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Existence and multiplicity of positive solutions to a system of fractional difference equations with parameters

Abstract

We consider a fractional difference-sum boundary problem for a system of fractional difference equations with parameters. Using the Banach fixed point theorem, we prove the existence and uniqueness of solutions. We also prove the existence of at least one and two solutions by using the Krasnoselskii’s fixed point theorem for a cone map. Finally, we give some examples to illustrate our results.

Introduction

Fractional difference calculus is quite new to researchers. It has been used in mathematical models that explain many real-life situations, for example, economics, electrical networks, and queuing problems (see [13] and the references therein). Basic definitions and properties of fractional difference calculus were presented in [4], and discrete fractional boundary value problems have been found in [533]. However, the studies of a system of fractional boundary value problems are quite rare (see [3442]).

For an extension of the research work in this area, in this paper, we study the following system of fractional difference equations with parameters:

$$\begin{aligned} \begin{gathered} -\Delta ^{\alpha _{1}} u_{1}(t) = \lambda _{1}F_{1} \bigl[t+\alpha _{1}-1,t+ \alpha _{2}-1,u_{1}(t+\alpha _{1}-1), u_{2}(t+\alpha _{2}-1) \bigr], \\ -\Delta ^{\alpha _{2}} u_{2}(t) = \lambda _{2}F_{2} \bigl[t+\alpha _{1}-1,t+ \alpha _{2}-1,u_{1}(t+ \alpha _{1}-1), u_{2}(t+\alpha _{2}-1) \bigr], \end{gathered} \end{aligned}$$
(1.1)

subject to nonlocal fractional difference-sum boundary conditions of the form

$$\begin{aligned} \begin{gathered} \Delta ^{-\beta _{1}} u_{1}( \alpha _{1}+\beta _{1}-3) = \Delta ^{ \gamma _{1}} u_{1}(\alpha _{1}-\gamma _{1}-2) = 0, \\ \Delta ^{-\beta _{2}} u_{2}(\alpha _{2}+\beta _{2}-3) = \Delta ^{ \gamma _{2}} u_{2}(\alpha _{2}-\gamma _{2}-2) = 0, \\ u_{1}(T+\alpha _{1}) = \chi _{1} u_{1}(\eta _{1}), \quad \eta _{1} \in \mathbb{N}_{\alpha _{1}-2,T+\alpha _{1}-1}, \\ u_{2}(T+\alpha _{2}) = \chi _{2}u_{2}( \eta _{2}), \quad \eta _{2} \in \mathbb{N}_{\alpha _{2}-2,T+\alpha _{2}-1}, \end{gathered} \end{aligned}$$
(1.2)

where \(t\in \mathbb{N}_{0,T}:=\{0,1,\ldots,T\}\), \(\alpha _{i}\in (2,3]\), \(\beta _{i},\gamma _{i}\in (0,1)\), \(i=1,2\). Moreover, we suppose that the following assumptions hold:

$$\begin{aligned} &(A1) \quad F_{i}\in C \bigl(\mathbb{N}_{\alpha _{1}-1,T+\alpha _{1}-1} \times \mathbb{N}_{\alpha _{2}-1,T+\alpha _{2}-1}\times (0,\infty ) \times (0,\infty ), (0,\infty ) \bigr); \\ &\hphantom{(A1)\quad } \text{where }\mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}:=\{ \alpha _{i}-1, \alpha _{i}, \alpha _{i}+1, \ldots, T+ \alpha _{i}-1 \}; \\ &(A2) \quad 0< \chi _{i}\eta _{i}^{\underline{\alpha _{i}-1}} < (T+\alpha _{i})^{ \underline{\alpha _{i}-1}}; \\ &(A3) \quad \lambda _{1},\lambda _{2} \text{ are positive parameters}; \\ &(A4) \quad F_{i} ( t_{1},t_{2},u_{1},u_{2} ) >0 \text{ for } u_{1},u_{2}>0, t_{i} \in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}. \end{aligned}$$

For convenience, we use the following notations (\(i=1,2\)):

$$\begin{aligned}& F_{i}^{0} = \lim_{u_{1},u_{2}\rightarrow 0^{+}} \biggl[ \max _{ t_{i} \in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}} \frac{F_{i} ( t_{1},t_{2},u_{1},u_{2} )}{u_{1}+u_{2}} \biggr], \\& F_{i}^{\infty } = \lim_{u_{1},u_{2}\rightarrow \infty } \biggl[ \min _{ t_{i} \in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}} \frac{F_{i} ( t_{1},t_{2},u_{1},u_{2} )}{u_{1}+u_{2}} \biggr]. \end{aligned}$$

We organize our paper as follows. In Sect. 2, we recall some definitions and basic lemmas and present some properties of the fractional difference operators. In this section, we also derive a representation for the solution to (1.1)–(1.2) by converting the problem to equivalent summation equations. In Sect. 3, we prove the existence and uniqueness result for problem (1.1)–(1.2) by using the Banach fixed point theorem. In Sect. 4, we prove the existence of at least one and two solutions for problem (1.1)–(1.2) by using the Krasnoselskii fixed point theorem in a cone map. In the last section, we provide some examples to illustrate our results.

Theorem 1.1

([43], Krasnoselskii’s fixed point theorem)

Let E be a Banach space, and let \(K\subset E\)be a cone. Let \(\varOmega _{1}\)and \(\varOmega _{2}\)be open subsets of E such that \(0\in \varOmega _{1}\)and \(\overline{\varOmega }_{1}\subset \varOmega _{2}\), and let

$$ A:K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})\longrightarrow K $$

be a completely continuous operator such that

  1. (i)

    \(\|Au\|\leqslant \|u\|\), \(u\in K\cap \partial \varOmega _{1}\), and \(\|Au\|\geqslant \|u\|\), \(u\in K\cap \partial \varOmega _{2}\), or

  2. (ii)

    \(\|Au\|\geqslant \|u\|\), \(u\in K\cap \partial \varOmega _{1}\), and \(\|Au\|\leqslant \|u\|\), \(u\in K\cap \partial \varOmega _{2}\).

Then A has a fixed point in \(K\cap (\overline{\varOmega }_{2}\setminus \varOmega _{1})\).

Theorem 1.2

([44], 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 1.3

([44])

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

Preliminaries

In this section, 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 \(t^{\underline{\alpha }}=0\).

Definition 2.2

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

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

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 αth-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 \(0\leq {N-1}<\alpha \leq N\).

Lemma 2.1

([5])

For \(0\leq N-1<\alpha \leq N\),

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

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

The following is a solution of a linear variant of the boundary value problem (1.1).

Lemma 2.2

Suppose that \((A1)\)\((A3)\)hold. For \(i\in \{1,2\}\), let \(\alpha _{i}\in (2, 3] \)and \(\beta _{i}\), \(\gamma _{i}\in (0, 1)\)be given constants, and let \(h_{i}\in C (\mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}, \mathbb{R}^{+} )\)be given functions. 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)
$$\begin{aligned}& \Delta ^{-\beta _{i}}u_{i}(\alpha _{i}+\beta _{i}-3) = \Delta ^{ \gamma _{i}}u_{i}(\alpha _{i}-\gamma _{i}-2) = 0, \end{aligned}$$
(2.2)
$$\begin{aligned}& u_{i}(T+\alpha _{i}) = \chi _{i}u_{i}( \eta _{i}), \eta _{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}-1} \end{aligned}$$
(2.3)

has the unique solution

$$\begin{aligned} u_{i}(t_{i})&= \frac{t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] \varGamma (\alpha _{i}) } \Biggl\{ \sum _{s=0}^{T}\bigl(T+\alpha _{i}-\sigma (s) \bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1) \\ &\quad {}- \chi _{i} \sum_{s=0}^{\eta _{i}-\alpha _{i} } \bigl(\eta _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+ \alpha _{i}-1) \Biggr\} \\ &\quad {}-\frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t_{i}-\alpha _{i}} \bigl(t_{i}- \sigma (s)\bigr)^{\underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1) \end{aligned}$$
(2.4)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\).

Proof

Using Lemma 2.1 and the fractional sum of order \(\alpha _{i}\in (2,3]\), \(i\in \{1,2\}\), for (2.1), we obtain

$$\begin{aligned} u_{i}(t_{i})=C_{1i}t_{i}^{\underline{\alpha _{i}-1}}+C_{2i}t_{i}^{ \underline{\alpha _{i}-2}}+C_{3i}t_{i}^{\underline{\alpha _{i}-3}} - \frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t_{i}-\alpha _{i}} \bigl(t_{i}- \sigma (s)\bigr)^{\underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1) \end{aligned}$$
(2.5)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\).

Next, applying the fractional sum of order \(\beta _{i}\in (0,1)\), \(i\in \{1,2\}\), to (2.5), we have

$$\begin{aligned} \Delta ^{-\beta _{i}}u_{i}(t_{i}) =& \frac{1}{\varGamma (\beta _{i})} \sum_{s=\alpha _{i}-3}^{t_{i}-\beta _{i}} \bigl(t_{i}-\sigma (s)\bigr)^{ \underline{\beta _{i}-1}} \bigl[ C_{1i}s^{\underline{\alpha _{i}-1}}+C_{2i}s^{ \underline{\alpha _{i}-2}}+C_{3i}s^{\underline{\alpha _{i}-3}} \bigr] \\ &{}-\frac{1}{\varGamma (\alpha _{i})\varGamma (\beta _{i})} \sum_{r=\alpha _{i}}^{t_{i}- \beta _{i}}\sum _{s=0}^{r-\alpha _{i}} \bigl(t_{i}-\sigma (r) \bigr)^{ \underline{\beta _{i}-1}} \bigl(r-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}} h_{i}(s+ \alpha _{i}-1) \end{aligned}$$
(2.6)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-3,T+\alpha _{i}+\beta _{i}}\).

Taking the fractional difference of order \(\gamma _{i}\in (0,1)\), \(i\in \{1,2\}\), of (2.5), we obtain

$$\begin{aligned} \Delta ^{\gamma _{i}}u_{i}(t_{i}) =& \frac{1}{\varGamma (-\gamma _{i})} \sum_{s=\alpha _{i}-3}^{t_{i}+\gamma _{i}} \bigl(t_{i}-\sigma (s)\bigr)^{ \underline{-\gamma _{i}-1}} \bigl[ C_{1i}s^{\underline{\alpha _{i}-1}}+C_{2i}s^{ \underline{\alpha _{i}-2}}+C_{3i}s^{\underline{\alpha _{i}-3}} \bigr] \\ &{}-\frac{1}{\varGamma (\alpha _{i})\varGamma (-\gamma _{i})} \sum_{r= \alpha _{i}}^{t_{i}+\gamma _{i}} \sum_{s=0}^{r-\alpha _{i}} \bigl(t_{i}- \sigma (r)\bigr)^{\underline{-\gamma -1}} \bigl(r-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1) \end{aligned}$$
(2.7)

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

Using the boundary condition \(\Delta ^{-\beta _{i}}u_{i}(\alpha _{i}+\beta _{i}-3)=0\) in (2.2), we find that \(C_{3i}=0\).

Then we have

$$\begin{aligned} u_{i}(t_{i})=C_{1i}t_{i}^{\underline{\alpha _{i}-1}}+C_{2i}t_{i}^{ \underline{\alpha _{i}-2}} -\frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t_{i}- \alpha _{i}} \bigl(t_{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}} h_{i}(s+ \alpha _{i}-1). \end{aligned}$$
(2.8)

From the boundary condition \(\Delta ^{\gamma _{i}}u_{i}(\alpha _{i}-\gamma _{i}-2)=0\) in (2.2) we have \(C_{2i}=0\).

Therefore

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

By using the boundary condition (2.3) we obtain

$$\begin{aligned} C_{1i}& = \frac{1}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] \varGamma (\alpha _{i}) } \Biggl\{ \sum _{s=0}^{T}\bigl(T+\alpha _{i}-\sigma (s) \bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+\alpha _{i}-1) \\ &\quad {}- \chi _{i} \sum_{s=0}^{\eta _{i}-\alpha _{i} } \bigl(\eta _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} h_{i}(s+ \alpha _{i}-1) \Biggr\} . \end{aligned}$$
(2.10)

Finally, substituting \(C_{1i}\) into (2.9), we obtain (2.4). The proof is complete. □

Corollary 2.1

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

$$\begin{aligned} u_{i}(t_{i}) = \sum _{s=0}^{T} G_{i}(t_{i},s) h_{i}(s+\alpha _{i}-1) \end{aligned}$$
(2.11)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\), \(i=1,2\), where

$$ G_{i}(t_{i},s):=\frac{1}{\varGamma (\alpha _{i})} \textstyle\begin{cases} g_{1}(t_{i},s), & s\in \mathbb{N}_{0,t_{i}-\alpha _{i}}\cap \mathbb{N}_{0,\eta _{i}-\alpha _{i}}, \\ g_{2}(t_{i},s), & s\in \mathbb{N}_{t_{i}-\alpha _{i}+1,\eta _{i}- \alpha _{i}}, \\ 3_{2}(t_{i},s), & s\in \mathbb{N}_{\eta _{i}-\alpha _{i}+1,t_{i}- \alpha _{i}}, \\ g_{4}(t_{i},s), & s\in \mathbb{N}_{t_{i}-\alpha _{i}+1,T}\cap \mathbb{N}_{\eta _{i}-\alpha _{i}+1,T}, \end{cases}$$
(2.12)

with

$$\begin{aligned} \begin{gathered} g_{1}(t_{i},s) := \frac{t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] } \bigl\{ \bigl(T+\alpha _{i}-\sigma (s) \bigr)^{\underline{\alpha _{i}-1}} - \chi _{i}\bigl(\eta _{i}-\sigma (s) \bigr)^{\underline{\alpha _{i}-1}} \bigr\} \\ \hphantom{g_{1}(t_{i},s) :=}{}-\bigl(t_{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}}, \\ g_{2}(t_{i},s) := \frac{t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] } \bigl\{ \bigl(T+ \alpha _{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}} - \chi _{i}\bigl( \eta _{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}} \bigr\} , \\ g_{3}(t_{i},s) := \frac{t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] } \bigl(T+\alpha _{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}}-\bigl(t_{i}- \sigma (s) \bigr)^{\underline{\alpha _{i}-1}}, \\ g_{4}(t_{i},s) := \frac{t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] } \bigl(T+\alpha _{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}}. \end{gathered} \end{aligned}$$
(2.13)

Green’s function (2.12) has the following properties.

Proposition 2.1

([13])

For \(i=1,2\), let \(G_{i}(t_{i},s)\)be Green’s function given in (2.12)(2.13). Then for all \((t_{i},s)\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\times \mathbb{N}_{0,T}\),

$$ G_{i}(t_{i},s)\geq 0. $$

Proposition 2.2

([13])

For \(i=1,2\), let \(G_{i}(t_{i},s)\)be Green’s function given in (2.12)(2.13). Suppose that for given \(\eta _{i}\in \mathbb{N}_{\alpha _{i}-2,T+\alpha _{i}-1}\)and \(\alpha _{i} \in (2,3]\), \(\chi _{i}\)satisfies the inequality

$$\begin{aligned} 0\leq \chi _{i}\leq \min_{(t_{i},s)\in \mathbb{N}_{\alpha _{i}-3,T+ \alpha _{i}}\times \mathbb{N}_{0,T}} \biggl\lbrace \frac{(T+\alpha _{i})^{\underline{\alpha _{i}-1}}}{\eta _{i}^{\underline{\alpha _{i}-1}}} - \frac{t_{i}^{\underline{\alpha _{i}-2}}(T+\alpha _{i}-\sigma (s))^{\underline{\alpha _{i}-1}} }{\eta _{i}^{\underline{\alpha _{i}-1}}(t_{i}-\sigma (s))^{\underline{\alpha _{i}-2}} } \biggr\rbrace . \end{aligned}$$

Then

$$\begin{aligned} \max_{(t_{i},s)\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\times \mathbb{N}_{0,T}} G_{i}(t_{i},s)=G_{i}(s+ \alpha -1,s). \end{aligned}$$

Proposition 2.3

([13])

For \(i=1,2\), let \(G_{i}(t_{i},s)\)be Green’s function given in (2.12)(2.13). Then

$$\begin{aligned} \min_{t_{i}\in [\frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+\alpha _{i})] } G_{i}(t_{i},s)\geq \theta _{i} \max_{(t_{i},s)\in \mathbb{N}_{ \alpha _{i}-3,T+\alpha _{i}}\times \mathbb{N}_{0,T}}G_{i}(t_{i},s)= \theta _{i} G_{i}(s+\alpha -1,s), \end{aligned}$$

where

$$\begin{aligned} \theta _{i}&:=\min \biggl\lbrace \frac{ ( \frac{1}{4}(T+\alpha _{i}) )^{\underline{\alpha _{i}-1}}}{ ( T+\alpha _{i} )^{\underline{\alpha _{i}-1}}} , \\ &\quad \frac{1}{ ( \frac{3}{4}(T+\alpha _{i}) )^{\underline{\alpha _{i}-1}}} \biggl[ \biggl( \frac{3}{4}(T+\alpha _{i}) \biggr)^{ \underline{\alpha _{i}-1}} - \frac{ ( \frac{3}{4}(T+\alpha _{i}) )^{\underline{\alpha _{i}-1}} ( T+\alpha _{i} )^{\underline{\alpha _{i}-1}}}{ ( T+\alpha _{i} )^{\underline{\alpha _{i}-1}}} \biggr] \biggr\rbrace , \end{aligned}$$
(2.14)

and \(\theta _{i}\)satisfy the inequality \(0<\theta _{i}<1\).

Existence and uniqueness of solution

In this section, we apply the Banach fixed point theorem to prove the existence and uniqueness result for problem (1.1)–(1.2). For each \(i,j \in \{1,2\}\), we let \(E_{i}=C ( \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}, \mathbb{R} )\) be the Banach space for all functions on \(\mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\) with the norm \(\|u_{i}\|=\max_{t_{i} \in \mathbb{N}_{\alpha _{i}-3,T+ \alpha _{i}}}|u_{i}(t_{i})|\). The product space \(\mathcal{U}=E_{1}\times E_{2}\) is a Banach space with the norm

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

Next, define the operator \({\mathcal{T}}:{\mathcal{U}}\rightarrow {\mathcal{U}}\) by

$$\begin{aligned} \bigl({\mathcal{T}}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) := \bigl( \bigl({ \mathcal{T}}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) , \bigl({\mathcal{T}}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr), \end{aligned}$$
(3.1)

and for \(i,j\in \{1,2\}\), \(i\neq j\),

$$\begin{aligned} &\bigl({\mathcal{T}}_{i}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \\ &\quad := \frac{t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] \varGamma (\alpha _{i}) } \\ &\qquad {}\times \Biggl\{ \sum_{s=0}^{T} \bigl(T+\alpha _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} F_{i} \bigl[ s+\alpha _{i}-1,t_{j},u_{i}(s+ \alpha _{i}-1),u_{j}(t_{j}) \bigr] \\ &\qquad {}- \chi _{i} \sum_{s=0}^{\eta _{i}-\alpha _{i} } \bigl(\eta _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} F_{i} \bigl[ s+\alpha _{i}-1,t_{j},u_{i}(s+ \alpha _{i}-1),u_{j}(t_{j}) \bigr] \Biggr\} \\ &\qquad {}-\frac{1}{\varGamma (\alpha _{i})}\sum_{s=0}^{t_{i}-\alpha _{1}} \bigl(t_{i}- \sigma (s)\bigr)^{\underline{\alpha _{i}-1}} F_{i} \bigl[ s+ \alpha _{i}-1,t_{j},u_{i}(s+ \alpha _{i}-1),u_{j}(t_{j}) \bigr] \end{aligned}$$
(3.2)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\) and \(t_{j}\in \mathbb{N}_{\alpha _{j}-1,T+\alpha _{j}-1}\). Obviously, problem (1.1)-(1.2) has a solution if and only if the operator \({\mathcal{T}}\) has a fixed point.

Theorem 3.1

Suppose that \((A1)\)\((A4)\)hold. In addition, suppose there exist constants \(M_{i},N_{i}>0\)for \(i=1,2\)such that

$$ \bigl\vert F_{i} [ t_{1},t_{2},u_{1},u_{2} ] -F_{i} [ t_{1},t_{2},v_{1},v_{2} ] \bigr\vert \leq M_{i} \vert u_{1}-v_{1} \vert +N_{i} \vert u_{2}-v_{2} \vert $$
(3.3)

for all \(t_{i}\in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}\)and \((u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{U}\). If

$$\begin{aligned} L_{1}\varOmega _{1}+L_{2} \varOmega _{2} < &1, \end{aligned}$$
(3.4)

where

$$\begin{aligned}& L_{i} = \max \{M_{i},N_{i}\}, \\& \varOmega _{i} = \frac{\lambda _{i}}{\varGamma (\alpha _{i}+1)} \biggl[ \biggl\vert \frac{(T+\alpha _{i})^{\underline{\alpha _{i}}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}}}}{(T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}}} \biggr\vert (T+\alpha _{i})^{\underline{\alpha _{i}-1}} +(T+ \alpha _{i})^{ \underline{\alpha _{i}}} \biggr], \end{aligned}$$

then problem (1.1)(1.2) has a unique solution.

Proof

Let us prove that \({\mathcal{T}}\) is a contraction mapping. For \(i,j\in \{1,2\}\), \(i\neq j\), denote

$$\begin{aligned} {\mathcal{F}}_{i} \vert u-v \vert (s,t_{j})&:= \bigl\vert F_{i} \bigl[ s+\alpha _{i}-1,t_{j},u_{i}(s+ \alpha _{i}-1),u_{j}(t_{j}) \bigr] \\ &\quad {} -F_{i} \bigl[ s+\alpha _{i}-1,t_{j},v_{i}(s+ \alpha _{i}-1),v_{j}(t_{j}) \bigr] \bigr\vert . \end{aligned}$$

For \(t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\), \(t_{j}\in \mathbb{N}_{\alpha _{j}-1,T+\alpha _{j}-1}\), and \((u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{U}\), we find that

$$\begin{aligned}& \bigl\vert \bigl({\mathcal{T}}_{i}(u_{1},u_{2}) \bigr) (t_{1},t_{2})- \bigl({\mathcal{T}}_{i}(v_{1},v_{2}) \bigr) (t_{1},t_{2}) \bigr\vert \\& \quad \leq \frac{\lambda _{i}t_{i}^{\underline{\alpha _{i}-1}}}{ [ (T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} ] \varGamma (\alpha _{i}) } \Biggl\vert \sum_{s=0}^{T} \bigl(T+\alpha _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} { \mathcal{F}}_{i} \vert u-v \vert (s,t_{j}) \\& \qquad {}- \chi _{i} \sum_{s=0}^{\eta _{i}-\alpha _{i} } \bigl(\eta _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} {\mathcal{F}}_{i} \vert u-v \vert (s,t_{j}) \Biggr\vert + \frac{\lambda _{i}}{\varGamma (\alpha _{i})}\sum_{s=0}^{t_{i}-\alpha _{1}} \bigl(t_{i}- \sigma (s)\bigr)^{\underline{\alpha _{i}-1}} {\mathcal{F}}_{i} \vert u-v \vert (s,t_{j}) \\& \quad \leq \frac{\lambda _{i} [ M_{i} \vert u_{1}+v_{1} \vert +N_{i} \vert u_{2}-v_{2} \vert ] }{\varGamma (\alpha _{i})} \Biggl[ \frac{(T+\alpha _{i})^{\underline{\alpha _{i}-1}}}{ t_{i}^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}} } \\& \qquad {}\times \Biggl\vert \sum_{s=0}^{T} \bigl(T+\alpha _{i}-\sigma (s)\bigr)^{ \underline{\alpha _{i}-1}} - \chi _{i} \sum_{s=0}^{\eta _{i}-\alpha _{i} }\bigl(\eta _{i}-\sigma (s)\bigr)^{\underline{\alpha _{i}-1}} \Biggr\vert + \sum _{s=0}^{t_{i}- \alpha _{1}}\bigl(t_{i}-\sigma (s) \bigr)^{\underline{\alpha _{i}-1}} \Biggr] \\& \quad \leq \frac{\lambda _{i}L_{i} [ \vert u_{1}+v_{1} \vert + \vert u_{2}-v_{2} \vert ] }{\varGamma (\alpha _{i}+1)} \biggl[ \biggl\vert \frac{(T+\alpha _{i})^{\underline{\alpha _{i}}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}}}}{(T+\alpha _{i})^{\underline{\alpha _{i}-1}}-\chi _{i} \eta _{i}^{\underline{\alpha _{i}-1}}} \biggr\vert (T+\alpha _{i})^{\underline{\alpha _{i}-1}} +(T+\alpha _{i})^{ \underline{\alpha _{i}}} \biggr] \\& \quad \leq L_{i}\varOmega _{i} \bigl\Vert (u_{1},v-1,u_{2}-v_{2}) \bigr\Vert _{ \mathcal{U}}. \end{aligned}$$

Therefore

$$\begin{aligned} \bigl\Vert \bigl({\mathcal{T}}(u_{1},u_{2}) \bigr)- \bigl({\mathcal{T}}(v_{1},v_{2}) \bigr) \bigr\Vert _{ \mathcal{U}} =& \bigl\Vert \bigl( {\mathcal{T}}_{1}(u_{1},u_{2})-{ \mathcal{T}}_{1}(v_{1},v_{2}) , { \mathcal{T}}_{2}(u_{1},u_{2})-{ \mathcal{T}}_{2}(v_{1},v_{2}) \bigr) \bigr\Vert _{ \mathcal{U}} \\ =& \bigl\Vert {\mathcal{T}}_{1}(u_{1},u_{2})-{ \mathcal{T}}_{1}(v_{1},v_{2}) \bigr\Vert + \bigl\Vert {\mathcal{T}}_{2}(u_{1},u_{2})-{ \mathcal{T}}_{2}(v_{1},v_{2}) \bigr\Vert \\ \leq & ( L_{1}\varOmega _{1}+L_{2}\varOmega _{2} ) \bigl\Vert (u_{1},v-1,u_{2}-v_{2}) \bigr\Vert _{ \mathcal{U}}. \end{aligned}$$

By (3.4) we find that \({\mathcal{T}}\) is a contraction. Therefore by the Banach fixed point theorem \({\mathcal{T}}\) has a fixed point, which is a unique solution of problem (1.1)–(1.2). □

Existence and multiplicity of positive solutions

In this section, we apply the Krasnoselskii fixed point theorem for a cone map to prove the existence and multiplicity of positive solutions for problem (1.1)–(1.2). The product space \(\mathcal{U}\) and the norm \(\|(u_{1},u_{2})\|_{\mathcal{U}}\) are defined in Sect. 3. Moreover, we define the cone \(\mathcal{P} \subset \mathcal{U}\) as

$$\begin{aligned} \mathcal{P}:= \Bigl\lbrace (u_{1},u_{2})\in \mathcal{U}: u_{1},u_{2} \geq 0 \text{ and } \min_{t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+ \alpha _{i}}} \bigl[ u_{1}(t_{1})+u_{2}(t_{2}) \bigr] \geq \theta \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}} \Bigr\rbrace , \end{aligned}$$

where \(\theta :=\min \{\theta _{1},\theta _{2}\}\) with \(\theta _{1}\), \(\theta _{2}\) defined in (2.14).

Next, by Corollary 2.1 we also define the operator \({\mathcal{T}}:{\mathcal{U}}\rightarrow {\mathcal{U}}\) by

$$\begin{aligned} \bigl({\mathcal{T}}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) =& \bigl( \bigl({ \mathcal{T}}_{1}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) , \bigl({\mathcal{T}}_{2}(u_{1},u_{2}) \bigr) (t_{1},t_{2}) \bigr) \end{aligned}$$
(4.1)

and for \(i,j\in \{1,2\}\), \(i\neq j\),

$$\begin{aligned} \bigl({\mathcal{T}}_{i}(u_{1},u_{2}) \bigr) (t_{1},t_{2}):=& \lambda _{i} \sum _{s=0}^{T}G_{i}(t_{i},s) F_{i} \bigl[ s+\alpha _{i}-1,t_{j},u_{i}(s+ \alpha _{i}-1),u_{j}(t_{j}) \bigr] \end{aligned}$$
(4.2)

for \(t_{i}\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\) and \(t_{j}\in \mathbb{N}_{\alpha _{j}-1,T+\alpha _{j}-1}\). The positive solutions of problem (1.1)–(1.2) and the fixed points of the operator \({\mathcal{T}}\) in the cone \(\mathcal{P}\) coincide.

Lemma 4.1

If \((A1)\)\((A4)\)hold, then \(\mathcal{T}(\mathcal{P})\subset \mathcal{P}\), and \(\mathcal{T}:\mathcal{P}\rightarrow \mathcal{P}\)is a completely continuous operator.

Proof

The continuity of \(\mathcal{T}\) is obvious. To prove \(\mathcal{T}(\mathcal{P})\subset \mathcal{P}\), we choose \((u_{1},u_{2}) \in \mathcal{P}\). Since, for \(i=1,2\), \(G_{i}(t_{i},s)\leq G_{i}(s+\alpha _{i}-1,s)\) for \(s \in \mathbb{N}_{0,T} \) and \(G_{i}(t_{i},s)\geq \theta _{i} G_{i}(s+\alpha -1,s)\) for \(t_{i} \in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+\alpha _{i}) ] \), we have

$$\begin{aligned} &\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+ \alpha _{i}) ] } {\mathcal{T}}_{1}(u_{1},u_{2}) (t_{1},t_{2}) \\ &\quad \geq \lambda _{1}\theta _{1} \sum _{s=0}^{T} G_{i}(s+\alpha _{i}-1,s) F_{1} \bigl[s+\alpha _{i}-1,t_{2},u_{1}(s+ \alpha _{i}-1),u_{2}(t_{2}) \bigr] \\ &\quad \geq \theta _{1} \bigl\Vert {\mathcal{T}}_{1}(u_{1},u_{2}) \bigr\Vert . \end{aligned}$$

Similarly, \(\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}), \frac{3}{4}(T+\alpha _{i}) ] } {\mathcal{T}}_{2}(u_{1},u_{2})(t_{1},t_{2}) \geq \theta _{2}\|{\mathcal{T}}_{2}(u_{1},u_{2})\|\).

Thus

$$\begin{aligned} &\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+ \alpha _{i}) ] } \bigl( {\mathcal{T}}_{1}(u_{1},u_{2}) (t_{1},t_{2})+{ \mathcal{T}}_{2}(u_{1},u_{2}) (t_{1},t_{2}) \bigr) \\ &\quad \geq \min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}), \frac{3}{4}(T+\alpha _{i}) ] } {\mathcal{T}}_{1}(u_{1},u_{2}) (t_{1},t_{2}) + \min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+ \alpha _{i}) ] } { \mathcal{T}}_{2}(u_{1},u_{2}) (t_{1},t_{2}) \\ &\quad \geq \theta \bigl\Vert \bigl( {\mathcal{T}}_{1}(u_{1},u_{2}),{ \mathcal{T}}_{2}(u_{1},u_{2}) \bigr) \bigr\Vert _{{\mathcal{U}}}. \end{aligned}$$

Since \(G_{i}(t_{i},s)\geq 0\) for all \((t_{i},s)\in \mathbb{N}_{\alpha _{i}-3,T+\alpha _{i}}\times \mathbb{N}_{0,T}\) and (A1)–(A4) hold, we conclude that \(\mathcal{T}(\mathcal{P})\subset \mathcal{P}\). It is easy to show that \(\mathcal{T}\) is uniformly bounded. By the Arzelá–Ascoli theorem we can conclude that \(\mathcal{T}:\mathcal{P}\rightarrow \mathcal{P}\) is a completely continuous operator. □

Theorem 4.1

Suppose assumptions \((A1)\)\((A4)\)hold. Then for \(\lambda _{1},\lambda _{2}>0\), problem (1.1)(1.2) has at least one positive solution in the following cases:

$$\begin{aligned} &(a) \quad F_{1}^{0}=F_{2}^{0}=0, \textit{ and either }F_{1}^{\infty }=\infty \textit{ or } F_{2}^{\infty }=\infty \textit{ (superlinear);} \\ &(b) \quad F_{1}^{\infty }=F_{2}^{\infty }=0, \textit{ and either }F_{1}^{0}= \infty \textit{ or } F_{2}^{0}=\infty \textit{ (sublinear)}. \end{aligned}$$

Proof

\((a)\) Since, \(F_{i}^{0}=0\), \(i=1,2\), we choose \(K_{1}>0\) such that \(F_{i}[t_{1},t_{2},u_{1},u_{2}]\leq \varepsilon (u_{1}+u_{2})\) for \(0< u_{1}+u_{2}\leq K_{1}\) and \(t_{i}\in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}\), where the constant \(\varepsilon >0\) satisfies

$$\begin{aligned} 2\varepsilon \lambda _{i}\sum_{s=0}^{T} G_{i}(s+\alpha _{i}-1,s) \leq 1. \end{aligned}$$

Set \(\varOmega _{1}= \lbrace (u_{1},u_{2}) \in \mathcal{U}: \|(u_{1},u_{2}) \|_{\mathcal{U}}< K_{1} \rbrace \). If \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{1}\) and \(\|(u_{1},u_{2})\|_{\mathcal{U}}=K_{1}\), then we have

$$\begin{aligned} \mathcal{T}_{1}(u_{1},u_{2}) (t_{1},t_{2}) \leq &\lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s-\alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr] \\ \leq &\varepsilon \lambda _{1}\sum_{s=0}^{T}G_{1}(s+ \alpha _{i}-1,s) (u_{1}+u_{2}) \\ \leq &\varepsilon \lambda _{1} \bigl( \Vert u_{1} \Vert + \Vert u_{2} \Vert \bigr)\sum _{s=0}^{T}G_{1}(s+ \alpha _{i}-1,s) \\ \leq &\frac{1}{2} \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}. \end{aligned}$$

Similarly, \(\mathcal{T}_{2}(u_{1},u_{2})(t_{1},t_{2}) \leq \frac{1}{2} \|(u_{1},u_{2}) \|_{\mathcal{U}}\). Therefore

$$ \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \bigl( \mathcal{T}_{1}(u_{1},u_{2}), \mathcal{T}_{2}(u_{1},u_{2}) \bigr) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \mathcal{T}_{1}(u_{1},u_{2}) \bigr\Vert + \bigl\Vert \mathcal{T}_{2}(u_{1},u_{2}) \bigr\Vert \leq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}} $$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{1}\).

If \(F_{1}^{\infty }=\infty \), then there exists \(\hat{K}>0\) such that \(F_{1}[t_{1},t_{2},u_{1},u_{2}] \geq \hat{\varepsilon }(u_{1}+u_{2})\) for \(u_{1}+u_{2}\geq \hat{K}_{1}\) and \(t_{i}\in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}\), where the constant \(\hat{\varepsilon }>0\) satisfies

$$\begin{aligned} \hat{\varepsilon }\lambda _{1}\sum_{s=0}^{T} G_{1}(s+\alpha _{1}-1,s) \geq 1. \end{aligned}$$

Let \(K_{2}=\max \lbrace 2K_{1},\frac{\hat{K}}{\theta _{1}} \rbrace \) and set \(\varOmega _{2}= \lbrace (u_{1},u_{2}) \in \mathcal{U}: \|(u_{1},u_{2}) \|_{\mathcal{U}}< K_{2} \rbrace \). For \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{2}\), we have \(\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}), \frac{3}{4}(T+\alpha _{i}) ] } ( (u_{1}(t_{1})+u_{2}(t_{2}) ) \geq \theta _{i}\|(u_{1},u_{2})\|_{\mathcal{U}}\geq \hat{K} \). For all \(t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+\alpha _{i}) ] \), we get

$$\begin{aligned} &\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+ \alpha _{i}) ] } \mathcal{T}_{1}(u_{1},u_{2}) (t_{1},t_{2}) \\ &\quad \geq \min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}), \frac{3}{4}(T+\alpha _{i}) ] } \lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s-\alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr] \\ &\quad \geq \hat{\varepsilon }\lambda _{1}\sum _{s=0}^{T}G_{1}(s+\alpha _{i}-1,s) (u_{1}+u_{2}) \\ &\quad \geq \hat{\varepsilon }\lambda _{1}\theta _{1} \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{ \mathcal{U}}\sum _{s=0}^{T}G_{1}(s+\alpha _{i}-1,s) \\ &\quad \geq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}. \end{aligned}$$

Therefore

$$ \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \bigl( \mathcal{T}_{1}(u_{1},u_{2}), \mathcal{T}_{2}(u_{1},u_{2}) \bigr) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \mathcal{T}_{1}(u_{1},u_{2}) \bigr\Vert + \bigl\Vert \mathcal{T}_{2}(u_{1},u_{2}) \bigr\Vert \geq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}} $$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{2}\). An analogous estimate holds for \(F_{2}^{\infty }=\infty \).

By Theorem 1.1(i), \(\mathcal{T}\) has a fixed point \((u_{1},u_{2}) \in \mathcal{P}\cap \partial ( \bar{\varOmega }_{2} \setminus \varOmega _{1} ) \) such that \(K_{1}\leq \|(u_{1},u_{2})\|_{\mathcal{U}}\leq K_{2}\) and problem (1.1)–(1.2) has a positive solution.

\((b)\) If \(F_{1}^{0}=\infty \), then we choose \(K_{1}>0\) such that \(F_{1}[t_{1},t_{2},u_{1},u_{2}] \geq \tilde{\varepsilon } (u_{1}+u_{2})\) for \(0< u_{1}+u_{2}\leq K_{2}\) and \(t_{i}\in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}-1}\), where the constant \(\tilde{\varepsilon }>0\) satisfies

$$\begin{aligned} \tilde{\varepsilon }\lambda _{1}\theta _{1}\sum _{s=0}^{T} G_{1}(s+ \alpha _{1}-1,s) \geq 1. \end{aligned}$$

If \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{1}\) and \(\|(u_{1},u_{2})\|_{\mathcal{U}}=K_{1} \), then for all \(t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+\alpha _{i}) ] \), we have

$$\begin{aligned} &\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+ \alpha _{i}) ] } \mathcal{T}_{1}(u_{1},u_{2}) (t_{1},t_{2}) \\ &\quad \geq \min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}), \frac{3}{4}(T+\alpha _{i}) ] } \lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s-\alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr] \\ &\quad \geq \tilde{\varepsilon }\lambda _{1}\sum _{s=0}^{T}G_{1}(s+\alpha _{i}-1,s) (u_{1}+u_{2}) \\ &\quad \geq \tilde{\varepsilon }\lambda _{1}\theta _{1} \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{ \mathcal{U}}\sum _{s=0}^{T}G_{1}(s+\alpha _{i}-1,s) \\ &\quad \geq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}. \end{aligned}$$

Therefore

$$ \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \bigl( \mathcal{T}_{1}(u_{1},u_{2}), \mathcal{T}_{2}(u_{1},u_{2}) \bigr) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \mathcal{T}_{1}(u_{1},u_{2}) \bigr\Vert + \bigl\Vert \mathcal{T}_{2}(u_{1},u_{2}) \bigr\Vert \geq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}} $$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{2}\). An analogous estimate holds for \(F_{2}^{0}=\infty \).

For \(i=1,2\), set \(F_{i}^{*}(t_{i})=\max_{0\leq u_{1}+u_{2}\leq t_{i}} F_{i}[t_{1},t_{2},u_{1},u_{2}]\). Then \(F_{i}^{*}\) are nondecreasing in their respective arguments. In addition, from \(F_{i}^{\infty }=0\) we see that \(\lim_{t_{i}\rightarrow \infty } \frac{F_{i}^{*}(t_{i})}{t_{i}}=0\). Hence there exist \(K_{2}>2K_{1}\) such that \(F_{i}^{*}(t_{i})\leq \varepsilon t_{i}\), where the constant \(\varepsilon >0\) satisfies

$$\begin{aligned} 2\varepsilon \lambda _{i}\sum_{s=0}^{T} G_{i}(s+\alpha _{i}-1,s) \leq 1. \end{aligned}$$

If \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{2}\) and \(\|(u_{1},u_{2})\|_{\mathcal{U}}=K_{2}\), then we have

$$\begin{aligned} \mathcal{T}_{1}(u_{1},u_{2}) (t_{1},t_{2}) \leq &\lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s-\alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr] \\ \leq &\lambda _{1}\sum_{s=0}^{T}G_{1}(s+ \alpha _{i}-1,s) F_{1}^{*}(K_{2}) \\ \leq &\varepsilon \lambda _{1}K_{2} \sum _{s=0}^{T}G_{1}(s+\alpha _{i}-1,s) \\ \leq &\frac{1}{2} \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}. \end{aligned}$$

Similarly, \(\mathcal{T}_{2}(u_{1},u_{2})(t_{1},t_{2}) \leq \frac{1}{2} \|(u_{1},u_{2}) \|_{\mathcal{U}}\). Therefore

$$ \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \bigl( \mathcal{T}_{1}(u_{1},u_{2}), \mathcal{T}_{2}(u_{1},u_{2}) \bigr) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \mathcal{T}_{1}(u_{1},u_{2}) \bigr\Vert + \bigl\Vert \mathcal{T}_{2}(u_{1},u_{2}) \bigr\Vert \leq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}} $$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{2}\).

By Theorem 1.1(ii), \(\mathcal{T}\) has a fixed point, and thus problem (1.1)–(1.2) has a positive solution \((u_{1},u_{2}) \in \mathcal{P}\cap \partial ( \bar{\varOmega }_{2} \setminus \varOmega _{1} ) \). □

Theorem 4.2

Suppose assumptions \((A1)\)\((A4)\)hold.

\((a)\):

If \(F_{1}^{0}=F_{2}^{0}=F_{1}^{\infty }=F_{2}^{\infty }=0\), then there exists \(\delta _{1}>0\)such that problem (1.1)(1.2) has at least two positive solutions for all \(\lambda _{1},\lambda _{2} \geq \delta _{1}\).

\((b)\):

If either \(F_{1}^{0}=\infty \)or \(F_{2}^{0}=\infty \)and either \(F_{1}^{\infty }=0\)or \(F_{2}^{\infty }=0\), then there exists \(\delta _{2}>0\)such that problem (1.1)(1.2) has at least two positive solutions for all \(\lambda _{1},\lambda _{2} \leq \delta _{2}\).

Proof

\((a)\) For \((u_{1},u_{2}) \in \mathcal{P}\) such that \(\|(u_{1},u_{2})\|_{\mathcal{U}}=\ell \), let

$$\begin{aligned} m(\ell )&=\min \Biggl\lbrace \lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s-\alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr], \\ &\quad \lambda _{2}\sum_{s=0}^{T} G_{2}(t_{1},s) F_{2}\bigl[t_{1},s- \alpha _{2}-1,u_{1}(t_{1}),u_{2}(s- \alpha _{2}-1)\bigr] \Biggr\rbrace . \end{aligned}$$

By assumption \(m(\ell )>0\) for \(\ell >0\). Choose two numbers \(0< K_{3}< K_{4}\), and let

$$\begin{aligned}& \delta _{1} = \max \biggl\lbrace \frac{K_{3}}{2m(K_{3})} , \frac{K_{4}}{2m(K_{4})} \biggr\rbrace , \\& \varOmega _{i} = \bigl\lbrace (u_{1},u_{2}) \in \mathcal{U}: \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}< K_{i} \bigr\rbrace \quad (i=1,2,3,4). \end{aligned}$$

Then, for \(\lambda _{1},\lambda _{2} \geq \delta _{1}\) and \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{i}\) (\(i=3,4\)) such that \(\|(u_{1},u_{2})\|_{\mathcal{U}}=K_{i}\), we have

$$\begin{aligned} &\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}),\frac{3}{4}(T+ \alpha _{i}) ] } \mathcal{T}_{1}(u_{1},u_{2}) (t_{1},t_{2}) \\ &\quad \geq \tilde{\varepsilon }\lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s-\alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr] \\ &\quad \geq \lambda _{1}m(K_{i}) \\ &\quad \geq \frac{K_{i}}{2}\quad (i=3,4). \end{aligned}$$

Similarly, \(\min_{ t_{i}\in [ \frac{1}{4}(T+\alpha _{i}), \frac{3}{4}(T+\alpha _{i}) ] } \mathcal{T}_{2}(u_{1},u_{2})(t_{1},t_{2}) \geq \frac{K_{i}}{2}\) (\(i=3,4\)).

This implies that

$$\begin{aligned} \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}= \bigl\Vert \mathcal{T}_{1}(u_{1},u_{2}) \bigr\Vert + \bigl\Vert \mathcal{T}_{2}(u_{1},u_{2}) \bigr\Vert \geq K_{i}= \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{ \mathcal{U}} \end{aligned}$$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{i}\) (\(i=3,4\)).

Since \(F_{1}^{0}=F_{2}^{0}=F_{1}^{\infty }=F_{2}^{\infty }=0\), it follows from the proof of Theorem 4.1(a,b) that we can choose \(K_{1}<\frac{K_{3}}{2}\) and \(K_{2}>2K_{4}\) such that

$$\begin{aligned} \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}\leq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{ \mathcal{U}} \end{aligned}$$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{i}\) (\(i=1,2\)).

Applying Theorem 1.1 to \(\varOmega _{1}\), \(\varOmega _{3}\) and \(\varOmega _{3}\), \(\varOmega _{4}\), we have a positive solution \((u_{1},u_{2})\) such that \(K_{1}\leq \|(u_{1},u_{2})\|_{\mathcal{U}} \leq K_{3}\), and another positive solution \((v_{1},v_{2})\) such that \(K_{4}\leq \|(v_{1},v_{2})\|_{\mathcal{U}} \leq K_{2}\). Since \(K_{3}< K_{4}\), these two solutions are distinct.

\((b)\) For \((u_{1},u_{2}) \in \mathcal{P}\) and \(\|(u_{1},u_{2})\|_{\mathcal{U}}=L\), let

$$\begin{aligned} M(L)&=\max \Biggl\lbrace \lambda _{1}\sum _{s=0}^{T} G_{1}(t_{1},s) F_{1}\bigl[s- \alpha _{1}-1,t_{2},u_{1}(s- \alpha _{1}-1),u_{2}(t_{2})\bigr], \\ &\quad \lambda _{2}\sum_{s=0}^{T} G_{2}(t_{1},s) F_{2}\bigl[t_{1},s- \alpha _{2}-1,u_{1}(t_{1}),u_{2}(s- \alpha _{2}-1)\bigr] \Biggr\rbrace . \end{aligned}$$

By assumption \(M(L)>0\) for \(L>0\). Choose two numbers \(0< K_{3}< K_{4}\), and let

$$\begin{aligned} \delta _{2} =&\max \biggl\lbrace \frac{K_{3}}{2M(K_{3})} , \frac{K_{4}}{2M(K_{4})} \biggr\rbrace . \end{aligned}$$

Then, for \(\lambda _{1},\lambda _{2} \leq \delta _{2}\) and \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{i}\) (\(i=3,4\)) such that \(\|(u_{1},u_{2})\|_{\mathcal{U}}=K_{i}\), we have

$$\begin{aligned} \mathcal{T}_{1}(u_{1},u_{2}) (t_{1},t_{2}\leq \lambda _{1}M(H_{i}) \leq \frac{K_{i}}{2} \quad \text{and}\quad \mathcal{T}_{2}(u_{1},u_{2}) (t_{1},t_{2}) \leq \lambda _{2}M(H_{i}) \leq \frac{K_{i}}{2} \quad (i=3,4). \end{aligned}$$

This implies that

$$\begin{aligned} \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}\leq K_{i}= \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{U}} \end{aligned}$$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{i}\) (\(i=3,4\)).

Since either \(F_{1}^{0}=\infty \) or \(F_{2}^{0}=\infty \) and either \(F_{1}^{\infty }=0\) or \(F_{2}^{\infty }=0\), it follows from the proof of Theorem 4.1(a,b) that we can choose \(K_{1}<\frac{K_{3}}{2}\) and \(K_{2}>2K_{4}\) such that

$$\begin{aligned} \bigl\Vert \mathcal{T}(u_{1},u_{2}) \bigr\Vert _{\mathcal{U}}\geq \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{ \mathcal{U}} \end{aligned}$$

for \((u_{1},u_{2}) \in \mathcal{P}\cap \partial \varOmega _{i}\) (\(i=1,2\)).

Once again, we obtain the existence of two distinct positive solutions. □

By the same arguments as in Theorem 4.2 we obtain the following theorem.

Theorem 4.3

Suppose assumptions \((A1)\)\((A4)\)hold.

\((a)\):

If \(F_{1}^{0}=F_{2}^{0}=0\)or \(F_{1}^{\infty }=F_{2}^{\infty }=0\), then there exists \(\delta _{3}>0\)such that problem (1.1)(1.2) has at least two positive solutions for all \(\lambda _{1},\lambda _{2} \geq \delta _{3}\).

\((b)\):

If \(F_{1}^{0}=\infty \)or \(F_{2}^{0}=\infty \), or if \(F_{1}^{\infty }=\infty \)or \(F_{2}^{\infty }=\infty \), then there exists \(\delta _{4}>0\)such that problem (1.1)(1.2) has at least two positive solutions for all \(\lambda _{1},\lambda _{2} \leq \delta _{2}\).

Examples

In this section, we provide some examples to illustrate our main results.

Example 1

Consider the following system of fractional difference equations with parameters:

$$\begin{aligned} \begin{gathered} -\Delta ^{\frac{5}{2}} u_{1}(t) = \frac{e^{- (t+\frac{4}{3} ) } ( \vert u_{2} \vert +1 ) }{400 (t+\frac{34}{3} )^{2} ( 1+\sin ^{2}u_{2}\pi ) }+ \frac{e^{- (t+\frac{3}{2} )\pi }u_{1} (t+\frac{3}{2} ) }{100e+20\sin ^{2} (t+\frac{3}{2} )\pi }, \\ -\frac{1}{2}\Delta ^{\frac{7}{3}} u_{2}(t) = \frac{e^{- (t+\frac{3}{2} )} ( \vert u_{1} \vert +e^{-\sin ^{2} (t+\frac{3}{2} )\pi } ) }{1000(e^{t+\frac{3}{2}}+10)^{2} ( \vert u_{1} \vert +\cos ^{2} (t+\frac{3}{2} )\pi ) }\\ \hphantom{-\frac{1}{2}\Delta ^{\frac{7}{3}} u_{2}(t) =}{}+ \frac{\arctan (\sin ^{2} (t+\frac{4}{3} )\pi ) u_{2} (t+\frac{4}{3} ) }{100\pi (t+\frac{13}{3} )^{2}} \end{gathered} \end{aligned}$$
(5.1)

for \(t\in {\mathbb{N}}_{0,5}\), subject to nonlocal fractional difference-sum conditions

$$\begin{aligned} \begin{gathered} \Delta ^{-\frac{1}{3}}u_{1} \biggl( - \frac{1}{6} \biggr) = \Delta ^{ \frac{1}{5}}u_{1} \biggl( \frac{3}{10} \biggr) = 0, \\ \Delta ^{-\frac{1}{4}}u_{2} \biggl( -\frac{5}{12} \biggr) = \Delta ^{ \frac{2}{5}}u_{2} \biggl( -\frac{1}{15} \biggr) = 0, \\ u_{1} \biggl( \frac{15}{2} \biggr) = \frac{1}{2}u_{1} \biggl( \frac{9}{2} \biggr), \\ u_{2} \biggl( \frac{22}{3} \biggr) = \frac{3}{4}u_{2} \biggl( \frac{13}{3} \biggr). \end{gathered} \end{aligned}$$
(5.2)

Here \(\alpha _{1}=\frac{5}{2}\), \(\alpha _{2}=\frac{7}{3}\), \(\beta _{1}= \frac{1}{3}\), \(\beta _{2}=\frac{1}{4}\), \(\gamma _{1}=\frac{1}{5}\), \(\gamma _{2}= \frac{2}{5}\), \(\lambda _{1}=1\), \(\lambda _{2}=2\), \(\chi _{1}=\frac{1}{2}\), \(\chi _{2}=\frac{3}{4}\), \(\eta _{1}=\frac{9}{2}\), \(\eta _{2}=\frac{13}{3}\), \(T=5\), and for \(t_{1}\in {\mathbb{N}}_{-\frac{1}{2},\frac{15}{2}}\) and \(t_{2}\in {\mathbb{N}}_{-\frac{2}{3},\frac{22}{3}}\),

$$\begin{aligned} &F_{1} [t_{1},t_{2},u_{1} ,u_{2} ] = \frac{e^{-t_{2} } ( \vert u_{2} \vert +1 ) }{400 (t_{2}+10 )^{2} ( 1+\sin ^{2}u_{2}\pi ) }+ \frac{e^{-t_{1}\pi }u_{1} (t_{1} ) }{100e+20\sin ^{2}t_{1}\pi }, \\ &F_{2} [t_{1},t_{2},u_{1},u_{2} ] = \frac{e^{-t_{1}} ( \vert u_{1} \vert +e^{-\sin ^{2}t_{1}\pi } ) }{1000(e^{t_{1}}+10)^{2} ( \vert u_{1} \vert +\cos ^{2}t_{1}\pi ) }+ \frac{\arctan (\sin ^{2}t_{2}\pi ) u_{2} (t_{2} ) }{100\pi (t_{2}+3 )^{2}}. \end{aligned}$$

We get that

$$\begin{aligned} & \bigl\vert F_{1} [t_{1},t_{2},u_{1} ,u_{2} ] -F_{1} [t_{1},t_{2},v_{1} ,v_{2} ] \bigr\vert \\ &\quad \leq \frac{e^{-t_{2}}}{400(t_{2}+10)^{2}} \biggl\vert \frac{ \vert u_{2} \vert +1}{1+\sin ^{2}u_{2}\pi }- \frac{ \vert v_{2} \vert +1}{1+\sin ^{2}v_{2}\pi } \biggr\vert + \frac{e^{-t_{1}\pi }}{100e+20\sin ^{2}t_{1}\pi } \bigl\vert \Delta ^{ \frac{1}{3}}u_{1}-\Delta ^{\frac{1}{3}}v_{1} \bigr\vert \\ &\quad \leq \frac{1}{45\mbox{,}511} \vert u_{2}-v_{2} \vert +\frac{1}{100e} \vert u_{1}-v_{1} \vert , \\ & \bigl\vert F_{2} [t_{1},t_{2},u_{1} ,u_{2} ] -F_{2} [t_{1},t_{2},v_{1} ,v_{2} ] \bigr\vert \\ &\quad \leq \frac{e^{-t_{1}}}{1000(e^{t_{1}}+10)^{2}} \biggl\vert \frac{ \vert u_{1} \vert }{1+ \vert u_{1} \vert }- \frac{ \vert v_{1} \vert }{1+ \vert v_{1} \vert } \biggr\vert + \frac{\arctan (1)}{100\pi (t_{2}+3)^{2}}| u_{2}-v_{2} )| \\ &\quad \leq \frac{1}{12\mbox{,}100} \vert u_{1}-v_{1} \vert +\frac{9}{19\mbox{,}600} \vert u_{2}-v_{2} \vert . \end{aligned}$$

So, (3.3) holds with \(M_{1}=0.00002197\), \(M_{2}=0.00008264\), \(N_{1}=0.00368\), and \(N_{2}=0.000459\).

Finally, we find that

$$\begin{aligned}& L_{1}=\max \lbrace M_{1},N_{1} \rbrace =0.00368,\qquad L_{2} = \max \lbrace M_{2},N_{2} \rbrace =0.000459 \quad \text{and} \\& \varOmega _{1}=75.4559, \qquad \varOmega _{2}=136.2638. \end{aligned}$$

Therefore we obtain

$$ L_{1}\varOmega _{1}+L_{2}\varOmega _{2} = 0.3402 < 1. $$

Hence, by Theorem 3.1, problem (5.1)–(5.2) has a unique solution.

Example 2

Consider the following system of fractional difference equations with parameters:

$$\begin{aligned} \begin{gathered} -\Delta ^{\frac{5}{2}} u_{1}(t) = \biggl[ u_{1} \biggl(t+\frac{3}{2} \biggr) + u_{1} \biggl(t+\frac{4}{3} \biggr) \biggr]^{6}, \\ -\frac{1}{2}\Delta ^{\frac{7}{3}} u_{2}(t) = \biggl[u_{1} \biggl(t+ \frac{3}{2} \biggr) + u_{1} \biggl(t+\frac{4}{3} \biggr) \biggr] ^{4}, \quad t \in { \mathbb{N}}_{0,5}, \end{gathered} \end{aligned}$$
(5.3)

for \(t\in {\mathbb{N}}_{0,5}\), subject to nonlocal fractional difference-sum conditions (5.2).

For all \(t_{1}\in {\mathbb{N}}_{-\frac{1}{2},\frac{15}{2}}\), \(t_{2}\in { \mathbb{N}}_{-\frac{2}{3},\frac{22}{3}}\), \(u_{1},u_{2}>0\), we have \(F_{1}[t_{1},t_{2},u_{1}u_{2}]= [ u_{1} (t_{1} ) + u_{1} (t_{2} ) ]^{6}>0\) and \(F_{2}[t_{1},t_{2},u_{1}u_{2}]= [ u_{1} (t_{1} ) + u_{1} (t_{2} ) ]^{4}>0 \). We find that

$$\begin{aligned}& 0 < 4.3619 = \chi _{1}\eta _{1}^{\underline{\alpha _{1}-1}} < (T+ \alpha _{1})^{\underline{\alpha _{1}-1}}=19.4922, \\& 0 < 5.0162 = \chi _{2}\eta _{2}^{\underline{\alpha _{2}-1}} < (T+ \alpha _{2})^{\underline{\alpha _{2}-1}}=13.8057. \end{aligned}$$

By direct calculation we have \(F_{1}^{0}=F_{2}^{0}=0\) and \(F_{1}^{\infty }=F_{2}^{\infty }=\infty \). Then, by Theorem 4.1(a), problem (5.3) and (5.2) has at least one positive solution.

Example 3

Consider the following system of fractional difference equations with parameters:

$$\begin{aligned} \begin{gathered} -\Delta ^{\frac{5}{2}} u_{1}(t) = \biggl[ u_{1} \biggl(t+\frac{3}{2} \biggr) + u_{1} \biggl(t+\frac{4}{3} \biggr) \biggr]^{\frac{1}{3}}, \\ -\frac{1}{2}\Delta ^{\frac{7}{3}} u_{2}(t) = \biggl[u_{1} \biggl(t+ \frac{3}{2} \biggr) + u_{1} \biggl(t+\frac{4}{3} \biggr) \biggr] ^{ \frac{1}{4}}, \quad t\in { \mathbb{N}}_{0,5}, \end{gathered} \end{aligned}$$
(5.4)

for \(t\in {\mathbb{N}}_{0,5}\), subject to nonlocal fractional difference-sum conditions (5.2).

For all \(t_{1}\in {\mathbb{N}}_{-\frac{1}{2},\frac{15}{2}}\), \(t_{2}\in { \mathbb{N}}_{-\frac{2}{3},\frac{22}{3}}\), \(u_{1},u_{2}>0\), we have \(F_{1}[t_{1},t_{2},u_{1}u_{2}]= [ u_{1} (t_{1} ) + u_{1} (t_{2} ) ]^{\frac{1}{3}}>0\) and \(F_{2}[t_{1},t_{2},u_{1}u_{2}]= [ u_{1} (t_{1} ) + u_{1} (t_{2} ) ]^{\frac{1}{4}}>0 \).

By the same argument as in Example 2 we have that \(\chi _{i}\eta _{i}^{\underline{\alpha _{i}-1}} < (T+\alpha _{i})^{ \underline{\alpha _{i}-1}}\), \(i=1,2\).

By direct calculation we have \(F_{1}^{0}=F_{2}^{0}=\infty \) and \(F_{1}^{\infty }=F_{2}^{\infty }=0\). Then, by Theorem 4.1(b), problem (5.4) and (5.2) has at least one positive solution.

References

  1. 1.

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

    MathSciNet  MATH  Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

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

    MathSciNet  MATH  Google Scholar 

  4. 4.

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

    MATH  Google Scholar 

  5. 5.

    Atici, F.M., Eloe, P.W.: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2(2), 165–176 (2007)

    MathSciNet  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, 31 pages (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. Numer. Simul. 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, 12 pages (2014)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Weidong, Lv.: 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, 12 pages (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, 12 pages (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, 9 pages (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, 13 pages (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.

    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 

  25. 25.

    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, 9 pages (2015)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    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 

  27. 27.

    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 

  28. 28.

    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 

  29. 29.

    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 

  30. 30.

    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, 14 pages (2017)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    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 

  32. 32.

    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 

  33. 33.

    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 

  34. 34.

    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, 15 pages (2012)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    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 

  36. 36.

    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 

  37. 37.

    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 

  38. 38.

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

    MathSciNet  MATH  Google Scholar 

  39. 39.

    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 

  40. 40.

    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 (2019)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    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 

  42. 42.

    Siricharuanun, P., Chasreechai, S., Sitthiwirattham, T.: On a coupled system of fractional sum-difference equations with p-Laplacian operator. Adv. Differ. Equ. 2020, Article ID 361 (2020)

    MathSciNet  Google Scholar 

  43. 43.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cone. Academic Press, Orlando (1988)

    MATH  Google Scholar 

  44. 44.

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

    MATH  Google Scholar 

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Siricharuanun, P., Chasreechai, S. & Sitthiwirattham, T. Existence and multiplicity of positive solutions to a system of fractional difference equations with parameters. Adv Differ Equ 2020, 445 (2020). https://doi.org/10.1186/s13662-020-02904-6

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MSC

  • 39A05
  • 39A12

Keywords

  • Positive solution
  • System of fractional difference equations
  • Green’s function
  • Cone