Theory and Modern Applications

# 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.

## 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 . Discrete fractional calculus can be applied in queuing problems, economics, logistic map, and electrical networks, see . The extension of discrete fractional calculus has helped to build up some of the basic theory in this area, see  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  and , respectively. Particularly, the boundary value problem for fractional difference equations with p-Laplacian operator was presented in . In addition, the existence results of systems of fractional boundary value problems were presented in .

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.

## 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

()

Assume that the following factorial functions are well defined:

1. (i)

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

2. (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

()

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.

1. (A1)

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

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

## 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:

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

2. (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}
3. (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:

1. (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:

1. (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). □

## 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.

## 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.

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### Acknowledgements

The first author of this research was supported by Kasetsart University. Furthermore, the last author of this research was supported by Suan Dusit University.

Not applicable.

## Funding

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-028.

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Correspondence to Thanin Sitthiwirattham.

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