} \biggr) ,\varPsi ^{\frac{3}{4}}u_{1} \biggl(t+ \frac{11}{12} \biggr) , u_{1} \biggl( t+ \frac{1}{6} \biggr) \biggr], \end{aligned} $$

(4.1)

subject to nonlocal fractional sum boundary conditions of the form

$$ \begin{aligned} &\Delta ^{\frac{2}{3} }_{C} u_{1} \biggl(-\frac{1}{2} \biggr) =0,\qquad u_{1} \biggl(\frac{67}{6} \biggr) =3\Delta ^{-\frac{2}{5} }e^{2\sin ( \frac{221}{40} \pi ) }u_{2} \biggl( \frac{221}{40} \biggr) , \\ &\Delta ^{\frac{3}{4} }_{C} u_{1} \biggl(- \frac{1}{4} \biggr) =0,\qquad u_{1} \biggl(\frac{89}{8} \biggr) =2\Delta ^{-\frac{1}{4} }e^{\cos ( \frac{41}{12} \pi ) }u_{1} \biggl( \frac{41}{12} \biggr) , \end{aligned} $$

(4.2)

where $t\in {\mathbb{N}}_{0,10}$. Functions $F_{1}$, $F_{2}$ are determined by

$$\begin{aligned}& F_{1} \biggl[ t_{1},t_{2},\Delta ^{\frac{1}{3} }u_{1} \biggl(t_{1}- \frac{2}{3} \biggr) ,\varPsi ^{\frac{1}{4}} u_{2} \biggl( t_{2}+ \frac{1}{4} \biggr) , u_{2} ( t_{2} ) \biggr] \\& \quad = 3t_{1}^{\underline{2}} \biggl[ 1+\frac{1}{50\text{,}000e^{7}}\sin ^{2} \biggl( \Delta ^{\frac{1}{3} }u_{1} \biggl(t_{1}-\frac{2}{3} \biggr) \biggr) \biggr] \\& \qquad {}+2t_{2}^{\underline{2}} \biggl[ 1+\frac{1}{40\text{,}000e^{8}} \cos ^{2} \biggl( \varPsi ^{\frac{1}{4}} u_{2} \biggl( t_{2}+\frac{1}{4} \biggr) \biggr) \biggr] \\& \qquad {}+ t_{2}^{\underline{2}} \biggl[ 1+\frac{1}{60\text{,}000e^{6}}\sin ^{2} \bigl( u_{2} ( t_{2} ) \bigr) \biggr], \\& F_{2} \biggl[ t_{1},t_{2},\Delta ^{\frac{2}{3} }u_{2} \biggl(t_{2}- \frac{1}{3} \biggr) ,\varPsi ^{\frac{3}{4}} u_{1} \biggl( t_{1}+ \frac{3}{4} \biggr) , u_{1} ( t_{1} ) \biggr] \\& \quad = 2t_{2}^{\underline{2}} \biggl[ 1+\frac{1}{60\text{,}000e^{6}}\sin ^{2} \biggl( \Delta ^{\frac{2}{3} }u_{2} \biggl(t_{2}-\frac{1}{3} \biggr) \biggr) \biggr] \\& \qquad {}+2t_{1}^{\underline{2}} \biggl[ 1+\frac{1}{50\text{,}000e^{7}} \cos ^{2} \biggl( \varPsi ^{\frac{3}{4}} u_{1} \biggl( t_{1}+\frac{3}{4} \biggr) \biggr) \biggr] \\& \qquad {}+ 3t_{1}^{\underline{2}} \biggl[ 1+\frac{1}{40\text{,}000e^{8}}\sin ^{2} \bigl( u_{1} ( t_{1} ) \bigr) \biggr], \end{aligned}$$

and

$$\begin{aligned}& \varPsi ^{\frac{3}{4}} u_{1} \biggl( t_{1}+ \frac{3}{4} \biggr) = \frac{1}{\varGamma ( \frac{3}{4} ) } \sum _{s=-\frac{19}{12}}^{t_{1}-\frac{3}{4}} \bigl(t_{1}-\sigma (s) \bigr)^{ \underline{\frac{3}{4}-1}} \frac{e^{-s}}{(t_{1}+10)^{3}} u_{1} \biggl( t_{1}+\frac{3}{4} \biggr), \end{aligned}$$

(4.3)

$$\begin{aligned}& \varPsi ^{\frac{1}{4}} u_{2} \biggl( t_{2}+ \frac{1}{4} \biggr)= \frac{1}{\varGamma ( \frac{1}{4} ) } \sum _{s=-\frac{9}{8}}^{t_{1}-\frac{1}{4}}\bigl(t_{2}-\sigma (s) \bigr)^{ \underline{\frac{1}{4}-1}} \frac{e^{-s}}{(t_{2}+20)^{2}} u_{2} \biggl( t_{2}+\frac{1}{4} \biggr). \end{aligned}$$

(4.4)

Here, $p=\frac{5}{2}$, $q=\frac{5}{3}$, $\alpha _{1}=\frac{1}{2}$, $\alpha _{2}= \frac{3}{4}$, $\beta _{1}=\frac{2}{3}$, $\beta _{2}=\frac{3}{8}$, $\gamma _{1}= \frac{1}{3}$, $\gamma _{2}=\frac{2}{3}$, $\omega _{1}=\frac{3}{4}$, $\omega _{2}=\frac{1}{4}$, $\theta _{1}=\frac{1}{4}$, $\theta _{2}= \frac{2}{5}$, $\eta _{1}=\frac{19}{6}$, $\eta _{2}=\frac{41}{8}$, $\lambda _{1}=2$, $\lambda _{2}=3$, $T=10$, $g_{1}(t_{1})=e^{\cos t_{1}\pi }$, $g_{2}(t_{2})=e^{2 \sin t_{2}\pi }$, $\varphi _{1}(t_{1},s)= \frac{e^{-s}}{(t_{1}+10)^{3}}$, $\varphi _{2}=\frac{e^{-s}}{(t_{2}+20)^{2}}$, and $\varphi _{1}^{o}=\frac{216}{166\text{,}375 e^{1/6} }\approx 0.0011$, $\varphi _{2}^{o}= \frac{64}{23\text{,}409 e^{1/8} }\approx 0.0024$.

Let $t_{1}\in {\mathbb{N}}_{-\frac{5}{6},\frac{67}{6}}$ and $t_{2}\in {\mathbb{N}}_{-\frac{7}{8},\frac{89}{8}}$. Taking $\chi _{1}=3$, $\chi _{2}=2$ and $1=\delta <\frac{1}{2-q}=3$, we have

$$\begin{aligned}& \chi _{1}\Delta _{C}^{\frac{1}{2} } \bigl( t_{1}^{\underline{1/2 }} \bigr)\leq 3t_{1}^{\underline{2}} \leq 3t_{1}^{\underline{2}}+3t_{2}^{ \underline{2}} \leq F_{1} [t_{1},t_{2},x,y,z ], \\& \chi _{2}\Delta _{C}^{\frac{3}{4} } \bigl( t_{2}^{\underline{3/4 }} \bigr)\leq 2t_{2}^{\underline{2}} \leq 2t_{2}^{\underline{2}}+ 5t_{1}^{ \underline{2}} \leq F_{2} [t_{1},t_{2},x,y,z ]. \end{aligned}$$

Thus, (H1) holds.

For $(u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}$, we have

$$\begin{aligned}& \bigl\vert F_{1} \bigl[t_{1},t_{2}, \Delta ^{\frac{1}{3} }u_{1},\varPsi ^{ \frac{1}{4}} u_{2}, u_{2} \bigr] - F_{1} \bigl[t_{1},t_{2},\Delta ^{ \frac{1}{3} }v_{1} ,\varPsi ^{\frac{1}{4}} v_{2}, v_{2} \bigr] \bigr\vert \\& \quad \leq \frac{3t_{1}^{\underline{2}}}{50\text{,}000e^{7}} \bigl\vert \Delta ^{ \frac{1}{3} }u_{1}- \Delta ^{\frac{1}{3} }v_{1} \bigr\vert + \frac{2t_{2}^{\underline{2}}}{40\text{,}000e^{8}} \bigl\vert \varPsi ^{\frac{1}{4}} u_{2}- \varPsi ^{\frac{1}{4}} v_{2} \bigr\vert + \frac{t_{2}^{\underline{2}}}{60\text{,}000e^{6}} \vert u_{2}-v_{2} \vert , \\& \bigl\vert F_{2} \bigl[t_{1},t_{2}, \Delta ^{\frac{2}{3} }u_{2},\varPsi ^{ \frac{3}{4}} u_{1}, u_{1} \bigr] - F_{2} \bigl[t_{1},t_{2},\Delta ^{ \frac{2}{3} }v_{2},\varPsi ^{\frac{3}{4}} v_{1}, v_{1} \bigr] \bigr\vert \\& \quad \leq \frac{2t_{2}^{\underline{2}}}{60\text{,}000e^{6}} \bigl\vert \Delta ^{ \frac{2}{3} }u_{2}- \Delta ^{\frac{2}{3} }v_{2} \bigr\vert + \frac{2t_{1}^{\underline{2}}}{50\text{,}000e^{7}} \bigl\vert \varPsi ^{\frac{3}{4}} u_{1}- \varPsi ^{\frac{3}{4}} v_{1} \bigr\vert + \frac{3t_{2}^{\underline{2}}}{40\text{,}000e^{8}} \vert u_{1}-v_{1} \vert . \end{aligned}$$

Thus, (H2) holds with $L_{1}=6.211\times 10^{-6}$, $L_{2}=9.307\times 10^{-6}$, $M_{1}=4.141 \times 10^{-6}$, $M_{2}=1.889\times 10^{-6}$, $N_{1}=2.856\times 10^{-6}$, and $N_{2}=4.653\times 10^{-6}$.

Since $\frac{1}{e}\leq g_{1}(t_{1})\leq e$ and $\frac{1}{e^{2}}\leq g_{2}(t_{2})\leq e^{2}$.

Thus, (H3) holds with $g_{1}=\frac{1}{e}$, $g_{2}=\frac{1}{e^{2}}$ and $G_{1}=e$, $G_{2}=e^{2}$.

Finally, we find that

$$\begin{aligned}& \varLambda \geq 0.029,\qquad \mathcal{K}_{1}=0.286,\qquad \mathcal{K}_{2}=0.574,\qquad \varOmega _{1}=3783.803, \\& \varOmega _{2}=31\text{,}848.989,\qquad \varTheta _{1}=39\text{,}305.323, \quad \mbox{and} \quad \varTheta _{2}=55\text{,}288.515. \end{aligned}$$

Therefore, we have

$$ \varPhi = \max \{ 0.446,0.410,0.130,0.030 \}=0.446< 1. $$

Hence, by Theorem 3.1, boundary value problem (4.1)–(4.2) has a unique solution.

Example 4.2

Consider the following fractional sum boundary value problem:

$$ \begin{aligned} &\Delta ^{\frac{1}{2}}_{C}\bigl[\phi _{\frac{5}{2}}\bigl(\Delta ^{\frac{2}{3}}_{C}u_{1} \bigr)\bigr](t) \\ &\quad = H_{1} \biggl[t+\frac{1}{6},t+ \frac{1}{8},\Delta ^{\frac{1}{3}}u_{1} \biggl(t+ \frac{5}{6} \biggr) ,\varPsi ^{\frac{1}{4}}u_{2} \biggl(t+ \frac{3}{8} \biggr) , u_{2} \biggl( t+ \frac{1}{8} \biggr) \biggr], \\ &\Delta ^{\frac{3}{4}}_{C}\bigl[\phi _{\frac{5}{2}}\bigl( \Delta ^{\frac{3}{8}}_{C}u_{2}\bigr)\bigr](t) \\ &\quad = H_{2} \biggl[t+\frac{1}{6},t+\frac{1}{8}, \Delta ^{\frac{2}{3}}u_{2} \biggl(t+\frac{11}{24} \biggr) ,\varPsi ^{\frac{3}{4}}u_{1} \biggl(t+ \frac{11}{12} \biggr) , u_{1} \biggl( t+ \frac{1}{6} \biggr) \biggr], \end{aligned} $$

(4.5)

where $t\in {\mathbb{N}}_{0,10}$, and the nonlocal fractional sum boundary conditions satisfy (4.2). Functions $H_{1}$, $H_{2}$ are determined by

$$\begin{aligned}& H_{1} \biggl[ t_{1},t_{2},\Delta ^{\frac{1}{3} }u_{1} \biggl(t_{1}- \frac{2}{3} \biggr) ,\varPsi ^{\frac{1}{4}} u_{2} \biggl( t_{2}+ \frac{1}{4} \biggr) , u_{2} ( t_{2} ) \biggr] \\& \quad = \frac{3t_{1}^{\underline{2}}}{500\text{,}000e^{7}}\sin ^{2} \biggl( \Delta ^{\frac{1}{3} }u_{1} \biggl(t_{1}- \frac{2}{3} \biggr) \biggr) \\& \qquad {}+ \frac{2t_{2}^{\underline{2}} }{400\text{,}000e^{8}} \biggl[ \cos ^{2} \biggl( \varPsi ^{\frac{1}{4}} u_{2} \biggl( t_{2}+\frac{1}{4} \biggr) \biggr) + \sin ^{2} \bigl( u_{2} ( t_{2} ) \bigr) \biggr], \\& H_{2} \biggl[ t_{1},t_{2},\Delta ^{\frac{2}{3} }u_{2} \biggl(t_{2}- \frac{1}{3} \biggr) ,\varPsi ^{\frac{3}{4}} u_{1} \biggl( t_{1}+ \frac{3}{4} \biggr) , u_{1} ( t_{1} ) \biggr] \\& \quad = \frac{2t_{2}^{\underline{2}}}{6\text{,}000\text{,}000e^{6}}\sin ^{2} \biggl( \Delta ^{\frac{2}{3} }u_{2} \biggl(t_{2}- \frac{1}{3} \biggr) \biggr) \\& \qquad {}+ \frac{t_{1}^{\underline{2}} }{5\text{,}000\text{,}000e^{7}} \biggl[ \cos ^{2} \biggl( \varPsi ^{\frac{3}{4}} u_{1} \biggl( t_{1}+\frac{3}{4} \biggr) \biggr) + \sin ^{2} \bigl( u_{1} ( t_{1} ) \bigr) \biggr], \end{aligned}$$

where $\varPsi ^{\frac{3}{4}} u_{1}$, $\varPsi ^{\frac{1}{4}} u_{2}$ are defined as (4.3) and (4.4), respectively.

Let $t_{1}\in {\mathbb{N}}_{-\frac{5}{6},\frac{67}{6}}$ and $t_{2}\in {\mathbb{N}}_{-\frac{7}{8},\frac{89}{8}}$. Using $g_{1}(t_{1},t_{2})=\frac{3t_{1}^{\underline{2}}}{500\text{,}000e^{7}}+ \frac{2t_{2}^{\underline{2}} }{400\text{,}000e^{8}}$ and $g_{2}(t_{1},t_{2})=\frac{2t_{2}^{\underline{2}}}{6\text{,}000\text{,}000e^{6}}+ \frac{t_{1}^{\underline{2}} }{5\text{,}000\text{,}000e^{7}}$, we have

$$ \mathcal{M}_{1}=0.000709 \quad \text{and}\quad \mathcal{M}_{2}=0.00272. $$

For $(u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}$, we have

$$\begin{aligned}& \bigl\vert F_{1} \bigl[t_{1},t_{2}, \Delta ^{\frac{1}{3} }u_{1},\varPsi ^{ \frac{1}{4}} u_{2}, u_{2} \bigr] - F_{1} \bigl[t_{1},t_{2},\Delta ^{ \frac{1}{3} }v_{1} ,\varPsi ^{\frac{1}{4}} v_{2}, v_{2} \bigr] \bigr\vert \\& \quad \leq \frac{3t_{1}^{\underline{2}}}{500\text{,}000e^{7}} \bigl\vert \Delta ^{ \frac{1}{3} }u_{1}- \Delta ^{\frac{1}{3} }v_{1} \bigr\vert + \frac{2t_{2}^{\underline{2}}}{40000e^{8}} \bigl[ \bigl\vert \varPsi ^{ \frac{1}{4}} u_{2}-\varPsi ^{\frac{1}{4}} v_{2} \bigr\vert + \vert u_{2}-v_{2} \vert \bigr] , \\& \bigl\vert F_{2} \bigl[t_{1},t_{2}, \Delta ^{\frac{2}{3} }u_{2},\varPsi ^{ \frac{3}{4}} u_{1}, u_{1} \bigr] - F_{2} \bigl[t_{1},t_{2},\Delta ^{ \frac{2}{3} }v_{2},\varPsi ^{\frac{3}{4}} v_{1}, v_{1} \bigr] \bigr\vert \\& \quad \leq \frac{2t_{2}^{\underline{2}}}{600\text{,}000e^{6}} \bigl\vert \Delta ^{ \frac{2}{3} }u_{2}- \Delta ^{\frac{2}{3} }v_{2} \bigr\vert + \frac{2t_{1}^{\underline{2}}}{500\text{,}000e^{7}} \bigl[ \bigl\vert \varPsi ^{ \frac{3}{4}} u_{1}-\varPsi ^{\frac{3}{4}} v_{1} \bigr\vert + \vert u_{1}-v_{1} \vert \bigr] . \end{aligned}$$

Thus, (H2) holds with $L_{1}=6.211\times 10^{-7}$, $L_{2}=9.307\times 10^{-7}$, $M_{1}=N_{1}=2.070 \times 10^{-7}$, and $M_{2}=N_{2}=9.447\times 10^{-8}$.

From Example 4.1, we get $\varLambda \geq 0.029$, $g_{1}=\frac{1}{e}$, $g_{2}=\frac{1}{e^{2}}$ and $G_{1}=e$, $G_{2}=e^{2}$.

Finally, we find that

$$\begin{aligned}& \mathcal{K}_{1}=5.385\times 10^{-7}, \qquad \mathcal{K}_{2}=8.261\times 10^{-7},\qquad \bar{\varOmega }_{1}=4993.134, \\& \bar{\varOmega }_{2}=33\text{,}202.614,\qquad \bar{\varTheta }_{1}=44\text{,}000.064\quad \text{and}\quad \bar{\varTheta }_{2}=77\text{,}432.180. \end{aligned}$$

Hence,

$$ \varUpsilon = \max \{ 0.285,0.019,0.489,0.155 \}=0.489< 1. $$

From Theorem 3.3, we can conclude that boundary value problem (4.5) and (4.2) has a unique solution.

5 Conclusions

We have proved existence and uniqueness results of the nonlocal fractional sum boundary value problem for a coupled system of fractional sum-difference equations with p-Laplacian operator (1.1)–(1.2) by using the Banach fixed point theorem. Our problem contains both Riemann–Liouville and Caputo fractional difference with five fractional differences and four fractional sums.