Skip to main content

Advertisement

Existence of positive solution to a coupled system of singular fractional difference equations via fractional sum boundary value conditions

Article metrics

  • 590 Accesses

Abstract

In this article, we study a coupled system of singular fractional difference equations with fractional sum boundary conditions. A sufficient condition of the existence of positive solutions is established by employing the upper and lower solutions of the system and using Schauder’s fixed point theorem. Finally, we provide an example to illustrate our results.

Introduction

Fractional difference calculus is a powerful tool for studying problems in many fields such as biology, mechanics, control systems, ecology, electrical networks and other areas (see [1,2,3,4,5,6,7,8,9,10] and the references therein). Particularly, this calculus can be used to study stability of discrete fractional systems [11] and impulsive fractional difference equations [12]. Recently, fractional differences have been utilized in several research works such as a study of fuzzy fractional discrete-time diffusion equation [13], and a study of an image encryption technique based on the fractional chaotic maps [14]. The study of approximating solutions of fractional equations is an important topic in this area. Recently, many researchers presented the method to find approximating solutions of some fractional integro-differential equations (see [15,16,17,18,19,20]).

Basic definitions and properties of fractional difference calculus were presented by Goodrich and Peterson [21]. In addition, there are other research works dealing with fractional difference boundary value problems which have helped to build up some of the basic theory of this area (see [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47] and references cited therein).

The boundary value problems for systems of fractional difference equations have been studied by some researchers; see [48,49,50,51,52,53] and references cited therein. For example, Pan et al. [48] proposed the system of fractional difference equations

$$\begin{aligned} \textstyle\begin{cases} -\Delta ^{\nu }y_{1}(t)=f (y_{1}(t+\nu _{1}),y_{2}(t+\mu -1) ), \\ -\Delta ^{\mu }y_{2}(t)=g (y_{1}(t+\nu _{1}),y_{2}(t+\mu -1) ), \end{cases}\displaystyle \end{aligned}$$
(1.1)

for \(t \in \mathbb{N}_{0,b+1}:=\{0,1,2,\dots ,b+1\}\) where \(b \in \mathbb{N}_{0}\), with the difference boundary conditions

$$\begin{aligned} \textstyle\begin{cases} y_{1}(\nu -2)=\Delta y_{1}(\nu +b)=0, \\ y_{2}(\mu -2)=\Delta y_{2}(\mu +b)=0, \end{cases}\displaystyle \end{aligned}$$
(1.2)

where \(1<\mu ,\nu \leq 2\), \(0<\beta \leq 1\), and \(f,g:\mathbb{R} \rightarrow \mathcal{R}\) are continuous functions.

Goodrich [51] studied the coupled system of fractional difference equations

$$\begin{aligned} \textstyle\begin{cases} -\Delta ^{-\nu }x(t)=\lambda _{1}f (t+\nu -1,y(t+\mu -1) ), \\ -\Delta ^{-\mu }y(t)=\lambda _{2}g (t+\mu -1,y(t+\nu -1) ), \end{cases}\displaystyle \end{aligned}$$
(1.3)

for \(t \in \mathbb{N}_{0,b+1}\), with the nonlinearities satisfying no growth conditions

$$\begin{aligned} \textstyle\begin{cases} x(\nu -2)=H_{1} ( \sum_{i=1}^{n}a_{i}y(\xi _{i}) ) , &x(\nu +b+1)=0, \\ y(\mu -2)=H_{2} ( \sum_{j=1}^{m}b_{i}x(\zeta _{i}) ) , &x(\mu +b+1)=0, \end{cases}\displaystyle \end{aligned}$$
(1.4)

where \(1<\nu \leq 2\), \(1<\mu \leq 2\), \(\lambda _{1},\lambda _{2}>0\), and \(H_{1},H_{2}\) are continuous functions.

In this paper, we aim to study the coupled system of singular fractional difference equations

$$\begin{aligned} \textstyle\begin{cases} -\Delta ^{\alpha _{1}} u_{1}(t)=F_{1} (t+\alpha _{1}-1,t+\alpha _{2}-1, \Delta ^{\beta _{1}}u_{1}(t+\alpha _{1}-\beta _{1}),u_{2}(t+\alpha _{2}-1) ), \\ -\Delta ^{\alpha _{2}} u_{2}(t)=F_{2} (t+\alpha _{1}-1,t+\alpha _{2}-1,u _{1}(t+\alpha _{1}-1),\Delta ^{\beta _{2}}u_{2}(t+\alpha _{2}-\beta _{2}) ), \end{cases}\displaystyle \end{aligned}$$
(1.5)

with fractional sum boundary conditions

$$\begin{aligned} \textstyle\begin{cases} u_{1}(\alpha _{1}-2)=0,\qquad u_{1}(T+\alpha _{1})=\lambda _{2}\Delta ^{-\theta _{2}} g_{2}(T+\alpha _{2}+\theta _{2})u_{2}(T+\alpha _{2}+\theta _{2}), \\ u_{2}(\alpha _{2}-2)=0,\qquad u_{2}(T+\alpha _{2})=\lambda _{1}\Delta ^{-\theta _{1}} g_{1}(T+\alpha _{1}+\theta _{1})u_{1}(T+\alpha _{1}+\theta _{1}), \end{cases}\displaystyle \end{aligned}$$
(1.6)

where \(t\in \mathbb{N}_{0,T}:=\{0,1,\dots ,T\}\), \(0<\lambda _{i}<\frac{ \varGamma (\alpha _{i})}{(T+\alpha _{i})^{\underline{\alpha _{i}-1}} \sum_{s=0}^{T}(T+\theta _{i}+1-\sigma (s))^{\underline{\theta _{i}-1}}g_{i}(s+ \alpha _{1}-1)(s+\alpha _{i}-1)^{\underline{\alpha _{i}-1}}}\), \(\alpha _{i}\in (1, 2], \beta _{i},\theta _{i}\in (0,1]\), \(g_{i}\in C (\mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}, \mathbb{R}^{+} ) \) are given functions, \(F_{i}:\mathbb{N}_{\alpha _{1}-1,T+\alpha _{1}-1} \times \mathbb{N}_{\alpha _{2}-1,T+\alpha _{1}}\times (0,+\infty ) \times (0,+\infty ) \rightarrow [0,+\infty )\) are are continuous and may be singular at \(u_{i}=0\) and \(t=\alpha _{i}-2,T+\alpha _{i}\) where \(i=1,2\).

This paper is organized as follows. In the next section, we present some definitions and basic lemmas. In Sect. 3, we prove the existence of solutions of the boundary value problem (1.5)–(1.6) by employing the upper and lower solutions of the system and Schauder’s fixed point theorem. An example and application of our results are presented in the last section.

Preliminaries

As the following, we provide some notations, definitions, and lemmas which are used in the main results.

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 \(t^{\underline{\alpha }}=0\).

Theorem 2.1

([22])

Assume the following factorial functions are well defined. If \(t\leq r\), then \(t^{\underline{\alpha }}\leq r^{\underline{\alpha }}\) for any \(\alpha >0\).

Definition 2.2

For \(\alpha >0\) and f 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

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

where \(t \in \mathbb{N}_{a+N-\alpha }\) and \(N \in \mathbb{N}\) is chosen so that \(0\leq {N-1}<\alpha \leq N\).

Theorem 2.2

([22])

Let \(0\leq N-1<\alpha \leq N\). Then

$$ \Delta ^{-\alpha }\Delta ^{\alpha }y(t)=y(t)+C_{1}t^{\underline{\alpha -1}} +C_{2}t^{\underline{\alpha -2}}+\cdots +C_{N}t^{\underline{ \alpha -N}}, $$

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

We next propose a lemma dealing with a solution of a linear variant of the boundary value problem (1.5).

Lemma 2.1

For \(i,j\in \{1,2\}\) and \(i\neq j\), let \(0<\varLambda <1, \mathcal{P}(h _{1},h_{2}),\mathcal{Q}(h_{1},h_{2})\geq \varLambda \), \(0<\lambda _{i}<\frac{ \varGamma (\alpha _{i})}{(T+\alpha _{i})^{\underline{\alpha _{i}-1}} \sum_{s=0}^{T}(T+\theta _{i}+1-\sigma (s))^{\underline{\theta _{i}-1}}g_{i}(s+ \alpha _{1}-1)(s+\alpha _{i}-1)^{\underline{\alpha _{i}-1}}}\), \(\alpha _{i}\in (1, 2], \theta _{i}\in (0, 1]\) be given constants, \(h_{i}\in C (\mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}, \mathbb{R} )\), \(g_{i}\in C (\mathbb{N}_{\alpha _{i}-2,T+ \alpha _{i}}, \mathbb{R}^{+} )\) given functions, and let \(\phi _{i}(u_{1},u_{2})\) be given functionals. The problem

$$\begin{aligned} &{-}\Delta ^{\alpha _{i}} u_{i}(t)=h_{i}(t+\alpha _{i}-1),\quad t\in \mathbb{N}_{0,T}, \end{aligned}$$
(2.1)
$$\begin{aligned} &u_{i}(\alpha _{i}-2)=0, \end{aligned}$$
(2.2)
$$\begin{aligned} &u_{i}(T+\alpha _{i})=\lambda _{j}\Delta ^{-\theta _{j}}g_{j}(T+\alpha _{j}+\theta _{j})u_{j}(T+\alpha _{j}+\theta _{j}) \end{aligned}$$
(2.3)

has the unique solution

$$\begin{aligned} u_{1}(t) = {}& t^{\underline{\alpha _{1}-1}} \Biggl\{ \frac{\lambda _{1}}{ \varLambda \varGamma (\theta _{1})}\sum _{s=0}^{T+1}\bigl(T+\theta _{1}+1- \sigma (s)\bigr)^{\underline{ \theta _{1}-1}} g_{1}(s+\alpha _{1}-1) (s+ \alpha _{1}-1)^{\underline{ \alpha _{1}-1}} {\mathcal{P}(h_{1},h_{2})} \\ &{}+\frac{\lambda _{2}}{\varLambda \varGamma (\theta _{2})}\sum_{s=0}^{T+1} \bigl(T+ \theta _{2}+1-\sigma (s)\bigr)^{\underline{\theta _{2}-1}} g_{2}(s+\alpha _{2}-1) (s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} {\mathcal{Q}(h _{1},h_{2})} \Biggr\} \\ &{}-\frac{1}{\varGamma (\alpha _{1})}\sum_{s=0}^{t-\alpha _{1}} \bigl(t-\sigma (s)\bigr)^{\underline{ \alpha _{1}-1}} h_{1}(s+\alpha _{1}-1),\quad t\in \mathbb{N}_{\alpha _{1}-2,T+\alpha _{1}}, \end{aligned}$$
(2.4)
$$\begin{aligned} u_{2}(t) = {}& t^{\underline{\alpha _{2}-1}} \biggl\{ \frac{(T+\alpha _{2})^{\underline{\alpha _{2}-1}}}{\varLambda } { \mathcal{P}(h_{1},h_{2})}+\frac{(T+ \alpha _{1})^{\underline{\alpha _{1}-1}}}{\varLambda } { \mathcal{Q}(h_{1},h _{2})} \biggr\} \\ &{}-\frac{1}{\varGamma (\alpha _{2})}\sum_{s=0}^{t-\alpha _{2}} \bigl(t-\sigma (s)\bigr)^{\underline{ \alpha _{2}-1}} h_{2}(s+\alpha _{2}-1),\quad t\in \mathbb{N}_{\alpha _{2}-2,T+\alpha _{2}}, \end{aligned}$$
(2.5)

where

$$\begin{aligned} &\varLambda = \frac{\lambda _{2}(T+\alpha _{2})^{ \underline{\alpha _{2}-1}}}{\varGamma (\alpha _{2})}\sum_{s=0}^{T+1} \bigl(T+ \theta _{1}+1-\sigma (s)\bigr)^{\underline{\theta _{1}-1}} g_{1}(s+\alpha _{1}-1) (s+\alpha _{1}-1)^{\underline{\alpha _{1}-1}} \\ &\phantom{\varLambda =}{}-\frac{\lambda _{1}(T+\alpha _{1})^{\underline{\alpha _{1}-1}}}{ \varGamma (\alpha _{1})}\sum_{s=0}^{T+1} \bigl(T+\theta _{2}+1-\sigma (s)\bigr)^{\underline{ \theta _{2}-1}} g_{2}(s+\alpha _{2}-1) (s+\alpha _{2}-1)^{\underline{ \alpha _{2}-1}}, \end{aligned}$$
(2.6)
$$\begin{aligned} &{\mathcal{P}(h_{1},h_{2})} = \frac{1}{\varGamma (\alpha _{1})}\sum _{s=0} ^{T}\bigl(T+\alpha _{1}- \sigma (s)\bigr)^{\underline{\alpha _{1}-1}} h_{1}(s+ \alpha _{1}-1)- \frac{\lambda _{2}}{\varGamma (\alpha _{2})\varGamma (\theta _{2})} \\ &\phantom{{\mathcal{P}(h_{1},h_{2})} =}{}\times\sum_{s=0}^{T}\sum _{\xi =s}^{T}\bigl(T+\theta _{2}-\sigma (\xi )\bigr)^{\underline{ \theta _{2}-1}}\bigl(\xi +\alpha _{2}-\sigma (s) \bigr)^{\underline{\alpha _{2}-1}} \\ &\phantom{{\mathcal{P}(h_{1},h_{2})} =}{}\times g_{2}(s+\alpha _{2}-1)h_{2}(s+ \alpha _{2}-1), \end{aligned}$$
(2.7)
$$\begin{aligned} &{\mathcal{Q}(h_{1},h_{2})} = -\frac{1}{\varGamma (\alpha _{2})}\sum _{s=0} ^{T}\bigl(T+\alpha _{2}- \sigma (s)\bigr)^{\underline{\alpha _{2}-1}} h_{2}(s+ \alpha _{2}-1)+ \frac{\lambda _{1}}{\varGamma (\alpha _{1})\varGamma (\theta _{1})} \\ & \phantom{{\mathcal{Q}(h_{1},h_{2})} =}{}\times\sum_{s=0}^{T}\sum _{\xi =s}^{T}\bigl(T+\theta _{1}-\sigma (\xi )\bigr)^{\underline{ \theta _{1}-1}}\bigl(\xi +\alpha _{1}-\sigma (s) \bigr)^{\underline{\alpha _{1}-1}} \\ & \phantom{{\mathcal{Q}(h_{1},h_{2})} =}{}\times g_{1}(s+\alpha _{1}-1)h_{1}(s+ \alpha _{1}-1). \end{aligned}$$
(2.8)

Proof

For \(i,j\in \{1,2\}\) where \(i\neq j\), using Lemma 2.2 and the fractional sum of order \(\alpha \in (1,2]\) for (2.1), we obtain

$$\begin{aligned} u_{i}(t)=C_{1i}t^{\underline{\alpha _{i}-1}}+C_{2i}t^{\underline{\alpha _{i}-2}} -\frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t-\alpha _{i}}\bigl(t- \sigma (s)\bigr)^{\underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1), \end{aligned}$$
(2.9)

for \(t\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\). Using the boundary condition (2.2), this implies that

$$\begin{aligned} C_{2i}=0. \end{aligned}$$
(2.10)

Then, we have

$$\begin{aligned} u_{i}(t)=C_{1i}t^{\underline{\alpha _{i}-1}} - \frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t-\alpha _{i}}\bigl(t- \sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1). \end{aligned}$$
(2.11)

Taking the fractional sum of order \(0<\theta _{i}\leq 1\) for (2.11), we obtain

$$\begin{aligned} &\Delta ^{-\theta _{i}}u(t) \\ &\quad=\frac{C_{1i}}{\varGamma (\theta _{i})}\sum_{s=\alpha _{i}-1}^{t-\theta _{i}} \bigl(t-\sigma (s)\bigr)^{\underline{\theta _{i}-1}} g_{i}(s) s^{\underline{ \alpha _{i}-1}} \\ &\qquad{}-\frac{1}{\varGamma (\theta _{i})\varGamma (\alpha _{i})}\sum_{\xi =\alpha _{i}}^{t-\theta _{i}}\sum _{s=0}^{\xi -\alpha _{i}} \bigl(t-\sigma (\xi ) \bigr)^{\underline{ \theta _{i}-1}}\bigl(\xi -\sigma (s)\bigr)^{\underline{\alpha _{i}-1}} \\ &\qquad{}\times g_{i}(s+ \alpha _{i}-1)h_{i}(s+\alpha _{i}-1), \end{aligned}$$
(2.12)

for \(t\in \mathbb{N}_{\alpha _{i}+\theta _{i}-2,T+\alpha _{i}+\theta _{i}}\). From the boundary condition (2.3), we find that

$$\begin{aligned} & C_{11}(T+\alpha _{1})^{\underline{\alpha _{1}-1}}-\frac{1}{\varGamma ( \alpha _{1})} \sum_{s=0}^{T}\bigl(T+\alpha _{1}-\sigma (s)\bigr)^{\underline{\alpha _{1}-1}} h_{1}(s+\alpha _{1}-1) \\ &\quad = \frac{\lambda _{2}C_{12}}{\varGamma (\theta _{2})}\sum_{s=\alpha _{2}-1} ^{T+\alpha _{2}} \bigl(T+\alpha _{2}+\theta _{2}-\sigma (s) \bigr)^{\underline{\theta _{2}-1}} g_{2}(s) s^{\underline{\alpha _{2}-1}}-\frac{\lambda _{2}}{ \varGamma (\alpha _{2})\varGamma (\theta _{2})} \\ &\qquad{} \times\sum_{\xi =\alpha _{2}}^{T+\alpha _{2}}\sum _{s=0}^{\xi -\alpha _{2}}\bigl(T+ \alpha _{2}+\theta _{2}-\sigma (\xi )\bigr)^{\underline{\theta _{2}-1}}\bigl( \xi -\sigma (s) \bigr)^{\underline{\alpha _{2}-1}} \\ &\qquad{}\times g_{2}(s+\alpha _{2}-1)h _{2}(s+\alpha _{2}-1), \end{aligned}$$
(2.13)

and

$$\begin{aligned} & C_{12}(T+\alpha _{2})^{\underline{\alpha _{2}-1}}-\frac{1}{\varGamma ( \alpha _{2})} \sum_{s=0}^{T}\bigl(T+\alpha _{2}-\sigma (s)\bigr)^{\underline{\alpha _{2}-1}} h_{2}(s+\alpha _{2}-1) \\ &\quad = \frac{\lambda _{1}C_{11}}{\varGamma (\theta _{1})}\sum_{s=\alpha _{1}-1} ^{T+\alpha _{1}} \bigl(T+\alpha _{1}+\theta _{1}-\sigma (s) \bigr)^{\underline{\theta _{1}-1}} g_{1}(s) s^{\underline{\alpha _{1}-1}}-\frac{\lambda _{1}}{ \varGamma (\alpha _{1})\varGamma (\theta _{1})} \\ &\qquad{} \times\sum_{\xi =\alpha _{1}}^{T+\alpha _{1}}\sum _{s=0}^{\xi -\alpha _{1}}\bigl(T+ \alpha _{1}+\theta _{1}-\sigma (\xi )\bigr)^{\underline{\theta _{1}-1}}\bigl( \xi -\sigma (s) \bigr)^{\underline{\alpha _{1}-1}} \\ &\qquad{}\times g_{1}(s+\alpha _{1}-1)h _{1}(s+\alpha _{1}-1). \end{aligned}$$
(2.14)

After solving the system of equations (2.13) and (2.14), we have

$$\begin{aligned} C_{11} ={} & \frac{\lambda _{1}}{\varLambda \varGamma (\theta _{1})}\sum_{s=0} ^{T+1}\bigl(T+\theta _{1}+1-\sigma (s)\bigr)^{\underline{\theta _{1}-1}} g_{1}(s+ \alpha _{1}-1) (s+\alpha _{1}-1)^{\underline{\alpha _{1}-1}} {\mathcal{P}(h _{1},h_{2})} \\ &{}+\frac{\lambda _{2}}{\varLambda \varGamma (\theta _{2})}\sum_{s=0}^{T+1} \bigl(T+ \theta _{2}+1-\sigma (s)\bigr)^{\underline{\theta _{2}-1}} \\ &{}\times g_{2}(s+\alpha _{2}-1) (s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} {\mathcal{Q}(h _{1},h_{2})}, \end{aligned}$$
(2.15)

and

$$\begin{aligned} C_{12}= \frac{(T+\alpha _{2})^{\underline{\alpha _{2}-1}}}{\varLambda } {\mathcal{P}(h_{1},h_{2})}+ \frac{(T+\alpha _{1})^{\underline{\alpha _{1}-1}}}{\varLambda } {\mathcal{Q}(h_{1},h_{2})}, \end{aligned}$$
(2.16)

where \(\varLambda ,{\mathcal{P}(h_{1},h_{2})}\) and \({\mathcal{Q}(h_{1},h _{2})}\) are defined in (2.6)–(2.8), respectively.

Finally, substituting \(C_{11}\) and \(C_{12}\) into (2.11), we obtain (2.4) and (2.5). The proof of this lemma is complete. □

Corollary 2.1

Problem (2.1)(2.3) has the unique solution which is of the from

$$\begin{aligned} u_{i}(t_{i})={} &\sum _{s=0}^{T} G_{i1}(t_{i},s) g_{1}(s+\alpha _{1}-1)h _{1}(s+\alpha _{1}-1) \\ &{}- \sum_{s=0}^{T} G_{i2}(t_{i},s) g_{2}(s+\alpha _{2}-1)h_{2}(s+\alpha _{2}-1) \end{aligned}$$
(2.17)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\), where

$$\begin{aligned} &G_{11}(t_{1},s)=\frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda } \sum _{\xi =0}^{T} \mathcal{H}_{11}(\xi -\alpha _{1}-1,s)+\mathcal{K} _{1}(t_{1},s), \end{aligned}$$
(2.18)
$$\begin{aligned} &G_{12}(t_{1},s)=\frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda } \sum _{\xi =0}^{T} \mathcal{H}_{12}(\xi -\alpha _{2}-1,s), \end{aligned}$$
(2.19)
$$\begin{aligned} &G_{21}(t_{2},s)=\frac{t_{2}^{\underline{\alpha _{2}-1}}}{\varLambda } \sum _{\xi =0}^{T} \mathcal{H}_{21}(\xi -\alpha _{1}-1,s), \end{aligned}$$
(2.20)
$$\begin{aligned} &G_{22}(t_{2},s)=\frac{t_{2}^{\underline{\alpha _{2}-1}}}{\varLambda } \sum _{\xi =0}^{T} \mathcal{H}_{22}(\xi -\alpha _{2}-1,s)+\mathcal{K} _{2}(t_{2},s), \end{aligned}$$
(2.21)

with

$$\begin{aligned} &\mathcal{K}_{1}(t_{1},s)= \frac{1}{\varGamma (\alpha _{1})} \textstyle\begin{cases} ( \frac{\lambda _{1}\mathcal{A}_{1}(T+\alpha _{1}-\sigma (s))^{\underline{ \alpha _{1}-1}}}{\varLambda \varGamma (\theta _{1})} ) t_{1}^{\underline{ \alpha _{1}-1}}-(t_{1}-\sigma (s))^{\underline{\alpha _{1}-1}}, & s \in \mathbb{N}_{0,t_{1}-\alpha _{1}}, \\ ( \frac{\lambda _{1}\mathcal{A}_{1}(T+\alpha _{1}-\sigma (s))^{\underline{ \alpha _{1}-1}}}{\varLambda \varGamma (\theta _{1})} ) t_{1}^{\underline{ \alpha _{1}-1}}, & s\in \mathbb{N}_{t_{1}-\alpha _{1}+1,T}, \end{cases}\displaystyle \end{aligned}$$
(2.22)
$$\begin{aligned} &\mathcal{K}_{2}(t_{2},s)=\frac{1}{\varGamma (\alpha _{2})} \textstyle\begin{cases} [(T+\alpha _{1})^{\underline{\alpha _{1}-1}}(T+\alpha _{2}-\sigma (s))^{\underline{ \alpha _{2}-1}} ]t_{2}^{\underline{\alpha _{2}-1}} & \\ \quad{}+(t_{2}-\sigma (s))^{\underline{\alpha _{2}-1}}, &s\in \mathbb{N}_{0,t_{2}-\alpha _{2}}, \\ [(T+\alpha _{1})^{\underline{\alpha _{1}-1}}(T+\alpha _{2}-\sigma (s))^{\underline{ \alpha _{2}-1}} ] t_{2}^{\underline{\alpha _{2}-1}} , & s \in \mathbb{N}_{t_{2}-\alpha _{2}+1,T}, \end{cases}\displaystyle \end{aligned}$$
(2.23)
$$\begin{aligned} &\mathcal{H}_{11} (\xi +\alpha _{1}-1,s) \\ &\quad = \frac{\lambda _{1}}{\varGamma (\alpha _{1})\varGamma (\theta _{1})} \textstyle\begin{cases} (T+\theta _{1}-\sigma (\xi ))^{\underline{\theta _{1}-1}} [ (\varLambda +1)(T+\alpha _{1}-\sigma (s))^{\underline{\alpha _{1}-1}}\\ \quad{}\times (\xi -\alpha _{1}+1)^{\underline{\alpha _{1}-1}} + \frac{\lambda _{2}\mathcal{A}_{2}}{\varGamma (\theta _{2})} (\xi +\alpha _{1}-\sigma (s))^{\underline{\alpha _{1}-1}} ],\\ \quad s\in \mathbb{N}_{0,\xi }, \\ (T+\theta _{1}-\sigma (\xi ))^{\underline{\theta _{1}-1}}(\varLambda +1)(T+ \alpha _{1}-\sigma (s))^{\underline{\alpha _{1}-1}}\\ \quad{}\times (\xi -\alpha _{1}+1)^{\underline{ \alpha _{1}-1}}, \\ \quad s\in \mathbb{N}_{\xi +1,T}, \end{cases}\displaystyle \end{aligned}$$
(2.24)
$$\begin{aligned} &\mathcal{H}_{12}(\xi +\alpha _{2}-1,s) \\ &\quad =\frac{\lambda _{2}}{\varGamma (\alpha _{2})\varGamma (\theta _{2})} \textstyle\begin{cases} (T+\theta _{2}-\sigma (\xi ))^{\underline{\theta _{2}-1}} [ (T+ \alpha _{2}-\sigma (s))^{\underline{\alpha _{2}-1}}(\xi -\alpha _{2}+1)^{\underline{ \alpha _{2}-1}} \\ \quad{}+ \frac{\lambda _{1}\mathcal{A}_{1}}{\varLambda \varGamma (\theta _{1})} ( \xi +\alpha _{2}-\sigma (s))^{\underline{\alpha _{2}-1}} ], \\\quad s \in \mathbb{N}_{0,\xi } \\ (T+\theta _{2}-\sigma (\xi ))^{\underline{\theta _{2}-1}} [ (T+ \alpha _{2}-\sigma (s))^{\underline{\alpha _{2}-1}}(\xi -\alpha _{2}+1)^{\underline{ \alpha _{2}-1}},\\ \quad s\in \mathbb{N}_{\xi +1,T} \end{cases}\displaystyle \end{aligned}$$
(2.25)
$$\begin{aligned} &\mathcal{H}_{21}(\xi +\alpha _{1}-1,s) \\ &\quad =\frac{1}{\varGamma (\alpha _{1})} \textstyle\begin{cases} (T+\alpha _{2})^{\underline{\alpha _{2}-1}} (T+\alpha _{1}-\sigma (s))^{\underline{ \alpha _{1}-1}} \\ \quad{}+ \frac{\lambda _{1} (T+\alpha _{1})^{\underline{\alpha _{1}-1}}}{ \varGamma (\theta _{1})} (T+\theta _{1}-\sigma (\xi ))^{\underline{\theta _{1}-1}}(\xi +\alpha _{1}-\sigma (s))^{\underline{\alpha _{1}-1}} , \\ \quad s \in \mathbb{N}_{0,\xi }, \\ (T+\alpha _{2})^{\underline{\alpha _{2}-1}} (T+\alpha _{1}-\sigma (s))^{\underline{ \alpha _{1}-1}}, \\ \quad s\in \mathbb{N}_{\xi +1,T}, \end{cases}\displaystyle \end{aligned}$$
(2.26)
$$\begin{aligned} &\mathcal{H}_{22}(\xi +\alpha _{2}-1,s) \\ &\quad =\frac{1}{\varGamma (\alpha _{2})} \textstyle\begin{cases} (1-\varLambda )(T+\alpha _{1})^{\underline{\alpha _{1}-1}} (T+\alpha _{2}- \sigma (s))^{\underline{\alpha _{2}-1}} \\ \quad{}+ \frac{\lambda _{2} (T+\alpha _{2})^{\underline{\alpha _{2}-1}}}{ \varGamma (\theta _{2})} (T+\theta _{2}-\sigma (\xi ))^{\underline{\theta _{2}-1}}(\xi +\alpha _{2}-\sigma (s))^{\underline{\alpha _{2}-1}} , \\ \quad s \in \mathbb{N}_{0,\xi }, \\ (1-\varLambda ) (T+\alpha _{1})^{\underline{\alpha _{1}-1}} (T+\alpha _{2}- \sigma (s))^{\underline{\alpha _{2}-1}}, \\ \quad s\in \mathbb{N} _{\xi +1,T}, \end{cases}\displaystyle \end{aligned}$$
(2.27)

and

$$\begin{aligned} \mathcal{A}_{i}=\bigl(T+\theta _{i}+1-\sigma (s) \bigr)^{\underline{\theta _{i}-1}} g_{i}(s+\alpha _{i}-1) (s+\alpha _{i}-1)^{\underline{\alpha _{i}-1}}. \end{aligned}$$
(2.28)

Lemma 2.2

For \(i,j\in \{1,2\}\), \(i\neq j\) and letting \(0<\varLambda <1\), \(\mathcal{P}(h_{1},h_{2}),\mathcal{Q}(h_{1},h_{2})\geq \varLambda \), \(0<\lambda _{i}<\frac{\varGamma (\alpha _{i})}{(T+\alpha _{i})^{\underline{ \alpha _{i}-1}} \sum_{s=0}^{T}(T+\theta _{i}+1-\sigma (s))^{\underline{ \theta _{i}-1}}g_{i}(s+\alpha _{1}-1)(s+\alpha _{i}-1)^{\underline{\alpha _{i}-1}}}\), the Green’s functions are defined by (2.18)(2.20) and satisfy:

\((X1)\) :

\(G_{i1}(t_{i}),G_{i2}(t_{i}) > 0\) for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\);

\((X2)\) :

There exist two constants \(\omega _{i1},\omega _{i2}\) such that for all \((t_{i},s)\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\times \mathbb{N}_{0,T}\),

$$\begin{aligned} &\frac{t_{i}^{\underline{\alpha _{i}-1}}}{\varLambda }\sum_{\xi =0}^{T} \mathcal{H}_{ii}(\xi +\alpha _{i}-1,s) \leq G_{ii}(t_{i},s)\leq \omega _{ii} t_{i}^{\underline{\alpha _{i}-1}}, \end{aligned}$$
(2.29)
$$\begin{aligned} &\frac{t_{1}^{\underline{\alpha _{1}-1}}\lambda _{2}\varGamma (\alpha _{2})}{ \varLambda } \leq G_{12}(t_{1},s)\leq \omega _{12} t_{1}^{\underline{ \alpha _{1}-1}}, \end{aligned}$$
(2.30)
$$\begin{aligned} &\frac{t_{2}^{\underline{\alpha _{2}-1}}(T+\alpha _{2})^{\underline{ \alpha _{2}-1}}}{\varLambda } \leq G_{21}(t_{2},s)\leq \omega _{21} t _{2}^{\underline{\alpha _{2}-1}}; \end{aligned}$$
(2.31)
\((X3)\) :

\(u_{i}(t_{i}) \geq 0\) for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\).

Proof

\((X1)\) is obvious. Here we only prove \((X2)\)\((X3)\).

Since \(0<\varLambda <1\) and from the fact that \(\mathcal{H}_{i1}(\xi + \alpha _{1}-1,s),\mathcal{H}_{i2}(\xi +\alpha _{2}-1,s) \geq 0\) for all \((t_{i},s)\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{1}}\times \mathbb{N} _{0,T}\), we have

$$\begin{aligned} &\frac{t_{i}^{\underline{\alpha _{i}-1}}}{\varLambda }\sum_{\xi =0}^{T} \mathcal{H}_{ii}(\xi +\alpha _{i}-1,s) \leq G_{ii}(t_{i},s), \end{aligned}$$
(2.32)
$$\begin{aligned} &\frac{t_{1}^{\underline{\alpha _{1}-1}}\lambda _{2}\varGamma (\alpha _{2})}{ \varLambda }\leq G_{12}(t_{1},s), \end{aligned}$$
(2.33)
$$\begin{aligned} &\frac{t_{2}^{\underline{\alpha _{2}-1}}(T+\alpha _{2})^{\underline{ \alpha _{2}-1}}}{\varLambda }\leq G_{21}(t_{2},s). \end{aligned}$$
(2.34)

By the definition of \(\mathcal{K}_{i}(t_{i},s)\), we find that

$$\begin{aligned} &\mathcal{K}_{1}(t_{1},s)\leq \frac{\lambda _{1}t_{1}^{\underline{ \alpha _{1}-1}} }{\varGamma (\alpha _{1})\varGamma (\theta _{1})} \bigl[ (T+ \alpha _{1}-1 )^{\underline{\alpha _{1}-1}} \bigr]^{2} (T+ \theta _{1}-1 )^{\underline{\theta _{1}-1}} \quad \text{and} \end{aligned}$$
(2.35)
$$\begin{aligned} &\mathcal{K}_{2}(t_{2},s)\leq \frac{t_{2}^{\underline{ \alpha _{2}-1}} }{\varGamma (\alpha _{1})} (T+\alpha _{2}-1 ) ^{\underline{\alpha _{2}-1}} (T+\alpha _{1} )^{\underline{ \alpha _{1}-1}}. \end{aligned}$$
(2.36)

Letting

$$\begin{aligned} \omega _{11} ={}&\frac{\lambda _{1} }{\varGamma (\alpha _{1})\varGamma (\theta _{1})} \biggl[ \bigl[ (T+ \alpha _{1}-1 ) ^{\underline{\alpha _{1}-1}} \bigr]^{2} (T+\theta _{1}-1 ) ^{\underline{\theta _{1}-1}} \\ &{}+\max_{0\leq \xi \leq T}\frac{\mathcal{H}_{11}(\xi +\alpha _{1}-1,s) }{\varLambda } \biggr], \end{aligned}$$
(2.37)
$$\begin{aligned} \omega _{12} ={}&\frac{\lambda _{2} }{\varGamma (\alpha _{2})\varGamma (\theta _{2})} \max_{0\leq \xi \leq T} \frac{\mathcal{H}_{12}(\xi +\alpha _{2}-1,s)}{ \varLambda }, \end{aligned}$$
(2.38)
$$\begin{aligned} \omega _{22} ={}&\frac{1}{\varGamma (\alpha _{2})} \biggl[1+ (T+\alpha _{1} )^{\underline{\alpha _{1}-1}} (T+\alpha _{2}-1 ) ^{\underline{\alpha _{2}-1}} \\ &{}+ \max_{0\leq \xi \leq T} \frac{\mathcal{H}_{22}(\xi +\alpha _{2}-1,s)}{ \varLambda } \biggr], \end{aligned}$$
(2.39)
$$\begin{aligned} \omega _{21} ={}&\frac{1}{\varGamma (\alpha _{1})} \max_{0\leq \xi \leq T} \frac{ \mathcal{H}_{21}(\xi +\alpha _{1}-1,s)}{\varLambda }, \end{aligned}$$
(2.40)

we obtain, for all \((t_{i},s)\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{1}}\times \mathbb{N}_{0,T}\),

$$\begin{aligned} G_{i1}(t_{i},s) \leq \omega _{i1} t_{i}^{\underline{\alpha _{i}-1}} \quad \text{and}\quad G_{i2}(t_{i},s) \leq \omega _{i2} t_{i}^{\underline{ \alpha _{i}-1}}. \end{aligned}$$
(2.41)

Consequently, by (2.32)–(2.34) and (2.41), this implies that \((X2)\) holds.

Next, we claim that \((X3)\) holds. By (2.4)–(2.5) with the conditions \(0<\varLambda <1\), \(\mathcal{P}(h_{1},h_{2}), \mathcal{Q}(h _{1},h_{2})\geq \varLambda \) and \(0<\lambda _{i}<\frac{\varGamma (\alpha _{i})}{(T+\alpha _{i})^{\underline{\alpha _{i}-1}} \sum_{s=0}^{T}(T+ \theta _{i}+1-\sigma (s))^{\underline{\theta _{i}-1}}g_{i}(s+\alpha _{1}-1)(s+ \alpha _{i}-1)^{\underline{\alpha _{i}-1}}}\), we have

$$\begin{aligned} u_{1}(t_{1})= {}&\sum_{s=0}^{T} G_{11}(t_{1},s)g_{1}(s+\alpha _{1}-1) h _{1}(s+\alpha _{1}-1) \\ &{}-\sum_{s=0}^{T} G_{12}(t_{2},s)g_{2}(s+ \alpha _{2}-1) h_{2}(s+\alpha _{2}-1) \\ = {}&\frac{1}{\varGamma (\alpha _{1})} \Biggl[\frac{2\lambda _{1}t_{1}^{\underline{ \alpha _{1}-1}}}{\varGamma (\theta _{1})}\sum _{s=0}^{T}\sum_{\xi =0}^{T} \bigl(T+ \alpha _{1}-\sigma (s)\bigr)^{\underline{\alpha _{1}-1}}\bigl(T+\theta _{1}-\sigma (\xi )\bigr)^{\underline{\theta _{1}-1}} \\ &{}\times(\xi +\alpha _{1}-1)^{\underline{\alpha _{1}-1}}-\bigl(t_{1}-\sigma (s)\bigr)^{\underline{ \alpha _{1}-1}} \Biggr]+\frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda }\mathcal{P}(h_{1},h_{2}) \\ \geq {}& \bigl[ t_{1}^{\underline{\alpha _{1}-1}}-\bigl(t_{1}-\sigma (s) \bigr)^{\underline{ \alpha _{1}-1}} \bigr]+\frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda } \mathcal{P}(h_{1},h_{2}) \geq 0, \end{aligned}$$
(2.42)

and

$$\begin{aligned} u_{2}(t_{2})= {}&\sum_{s=0}^{T} G_{21}(t_{1},s)g_{1}(s+\alpha _{1}-1)h _{1}(s+\alpha _{1}-1) \\ &{}-\sum_{s=0}^{T} G_{22}(t_{2},s)g_{2}(s+ \alpha _{2}-1) h_{2}(s+\alpha _{2}-1) \\ = {}&\frac{t_{2}^{\underline{\alpha _{2}-1}}}{\varLambda }\mathcal{Q}(h _{1},h_{2})- \frac{(t_{2}-\sigma (s))^{\underline{\alpha _{2}-1}}}{ \varGamma (\alpha _{2})} \\ \geq {}&t_{2}^{\underline{\alpha _{2}-1}}-\bigl(t_{2}-\sigma (s) \bigr)^{\underline{ \alpha _{2}-1}} \geq 0, \end{aligned}$$
(2.43)

so \((X3)\) holds. The proof is complete. □

The following theorems [54] are provided to study the existence of positive solution to the boundary value problem (1.5) in the next section.

Theorem 2.3

(Arzelá–Ascoli theorem)

A set of functions in \(C[a,b]\) with the sup norm is relatively compact if and only it is uniformly bounded and equicontinuous on \([a,b]\).

Theorem 2.4

If a set is closed and relatively compact, then it is compact.

Theorem 2.5

(Schauder’s fixed point theorem)

Let T be a continuous and compact mapping of a Banach space E into itself such that the set

$$\begin{aligned} \{ x\in E: x=\eta Tx, \textit{for some }0\leq \eta \leq 1 \} \end{aligned}$$

is bounded. Then T has a fixed point.

Main results

In this section, we aim to establish the existence result for problem (1.5)–(1.6). For each \(i,j \in \{1,2\}\) where \(i\neq j\), we let \(E_{i}:C ( \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}, \mathbb{R} )\) be the Banach space for all functions on \(\mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\).

Therefore, the product space \(\mathcal{C}=E_{1}\times E_{2}\) is a Banach space. We consider the spaces

$$\begin{aligned} \mathcal{C}_{i}= \bigl\{ (u_{1},u_{2}) \in { \mathcal{C}}: \Delta ^{\beta _{i}}u_{i}(t_{i}-\beta _{i}+1) \in \mathcal{C} \bigr\} , \end{aligned}$$

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\), and define the norm by

$$\begin{aligned} \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{C}_{i}}= \bigl\Vert \Delta ^{\beta _{i}}u_{i} \bigr\Vert _{E _{i}}+ \Vert u_{j} \Vert _{E_{j}}, \end{aligned}$$

where

$$\begin{aligned} &\bigl\Vert \Delta ^{\beta _{i}}u_{i} \bigr\Vert _{E_{i}}= \max_{t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}} \bigl\vert \Delta ^{\beta _{i}}u_{i}(t_{i}-\beta _{i}+1) \bigr\vert \quad\mbox{and} \\ &\Vert u_{j} \Vert _{E _{j}}= \max _{t_{j}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}} \bigl\vert u_{j}(t_{j}) \bigr\vert . \end{aligned}$$

Let \({\mathcal{U}}=\mathcal{C}_{1}\cap \mathcal{C}_{2}\). Obviously, the space \(( {\mathcal{U}},\|(u_{1},u_{2})\|_{\mathcal{U}} )\) is also a Banach space with the norm

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

A positive solution of problem (1.5)–(1.6) is a pair of functions \((x_{1},x_{2})\in \mathcal{U}\) satisfying (1.5)–(1.6) with \(x_{i}(t_{i}) \geq 0\) for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\) and \((x_{1},x_{2}) \neq (0,0)\).

From Lemmas 2.1 and 2.2, we obtain the following lemma.

Lemma 3.1

For \(t_{i}\in {\mathbb{N}}_{\alpha _{i}-2,T+\alpha _{i}}\), \(i,j\in \{1,2 \}\) and \(i \neq j\). If \((u_{1},u_{2} )\in \mathcal{U}\) satisfy

  1. (i)

    \(u_{i}(\alpha _{i}-2)=0, u_{i}(T+\alpha _{i})=\lambda _{j} \Delta ^{-\theta _{j}}g_{j}(T+\alpha _{j}+\theta _{j})u_{j}(T+\alpha _{j}+ \theta _{j})\);

  2. (ii)

    \(\Delta ^{\alpha _{i}}u_{i}(t_{i})\leq 0\) for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\).

Then \(u_{i}(t_{i})\geq 0\).

In what follows, we give the definitions of the lower and upper solution of problem (1.5)–(1.6).

Definition 3.1

A pair of functions \((\chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) )\in \mathcal{U}\) is called a lower solution of problem (1.5)–(1.6) if it satisfies

$$\begin{aligned} &{-}\Delta ^{\alpha _{i}} \chi _{i}^{*}(t_{i}) \leq F_{i} \bigl(t_{1},t_{2}, \chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) \bigr), \\ &\chi _{i}^{*}(\alpha _{i}-2)\geq 0, \\ &\chi _{i}^{*}(T+\alpha _{i})\geq \lambda _{j}\Delta ^{-\theta _{j}}g_{j}(T+ \alpha _{j}+\theta _{j}) \chi _{j}^{*}(T+ \alpha _{j}+\theta _{j}). \end{aligned}$$

Definition 3.2

A pair of functions \((\bar{\chi }_{1}^{*}(t_{1}),\bar{\chi }_{2} ^{*}(t_{2}) )\in \mathcal{U}\) is called an upper solution of problem (1.5)–(1.6), if it satisfies

$$\begin{aligned} &{-}\Delta ^{\alpha _{i}} \bar{\chi }_{i}^{*}(t_{i}) \geq F_{i} \bigl(t _{1},t_{2},\bar{\chi }_{1}^{*}(t_{1}),\bar{\chi }_{2}^{*}(t_{2}) \bigr), \\ &\bar{\chi }_{i}^{*}(\alpha _{i}-2)\leq 0, \\ &\bar{\chi }_{i}^{*}(T+\alpha _{i})\leq \lambda _{j}\Delta ^{-\theta _{j}}g_{j}(T+\alpha _{j}+\theta _{j}) \bar{\chi }_{j}^{*}(T+ \alpha _{j}+ \theta _{j}). \end{aligned}$$

The following assumptions are set throughout this paper: for \(i,j\in \{1,2\}\) and \(i\neq j\),

\((H1)\) :

\(0<\varLambda <1\) and \(\sum_{\xi =0}^{T}{\mathcal{H}}_{i,j}( \xi +\alpha _{i}-1,s)\geq 0\) for all \(s \in \mathbb{N}_{0,T}\).

\((H2)\) :

\(F_{i}\in C ( \mathbb{N}_{\alpha _{1}-1,T+\alpha _{1}-1} \times \mathbb{N}_{\alpha _{2}-1,T+\alpha _{2}-1}\times (0,+\infty ) \times (0,+\infty ),[0,+\infty ) )\) are decreasing in third and fourth variables, and

$$ F_{i} \bigl(t_{i},t_{j},t_{i}^{\underline{\alpha _{i}-1}},t_{j}^{\underline{ \alpha _{j}-1}} \bigr)\in l^{1}. $$
\((H3)\) :

For all \(\ell \in (0,1)\), there exist constants \(0<\rho _{i}<1\) such that, for any \((t_{1},t_{2},v_{1},v_{2})\in \mathbb{N}_{\alpha _{1}-1,T+ \alpha _{1}-1}\times \mathbb{N}_{\alpha _{2}-1,T+\alpha _{2}-1}\times (0,+ \infty ) \times (0,+\infty )\),

$$ F_{i} (t_{1},t_{2},\ell v_{1},\ell v_{2} )\leq \ell ^{-\rho _{i}} F_{i} (t_{1},t_{2},v_{1},v_{2} ). $$
\((H4)\) :

\(\varsigma _{i}\leq g_{i}(t_{i}) \leq {\mathcal{G}}_{i} \) for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\).

Remark

Conditions \((H2)\)\((H3)\) imply that \(F_{i}\) have a power singularity at \(u_{i}=0\) for \(i=1,2\).

Theorem 3.1

Suppose that \((H{1})\)\((H4)\) hold. Then problem (1.5)(1.6) has at least one positive solution \((u_{1}^{*},u_{2}^{*} )\), which satisfies

$$\begin{aligned} \bigl( \varsigma {\mathcal{L}}^{-1}t_{1}^{\underline{\alpha _{1}-1}}, \varsigma {\mathcal{L}}^{-1}t_{2}^{\underline{\alpha _{2}-1}} \bigr) \leq \bigl(u_{1}^{*},u_{2}^{*} \bigr) \leq \bigl( \mathcal{G} {\mathcal{L}} t_{1}^{\underline{\alpha _{1}-1}},\mathcal{G} { \mathcal{L}} t_{2} ^{\underline{\alpha _{2}-1}} \bigr), \end{aligned}$$
(3.1)

where \({\mathcal{G}}:= \max \lbrace {\mathcal{G}}_{1}, {\mathcal{G}}_{2} \rbrace , \varsigma:=\max \lbrace \varsigma _{1},\varsigma _{2} \rbrace , {\mathcal{L}}^{\rho }:=\max \lbrace {\mathcal{L}}_{1}^{\rho _{1}},{\mathcal{L}}_{2}^{\rho _{2}} \rbrace \),

$$\begin{aligned} {\mathcal{L}}:={} &\max \Biggl\{ 1, \Biggl\vert \omega _{i1} \sum _{s=0}^{T} \tilde{\mathcal{F}}_{1}(s)- \omega _{i2} \sum_{s=0}^{T} \tilde{\mathcal{F}}_{2}(s) \Biggr\vert ^{\frac{1}{1-\rho }}, \\ & \biggl\vert \frac{\varLambda \varGamma (T+2)}{\sum_{s=0}^{T}\sum_{\xi =0}^{T} \mathcal{H}_{11}\tilde{\mathcal{F}}_{1}(s) -\sum_{s=0}^{T}\sum_{ \xi =0}^{T} \mathcal{H}_{12} \tilde{\mathcal{F}}_{2}(s)} \biggr\vert ^{\frac{1}{1- \rho }}, \\ & \biggl\vert \frac{\varLambda \varGamma (T+2)}{\sum_{s=0}^{T}\sum_{\xi =0}^{T} \mathcal{H}_{21} \tilde{\mathcal{F}}_{1}(s) - \sum_{s=0}^{T}\sum_{ \xi =0}^{T} \mathcal{H}_{22}\tilde{\mathcal{F}}_{2}(s)} \biggr\vert ^{\frac{1}{1- \rho }} \Biggr\} , \end{aligned}$$
(3.2)

with

$$\begin{aligned} &\tilde{\mathcal{F}}_{1}(s):=F_{1} \bigl(s+ \alpha _{1}-1, \alpha _{2}-1,(s+\alpha _{1}-1)^{\underline{\alpha _{1}-1}},( \alpha _{2}-1)^{\underline{ \alpha _{2}-1}} \bigr), \\ &\tilde{\mathcal{F}}_{2}(s):=F_{2} \bigl(\alpha _{1}-1,s+\alpha _{2}-1,( \alpha _{1}-1)^{\underline{\alpha _{1}-1}},(s+ \alpha _{2}-1)^{\underline{ \alpha _{2}-1}} \bigr), \end{aligned}$$

and \(\omega _{i,j}, {\mathcal{H}}_{i,j}, i,j=1,2 \) are defined in the previous section. In particular, if \({\mathcal{L}}=1\), then \((t_{1}^{\underline{\alpha _{1}-1}},t_{2}^{ \underline{\alpha _{2}-1}} ) \) is a positive solution of problem (1.5)(1.6).

Proof

Define the cone

$$\begin{aligned} \mathcal{P}={}& \bigl\{ (u_{1},u_{2} ) \in \mathcal{U}: {{ \mathcal{L}} _{i}^{-1}} t_{i}^{\underline{\alpha _{i}-1}} \leq \Delta ^{\beta _{i}}u _{i}(t_{i}-\beta _{i}+1) \leq {\mathcal{L}}_{i} t_{i}^{\underline{ \alpha _{i}-1}} \\ &\text{and } {{\mathcal{L}}_{i}^{-1}} t_{j}^{ \underline{\alpha _{j}-1}} \leq u_{j}(t_{j}) \leq {\mathcal{L}}_{i} t _{j}^{\underline{\alpha _{j}-1}}\text{ for } i,j\in \{1,2\} \text{ and }i \neq j \bigr\} , \end{aligned}$$

and the operator \(\mathcal{T}:{\mathcal{U}}\times {\mathcal{U}}\rightarrow {\mathcal{U}}\) by

$$\begin{aligned} \bigl(\mathcal{T}(u_{1},u_{2}) \bigr) (t_{1},t_{2})= \bigl( \bigl(\mathcal{T} _{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2}), \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr), \end{aligned}$$
(3.3)

for all \((u_{1},u_{2})\in \mathcal{P}\) and

$$\begin{aligned} &\bigl(\mathcal{T}_{i}(u_{1},u_{2})\bigr) (t_{1},t_{2}) \\ &\quad =\sum_{s=0}^{T}G_{i1}(t_{i},s) g_{1}(s+\alpha _{1}-1)F_{1} \bigl(s+ \alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+\alpha _{1}-\beta _{1}),u _{2}(t_{2}) \bigr) \\ &\qquad{}-\sum_{s=0}^{T}G_{i2}(t_{i},s) g_{2}(s+\alpha _{2}-1)F_{2} \bigl(t _{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+\alpha _{2}- \beta _{2}) \bigr), \end{aligned}$$
(3.4)

where \(G_{i1}(t_{i},s)\) and \(G_{i2}(t_{i},s)\) are defined in (2.17)–(2.20).

Firstly, we claim that \(\mathcal{T}\) is well defined and \(\mathcal{T}( \mathcal{P})\subset \mathcal{P}\). By Lemma 2.1 and \((H1)\)\((H4)\), we obtain

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}_{i}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\ &\quad = \Biggl\vert \sum_{s=0}^{T}G_{i1}(t_{i},s) g_{1}(s+\alpha _{1}-1)F_{1} \bigl(s+\alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+\alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\sum_{s=0}^{T}G_{i2}(t_{i},s) g_{2}(s+\alpha _{2}-1)F_{2} \bigl(t _{1},s+\alpha _{2}-1,u_{1}(t_{1}),u_{2}(s+ \alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad\leqslant \Biggl\vert \omega _{i1}t_{i}^{\underline{\alpha _{i}-1}} \mathcal{G}_{1} \sum_{s=0}^{T}F_{1} \bigl( s+\alpha _{1}-1,t_{2}, {\mathcal{L}}_{1}^{-1}(s+ \alpha _{1}-1)^{\underline{\alpha _{1}-1}},{\mathcal{L}}_{1}^{-1}t_{2} ^{\underline{\alpha _{2}-1}} \bigr) \\ &\qquad{}-\omega _{i2}t_{i}^{\underline{\alpha _{i}-1}}\mathcal{G}_{2} \sum_{s=0}^{T}F_{2} \bigl(t_{1},s+\alpha _{2}-1, {\mathcal{L}}_{2}^{-1}t_{1} ^{\underline{\alpha _{1}-1}},{\mathcal{L}}_{2}^{-1}(s+\alpha _{2}-1)^{\underline{ \alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad \leqslant t_{i}^{\underline{\alpha _{i}-1}}\mathcal{G} \Biggl\vert \omega _{i1}\sum_{s=0}^{T}{ \mathcal{L}}_{1}^{\rho _{1}} F_{1} \bigl( s+\alpha _{1}-1,t _{2}, (s+\alpha _{1}-1)^{\underline{\alpha _{1}-1}},t_{2}^{\underline{ \alpha _{2}-1}} \bigr) \\ &\qquad{}-\omega _{i2}\sum_{s=0}^{T}{ \mathcal{L}}_{2}^{\rho _{2}} F_{2} \bigl(t_{1},s+ \alpha _{2}-1, t_{1}^{\underline{\alpha _{1}-1}},(s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad \leqslant t_{i}^{\underline{\alpha _{i}-1}}\mathcal{G} {\mathcal{L}}^{\rho } \Biggl\vert \omega _{i1}\sum_{s=0}^{T} F_{1} \bigl( s+\alpha _{1}-1,\alpha _{2}-1, (s+ \alpha _{1}-1)^{\underline{\alpha _{1}-1}},(\alpha _{2}-1)^{\underline{ \alpha _{2}-1}} \bigr) \\ &\qquad{}-\omega _{i2}\sum_{s=0}^{T} F_{2} \bigl(\alpha _{1}-1,s+\alpha _{2}-1, ( \alpha _{1}-1)^{\underline{\alpha _{1}-1}},(s+\alpha _{2}-1)^{\underline{ \alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad =t_{i}^{\underline{\alpha _{i}-1}}\mathcal{G} {\mathcal{L}}. \end{aligned}$$
(3.5)

On the other hand, we have

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\ &\quad \geq\frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda } \Biggl\vert \sum_{s=0} ^{T}\sum _{\xi =0}^{T}\mathcal{H}_{11}(\xi +\alpha _{1}-1,s)g_{1}(s+ \alpha _{1}-1) \\ &\qquad{}\times F_{1} \bigl( s+\alpha _{1}-1,t_{2}, { \mathcal{L}}_{1}(s+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},{ \mathcal{L}}_{1}t_{2}^{\underline{\alpha _{2}-1}} \bigr) \\ &\qquad{}-\sum_{s=0}^{T}\sum _{\xi =0}^{T} \mathcal{H}_{12}(\xi +\alpha _{2}-1,s)g _{1}(s+\alpha _{1}-1) \\ &\qquad{}\times F_{2} \bigl(t_{1},s+\alpha _{2}-1, { \mathcal{L}}_{2}t_{1}^{\underline{ \alpha _{1}-1}},{\mathcal{L}}_{2}(s+ \alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad \geq\frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda } \Biggl\vert {\mathcal{L}} _{1}^{-\rho _{1}} \varsigma _{1}\sum_{s=0}^{T}\sum _{\xi =0}^{T} \mathcal{H}_{11}(\xi +\alpha _{1}-1,s)g_{1}(s+\alpha _{1}-1) \\ &\qquad{}\times F_{1} \bigl( s+\alpha _{1}-1,T+\alpha _{2}-1, (s+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(T+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \\ &\qquad{}-{\mathcal{L}}_{2}^{-\rho _{2}}\varsigma _{2}\sum _{s=0}^{T}\sum _{ \xi =0}^{T} \mathcal{H}_{12}(\xi +\alpha _{2}-1,s) g_{2}(s+\alpha _{2}-1) \\ &\qquad{}\times F_{2} \bigl(T+\alpha _{1}-1,s+\alpha _{2}-1, (T+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad \geq t_{1}^{\underline{\alpha _{1}-1}}\cdot \frac{\varsigma {\mathcal{L} ^{-1}}}{\varGamma (T+2)} \end{aligned}$$
(3.6)

and

$$\begin{aligned} &\bigl\vert \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\ &\quad \geq\frac{t_{2}^{\underline{\alpha _{2}-1}}}{\varLambda } \Biggl\vert {\mathcal{L}} _{1}^{-\rho _{1}}\varsigma _{1}\sum_{s=0}^{T}\sum _{\xi =0}^{T} \mathcal{H}_{21}(\xi +\alpha _{1}-1,s) \\ &\qquad{}\times F_{1} \bigl( s+\alpha _{1}-1,T+\alpha _{2}-1, (s+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(T+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \\ &\qquad {}-{\mathcal{L}}_{2}^{-\rho _{2}}\varsigma _{2}\sum _{s=0}^{T}\sum _{ \xi =0}^{T} \mathcal{H}_{22}(\xi +\alpha _{2}-1,s) \\ &\qquad{}\times F_{2} \bigl(T+\alpha _{1}-1,s+\alpha _{2}-1, (T+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad \geq t_{2}^{\underline{\alpha _{2}-1}}\cdot \frac{\varsigma {\mathcal{L} ^{-1}}}{\varGamma (T+2)}. \end{aligned}$$
(3.7)

Next, taking the fractional difference of order \(0<\beta _{i}\leq 1\) for (3.4), we have

$$\begin{aligned} & \Delta^{\beta _{i}}\bigl(\mathcal{T}_{i}(u_{1},u_{2}) \bigr) (t_{i}-\beta _{i}+1,t _{j}) \\ &\quad =\sum_{s=0}^{T} \bigl[\Delta ^{\beta _{i}} G_{i1}(t_{i},s) \bigr] g_{1}(s+ \alpha _{1}-1)F_{1} \bigl(s+\alpha _{1}-1,t_{2}, \Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\sum_{s=0}^{T} \bigl[\Delta ^{\beta _{i}} G_{i2}(t_{i},s) \bigr] g_{2}(s+ \alpha _{2}-1) \\ &\qquad{}\times F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+ \alpha _{2}-\beta _{2}) \bigr). \end{aligned}$$
(3.8)

By the same arguments as before and since \(\Delta ^{\beta _{i}}{\mathcal{K} _{i}}(t_{i},s)\leq {\mathcal{K}_{i}}(t_{i},s)\), we obtain

$$\begin{aligned} &\bigl\vert \Delta ^{\beta _{1}} \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t _{1}-\beta _{1}+1,t_{2}) \bigr\vert \\ &\quad=\Biggl|\sum_{s=0}^{T} \Biggl[ \Delta ^{\beta _{1}}{\mathcal{K}_{1}}(t_{1},s)+\frac{1}{ \varLambda \varGamma (-\beta _{1})} \Biggl( \sum_{p=\alpha _{1}-1}^{t_{1}+1}\bigl(t _{1}-\beta _{1}+1-\sigma (p)\bigr)^{\underline{-\beta _{1}-1}}p^{\underline{ \alpha _{1}-1}} \Biggr) \\ &\qquad{}\times\sum_{\xi =0}^{T}{\mathcal{H}_{11}}( \xi +\alpha _{1}-1,s) \Biggr] g _{1}(s+\alpha _{1}-1) F_{1} \bigl(s+\alpha _{1}-1,t_{2}, \Delta ^{\beta _{1}}u_{1}(s+\alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\frac{1}{\varLambda \varGamma (-\beta _{1})} \Biggl( \sum_{p=\alpha _{1}-1} ^{t_{1}+1}\bigl(t_{1}-\beta _{1}+1-\sigma (p) \bigr)^{\underline{-\beta _{1}-1}}p ^{\underline{\alpha _{1}-1}} \Biggr)\sum_{s=0}^{T} \sum_{\xi =0}^{T} {\mathcal{H}_{12}}( \xi +\alpha _{2}-1,s) ] \\ &\qquad{}\times g_{2}(s+\alpha _{2}-1) F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+\alpha _{2}-\beta _{2}) \bigr)\Biggr| \\ &\quad\leq\Biggl|\sum_{s=0}^{T} \Biggl[ { \mathcal{K}_{1}}(t_{1},s)+\frac{(t_{1}+1)^{\underline{ \alpha _{1}-1}}(t_{1}+\alpha _{1}-\beta _{1}+2)^{\underline{-\beta _{1}}}}{ \varLambda \varGamma (1-\beta _{1})}\sum _{\xi =0}^{T}{\mathcal{H}_{11}}( \xi +\alpha _{1}-1,s) \Biggr] \\ &\qquad{}\times g_{1}(s+\alpha _{1}-1) F_{1} \bigl(s+\alpha _{1}-1,t_{2}, \Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\frac{(t_{1}+1)^{\underline{\alpha _{1}-1}}(t_{1}+\alpha _{1}-\beta _{1}+2)^{\underline{-\beta _{1}}}}{\varLambda \varGamma (1-\beta _{1})}\sum_{s=0}^{T}\sum _{\xi =0}^{T}{\mathcal{H}_{12}}(\xi +\alpha _{2}-1,s) ] \\ &\qquad{}\times g_{2}(s+\alpha _{2}-1) F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+\alpha _{2}-\beta _{2}) \bigr)\Biggr| \\ &\quad \leq \Biggl\vert {\mathcal{G}_{1}}\sum _{s=0}^{T}G_{11}(t_{1},s) F_{1} \bigl(s+ \alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+\alpha _{1}-\beta _{1}),u _{2}(t_{2}) \bigr) \\ &\qquad{}- {\mathcal{G}_{2}}\sum_{s=0}^{T}G_{12}(t_{1},s) F_{2} \bigl(t_{1},s+ \alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+\alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad \leq\mathcal{L}^{*}{\mathcal{G}} t_{1}^{\underline{\alpha _{1}-1}}, \end{aligned}$$
(3.9)

and

$$\begin{aligned} &\bigl\vert \Delta ^{\beta _{2}} \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t _{1},t_{2}-\beta _{2}+1) \bigr\vert \\ &\quad \leq \Biggl\vert \frac{(t_{2}+1)^{\underline{\alpha _{2}-1}}(t_{2}+\alpha _{2}- \beta _{2}+2)^{\underline{-\beta _{2}}}}{\varGamma (1-\beta _{1})}\sum_{s=0} ^{T} \sum_{\xi =0}^{T}{ \mathcal{H}_{21}}(\xi +\alpha _{1}-1,s) \\ &\qquad{} \times g_{1}(s+\alpha _{1}-1) F_{1} \bigl(s+\alpha _{1}-1,t_{2}, \Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\sum_{s=0}^{T} \Biggl[{ \mathcal{K}_{2}}(t_{2},s)+\frac{(t_{2}+1)^{\underline{ \alpha _{2}-1}}(t_{2}+\alpha _{2}-\beta _{2}+2)^{\underline{-\beta _{2}}}}{ \varLambda \varGamma (1-\beta _{1})} \sum _{\xi =0}^{T}{\mathcal{H}_{22}}( \xi +\alpha _{2}-1,s) \Biggr] \\ &\qquad{}\times g_{2}(s+\alpha _{2}-1) F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+\alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad \leq \Biggl\vert \mathcal{G}_{1}\sum _{s=0}^{T}G_{21}(t,s) F_{1} \bigl(s+\alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t _{2}) \bigr) \\ &\qquad{}-\mathcal{G}_{2}\sum_{s=0}^{T}G_{22}(t,s) F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+\alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad \leq\mathcal{L}\mathcal{G} t_{2}^{\underline{\alpha _{2}-1}}. \end{aligned}$$
(3.10)

On the other hand, we have

$$\begin{aligned} &\bigl\vert \Delta ^{\beta _{1}} \bigl(\mathcal{T}_{1}(u_{1},u_{2}) \bigr) (t _{1}-\beta _{1}+1,t_{2}) \bigr\vert \\ &\quad \geq \frac{\varGamma (\alpha _{1})(t_{1}+\alpha _{1}-\beta _{1}+2)^{\underline{- \beta _{1}}}}{\varLambda \varGamma (1-\beta _{1})} \Biggl\vert \sum_{s=0}^{T} \sum_{\xi =0}^{T}{\mathcal{H}_{11}}( \xi +\alpha _{1}-1,s) \\ &\qquad{}\times g_{1}(s+\alpha _{1}-1) F_{1} \bigl(s+\alpha _{1}-1,t_{2}, \Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\sum_{s=0}^{T}\sum _{\xi =0}^{T} \mathcal{H}_{12} g_{2}(s+\alpha _{2}-1)F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u _{2}(s+\alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad \geq \frac{t_{1}^{\underline{\alpha _{1}-1}}}{\varLambda \varGamma (T+2)} \Biggl\vert \varsigma _{1}\sum _{s=0}^{T} \sum_{\xi =0}^{T}{ \mathcal{H}_{11}}( \xi +\alpha _{1}-1,s) \\ &\qquad{}\times F_{1} \bigl(s+\alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{} -\varsigma _{2}\sum_{s=0}^{T} \sum_{\xi =0}^{T} \mathcal{H}_{12} F _{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+ \alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad \geq t_{1}^{\underline{\alpha _{1}-1}}\varsigma \mathcal{L}^{-1}, \end{aligned}$$
(3.11)

and

$$\begin{aligned} &\bigl\vert \Delta ^{\beta _{2}} \bigl(\mathcal{T}_{2}(u_{1},u_{2}) \bigr) (t _{1},t_{2}-\beta _{2}+1) \bigr\vert \\ &\quad \geq \frac{t_{2}^{\underline{\alpha _{2}-1}}}{\varLambda \varGamma (T+2)} \Biggl\vert \varsigma _{1}\sum _{s=0}^{T} \sum_{\xi =0}^{T}{ \mathcal{H}_{21}}( \xi +\alpha _{1}-1,s) \\ &\qquad{}\times F_{1} \bigl(s+\alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\varsigma _{2}\sum_{s=0}^{T} \sum_{\xi =0}^{T} \mathcal{H}_{22}( \xi +\alpha _{2}-1,s)F_{2} \bigl(t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u_{2}(s+ \alpha _{2}-\beta _{2}) \bigr) \Biggr\vert \\ &\quad \geq t_{2}^{\underline{\alpha _{2}-1}}\varsigma \mathcal{L}^{-1}. \end{aligned}$$
(3.12)

Thus it follows from (2.5)–(2.7) and (2.9)–(2.12) that \(\mathcal{T}\) is well defined and \(\mathcal{T}(\mathcal{P})\subset \mathcal{P}\).

Furthermore, by Lemma 2.2, we obtain

$$\begin{aligned} &{-}\Delta ^{\alpha _{i}} {\mathcal{T}}_{i} ( u_{1},u_{2} ) (t _{1},t_{2})= F_{i} \bigl(t_{1},t_{2}, \Delta ^{\beta _{i}}{\mathcal{T}} _{i} ( u_{1},u_{2} ) (t_{1},t_{2}),{\mathcal{T}}_{i} ( u_{1},u_{2} ) (t_{1},t_{2}) \bigr), \\ &{\mathcal{T}}_{i} ( u_{1},u_{2} ) (\alpha _{i}-2,t_{j})= 0, \\ &{\mathcal{T}}_{i} ( u_{1},u_{2} ) (T+\alpha _{i},t_{j})= \lambda _{j}\Delta ^{-\theta _{j}} g_{j}(T+\alpha _{j}+\theta _{j}) {\mathcal{T}} _{j} ( u_{1},u_{2} ) (t_{i},T+\alpha _{j}+\theta _{j}). \end{aligned}$$
(3.13)

We let

$$\begin{aligned} &\chi _{i}(t_{i})=\min \bigl\lbrace t_{i}^{\underline{\alpha _{i}-1}}, \mathcal{T}_{i} \bigl( t_{i}^{\underline{\alpha _{i}-1}},t_{j}^{\underline{ \alpha _{j}-1}} \bigr) \bigr\rbrace , \end{aligned}$$
(3.14)
$$\begin{aligned} &\bar{\chi }_{i}(t_{i}) =\max \bigl\lbrace t_{i}^{\underline{\alpha _{i}-1}},\mathcal{T}_{i} \bigl( t_{i}^{\underline{\alpha _{i}-1}},t _{j}^{\underline{\alpha _{j}-1}} \bigr) \bigr\rbrace . \end{aligned}$$
(3.15)

Since \(( t_{1}^{\underline{\alpha _{1}-1}},t_{2}^{\underline{ \alpha _{2}-1}} ), ({\mathcal{T}}_{1} ( t_{1}^{\underline{ \alpha _{1}-1}},t_{2}^{\underline{\alpha _{2}-1}} ),{\mathcal{T}} _{2} ( t_{1}^{\underline{\alpha _{1}-1}},t_{2}^{\underline{\alpha _{2}-1}} ) )\in {\mathcal{P}}\), we have

$$\begin{aligned} \begin{aligned} & (\chi _{1},\chi _{2} ), (\bar{\chi }_{1}, \bar{\chi _{2}} )\in {\mathcal{P}}, \\ &\chi _{1}\leq t_{1}^{\underline{\alpha _{1}-1}} \leq \bar{\chi }_{1}\quad \text{and}\quad\chi _{2}\leq t_{2}^{\underline{\alpha _{2}-1}} \leq \bar{\chi }_{2}. \end{aligned} \end{aligned}$$
(3.16)

Let

$$\begin{aligned} \bigl(\chi ^{*}_{1},\chi ^{*}_{2} \bigr) = \bigl( \mathcal{T}_{1} ( \chi _{1},\chi _{2} ), \mathcal{T}_{2} ( \chi _{1}, \chi _{2} ) \bigr) \quad\text{and}\quad \bigl( \bar{\chi }^{*}_{1}, \bar{ \chi }^{*}_{2} \bigr) = \bigl( \mathcal{T}_{1} ( \bar{\chi } _{1},\bar{\chi }_{2} ), \mathcal{T}_{2} ( \bar{\chi }_{1},\bar{ \chi }_{2} ) \bigr). \end{aligned}$$
(3.17)

Then, by (3.14)–(3.17) and \((H3)\), we obtain

$$\begin{aligned} &\bigl( \bar{\chi }^{*}_{1},\bar{\chi }^{*}_{2} \bigr)\leq \bigl( \mathcal{T}_{1} \bigl(t_{1}^{\underline{\alpha _{1}-1}} ,t_{2}^{\underline{ \alpha _{2}-1}} \bigr), \mathcal{T}_{2} \bigl(t_{1}^{\underline{ \alpha _{1}-1}} ,t_{2}^{\underline{\alpha _{2}-1}} \bigr) \bigr) \\ &\phantom{\bigl( \bar{\chi }^{*}_{1},\bar{\chi }^{*}_{2} \bigr)}\leq \bigl( \mathcal{T}_{1} (\chi _{1},\chi _{2} ), \mathcal{T}_{2} (\chi _{1},\chi _{2} ) \bigr) = \bigl(\chi ^{*}_{1},\chi ^{*}_{2} \bigr) \leq ( \bar{\chi }_{1},\bar{ \chi }_{2} ), \end{aligned}$$
(3.18)
$$\begin{aligned} &\bigl(\chi ^{*}_{1},\chi ^{*}_{2} \bigr)\geq \bigl( \mathcal{T} _{1} \bigl(t_{1}^{\underline{\alpha _{1}-1}} ,t_{2}^{\underline{\alpha _{2}-1}} \bigr), \mathcal{T}_{2} \bigl(t_{1}^{ \underline{\alpha _{1}-1}} ,t_{2}^{\underline{\alpha _{2}-1}} \bigr) \bigr) \geq (\chi _{1},\chi _{2} ). \end{aligned}$$
(3.19)

So, it follows from (3.13) and (3.16)–(3.19) that

$$\begin{aligned} \begin{aligned} &\Delta ^{\alpha _{i}}\chi _{i}^{*}(t_{i}) +F_{i} \bigl(t_{1},t_{2},\chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) \bigr) \\ &\quad=\Delta ^{\alpha _{i}}{\mathcal{T}}_{i}(\chi _{1},\chi _{2}) (t_{1},t_{2})+F _{i} \bigl(t_{1},t_{2},\chi _{1}^{*}(t_{1}), \chi _{2}^{*}(t_{2}) \bigr) \\ &\quad =-F_{i} \bigl(t_{1},t_{2},\chi _{1}(t_{1}),\chi _{2}(t_{2}) \bigr)+F _{i} \bigl(t_{1},t_{2},\chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) \bigr) \\ &\quad \leq-F_{i} (t_{1},t_{2},\chi _{1},\chi _{2} )+F_{i} \bigl(t _{1},t_{2},\chi _{1}(t_{1}),\chi _{2}(t_{2}) \bigr)=0, \\ &\chi _{i}^{*}(\alpha _{i}-2)=0, \\ &\chi _{i}^{*}(T+\alpha _{i})=\lambda _{j}\Delta ^{-\theta _{j}} g_{j}(T+ \alpha _{j}+\theta _{j}) {\chi ^{*}_{j}}(T+ \alpha _{j}+\theta _{j}), \end{aligned} \end{aligned}$$
(3.20)

and

$$\begin{aligned} \begin{aligned} &\Delta ^{\alpha _{i}}\bar{\chi }_{i}^{*}(t_{1},t_{2}) +F_{i} \bigl(t _{1},t_{2},\bar{\chi }_{1}^{*}(t_{1}),\bar{\chi }_{2}^{*}(t_{2}) \bigr) \\ &\quad =\Delta ^{\alpha _{i}}{\mathcal{T}}_{i}(\bar{\chi }_{1},\bar{\chi } _{2}) (t_{1},t_{2})+F_{i} \bigl(t_{1},t_{2},\bar{\chi }_{1}^{*}(t_{1}), \bar{ \chi }_{2}^{*} (t_{2}) \bigr) \\ &\quad =-F_{i} (t_{1},t_{2},\bar{\chi }_{1},\bar{\chi }_{2} )+F _{i} \bigl(t_{1},t_{2},\bar{\chi }_{1}^{*}(t_{1}), \bar{\chi }_{2}^{*}(t _{2}) \bigr) \\ &\quad \geq-F_{i} (t_{1},t_{2},\bar{\chi }_{1},\bar{\chi }_{2} )+F _{i} \bigl(t_{1},t_{2},\bar{\chi }_{1}(t_{1}), \bar{\chi }_{2}(t_{2}) \bigr)=0, \\ &\bar{\chi }_{i}^{*}(\alpha _{i}-2)=0, \\ &\bar{ \chi }_{i}^{*}(T+\alpha _{i})=\lambda _{j}\Delta ^{-\theta _{j}} g _{j}(T+\alpha _{j}+\theta _{j}) {\bar{\chi }^{*}_{j}}(T+ \alpha _{j}+ \theta _{j}). \end{aligned} \end{aligned}$$
(3.21)

Thus, it follows from (3.18)–(3.21) that \((\bar{\chi }_{1}^{*},\bar{\chi }_{2}^{*} ), ( \chi _{1}^{*},\chi _{2}^{*} )\) are lower and upper solutions of problem (1.5)–(1.6), and \((\bar{\chi }_{1}^{*},\bar{ \chi }_{2}^{*} ), ( \chi _{1}^{*},\chi _{2}^{*} ) \in {\mathcal{P}}\).

Define the function \(\mathcal{F}^{*}_{i}\) and the operator \(\mathcal{T}^{*}\) in \(\mathcal{U}\) by

$$\begin{aligned} &\mathcal{F}^{*}_{i} (t_{1},t_{2},x,y )= \textstyle\begin{cases} F_{i} (t_{1},t_{2},\bar{\chi }_{1}^{*},\bar{\chi }_{2}^{*} ), &(x,y)< (\bar{\chi }_{1}^{*},\bar{\chi }_{2}^{*} ), \\ F_{i} (t_{1},t_{2},x,y ), & (\bar{\chi }_{1}^{*},\bar{\chi }_{2}^{*} )\leq (x,y) \leq (\chi _{1}^{*},\chi _{2}^{*} ), \\ F_{i} (t_{1},t_{2},\chi _{1}^{*},\chi _{2}^{*} ), &(x,y)> (\chi _{1}^{*},\chi _{2}^{*} ), \end{cases}\displaystyle \end{aligned}$$
(3.22)
$$\begin{aligned} &{\mathcal{T}}^{*} (u_{1},u_{2} ) (t_{1},t_{2})= \bigl( {\mathcal{T}}^{*}_{1} (u_{1},u_{2} ) (t_{1},t_{2}), { \mathcal{T}} ^{*}_{2} (u_{1},u_{2} ) (t_{1},t_{2}) \bigr), \end{aligned}$$
(3.23)

where

$$\begin{aligned} &{\mathcal{T}}^{*}_{i} (u_{1},u_{2} ) (t _{1},t_{2}) \\ &\quad =\sum_{s=0}^{T}G_{i1}(t_{i},s) g_{1} (s+\alpha _{1}-1)\mathcal{F} ^{*}_{1} \bigl( s+\alpha _{1}-1,t_{2},\Delta ^{\beta _{1}}u_{1}(s+ \alpha _{1}-\beta _{1}),u_{2}(t_{2}) \bigr) \\ &\qquad{}-\sum_{s=0}^{T}G_{i2}(t_{i},s) g_{2}( s+\alpha _{2}-1)\mathcal{F} ^{*}_{2} \bigl( t_{1},s+\alpha _{2}-1,u_{1}(t_{1}), \Delta ^{\beta _{2}}u _{2}(s+\alpha _{2}-\beta _{2}) \bigr). \end{aligned}$$
(3.24)

It follows from the assumption that \(\mathcal{F}^{*}_{i}:{\mathbb{N}} _{\alpha _{1}-1,T+\alpha _{1}-1}\times {\mathbb{N}}_{\alpha _{2}-1,T+ \alpha _{2}-1}\times [0,\infty )\times [0,\infty )\rightarrow [0, \infty )\) are continuous. Consider the following problem:

$$\begin{aligned} &{-}\Delta ^{\alpha _{i}} u_{i}(t_{i})= \mathcal{F}^{*}_{i} \bigl(t_{1},t _{2},\Delta ^{\beta _{i}}u_{i} (s+\alpha _{i}-\beta _{i}) ,u_{j}(t_{j}) \bigr), \\ &u_{i}(\alpha _{i}-2)= 0, \\ &u_{i}(T+\alpha _{i})= \lambda _{j}\Delta ^{-\theta _{j}} g_{j}(T+\alpha _{j}+\theta _{j}) u_{j}(T+\alpha _{j}+\theta _{j}). \end{aligned}$$
(3.25)

For \(i,j\in \{1,2\}, i\neq j\) and for all \((u_{1},u_{2} ) \in \mathcal{U}\), by (3.22) we obtain

$$\begin{aligned} & \bigl\vert {\mathcal{T}}_{i}^{*} (u_{1},u_{2} ) (t_{1},t_{2}) \bigr\vert \\ &\quad\leq t_{i}^{\underline{\alpha _{i}-1}} \Biggl\vert \omega _{i1}{ \mathcal{G} _{1}}\sum_{s=0}^{T}{ \mathcal{F}^{*}_{1}} \bigl(s+\alpha _{1}-1,t_{2}, \Delta ^{\beta _{1}}u_{1},u_{2} \bigr) \\ &\qquad{}-\omega _{i2}{\mathcal{G}_{2}}\sum_{s=0}^{T}{ \mathcal{F}^{*}_{2}} \bigl(t_{1},s+\alpha _{2}-1,u_{1}, \Delta ^{\beta _{2}}u_{2} \bigr) \Biggr\vert \\ &\quad\leq (T+\alpha _{i})^{\underline{\alpha _{i}-1}}{\mathcal{G}} \Biggl\vert \omega _{i1}\sum_{s=0}^{T}{ \mathcal{F}^{*}_{1}} \bigl(s+\alpha _{1}-1,t _{2},\bar{\chi }^{*}_{1},\bar{\chi }^{*}_{2} \bigr) -\omega _{i2}\sum _{s=0}^{T}{\mathcal{F}^{*}_{2}} \bigl(t_{1},s+\alpha _{2}-1,\bar{\chi } ^{*}_{1},\bar{\chi }^{*}_{2} \bigr) \Biggr\vert \\ &\quad \leq (T+\alpha _{i})^{\underline{\alpha _{i}-1}}{\mathcal{G}} \Biggl\vert \omega _{i1}\sum_{s=0}^{T}{ \mathcal{F}^{*}_{1}} \bigl(s+\alpha _{1}-1,t _{2},{\mathcal{L}}_{1}^{-1}(s+\alpha _{1}-1)^{\underline{\alpha _{1}-1}}, {\mathcal{L}}_{2}^{-1}t_{2}^{\underline{\alpha _{2}-1}} \bigr) \\ &\qquad{}-\omega _{i2}\sum_{s=0}^{T}{ \mathcal{F}^{*}_{2}} \bigl(t_{1},s+\alpha _{2}-1,{\mathcal{L}}_{1}^{-1}t_{1}^{\underline{\alpha _{1}-1}}, {\mathcal{L}} _{2}^{-1}(s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad\leq (T+\alpha _{i})^{\underline{\alpha _{i}-1}}{\mathcal{G}} \Biggl\vert \omega _{i1}{\mathcal{L}}_{1}^{\rho _{1}}\sum _{s=0}^{T}{\mathcal{F}^{*} _{1}} \bigl(s+\alpha _{1}-1,\alpha _{2}-1,(s+ \alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \\ &\qquad{}-\omega _{i2}{\mathcal{L}}_{2}^{\rho _{1}}\sum _{s=0}^{T}{\mathcal{F} ^{*}_{2}} \bigl(\alpha _{1}-1,s+\alpha _{2}-1,(\alpha _{1}-1)^{\underline{ \alpha _{1}-1}}, (s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \Biggr\vert \\ &\quad\leq (T+\alpha _{i})^{\underline{\alpha _{i}-1}}{\mathcal{G}} {\mathcal{L}}. \end{aligned}$$
(3.26)

By the same argument, we obtain \(\vert \Delta ^{\beta _{i}}{\mathcal{T}} _{i}^{*} (u_{1},u_{2} )(t_{1},t_{2}) \vert \leq (T+ \alpha _{i})^{\underline{\alpha _{i}-1}}{\mathcal{G}}{\mathcal{L}}\).

Thus,

$$ \bigl\Vert {\mathcal{T}}^{*} \bigr\Vert _{\mathcal{C}_{i}} = \bigl\Vert \Delta ^{\beta _{i}}{\mathcal{T}} _{i}^{*} \bigr\Vert _{E_{i}}+ \bigl\Vert {\mathcal{T}}_{j}^{*} \bigr\Vert _{E_{j}} \leq \bigl[ (T+ \alpha _{1})^{\underline{\alpha _{1}-1}}+(T+ \alpha _{2})^{\underline{ \alpha _{2}-1}} \bigr] {\mathcal{G}} {\mathcal{L}} :={ \mathcal{M}}. $$

Consequently, we have

$$ \bigl\Vert {\mathcal{T}}^{*} \bigr\Vert _{\mathcal{U}} = \max \bigl\lbrace \bigl\Vert {\mathcal{T}} ^{*} \bigr\Vert _{\mathcal{C}_{1}}, \bigl\Vert {\mathcal{T}}^{*} \bigr\Vert _{\mathcal{C}_{2}} \bigr\rbrace \leq {\mathcal{M}}, $$

which implies that \({\mathcal{T}}^{*}\) is uniformly bounded. Moreover, it follows from the continuity of \({\mathcal{F}}^{*}_{i}\) and the uniform continuity of \(G_{i1}(t_{i},s),G_{i2}(t_{i},s) \) and \((H2)\) that \(\mathcal{T}^{*}:{\mathcal{U}}\times {\mathcal{U}}\rightarrow {\mathcal{U}}\) is continuous.

Let \({\mathcal{E}}\subset {\mathcal{U}}\times {\mathcal{U}}\) be bounded. By the Arzelá–Ascoli theorem and Theorem 2.4, we easily know that \({\mathcal{T}^{*}}({\mathcal{E}})\) is equicontinuous. Therefore \({\mathcal{T}^{*}}\) is completely continuous. Hence, by using Schauder’s fixed point theorem, \({\mathcal{T}^{*}}\) has at least one fixed point \(( u_{1}^{*},u_{2}^{*} )\) such that \(( u_{1}^{*},u _{2}^{*} )={\mathcal{T}^{*}} ( u_{1}^{*},u_{2}^{*} )\).

Next, we will show that

$$\begin{aligned} \bigl( \bar{\chi }_{1}^{*}(t_{1}),\bar{\chi }_{2}^{*}(t_{2}) \bigr) \leq \bigl( u_{1}^{*}(t_{1}),u_{2}^{*}(t_{2}) \bigr) \leq \bigl( \chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) \bigr),\quad t_{i}\in { \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}}. \end{aligned}$$
(3.27)

Firstly, we will prove that \(( u_{1}^{*},u_{2}^{*} ) \leq ( \chi _{1}^{*},\chi _{2}^{*} )\). Suppose \(( u _{1}^{*},u_{2}^{*} ) > ( \chi _{1}^{*},\chi _{2}^{*} )\). According to the definition of \({\mathcal{F}^{*}_{i}}\), we have

$$\begin{aligned} -\Delta ^{\alpha _{i}} u_{i}^{*}(t_{i})= \mathcal{F}^{*}_{i} \bigl(t_{1},t _{2},u_{1}^{*}(t_{1}),u_{2}^{*}(t_{2}) \bigr)=\mathcal{F}_{i} \bigl(t_{1},t _{2},\chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) \bigr). \end{aligned}$$
(3.28)

On the other hand, since \(( \chi _{1}^{*},\chi _{2}^{*} )\) is an upper solution of problem (1.5), we have

$$\begin{aligned} -\Delta ^{\alpha _{i}} \chi _{i}^{*}(t_{i}) \geq \mathcal{F}_{i} \bigl(t _{1},t_{2},\chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) \bigr). \end{aligned}$$
(3.29)

Letting \(z_{i}(t_{i})=\chi _{i}^{*}(t_{i})-u_{i}^{*}(t_{i})\), and from (3.28)–(3.29), it implies that

$$\begin{aligned} \Delta ^{\alpha _{i}} z_{i}(t_{i})= \Delta ^{\alpha _{i}}\chi _{i}^{*}(t _{i})-\Delta ^{\alpha _{i}}u_{i}^{*}(t_{i})\leq 0. \end{aligned}$$
(3.30)

Furthermore, since \(( \chi _{1}^{*},\chi _{2}^{*} )\) is an upper solution of problem (1.5) and \(( u_{1}^{*},u_{2} ^{*} ) \) is a fixed point of \({\mathcal{T}^{*}}\), we have

$$\begin{aligned} z_{i}(\alpha _{i}-2)=0,\qquad z_{i}(T+\alpha _{i})=\lambda _{j}\Delta ^{-\theta _{j}} g_{j}(T+\alpha _{j}+\theta _{j}) z_{j}(T+\alpha _{j}+\theta _{j}). \end{aligned}$$
(3.31)

By Lemma 3.1, we have

$$ z_{i}(t_{1},t_{2})\geq 0. $$

So, \(( u_{1}^{*}(t_{1}),u_{2}^{*}(t_{2}) )\leq ( \chi _{1}^{*}(t_{1}),\chi _{2}^{*}(t_{2}) )\) for all \(t_{i} \in {\mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}}\), which contradicts \(( u_{1}^{*},u_{2}^{*} )> ( \chi _{1}^{*},\chi _{2} ^{*} )\). Therefore we have \(( u_{1}^{*},u_{2}^{*} ) \leq ( \chi _{1}^{*},\chi _{2}^{*} )\) for all \(t_{i} \in {\mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}}\).

In the same argument, we have \(( u_{1}^{*}(t_{1}),u_{2}^{*}(t _{2}) )\geq ( \bar{\chi }_{1}^{*}(t_{1}),\bar{\chi }_{2} ^{*}(t_{2}) )\) for all \(t_{i} \in {\mathbb{N}_{\alpha _{i}-2,T+ \alpha _{i}}}\).

Thus (3.27) holds. Hence \(( u_{1}^{*}(t_{1}),u_{2}^{*}(t _{2}) )\) is a positive solution of problem (1.5)–(1.6). From \(( (\bar{\chi }_{1} ^{*},\bar{\chi }_{2}^{*} ), ( \chi _{1}^{*},\chi _{2}^{*} ) )\in {\mathcal{P}}\) and (3.27), we obtain

$$ \bigl( \varsigma {\mathcal{L}}^{-1}t_{1}^{\underline{\alpha _{1}-1}}, \varsigma {\mathcal{L}}^{-1}t_{2}^{\underline{\alpha _{2}-1}} \bigr) \leq \bigl(u_{1}^{*},u_{2}^{*} \bigr) \leq \bigl({\mathcal{G}} {\mathcal{L}} t_{1}^{\underline{\alpha _{1}-1}},{\mathcal{G}} { \mathcal{L}} t_{2} ^{\underline{\alpha _{2}-1}} \bigr). $$

This completes the proof. □

An example

In this section, in order to illustrate our result, we consider the coupled system of singular fractional difference equations with fractional sum boundary conditions

$$\begin{aligned} &{-}\Delta ^{\frac{4}{3}} u_{1}(t)= F_{1} \biggl(t+ \frac{1}{3},t+ \frac{1}{2},\Delta ^{\frac{1}{2}}u_{1} \biggl( t+\frac{5}{6} \biggr) ,u _{2} \biggl( t+ \frac{1}{2} \biggr) \biggr), \\ &{-}\Delta ^{\frac{3}{2}} u_{2}(t) = F_{2} \biggl(t+ \frac{1}{3},t+ \frac{1}{2},u_{1} \biggl( t+ \frac{1}{3} \biggr) \Delta ^{\frac{1}{3}}u _{1} \biggl( t+ \frac{7}{6} \biggr) \biggr),\quad t\in {\mathbb{N}}_{0,10}, \\ &u_{1} \biggl( -\frac{2}{3} \biggr)= 0,\qquad u_{1} \biggl( \frac{34}{3} \biggr) = 9\Delta ^{-\frac{3}{4}} (g _{2} u_{2} ) \biggl( \frac{49}{4} \biggr), \\ &u_{2} \biggl( -\frac{1}{2} \biggr)= 0,\qquad u_{2} \biggl( \frac{23}{2} \biggr) = 4\Delta ^{-\frac{2}{3}} (g _{1}u_{1} ) ( 12 ), \end{aligned}$$

where \(a_{i},b_{i},x_{i},y_{i}>0\), \(0< x_{i}+\frac{1}{3}a_{i}<1, 0<y _{i}+\frac{1}{2}b_{i}<1, i=1,2\), and, for \(t_{1}\in {\mathbb{N}}_{- \frac{2}{3},\frac{34}{3}}, t_{2}\in {\mathbb{N}}_{-\frac{1}{2}, \frac{23}{2}}\),

$$\begin{aligned} &F_{1} \bigl(t_{1},t_{2},\Delta ^{\frac{1}{2}}u_{1} ,u_{2} \bigr) =t_{1} ^{-x_{1}} \bigl(\Delta ^{\frac{1}{2}}u_{1} \bigr)^{-a_{1}}+t_{2}^{-y _{1}}u_{2}^{-b_{1}}, \\ &F_{2} \bigl(t_{1},t_{2},u_{1},\Delta ^{\frac{1}{3}}u_{2} \bigr) =t_{1} ^{-x_{2}}u_{1}^{-a_{2}}+t_{2}^{-y_{2}} \bigl(\Delta ^{\frac{1}{3}}u _{2} \bigr)^{-b_{2}}, \\ &g_{1}(t_{1}) = \frac{1}{200e+10\sin ^{2}2\pi t_{1}}\quad\text{and}\quad g_{2}(t_{2}) = \frac{1}{100\pi +20\cos ^{2}2\pi t_{2}}. \end{aligned}$$

Here \(\alpha _{1}=\frac{4}{3}, \alpha _{2}=\frac{3}{2}, \beta _{1}= \frac{1}{2}, \beta _{2}=\frac{1}{3}, \theta _{1}=\frac{2}{3}, \theta _{2}=\frac{3}{4}, T=10\). We can find that

$$\begin{aligned} 0< \lambda _{1}< 24.524,\qquad 0< \lambda _{2}< 9.651,\qquad \varLambda =0.248< 1, \end{aligned}$$

Clearly, \(\sum_{\xi =0}^{T}={\mathcal{H}}_{ij}(\xi +\alpha _{i}-1,s) \geq 0\) for all \(s\in {\mathbb{N}}_{0,10}\). So, \((H{1})\) holds.

For \(t_{1}\in {\mathbb{N}}_{-\frac{2}{3},\frac{34}{3}}, t_{2}\in {\mathbb{N}}_{-\frac{1}{2},\frac{23}{2}}\), we obtain that \(F_{1},F _{2}\) are decreasing in \(u_{i},\Delta ^{\alpha _{i}}u_{i}\), and

$$\begin{aligned} &F_{1} \bigl(s+\alpha _{1}-1,s+\alpha _{2}-1,(s+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \\ &\quad = \biggl( s+\frac{1}{3} \biggr)^{-x_{1}} \biggl[ \biggl( s+ \frac{1}{3} \biggr)^{\underline{\frac{1}{3}}} \biggr]^{-a_{1}} + \biggl( s+ \frac{1}{2} \biggr)^{-y_{1}} \biggl[ \biggl( s+\frac{1}{2} \biggr) ^{\underline{\frac{1}{2}}} \biggr]^{-b_{1}} \\ &\quad \leq \biggl( s+\frac{1}{3} \biggr)^{- ( x_{1}+\frac{1}{3}a _{1} ) } + \biggl( s+ \frac{1}{2} \biggr)^{- ( y_{1}+ \frac{1}{2}b_{1} ) } \in l^{1}, \\ &F_{2} \bigl(s+\alpha _{1}-1,s+\alpha _{2}-1,(s+\alpha _{1}-1)^{\underline{ \alpha _{1}-1}},(s+\alpha _{2}-1)^{\underline{\alpha _{2}-1}} \bigr) \\ &\quad = \biggl( s+\frac{1}{3} \biggr)^{-x_{2}} \biggl[ \biggl( s+ \frac{1}{3} \biggr)^{\underline{\frac{1}{3}}} \biggr]^{-a_{2}} + \biggl( s+ \frac{1}{2} \biggr)^{-y_{2}} \biggl[ \biggl( s+\frac{1}{2} \biggr) ^{\underline{\frac{1}{2}}} \biggr]^{-b_{2}} \\ &\quad \leq \biggl( s+\frac{1}{3} \biggr)^{- ( x_{2}+\frac{1}{3}a _{2} ) } + \biggl( s+ \frac{1}{2} \biggr)^{- ( y_{2}+ \frac{1}{2}b_{2} ) } \in l^{1}. \end{aligned}$$

Therefore, \((H{2})\) holds.

For all \(\ell \in (0,1)\) and \((t_{1},t_{2},v_{1},v_{2})\in {\mathbb{N}} _{-\frac{2}{3},\frac{34}{3}}\times {\mathbb{N}}_{-\frac{1}{2}, \frac{23}{2}}\times (0,\infty ) \times (0,\infty )\), we have

$$ F_{i} (t_{1},t_{2},\ell v_{1},\ell v_{2} )\leq \ell ^{-\max \{a_{i},b_{i}\}} F_{i} (t_{1},t_{2},v_{1},v_{2} ), $$

Thus, \((H{3})\) holds. Also, \((H{4})\) holds for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}}\) where

$$\begin{aligned} &\varsigma _{1}=0.00181\leq g_{1}(t_{1}) \leq 0.00184= {\mathcal{G}} _{i}\quad\text{and} \\ &\varsigma _{2}=0.00299\leq g_{2}(t_{2}) \leq 0.00318= {\mathcal{G}}_{2}. \end{aligned}$$

Hence, by Theorem 3.1, this problem has at least one positive solution \((u_{1}^{*},u_{2}^{*})\).

For a numerical example to show the existence of a positive solution, we give

$$\begin{aligned} F_{1}(t_{1})=\frac{1}{2}t_{1}^{-\frac{1}{2}}\quad \mbox{and}\quad F_{2}(t_{2})= \frac{1}{500}t_{2}^{-\frac{1}{3}}. \end{aligned}$$

We can find that \(\lambda _{1}=2.1, \lambda _{2}=3.8, \varLambda =0.041\), \({\mathcal{P}(F_{1},F_{2})}=6.697 \) and \({\mathcal{Q}(F_{1},F_{2})}=0.054\), then we have

$$\begin{aligned} &u_{1}(t_{1})=\frac{7.054\varGamma (t_{1}+1)}{\varGamma }(t_{1}+0.666)-1.120 \sum_{s=0}^{t_{1}-\frac{4}{3}}\frac{\varGamma (t_{1}-s)}{2(s+\frac{1}{3})(\frac{1}{2})\varGamma (t_{1}-s-\frac{1}{3})}, \\ & u_{2}(t_{2})=\frac{573.023\varGamma (t_{2}+1)}{\varGamma }(t_{2}+0.500)-1.128 \sum_{s=0}^{t_{2}-\frac{3}{2}}\frac{\varGamma (t_{2}-s)}{(s+\frac{1}{2})( \frac{1}{3})\varGamma (t_{2}-s-\frac{1}{2})} \end{aligned}$$

for \(t_{1}\in {\mathbb{N}}_{-\frac{2}{3},\frac{34}{3}}, t_{2}\in {\mathbb{N}}_{-\frac{1}{2},\frac{23}{2}}\). Therefore, we obtain

$$\begin{aligned} &u_{1} \biggl(-\frac{2}{3} \biggr) =0,\qquad u_{1} \biggl(\frac{1}{3} \biggr) =6.299,\qquad u_{1} \biggl( \frac{4}{3} \biggr) =7.533,\qquad u_{1} \biggl(\frac{7}{3} \biggr) =8.211, \\ &u_{1} \biggl(\frac{10}{3} \biggr) =8.636,\qquad u_{1} \biggl(\frac{13}{3} \biggr) =8.914, \qquad u_{1} \biggl( \frac{16}{3} \biggr) =9.097,\qquad u_{1} \biggl(\frac{19}{3} \biggr) =9.211, \\ &u_{1} \biggl(\frac{22}{3} \biggr) =9.275,\qquad u_{1} \biggl(\frac{25}{3} \biggr) =9.299,\qquad u_{1} \biggl( \frac{28}{3} \biggr) =9.291,\qquad u_{1} \biggl(\frac{31}{3} \biggr) =9.257, \\ &u_{1} \biggl(\frac{34}{3} \biggr) =9.201,\quad\text{and} \\ &u_{2} \biggl(-\frac{1}{2} \biggr) =0,\qquad u_{2} \biggl(\frac{1}{2} \biggr) =507.828,\qquad u_{2} \biggl(\frac{3}{2} \biggr) =761.740,\qquad u_{2} \biggl( \frac{5}{2} \biggr) =952.173, \\ &u_{2} \biggl(\frac{7}{2} \biggr) =1110.866,\qquad u_{2} \biggl(\frac{9}{2} \biggr) =1249.722,\qquad u_{2} \biggl( \frac{111}{2} \biggr) =1347.692, \\ &u_{2} \biggl(\frac{13}{2} \biggr) =1489.247,\qquad u_{2} \biggl(\frac{15}{2} \biggr) =1595.619,\qquad u_{2} \biggl( \frac{17}{2} \biggr) =1695.343, \\ &u_{2} \biggl(\frac{19}{2} \biggr) =1789.526,\qquad u_{2} \biggl(\frac{21}{2} \biggr) =51878.999, \qquad u_{2} \biggl( \frac{23}{2} \biggr) =1964.406. \end{aligned}$$

In Fig. 1, the graphs of solutions \(u_{1}\) and \(u_{2}\) are plotted in a two-dimensional space.

Figure 1
figure1

The graph of \(u_{1}(t_{1})\) where \(t_{1}\in {\mathbb{N}} _{\frac{1}{3},\frac{34}{3}} \) and \(u_{2}(t_{2})\) where \(t_{2}\in {\mathbb{N}}_{\frac{1}{2},\frac{23}{2}}\)

References

  1. 1.

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

  2. 2.

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

  3. 3.

    Baleanu, D., Rezapour, S., Salehi, S.: A k-dimensional system of fractional finite difference equations. Abstr. Appl. Anal. 2014, Article ID 312578 (2014)

  4. 4.

    Agarwal, R.P., Baleanu, D., Rezapour, S., Salehi, S.: The existence of solutions for some fractional finite difference equations via sum boundary conditions. Adv. Differ. Equ. 2014, 282 (2014)

  5. 5.

    Baleanu, D., Rezapour, S., Salehi, S.: On some self-adjoint fractional finite difference equations. J. Comput. Anal. Appl. 19, 59–67 (2015)

  6. 6.

    Rezapour, S., Salehi, S.: On the existence of solution for a k-dimensional system of three points Nabla fractional finite difference equations. Bull. Iran. Math. Soc. 41(6), 1433–1444 (2015)

  7. 7.

    Baleanu, D., Rezapour, S., Salehi, S.: A fractional finite difference inclusion. J. Comput. Anal. Appl. 20(5), 834–842 (2016)

  8. 8.

    Ghorbanian, V., Rezapour, S.: On a system of fractional finite difference inclusions. Adv. Differ. Equ. 2017, 325 (2017)

  9. 9.

    Ghorbanian, V., Rezapour, S.: A two-dimensional system of Delta–Nabla fractional difference inclusions. Novi Sad J. Math. 47(1), 143–163 (2017)

  10. 10.

    Ghorbanian, V., Rezapour, S., Salehi, S.: On the existence of solution for a sum fractional finite difference inclusion. In: Tas, K., Baleanu, D., Tenreiro Machado, J.A. (eds.) Mathematical Methods in Engineering, pp. 145–160. Springer, Berlin (2018)

  11. 11.

    Wu, G.C., Baleanu, D., Luo, W.H.: Lyapunov functions for Riemann–Liouville-like fractional difference equations. Appl. Math. Comput. 314, 228–236 (2017)

  12. 12.

    Wu, G.C., Baleanu, D.: Stability analysis of impulsive fractional difference equations. Fract. Calc. Appl. Anal. 21, 354–375 (2018)

  13. 13.

    Huang, L.L., Baleanu, D., Mo, Z.W., Wu, G.C.: Fractional discrete-time diffusion equation with uncertainty: applications of fuzzy discrete fractional calculus. Physica A 508, 166–175 (2018)

  14. 14.

    Bai, Y.R., Baleanu, D., Wu, G.C.: A novel shuffling technique based on fractional chaotic maps. Optik 168, 553–562 (2018)

  15. 15.

    Baleanu, D., Mousalou, A., Rezapour, S.: A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo–Fabrizio derivative. Adv. Differ. Equ. 2017, 51 (2017)

  16. 16.

    Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017, 145 (2017)

  17. 17.

    Aydogan, S.M., Baleanu, D., Mousalou, A., Rezapour, S.: On approximate solutions for two higher-order Caputo–Fabrizio fractional integro-differential equations. Adv. Differ. Equ. 2017, 221 (2017)

  18. 18.

    Kojabad, E.A., Rezapour, S.: Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials. Adv. Differ. Equ. 2017, 351 (2017)

  19. 19.

    Aghazadeh, N., Ravash, E., Rezapour, S.: Numerical solutions for a k-dimensional system of fractional differential equations by using Alpert’s multiwavelets. J. Adv. Math. Stud. 10(3), 295–313 (2017)

  20. 20.

    Aghazadeh, N., Ravash, E., Rezapour, S.: Existence results and numerical solutions for a multi-term fractional integro-differential equation. Kragujev. J. Math. 43(3), 451–463 (2019)

  21. 21.

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

  22. 22.

    Atici, F.M., Eloe, P.W.: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2, 2 (2007)

  23. 23.

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

  24. 24.

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

  25. 25.

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

  26. 26.

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

  27. 27.

    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)

  28. 28.

    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)

  29. 29.

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

  30. 30.

    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)

  31. 31.

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

  32. 32.

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

  33. 33.

    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)

  34. 34.

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

  35. 35.

    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)

  36. 36.

    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)

  37. 37.

    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)

  38. 38.

    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)

  39. 39.

    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)

  40. 40.

    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)

  41. 41.

    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)

  42. 42.

    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)

  43. 43.

    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)

  44. 44.

    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)

  45. 45.

    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)

  46. 46.

    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)

  47. 47.

    Chasreechai, S., Sitthiwirattham, T.: On nonlocal boundary value problems for hybrid fractional sum-difference equations involving different orders. J. Nonlinear Funct. Anal. 2018, Article ID 15 (2018)

  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)

  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)

  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)

  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)

  52. 52.

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

  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)

  54. 54.

    Griffel, D.H.: Applied Functional Analysis. Ellis Horwood Publishers, Chichester (1981)

Download references

Availability of data and materials

Not applicable.

Funding

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract No. KMUTNB-ART-60-39. The last author would also like to thank Suan Dusit University for the support.

Author information

The authors declare that they carried out all the work in this manuscript and read and approved the final manuscript.

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.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

MSC

  • 39A05
  • 39A12

Keywords

  • Positive solution
  • Singularity
  • Fractional difference equations
  • Boundary value problem