Skip to main content

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 [1]. Discrete fractional calculus can be applied in queuing problems, economics, logistic map, and electrical networks, see [24]. The extension of discrete fractional calculus has helped to build up some of the basic theory in this area, see [532] 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 [3339] and [4044], respectively. Particularly, the boundary value problem for fractional difference equations with p-Laplacian operator was presented in [4547]. In addition, the existence results of systems of fractional boundary value problems were presented in [4855].

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

([5])

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

([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.

  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.

References

  1. 1.

    Goodrich, C.S., Peterson, A.C.: Discrete Fractional Calculus. Springer, New York (2015)

    Google Scholar 

  2. 2.

    Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Wu, G.C., Baleanu, D.: Chaos synchronization of the discrete fractional logistic map. Signal Process. 102, 96–99 (2014)

    Google Scholar 

  4. 4.

    Wu, G.C., Baleanu, D., Xie, H.P., Chen, F.L.: Chaos synchronization of fractional chaotic maps based on stability results. Physica A 460, 374–383 (2016)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2009)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Atici, F.M., Eloe, P.W.: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 17, 445–456 (2011)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62(3), 1602–1611 (2011)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Jia, B., Erbe, L., Peterson, A.: Two monotonicity results for nabla and delta fractional differences. Arch. Math. 104, 589–597 (2015)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Jia, B., Erbe, L., Peterson, A.: Convexity for nabla and delta fractional differences. J. Differ. Equ. Appl. 21, 360–373 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ferreira, R.A.C.: Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 19, 712–718 (2013)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Ferreira, R.A.C., Goodrich, C.S.: Positive solution for a discrete fractional periodic boundary value problem. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 19, 545–557 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Goodrich, C.S.: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61, 191–202 (2011)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Goodrich, C.S.: On a discrete fractional three-point boundary value problem. J. Differ. Equ. Appl. 18, 397–415 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Erbe, L., Goodrich, C.S., Jia, B., Peterson, A.: Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions. Adv. Differ. Equ. 2016, 43 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Chen, Y., Tang, X.: Three difference between a class of discrete fractional and integer order boundary value problems. Commun. Nonlinear Sci. 19(12), 4057–4067 (2014)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Lv, W., Feng, J.: Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces. Adv. Differ. Equ. 2014, 184 (2014)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Lv, W.: Existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator. Adv. Differ. Equ. 2012, 163 (2012)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Kang, S.G., Li, Y., Chen, H.Q.: Positive solutions to boundary value problems of fractional difference equations with nonlocal conditions. Adv. Differ. Equ. 2014, 7 (2014)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Dong, W., Xu, J., Regan, D.O.: Solutions for a fractional difference boundary value problem. Adv. Differ. Equ. 2013, 319 (2013)

    MathSciNet  Google Scholar 

  20. 20.

    Holm, M.: Sum and difference compositions in discrete fractional calculus. CUBO 13(3), 153–184 (2011)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Sitthiwirattham, T., Tariboon, J., Ntouyas, S.K.: Existence results for fractional difference equations with three-point fractional sum boundary conditions. Discrete Dyn. Nat. Soc. 2013, Article ID 104276 (2013)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Sitthiwirattham, T., Tariboon, J., Ntouyas, S.K.: Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions. Adv. Differ. Equ. 2013, 296 (2013)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Sitthiwirattham, T.: Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three-point fractional sum boundary conditions. Math. Methods Appl. Sci. 38, 2809–2815 (2015)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Chasreechai, S., Kiataramkul, C., Sitthiwirattham, T.: On nonlinear fractional sum-difference equations via fractional sum boundary conditions involving different orders. Math. Probl. Eng. 2015, Article ID 519072 (2015)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Reunsumrit, J., Sitthiwirattham, T.: Positive solutions of three-point fractional sum boundary value problem for Caputo fractional difference equations via an argument with a shift. Positivity 20(4), 861–876 (2016)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Reunsumrit, J., Sitthiwirattham, T.: On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations. Math. Methods Appl. Sci. 39(10), 2737–2751 (2016)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Soontharanon, J., Jasthitikulchai, N., Sitthiwirattham, T.: Nonlocal fractional sum boundary value problems for mixed types of Riemann–Liouville and Caputo fractional difference equations. Dyn. Syst. Appl. 25, 409–414 (2016)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Laoprasittichok, S., Sitthiwirattham, T.: On a fractional difference-sum boundary value problems for fractional difference equations involving sequential fractional differences via different orders. J. Comput. Anal. Appl. 23(6), 1097–1111 (2017)

    MathSciNet  Google Scholar 

  29. 29.

    Kaewwisetkul, B., Sitthiwirattham, T.: On nonlocal fractional sum-difference boundary value problems for Caputo fractional functional difference equations with delay. Adv. Differ. Equ. 2017, 219 (2017)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Reunsumrit, J., Sitthiwirattham, T.: A new class of four-point fractional sum boundary value problems for nonlinear sequential fractional difference equations involving shift operators. Kragujev. J. Math. 42(3), 371–387 (2018)

    MathSciNet  Google Scholar 

  31. 31.

    Chasreechai, S., Sitthiwirattham, T.: Existence results of initial value problems for hybrid fractional sum-difference equations. Discrete Dyn. Nat. Soc. 2018, Article ID 5268528 (2018)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Chasreechai, S., Sitthiwirattham, T.: On separate fractional sum-difference boundary value problems with n-point fractional sum-difference boundary conditions via arbitrary different fractional orders. Mathematics 2019(7), Article ID 471 (2019)

    Google Scholar 

  33. 33.

    Khan, H., Chen, W., Khan, A., Khan, T.S., Al-Madlal, Q.M.: Hyers–Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator. Adv. Differ. Equ. 2018, 455 (2018)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Khan, H., Tunc, C., Chen, W., Khan, A.: Existence theorems and Hyers–Ulam stability for a class of hybrid fractional differential equations with p-Laplacian operator. J. Appl. Anal. Comput. 8(4), 1211–1226 (2018)

    MathSciNet  Google Scholar 

  35. 35.

    Khan, H., Chen, W., Sun, H.: Analysis of positive solution and Hyers–Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space. Math. Methods Appl. Sci. 41(9), 3430–3440 (2018)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Khan, H., Li, Y., Chen, W., Baleanu, D., Khan, A.: Existence theorems and Hyers–Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator. Bound. Value Probl. 2018, 157 (2017)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Jafari, H., Baleanu, D., Khan, H., Khan, R.A., Khan, A.: Existence criterion for the solutions of fractional order p-Laplacian boundary value problems. Bound. Value Probl. 2015, 164 (2015)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019)

    MathSciNet  Google Scholar 

  39. 39.

    Khan, H., Jarad, F., Abdeljawad, T., Khan, A.: A singular ABC-fractional differential equation with p-Laplacian operator. Chaos Solitons Fractals 129, 56–61 (2019)

    MathSciNet  Google Scholar 

  40. 40.

    Hu, L., Shuqin, Z.: Existence results for a coupled system of fractional differential equations with p-Laplacian operator and infinite-point boundary conditions. Bound. Value Probl. 2017, 88 (2017)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, 182 (2017)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Rao, S.N.: Multiple positive solutions for a coupled system of p-Laplacian fractional order three-point boundary value problems. Rocky Mt. J. Math. 49(2), 609–626 (2019)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Liu, Y., Xie, D., Bai, C., Yang, D.: Multiple positive solutions for a coupled system of fractional multi-point BVP with p-Laplacian operator. Adv. Differ. Equ. 2017, 168 (2017)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Rao, S.N., Meshari, A.: Existence of positive solutions for systems of nonlinear fractional differential equation with p-Laplacian. Asian-Eur. J. Math. 2019, 2050089 (2019)

    Google Scholar 

  45. 45.

    Sitthiwirattham, T.: Boundary value problem for p-Laplacian Caputo fractional difference equations with fractional sum boundary conditions. Math. Methods Appl. Sci. 39(6), 1522–1534 (2016)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Lv, W.: Solvability for discrete fractional boundary value problems with a p-Laplacian operator. Discrete Dyn. Nat. Soc. 2013, Article ID 679290 (2013)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Lv, W.: Existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator. Adv. Differ. Equ. 2012, 163 (2012)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Pan, Y., Han, Z., Sun, S., Zhao, Y.: The existence of solutions to a system of discrete fractional boundary value problems. Abstr. Appl. Anal. 2012, Article ID 707631 (2012)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Goodrich, C.S.: Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 217(9), 4740–4753 (2011)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Dahal, R., Duncan, D., Goodrich, C.S.: Systems of semipositone discrete fractional boundary value problems. J. Differ. Equ. Appl. 20(3), 473–491 (2014)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Goodrich, C.S.: Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions. J. Differ. Equ. Appl. 21(5), 437–453 (2015)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Goodrich, C.S.: Coupled systems of boundary value problems with nonlocal boundary conditions. Appl. Math. Lett. 41, 17–22 (2015)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Kunnawuttipreechachan, E., Promsakon, C., Sitthiwirattham, T.: Nonlocal fractional sum boundary value problems for a coupled system of fractional sum-difference equations. Dyn. Syst. Appl. 28(1), 73–92 (2019)

    Google Scholar 

  54. 54.

    Promsakon, C., Chasreechai, S., Sitthiwirattham, T.: Positive solution to a coupled system of singular fractional difference equations with fractional sum boundary value conditions. Adv. Differ. Equ. 2019, Article ID 218 (2017)

    MATH  Google Scholar 

  55. 55.

    Soontharanon, J., Chasreechai, S., Sitthiwirattham, T.: On a coupled system of fractional difference equations with nonlocal fractional sum boundary value conditions on the discrete half-line. Mathematics 2019(7), Article ID 256 (2019)

    Google Scholar 

Download references

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.

Availability of data and materials

Not applicable.

Funding

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

Author information

Affiliations

Authors

Contributions

All authors read and approved the final manuscript.

Corresponding author

Correspondence to Thanin Sitthiwirattham.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Siricharuanun, P., Chasreechai, S. & Sitthiwirattham, T. On a coupled system of fractional sum-difference equations with p-Laplacian operator. Adv Differ Equ 2020, 361 (2020). https://doi.org/10.1186/s13662-020-02826-3

Download citation

MSC

  • 39A05
  • 39A12

Keywords

  • p-Laplacian operator
  • Existence and uniqueness
  • Coupled system of fractional difference equations
  • Boundary value problem