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Positive solutions for a class of fractional difference systems with coupled boundary conditions

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Abstract

In this paper we use the fixed point index and nonnegative matrices to study the existence of positive solutions for a class of fractional difference systems with coupled boundary conditions.

Introduction

For \(a,b\in {\mathbf{R}}\), let \(\mathbf{N}_{a}=\{a,a+1,a+2,\ldots \}\) and \([a,b]_{\mathbf{N}_{a}}=\{a,a+1,a+2,\ldots ,b\}\) with \(b-a\in {\mathbf{N}}_{1}\). In this paper we study the existence of positive solutions for the following fractional difference system with coupled boundary conditions:

$$ \textstyle\begin{cases} -\Delta _{\nu -3}^{\nu }x(t)= f_{1}(t+\nu -1,x(t+\nu -1),y(t+\nu -1)),\quad t\in [0,T-1]_{\mathbf{N_{0}}}, \\ -\Delta _{\nu -3}^{\nu }y(t)= f_{2}(t+\nu -1,x(t+\nu -1),y(t+\nu -1)),\quad t\in [0,T-1]_{\mathbf{N_{0}}}, \\ x(\nu -3)=[\Delta _{\nu -3}^{\alpha }x(t)]|_{t=\nu -\alpha -2}=0, \qquad y( \nu -3)=[\Delta _{\nu -3}^{\alpha }y(t)]|_{t=\nu -\alpha -2}=0, \\ x(T+\nu -1)=a y(\xi +\nu ), \qquad y(T+\nu -1)=bx(\eta +\nu ), \end{cases} $$
(1.1)

where \(\nu \in (2,3]\), \(\alpha \in (0,1)\) are two real numbers, \(\Delta _{\nu -3}^{\nu }\), \(\Delta _{\nu -3}^{\alpha }\) are discrete fractional operators, \(\nu -\alpha -2>0\), \(\xi ,\eta \in [0,T-2]_{ \mathbf{N_{0}}}\), \(a,b>0\) with \(ab<\frac{(\xi +1)!(\eta +1)!}{\varGamma (\xi +\nu +1)\varGamma (\eta +\nu +1)} [\frac{\varGamma (T+\nu )}{T!} ] ^{2}\), and the nonlinearities \(f_{i}(t,x,y):[\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\times \mathbf{R}^{+}\times \mathbf{R}^{+}\to {\mathbf{R}^{+}}\) are continuous functions \((i=1,2,\mathbf{R}^{+}=[0,+ \infty ))\).

In recent years, the fractional calculus and fractional differential equations have been of great interest in the literature, and they have been widely applied in numerous diverse fields including electrical engineering, chemistry, mathematical biology, control theory, and the calculus of variations. For example, papers [1, 2] have introduced a fractional order model for infection of CD4+T cells in HIV, which can be depicted by the system

$$\textstyle\begin{cases} D^{\alpha _{1}}(T)=s-KVT-dT+bI, \\ D^{\alpha _{2}}(I)=KVT-(b+\delta )I, \\ D^{\alpha _{3}}(V)=N\delta I-cV, \end{cases} $$

where \(D^{\alpha _{i}}\) are fractional derivatives, \(i=1,2,3\). Till now, we have noted that by using the techniques of nonlinear analysis, a large number of results concerning the existence and multiplicity of solutions (or positive solutions) of nonlinear fractional differential equations can be found in the literature, we refer the reader to [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and the references cited therein. In [3], the authors studied the singular fractional p-Laplacian boundary value system

$$ \textstyle\begin{cases} D_{0+}^{\alpha }(\varphi _{p}(D_{0+}^{\gamma }u))(t)+\lambda ^{1/(q-1)}f(t,u(t),D _{0+}^{\mu _{1}}u(t), D_{0+}^{\mu _{2}}u(t),\ldots,D_{0+}^{\mu _{n-1}}u(t),v(t))=0,\\ \quad 0< t< 1, \\ D_{0+}^{\beta }(\varphi _{p}(D_{0+}^{\delta }v))(t)+\mu ^{1/(q-1)}g(t,u(t),D _{0+}^{\eta _{1}}u(t), D_{0+}^{\eta _{2}}u(t),\ldots,D_{0+}^{\eta _{m-1}}u(t))=0,\\ \quad 0< t< 1, \\ u(0)=D_{0+}^{\mu _{i}}u(0)=0,\qquad D_{0+}^{\gamma }u(0)=D_{0+}^{\gamma +\mu _{i}} u(0)=0, \quad i=1,2,\ldots,n-2, \\ D_{0+}^{\mu _{n-1}}u(1)=\chi \int _{0}^{\eta }h(t)D_{0+}^{\mu _{n-1}}u(t) \,dA(t), \\ v(0)=D_{0+}^{\eta _{i}}v(0)=0,\qquad D_{0+}^{\delta }v(0)= D_{0+}^{\delta + \eta _{i}} v(0)=0, \quad i=1,2,\ldots,m-2, \\ D_{0+}^{\eta _{m-1}}v(1)=\iota \int _{0}^{\vartheta }a(t) D_{0+}^{ \eta _{m-1}}v(t)\, dB(t). \end{cases} $$
(1.2)

Here, they used the mixed monotone methods to obtain the uniqueness of positive solutions for (1.2) and established an iterative sequence, which can converge uniformly to the unique solution.

In [4], the authors studied the system of nonlinear fractional differential equations with coupled integral boundary conditions

$$ \textstyle\begin{cases} D_{0+}^{\alpha }u(t)+\lambda f(t,u(t),v(t))=0,\quad 0< t< 1, \\ D_{0+}^{\beta }v(t)+\mu g(t,u(t),v(t))=0,\quad 0< t< 1, \\ u(0)=u^{(i)}(0)=0,\qquad u'(1)=\int _{0}^{1} v(s)\,d H(s),\quad i=1,2,\ldots,n-2, \\ v(0)=v^{(i)}(0)=0,\qquad v'(1)=\int _{0}^{1} u(s)\,d K(s),\quad i=1,2,\ldots,m-2, \end{cases} $$
(1.3)

where the nonlinear terms f, g are sign-changing nonsingular or singular functions. They used the Guo–Krasnosel’skii fixed point theorem to obtain the existence of positive solutions for (1.3), and they also presented intervals for parameters λ and μ for the positive solutions.

However, as is mentioned by Christopher S. Goodrich in [28], there has been little work done in fractional difference equations, we only refer to [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. For example, in [29] the authors studied discrete fractional calculus and offered some important properties of the fractional sum and the fractional difference operators. Also, they studied the uniqueness of solutions for the nonlinear fractional difference equation

$$ \textstyle\begin{cases} \Delta ^{\nu }y(t)=f(t+\nu -1,y(t+\nu -1)),\quad t=0,1,2,\ldots, \\ \Delta ^{\nu -1} y(t)|_{t=0}=a_{0}. \end{cases} $$
(1.4)

Christopher S. Goodrich has made a great contribution to the development of the theory for discrete fractional calculus and associated difference equations (see [31, 32, 35,36,37, 39, 44]), presented and summarized many excellent results in his monograph with A. Peterson [43] in this direction. For example, in [35, 36] the authors studied the following two fractional difference equations boundary value problems:

$$ \textstyle\begin{cases} -\Delta ^{\nu _{1}}y_{1}(t)=\lambda _{1}f_{1}(t+\nu _{1}-1,y_{1}(t+\nu _{1}-1),y_{2}(t+\nu _{2}-1)),\quad t\in [1,b+1], \\ -\Delta ^{\nu _{2}}y_{2}(t)=\lambda _{2}f_{2}(t+\nu _{2}-1,y_{1}(t+\nu _{1}-1),y_{2}(t+\nu _{2}-1)),\quad t\in [1,b+1], \\ y_{1}(\nu _{1}-2)=y_{1}(\nu _{1}+b+1)=y_{2}(\nu _{2}-2)=y_{2}(\nu _{2}+b+1)=0, \end{cases} $$
(1.5)

and

$$ \textstyle\begin{cases} -\Delta ^{\nu _{1}}y_{1}(t)=\lambda _{1}a_{1}(t+\nu _{1}-1)f_{1}(y_{1}(t+ \nu _{1}-1),y_{2}(t+\nu _{2}-1)),\quad t\in [0,b], \\ -\Delta ^{\nu _{2}}y_{2}(t)=\lambda _{2}a_{2}(t+\nu _{2}-1)f_{2}(y_{1}(t+ \nu _{1}-1),y_{2}(t+\nu _{2}-1)),\quad t\in [0,b], \\ y_{1}(\nu _{1}-2)=\psi _{1}(y_{1}),\qquad y_{1}(\nu _{1}+b)=\phi _{1}(y_{1}),\\ y_{2}(\nu _{2}-2)=\psi _{2}(y_{2}),\qquad y_{2}(\nu _{2}+b)=\phi _{2}(y_{2}), \end{cases} $$
(1.6)

where \(\nu _{1},\nu _{2}\in (1,2]\). They used the Guo–Krasnosel’skii fixed point theorem to obtain the existence of positive solutions for the above two problems, where the nonlinearities in (1.5) can be sign-changing.

Motivated by works aforementioned and some results from integer-order equations (including differential and difference equations, see [45,46,47,48,49,50,51,52,53,54,55]), we study the existence of positive solutions for the fractional difference systems (1.1). We use the fixed point index theory to establish our main results based on a priori estimates achieved by utilizing nonnegative matrices (see [10, 54, 55]) that involve some useful inequalities associated with the Green’s functions for (1.1). Moreover, our nonlinearities \(f_{i}\) (\(i=1,2\)) are allowed to grow superlinearly and sublinearly about the linear combinations of unknown functions x, y, see conditions (H1)–(H4) in Sect. 3.

Preliminaries

In this section, we first offer some necessary definitions from discrete fractional calculus. These materials can be found in some recent papers.

Definition 2.1

(see [43])

We define \(t^{\underline{ \nu }}:=\frac{\varGamma (t+1)}{\varGamma (t+1-\nu )}\) for any \(t,\nu \in {\mathbf{R}}\) for which the right-hand side is well-defined. We use the convention that if \(t+1-\nu \) is a pole of the gamma function and \(t + 1\) is not a pole, then \(t^{\underline{\nu }}=0\).

Definition 2.2

(see [43])

For \(\nu >0\), the νth fractional sum of a function f is

$$\Delta _{a}^{-\nu }f(t)=\frac{1}{\varGamma (\nu )}\sum _{s=a}^{t- \nu }(t-s-1)^{\underline{\nu -1}}f(s)\quad \text{for }t\in {\mathbf{N}} _{a+\nu }. $$

We also define the νth fractional difference for \(\nu >0\) by

$$\Delta _{a}^{\nu }f(t)=\Delta _{a}^{N} \Delta _{a}^{\nu -N}f(t)\quad \text{for }t\in { \mathbf{N}}_{a+N-\nu }, $$

where \(N\in \mathbf{N}\) with \(0\le N-1<\nu \le N\).

Lemma 2.3

(see [43])

Let \(N\in \mathbf{N}\) with \(0\le N-1<\nu \le N\). Then

$$\Delta _{0}^{-\nu }\Delta _{\nu -N}^{\nu }f(t)=f(t)+c_{1}t^{\underline{ \nu -1}}+c_{2}t^{\underline{\nu -2}}+ \cdots +C_{N}t^{\underline{ \nu -N}} \quad \textit{for } c_{i}\in { \mathbf{R}}, 1\le i\le N. $$

Lemma 2.4

(see [44, Lemma 4.1])

For all \(\nu \in {\mathbf{R}}\), we have \(\Delta _{a}^{\alpha }t^{\underline{ \nu }}=\frac{\varGamma (\nu +1)t^{\underline{\nu -\alpha }}}{\varGamma ( \nu +1-\alpha )}\) with \(\alpha >0\), if \(t^{\underline{\nu }}\), \(t^{\underline{ \nu -\alpha }}\) are well-defined.

Next, we use Lemmas 2.3 and 2.4 to calculate the Green’s functions associated with (1.1). For convenience, we let \(L= [\frac{ \varGamma (T+\nu )}{T!} ]^{2}-ab\frac{\varGamma (\xi +\nu +1)\varGamma ( \eta +\nu +1)}{(\xi +1)!(\eta +1)!}\), and

$$ \begin{aligned} G(t,s)=\frac{1}{\varGamma (\nu )}{\textstyle\begin{cases} \frac{t^{\underline{\nu -1}}(T+\nu -s-2)^{\underline{\nu -1}}}{(T+ \nu -1)^{\underline{\nu -1}}}-(t-s-1)^{\underline{\nu -1}},\quad 0\leq s< t- \nu +1\leq T-1;\\ \frac{t^{\underline{\nu -1}}(T+\nu -s-2)^{\underline{ \nu -1}}}{(T+\nu -1)^{\underline{\nu -1}}},\quad 0\leq t-\nu +1\leq s\leq T-1. \end{cases}\displaystyle } \end{aligned} $$
(2.1)

The following lemma is as in [40] (for completeness, we present its proof).

Lemma 2.5

Let \(\nu \in (2,3]\), \(\alpha \in (0,1)\), and \(h_{i}(t):[\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\to {\mathbf{R}}\) (\(i=1,2\)). Then the fractional difference system

$$ \textstyle\begin{cases} -\bigtriangleup _{\nu -3}^{\nu }x(t)=h_{1}(t+\nu -1),\quad t\in [0,T-1]_{ \mathbf{N_{0}}}, \\ -\bigtriangleup _{\nu -3}^{\nu }y(t)=h_{2}(t+\nu -1),\quad t\in [0,T-1]_{ \mathbf{N_{0}}}, \\ x(\nu -3)=[\bigtriangleup _{\nu -3}^{\alpha }x(t)]|_{t=\nu -\alpha -2}=0, \qquad y(\nu -3)=[\bigtriangleup _{\nu -3}^{\alpha }y(t)]|_{t=\nu -\alpha -2}=0, \\ x(T+\nu -1)=ay(\xi +\nu ),\qquad y(T+\nu -1)=bx(\eta +\nu ), \end{cases} $$
(2.2)

has the unique solution, which takes the form

$$ \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}= \begin{pmatrix} \sum_{s=0}^{T-1}H_{1}(t,s)h_{1}(s+\nu -1)+\sum_{s=0} ^{T-1}K_{1}(t,s)h_{2}(s+\nu -1) \\ \sum_{s=0}^{T-1}H_{2}(t,s)h_{2}(s+\nu -1)+\sum_{s=0} ^{T-1}K_{2}(t,s)h_{1}(s+\nu -1) \end{pmatrix}, $$
(2.3)

where

$$\begin{aligned}& \begin{aligned} &H_{1}(t,s)=G(t,s)+\frac{ab(\xi +\nu )^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L}G( \eta +\nu ,s), \\ & K_{1}(t,s)=\frac{{a(T+\nu -1)}^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L}G(\xi +\nu ,s), \end{aligned} \end{aligned}$$
(2.4)
$$\begin{aligned}& \begin{aligned} &H_{2}(t,s)=G(t,s)+\frac{ab(\eta +\nu )^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L}G(\xi +\nu ,s), \\ & K_{2}(t,s)=\frac{{b(T+\nu -1)}^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L}G(\eta +\nu ,s). \end{aligned} \end{aligned}$$
(2.5)

Proof

From Lemma 2.3 we have

$$\begin{aligned}& x(t)=-\frac{1}{\varGamma (\nu )}\sum_{s=0}^{t-\nu }(t-s-1)^{\underline{ \nu -1}}h_{1}(s+ \nu -1)+C_{1}t^{\underline{\nu -1}}+C_{2}t^{\underline{ \nu -2}} +C_{3}t^{\underline{\nu -3}}, \\& \quad (C_{i}\in {\mathbf{R}},i=1,2,3) \end{aligned}$$
(2.6)

and

$$\begin{aligned}& y(t)=-\frac{1}{\varGamma (\nu )}\sum_{s=0}^{t-\nu }(t-s-1)^{\underline{ \nu -1}}h_{2}(s+ \nu -1)+\overline{C}_{1}t^{\underline{\nu -1}}+ \overline{C}_{2}t^{\underline{\nu -2}} +\overline{C}_{3}t^{\underline{ \nu -3}}, \\& \quad (\overline{C}_{i}\in {\mathbf{R}},i=1,2,3). \end{aligned}$$
(2.7)

Substituting \(x(\nu -3)=y(\nu -3)=0\) into (2.6), (2.7), we obtain \(C_{3}=\overline{C}_{3}=0\). Because of

$$\begin{aligned} \bigtriangleup _{\nu -3}^{\alpha }x(t) =&C_{1} \bigtriangleup _{\nu -3} ^{\alpha }t^{\underline{\nu -1}}+C_{2} \bigtriangleup _{\nu -3}^{\alpha }t^{\underline{\nu -2}}-\bigtriangleup _{\nu -3}^{{-(\nu -\alpha )}}h _{1}(t+\nu -1) \\ =& C_{1}\frac{\varGamma (\nu )t^{\underline{\nu -\alpha -1}}}{\varGamma ( \nu -\alpha )}+C_{2}\frac{\varGamma (\nu -1)t^{ \underline{\nu -\alpha -2}}}{\varGamma (\nu -\alpha -1)} \\ &{}- \frac{1}{ \varGamma (\nu -\alpha )}\sum_{s=0}^{t-\nu +\alpha }(t-s-1)^{\underline{ \nu -\alpha -1}}h_{1}(s+ \nu -1), \end{aligned}$$

and using the boundary condition \([\bigtriangleup _{\nu -3}^{\alpha }x(t)]|_{t= \nu -\alpha -2}=0\) to obtain \(C_{2}=0\). Similarly, we have \(\overline{C}_{2}=0\). By virtue of the conditions \(x(T+\nu -1)=ay( \xi +\nu )\), \(y(T+\nu -1)=bx(\eta +\nu )\), we respectively obtain

$$ \begin{aligned} &{-}\frac{1}{\varGamma (\nu )}\sum_{s=0}^{T-1}(T+ \nu -s-2)^{\underline{ \nu -1}}h_{1}(s+\nu -1)+C_{1}(T+\nu -1)^{\underline{\nu -1}} \\ &\quad =-\frac{a}{\varGamma (\nu )}\sum_{s=0}^{\xi }(\xi + \nu -s-1)^{\underline{ \nu -1}}h_{2}(s+\nu -1)+a\overline{C}_{1}(\xi +\nu )^{\underline{ \nu -1}}, \end{aligned} $$

and

$$ \begin{aligned} &{-}\frac{1}{\varGamma (\nu )}\sum_{s=0}^{T-1}(T+ \nu -s-2)^{\underline{ \nu -1}}h_{2}(s+\nu -1)+\overline{C}_{1}(T+ \nu -1)^{ \underline{\nu -1}} \\ &\quad =-\frac{b}{\varGamma (\nu )}\sum_{s=0}^{\eta }(\eta + \nu -s-1)^{\underline{ \nu -1}}h_{1}(s+\nu -1)+bC_{1}(\eta +\nu )^{\underline{\nu -1}}. \end{aligned} $$

Note that

$$ \begin{aligned} & \begin{vmatrix} (T+\nu -1)^{\underline{\nu -1}} & -a(\xi +\nu )^{\underline{\nu -1}} \\ -b(\eta +\nu )^{\underline{\nu -1}} & (T+\nu -1)^{\underline{ \nu -1}} \end{vmatrix}\\ &\quad = \bigl((T+\nu -1)^{\underline{\nu -1}} \bigr)^{2}-ab(\xi +\nu )^{\underline{ \nu -1}}( \eta +\nu )^{\underline{\nu -1}} \\ &\quad = \biggl(\frac{\varGamma (T+ \nu )}{T!} \biggr)^{2}-\frac{ab\varGamma (\xi +\nu +1)\varGamma (\eta + \nu +1)}{(\xi +1)!(\eta +1)!} \\ &\quad =L>0, \end{aligned} $$

so we have

$$\begin{aligned} C_{1}={}&\frac{1}{L\varGamma (\nu )} \Biggl[(T+\nu -1)^{\underline{\nu -1}} \Biggl[\sum_{s=0}^{T-1}(T +\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1)\\ &{} -a\sum _{s=0}^{\xi }(\xi +\nu -s-1)^{\underline{\nu -1}}h_{2}(s+ \nu -1) \Biggr] \\ &{} +a(\xi +\nu )^{\underline{\nu -1}} \Biggl[\sum_{s=0} ^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{2}(s+\nu -1)\\ &{} - b\sum _{s=0} ^{\eta }(\eta +\nu -s-1)^{\underline{\nu -1}}h_{1}(s+ \nu -1) \Biggr] \Biggr], \\ \overline{C}_{1}={} &\frac{1}{L\varGamma (\nu )} \Biggl[(T+\nu -1)^{\underline{ \nu -1}} \Biggl[-b\sum_{s=0}^{\eta }( \eta +\nu -s-1)^{ \underline{\nu -1}}h_{1}(s+\nu -1)\\ &{} +\sum _{s=0}^{T-1}(T+\nu -s-2)^{\underline{ \nu -1}}h_{2}(s+ \nu -1) \Biggr] \\ &{} +b(\eta +\nu )^{\underline{\nu -1}} \Biggl[-a\sum_{s=0}^{\xi }( \xi +\nu -s-1)^{\underline{\nu -1}}h_{2}(s+ \nu -1)\\ &{} + \sum _{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1) \Biggr] \Biggr]. \end{aligned}$$

As a result, we have

$$\begin{aligned} x(t) =&-\frac{1}{\varGamma (\nu )}\sum_{s=0}^{t-\nu }(t-s-1)^{\underline{ \nu -1}}h_{1}(s+ \nu -1) \\ &{}+\frac{t^{\underline{\nu -1}}}{L\varGamma (\nu )} \Biggl[(T+\nu -1)^{\underline{ \nu -1}} \Biggl[\sum_{s=0}^{T-1}(T +\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1)\\ &{}-a\sum _{s=0}^{\xi }(\xi +\nu -s-1)^{\underline{\nu -1}}h_{2}(s+ \nu -1) \Biggr] \\ &{} +a(\xi +\nu )^{\underline{\nu -1}} \Biggl[\sum_{s=0} ^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{2}(s+\nu -1)\\ &{}- b\sum _{s=0} ^{\eta }(\eta +\nu -s-1)^{\underline{\nu -1}}h_{1}(s+ \nu -1) \Biggr] \Biggr] \\ =&-\frac{1}{\varGamma (\nu )}\sum_{s=0}^{t-\nu }(t-s-1)^{\underline{ \nu -1}}h_{1}(s+ \nu -1)\\ &{}+ \frac{t^{\underline{\nu -1}}(T+\upsilon -1)^{\underline{ \nu -1}}}{L\varGamma (\nu )}\sum_{s=0}^{T-1}(T+ \nu -s-2)^{\underline{ \nu -1}}h_{1}(s+\nu -1) \\ &{}-\frac{t^{\underline{\nu -1}}}{\varGamma (\nu )(T+\nu -1)^{\underline{ \nu -1}}}\sum_{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1)\\ &{}+ \frac{t^{\underline{\nu -1}}}{\varGamma (\nu )(T+\nu -1)^{\underline{ \nu -1}}}\sum _{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1) \\ &{}-\frac{ab(\xi +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{\eta }(\eta + \nu -s-1)^{\underline{\nu -1}}h _{1}(s+\nu -1)\\ &{}- \frac{a(T+\nu -1)^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L\varGamma (\nu )}\sum _{s=0}^{\xi }(\xi +\nu -s-1)^{\underline{ \nu -1}}h_{2}(s+ \nu -1) \\ &{}+\frac{a(\xi +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{2}(s+ \nu -1) \\ =&\sum_{s=0}^{T-1}G(t,s)h_{1}(s+ \nu -1)\\ &{}+\frac{abt^{\underline{\nu -1}}( \xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{\nu -1}}}{L \varGamma (\nu ) (T+\nu -1)^{\underline{\nu -1}}}\sum_{s=0}^{T-1}(T+ \nu -s-2)^{\underline{\nu -1}}h_{1}(s+\nu -1) \\ &{}-\frac{ab(\xi +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{\eta }(\eta + \nu -s-1)^{\underline{\nu -1}}h _{1}(s+\nu -1)\\ &{}- \frac{a(T+\nu -1)^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L\varGamma (\nu )}\sum _{s=0}^{\xi }(\xi +\nu -s-1)^{\underline{ \nu -1}}h_{2}(s+ \nu -1) \\ &{}+\frac{a(\xi +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{2}(s+ \nu -1) \\ =&\sum_{s=0}^{T-1}G(t,s)h_{1}(s+ \nu -1)\\ &{}+\frac{abt^{\underline{\nu -1}}( \xi +\nu )^{\underline{\nu -1}}}{L}\sum_{s=0}^{T-1}G( \eta +\nu ,s)h _{1}(s+\nu -1) \\ &{}+\frac{a(T+\nu -1)^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L} \sum_{s=0}^{T-1}G(\xi + \nu ,s)h_{2}(s+\nu -1) \\ =&\sum_{s=0}^{T-1} \biggl[G(t,s)+ \frac{abt^{\underline{\nu -1}}(\xi + \nu )^{\underline{\nu -1}}}{L}G(\eta +\nu ,s) \biggr]h_{1}(s+\nu -1)\\ &{}+ \sum _{s=0}^{T-1}\frac{a(T+\nu -1)^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L}G(\xi +\nu ,s)]h_{2}(s+\nu -1). \end{aligned}$$

Similarly, we can obtain

$$\begin{aligned} y(t) =&-\frac{1}{\varGamma (\nu )}\sum_{s=0}^{t-\nu }(t-s-1)^{\underline{ \nu -1}}h_{2}(s+ \nu -1) \\ &{} +\frac{t^{\underline{\nu -1}}}{L\varGamma (\nu )} \Biggl[(T+\nu -1)^{\underline{ \nu -1}} \Biggl[-b\sum_{s=0}^{\eta }( \eta +\nu -s-1)^{ \underline{\nu -1}}h_{1}(s+\nu -1)\\ &{}+ \sum _{s=0}^{T-1}(T+\nu -s-2)^{\underline{ \nu -1}}h_{2}(s+ \nu -1) \Biggr] \\ &{} +b(\eta +\nu )^{\underline{\nu -1}} \Biggl[-a\sum_{s=0}^{\xi }( \xi +\nu -s-1)^{\underline{\nu -1}}h_{2}(s+ \nu -1)\\ &{}+ \sum _{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1) \Biggr] \Biggr] \\ =&-\frac{1}{\varGamma (\nu )}\sum_{s=0}^{t-\nu }(t-s-1)^{\underline{ \nu -1}}h_{2}(s+ \nu -1)\\ &{}+ \frac{t^{\underline{\nu -1}}(T+\nu -1)^{\underline{ \nu -1}}}{L\varGamma (\nu )}\sum_{s=0}^{T-1}(T+ \nu -s-2)^{\underline{ \nu -1}}h_{2}(s+\nu -1) \\ &{}-\frac{t^{\underline{\nu -1}}}{\varGamma (\nu )(T+\nu -1)^{\underline{ \nu -1}}}\sum_{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{2}(s+ \nu -1)\\ &{}+ \frac{t^{\underline{\nu -1}}}{\varGamma (\nu )(T+\nu -1)^{\underline{ \nu -1}}}\sum _{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{2}(s+ \nu -1) \\ &{}-\frac{ab(\eta +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{\xi }(\xi + \nu -s-1)^{\underline{\nu -1}}h _{2}(s+\nu -1)\\ &{}- \frac{b(T+\nu -1)^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L\varGamma (\nu )}\sum _{s=0}^{\eta }(\eta +\nu -s-1)^{\underline{ \nu -1}}h_{1}(s+ \nu -1) \\ &{}+\frac{b(\eta +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{T-1}(T+\nu -s-2)^{\underline{\nu -1}}h_{1}(s+ \nu -1) \\ =&\sum_{s=0}^{T-1}G(t,s)h_{2}(s+\nu -1)\\ &{}+\frac{abt^{\underline{\nu -1}}( \xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{\nu -1}}}{L \varGamma (\nu )(T+\nu -1)^{\underline{\nu -1}}}\sum_{s=0}^{T-1}(T+ \nu -s-2)^{\underline{\nu -1}}h_{2}(s+\nu -1) \\ &{}-\frac{ab(\eta +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L \varGamma (\nu )}\sum_{s=0}^{\xi }(\xi + \nu -s-1)^{\underline{\nu -1}}h _{2}(s+\nu -1)\\ &{}- \frac{b(T+\nu -1)^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L\varGamma (\nu )}\sum _{s=0}^{\eta }(\eta +\nu -s-1)^{\underline{ \nu -1}}h_{1}(s+ \nu -1) \\ &{}+\frac{b(\eta +\nu )^{\underline{\nu -1}}t^{\underline{\nu -1}}(T+ \nu -1)^{\underline{\nu -1}}}{L\varGamma (\nu )}\sum_{s=0}^{T-1} \frac{(T+ \nu -s-2)^{\underline{v-1}}}{(T+\nu -1)}h_{1}(s+\nu -1) \\ =&\sum_{s=0}^{T-1}G(t,s)h_{2}(s+\nu -1)\\ &{}+\frac{abt^{\underline{\nu -1}}( \eta +\nu )^{\underline{\nu -1}}}{L}\sum_{s=0}^{T-1}G(\xi +\nu ,s)h _{2}(s+\nu -1) \\ &{}+\frac{b(T+\nu -1)^{\underline{\nu -1}}t^{\underline{\nu -1}}}{L} \sum_{s=0}^{T-1}G(\eta +\nu ,s)h_{1}(s+\nu -1) \\ =&\sum_{s=0}^{T-1} \biggl[G(t,s)+ \frac{abt^{\underline{\nu -1}}(\eta + \nu )^{\underline{\nu -1}}}{L}G(\xi +\nu ,s) \biggr]h_{2}(s+\nu -1)\\ &{}+ \sum _{s=0}^{T-1}\frac{b(T+\nu -1)^{\underline{\nu -1}}t^{\underline{ \nu -1}}}{L}G(\eta +\nu ,s)]h_{1}(s+\nu -1). \end{aligned}$$

This completes the proof. □

Lemma 2.6

(see [40, Theorems 2.2, 2.3 and Remark 2.4])

Let \(L_{1}=\frac{\nu -1}{T(T+\nu -1)^{\underline{\nu -1}}(T+\nu -2)}\) and \(\rho (s)=(T+\nu -s-2)^{\underline{\nu -1}}\) for \(s\in [0,T-1]_{ \mathbf{N}_{0}}\). Then we have

  1. (i)

    \(G(t,s)>0\), for \((t,s)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{ \nu -1}}\times [0,T-1]_{\mathbf{N}_{0}}\);

  2. (ii)

    for all \((t,s)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\times [0,T-1]_{\mathbf{N}_{0}}\), there holds

    $$\begin{aligned}& \frac{abL_{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{ \nu -1}}t^{\underline{\nu -1}} \rho (s)}{L\varGamma (\nu )}\\& \quad \le H_{1}(t,s) \le \frac{[L+ab(\xi +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}]\rho (s)}{L\varGamma (\nu )}, \\& \frac{aL_{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{ \nu -1}}t^{\underline{\nu -1}}\rho (s)}{L\varGamma (\nu )}\\& \quad \le K_{1}(t,s) \le \frac{a(T+\nu -1)^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}\rho (s)}{L\varGamma (\nu )}; \end{aligned}$$
  3. (iii)

    for all \((t,s)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\times [0,T-1]_{\mathbf{N}_{0}}\), there holds

    $$\begin{aligned}& \frac{abL_{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{ \nu -1}}t^{\underline{\nu -1}}\rho (s)}{L\varGamma (\nu )}\\& \quad \le H_{2}(t,s) \le \frac{[L+ab(\eta +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}]\rho (s)}{L\varGamma (\nu )}, \\& \frac{bL_{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{ \nu -1}}t^{\underline{\nu -1}}\rho (s)}{L\varGamma (\nu )}\\& \quad \le K_{2}(t,s) \le \frac{b(T+\nu -1)^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}\rho (s)}{L\varGamma (\nu )}. \end{aligned}$$

Lemma 2.7

Let \(\rho ^{*}(t)=(T+2\nu -t-3)^{ \underline{\nu -1}}\) for \(t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\). Then, for all \(s\in [0,T-1]_{\mathbf{N}_{0}}\), we have the following inequalities:

$$ h_{\mu _{1}} \rho (s)\le \sum_{t=\nu -1}^{T+\nu -2} H_{1}(t,s) \rho ^{*}(t) \le h_{\mu _{2}} \rho (s), $$
(2.8)

and

$$ k_{\mu _{1}} \rho (s)\le \sum_{t=\nu -1}^{T+\nu -2} K_{1}(t,s) \rho ^{*}(t) \le k_{\mu _{2}} \rho (s), $$
(2.9)

and

$$ h_{\mu _{3}} \rho (s)\le \sum_{t=\nu -1}^{T+\nu -2} H_{2}(t,s) \rho ^{*}(t) \le h_{\mu _{4}} \rho (s), $$
(2.10)

and

$$ k_{\mu _{3}} \rho (s)\le \sum_{t=\nu -1}^{T+\nu -2} K_{2}(t,s) \rho ^{*}(t) \le k_{\mu _{4}} \rho (s), $$
(2.11)

where

$$\begin{aligned}& \begin{pmatrix} h_{\mu _{1}} & h_{\mu _{2}} \\ k_{\mu _{1}} & k_{\mu _{2}} \\ h_{\mu _{3}} & h_{\mu _{4}} \\ k_{\mu _{3}} & k_{\mu _{4}} \end{pmatrix}\\& \quad = \begin{pmatrix} \sum_{t=0}^{T-1} \frac{abL_{1}(\xi +\nu )^{\underline{\nu -1}}( \eta +\nu )^{\underline{\nu -1}}{(t+\nu -1)}^{\underline{\nu -1}} \rho (t)}{L\varGamma (\nu )} & \sum_{t=0}^{T-1} \frac{[L+ab( \xi +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{\nu -1}}]\rho (t)}{L \varGamma (\nu )} \\ \sum_{t=0}^{T-1} \frac{aL_{1}(\xi +\nu )^{\underline{\nu -1}}( \eta +\nu )^{\underline{\nu -1}}(t+\nu -1)^{\underline{\nu -1}}\rho (t)}{L \varGamma (\nu )} & \sum_{t=0}^{T-1}\frac{a(T+\nu -1)^{\underline{ \nu -1}}(T+\nu -2)^{\underline{\nu -1}}\rho (t)}{L\varGamma (\nu )} \\ \sum_{t=0}^{T-1} \frac{abL_{1}(\xi +\nu )^{\underline{\nu -1}}( \eta +\nu )^{\underline{\nu -1}}(t+\nu -1)^{\underline{\nu -1}}\rho (t)}{L \varGamma (\nu )} & \sum_{t=0}^{T-1} \frac{[L+ab(\eta +\nu )^{\underline{ \nu -1}}(T+\nu -2)^{\underline{\nu -1}}]\rho (t)}{L\varGamma (\nu )} \\ \sum_{t=0}^{T-1} \frac{bL_{1}(\xi +\nu )^{\underline{\nu -1}}( \eta +\nu )^{\underline{\nu -1}}(t+\nu -1)^{\underline{\nu -1}}\rho (t)}{L \varGamma (\nu )} & \sum_{t=0}^{T-1}\frac{b(T+\nu -1)^{\underline{ \nu -1}}(T+\nu -2)^{\underline{\nu -1}}\rho (t)}{L\varGamma (\nu )} \end{pmatrix} . \end{aligned}$$

Proof

We only prove (2.8). Indeed, for all \(s\in [0,T-1]_{\mathbf{N}_{0}}\), from Lemma 2.6(ii) we have

$$\begin{aligned} \sum_{t=\nu -1}^{T+\nu -2} H_{1}(t,s) \rho ^{*}(t) &= \sum _{t=0}^{T-1} H _{1}(t-\nu +1,s) \rho ^{*}(t+\nu -1) \\ &\le \sum_{t=0}^{T-1} \frac{[L+ab( \xi +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{\nu -1}}]\rho (s)}{L \varGamma (\nu )} \rho ^{*}(t+\nu -1) \\ & \le \sum_{t=0}^{T-1} \frac{[L+ab( \xi +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{\nu -1}}]\rho (s)}{L \varGamma (\nu )} \rho (t)= h_{\mu _{2}} \rho (s). \end{aligned} $$

On the other hand, we obtain

$$\begin{aligned} \sum_{t=\nu -1}^{T+\nu -2} H_{1}(t,s) \rho ^{*}(t) &= \sum _{t=0}^{T-1} H _{1}(t-\nu +1,s) \rho ^{*}(t+\nu -1) \\ &\ge \sum_{t=0}^{T-1} \frac{abL _{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{\nu -1}} {(t+\nu -1)}^{\underline{\nu -1}} \rho (s)}{L\varGamma (\nu )} \rho ^{*}(t+ \nu -1) \\ & \ge \sum_{t=0}^{T-1} \frac{abL_{1}(\xi +\nu )^{\underline{ \nu -1}}(\eta +\nu )^{\underline{\nu -1}}{(t+\nu -1)}^{\underline{ \nu -1}} \rho (s)}{L\varGamma (\nu )} \rho (t)\\ &= h_{\mu _{1}} \rho (s). \end{aligned} $$

This completes the proof. □

Let E be the collection of all maps from \([\nu -3,T+\nu -2]_{ \mathbf{N}_{\nu -3}}\) to R with the norm \(\|z\|= \max_{t\in [\nu -3,T+\nu -2]_{\mathbf{N}_{\nu -3}}}|z(t)|\). Then \(( E,\|\cdot \|)\) is a Banach space, and \(P=\{z\in E: z(t)\ge 0, t \in [\nu -3,T+\nu -2]_{\mathbf{N}_{\nu -3}} \}\) is a cone on E. From Lemma 2.5, we know that the fractional difference system (1.1) can be expressed in the following form:

$$\begin{aligned} &\footnotesize \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} \\ &\footnotesize\quad = \begin{pmatrix} \sum_{s=0}^{T-1}H_{1}(t,s)f_{1}(s+\nu -1,x(s+\nu -1),y(s+ \nu -1)) +\sum_{s=0}^{T-1}K_{1}(t,s)f_{2}(s+\nu -1,x(s+\nu -1),y(s+ \nu -1)) \\ \sum_{s=0}^{T-1}H_{2}(t,s)f_{2}(s+\nu -1,x(s+ \nu -1),y(s+\nu -1)) +\sum_{s=0}^{T-1}K_{2}(t,s)f_{1}(s+\nu -1,x(s+ \nu -1),y(s+\nu -1)) \end{pmatrix} \\ &\footnotesize \quad := \begin{pmatrix} A_{1}(x,y)(t) \\ A_{2}(x,y)(t) \end{pmatrix}, \quad \textstyle\begin{array}{l}\forall t\in [\nu -3,T+\nu -2]_{\mathbf{N}_{\nu -3}}.\end{array}\displaystyle \end{aligned}$$
(2.12)

Then we define an operator \(A:P\times P\to P\times P\) as follows:

$$A(x,y) (t)=(A_{1},A_{2}) (x,y) (t), \quad \forall t\in [\nu -3,T+\nu -2]_{ \mathbf{N}_{\nu -3}}. $$

Then positive solutions for the fractional difference system (1.1) exist if and only if positive fixed points for A exist, i.e., if there exists \((\overline{x},\overline{y})\in P\) such that \(A(\overline{x},\overline{y})=(\overline{x},\overline{y})\), and \(A_{1}(\overline{x},\overline{y})(t)=\overline{x}(t)\), \(A_{2}( \overline{x},\overline{y})(t)=\overline{y}(t)\), from (2.12) we have \((\overline{x},\overline{y})(t)\) is a positive solution for (1.1), for \(t\in [\nu -3,T+\nu -2]_{ \mathbf{N}_{\nu -3}}\). Now, we turn to study the existence of fixed points for the operator A. In what follows, we provide two lemmas involving the fixed point index; for more details, we refer to the book [56].

Lemma 2.8

Let E be a real Banach space and P be a cone on E. Suppose that \(\varOmega \subset E\) is a bounded open set and that \(A:\overline{\varOmega }\cap P\to P\) is a continuous compact operator. If there exists \(\omega _{0}\in P\backslash \{0\}\) such that

$$\omega -A\omega \neq \lambda \omega _{0},\quad \forall \lambda \geq 0, \omega \in \partial \varOmega \cap P, $$

then \(i(A,\varOmega \cap P,P)=0\), where i denotes the fixed point index on P.

Lemma 2.9

Let E be a real Banach space and P be a cone on E. Suppose that \(\varOmega \subset E\) is a bounded open set with \(0\in \varOmega \) and that \(A:\overline{\varOmega }\cap P\to P\) is a continuous compact operator. If

$$\omega -\lambda A\omega \neq 0,\quad \forall \lambda \in [0,1], \omega \in \partial \varOmega \cap P, $$

then \(i(A,\varOmega \cap P,P)=1\).

Main results

In this section, we first provide some assumptions for our nonlinearities \(f_{i}\), \(i=1,2\). Here, we make an explanation: in \(P\times P\), if ( x 1 x 2 ) (or ) ( y 1 y 2 ) , we mean that \(x_{1}(t)\ge (\text{or } \le ) y(t)\), \(x_{2}(t) \ge (\text{or } \le ) y_{2}(t)\) for \(t\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\).

  1. (H1)

    There exist \(a_{1},b_{1},c_{1},d_{1}\ge 0\) and \(l_{1},l_{2}>0\) such that

    $$\begin{aligned}& \begin{gathered} \begin{pmatrix} f_{1}(t,x,y) \\ f_{2}(t,x,y) \end{pmatrix}\ge \begin{pmatrix} a_{1}x+b_{1}y-l_{1} \\ c_{1}x+d_{1}y-l_{2} \end{pmatrix},\\ \quad \forall (t,x,y)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}} \times \mathbf{R}^{+}\times \mathbf{R}^{+}, \end{gathered} \end{aligned}$$

    and

    $$\begin{aligned}& h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}< 1,\qquad h_{\mu _{3}}d_{1}+k_{\mu _{3}}b _{1}< 1,\\& \det \begin{pmatrix} h_{\mu _{1}}b_{1}+k_{\mu _{1}}d_{1} & h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}- 1 \\ h_{\mu _{3}}d_{1}+k_{\mu _{3}}b_{1}-1 & h_{\mu _{3}}c_{1}+k_{\mu _{3}}a _{1} \end{pmatrix}:=\kappa _{1}>0. \end{aligned}$$
  2. (H2)

    There exist \(a_{2} ,b_{2} ,c_{2} ,d_{2}\ge 0\) and \(r_{1} > 0\) such that

    $$\begin{aligned}& \begin{pmatrix} f_{1}(t,x,y) \\ f_{2}(t,x,y) \end{pmatrix}\le \begin{pmatrix} a_{2}x+b_{2}y \\ c_{2}x+d_{2}y \end{pmatrix}, \\& \quad \forall (t,x,y)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}} \times [0,r_{1}] \times [0,r_{1}], \end{aligned}$$

    and

    $$\begin{aligned}& h_{\mu _{2}}a_{2}+k_{\mu _{2}}c_{2}< 1,\qquad h_{\mu _{4}}d_{2}+k_{\mu _{4}}b _{2}< 1, \\& \det \begin{pmatrix} 1-h_{\mu _{2}}a_{2}-k_{\mu _{2}}c_{2} & -h_{\mu _{2}}b_{2}-k_{\mu _{2}}d _{2} \\ -h_{\mu _{4}}c_{2}-k_{\mu _{4}} a_{2} & 1-h_{\mu _{4}}d_{2}-k_{\mu _{4}}b _{2} \end{pmatrix}:=\kappa _{2}>0. \end{aligned}$$
  3. (H3)

    There exist \(a_{3} ,b_{3} ,c_{3} ,d_{3}\ge 0\) and \(r_{2} > 0\) such that

    $$\begin{aligned}& \begin{pmatrix} f_{1}(t,x,y) \\ f_{2}(t,x,y) \end{pmatrix}\ge \begin{pmatrix} a_{3}x+b_{3}y \\ c_{3}x+d_{3}y \end{pmatrix},\\& \quad \forall (t,x,y)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}} \times [0,r_{2}]\times [0,r_{2}], \end{aligned}$$

    and

    $$\begin{aligned}& h_{\mu _{1}}a_{3}+k_{\mu _{1}}c_{3}< 1,\qquad h_{\mu _{3}}d_{3}+k_{\mu _{3}}b _{3}< 1,\\& \det \begin{pmatrix} h_{\mu _{1}}b_{3}+k_{\mu _{1}}d_{3} & h_{\mu _{1}}a_{3}+k_{\mu _{1}}c_{3}- 1 \\ h_{\mu _{3}}d_{3}+k_{\mu _{3}}b_{3}-1 & h_{\mu _{3}}c_{3}+k_{\mu _{3}}a _{3} \end{pmatrix}:=\kappa _{3}>0. \end{aligned}$$
  4. (H4)

    There exist \(a_{4},b_{4},c_{4},d_{4}\ge 0\) and \(l_{3},l_{4}>0\) such that

    $$\begin{aligned}& \begin{pmatrix} f_{1}(t,x,y) \\ f_{2}(t,x,y) \end{pmatrix}\le \begin{pmatrix} a_{4}x+b_{4}y+l_{3} \\ c_{4}x+d_{4}y+l_{4} \end{pmatrix}, \\& \quad \forall (t,x,y)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}} \times \mathbf{R}^{+} \times \mathbf{R}^{+}, \end{aligned}$$

    and

    $$\begin{aligned}& h_{\mu _{2}}a_{4}+k_{\mu _{2}}c_{4} < 1, \qquad h_{\mu _{4}}d_{4}+k_{\mu _{4}}b _{4}< 1, \\& \det \begin{pmatrix} 1-h_{\mu _{2}}a_{4}-k_{\mu _{2}}c_{4} & -h_{\mu _{2}}b_{4}-k_{\mu _{2}}d _{4} \\ -h_{\mu _{4}}c_{4}-k_{\mu _{4}} a_{4} & 1-h_{\mu _{4}}d_{4}-k_{\mu _{4}}b _{4} \end{pmatrix}:=\kappa _{4}>0. \end{aligned}$$

Theorem 3.1

Suppose that (H1)–(H2) hold. Then the fractional difference system (1.1) has at least one positive solution.

Proof

Define a set

$$S_{1}=\bigl\{ (x,y)\in P\times P: (x,y)=A(x,y)+\lambda (\varphi _{0},\varphi _{0})\text{ for some }\lambda \ge 0\bigr\} , $$

where \(\varphi _{0}\in P\) is a fixed element. Then we will claim that \(S_{1}\) is a bounded set in \(P\times P\). In fact, if \((x,y)\in S_{1}\), we have \(x(t)=A_{1}(x,y)(t)+\lambda \varphi _{0}(t)\), \(y(t)=A_{2}(x,y)(t)+ \lambda \varphi _{0}(t)\) for \(t\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\). Together with (H1), we obtain

$$\footnotesize \begin{aligned} \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} &\ge \begin{pmatrix} A_{1}(x,y)(t) \\ A_{2}(x,y)(t) \end{pmatrix} \\ & \ge \begin{pmatrix} \sum_{s=0}^{T-1}H_{1}(t,s)(a_{1}x(s+\nu -1)+b_{1}y(s+\nu -1)-l _{1})+\sum_{s=0}^{T-1}K_{1}(t,s)(c_{1}x(s+\nu -1)+d_{1}y(s+ \nu -1)-l_{2}) \\ \sum_{s=0}^{T-1}H_{2}(t,s)(c_{1}x(s+ \nu -1)+d_{1}y(s+\nu -1)-l_{2})+\sum_{s=0}^{T-1}K_{2}(t,s)(a _{1}x(s+\nu -1)+b_{1}y(s+\nu -1)-l_{1}) \end{pmatrix} \end{aligned} $$

for \(t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\). Multiplying both sides of the above inequality by \(\rho ^{*}(t)\) and summing from \(\nu -1\) to \(T+\nu -2\), together with (2.8)–(2.11), we obtain

$$\tiny \begin{aligned} & \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix} \\ &\quad \ge \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) [ \sum_{s=0} ^{T-1}H_{1}(t,s)(a_{1}x(s+\nu -1)+b_{1}y(s+\nu -1)-l_{1})+\sum_{s=0}^{T-1}K_{1}(t,s)(c_{1}x(s+\nu -1)+d_{1}y(s+\nu -1)-l_{2}) ] \\ \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) [\sum_{s=0}^{T-1}H_{2}(t,s)(c_{1}x(s+\nu -1)+d_{1}y(s+\nu -1)-l_{2})+ \sum_{s=0}^{T-1}K_{2}(t,s)(a_{1}x(s+\nu -1)+b_{1}y(s+\nu -1)-l _{1}) ] \end{pmatrix} \\ & \quad \ge \begin{pmatrix} h_{\mu _{1}} \sum_{s=0}^{T-1}\rho (s)(a_{1}x(s+\nu -1)+b_{1}y(s+ \nu -1))+k_{\mu _{1}}\sum_{s=0}^{T-1}\rho (s)(c_{1}x(s+\nu -1)+d _{1}y(s+\nu -1))-(h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\sum_{s=0} ^{T-1}\rho (s) \\ h_{\mu _{3}} \sum_{s=0}^{T-1}\rho (s)(c _{1}x(s+\nu -1)+d_{1}y(s+\nu -1))+k_{\mu _{3}} \sum_{s=0}^{T-1} \rho (s)(a_{1}x(s+\nu -1)+b_{1}y(s+\nu -1))-(k_{\mu _{4}}l_{1}+h_{\mu _{4}}l_{2})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix} \\ &\quad = \begin{pmatrix} \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)[h_{\mu _{1}}(a_{1}x(s+\nu -1)+b _{1}y(s+\nu -1))+k_{\mu _{1}}(c_{1}x(s+\nu -1)+d_{1}y(s+\nu -1))]-(h _{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\sum_{s=0}^{T-1}\rho (s) \\ \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)[h_{\mu _{3}}(c_{1}x(s+ \nu -1)+d_{1}y(s+\nu -1))+k_{\mu _{3}}(a_{1}x(s+\nu -1)+b_{1}y(s+ \nu -1))]-(k_{\mu _{4}}l_{1}+h_{\mu _{4}}l_{2})\sum_{s=0}^{T-1} \rho (s) \end{pmatrix} \\ &\quad = \begin{pmatrix} h_{\mu _{1}} \sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{1}x(t)+b _{1}y(t))+k_{\mu _{1}}\sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(c _{1}x(t)+d_{1}y(t))-(h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\sum_{s=0}^{T-1}\rho (s) \\ h_{\mu _{3}} \sum_{t=\nu -1}^{T+ \nu -2}\rho ^{*}(t)(c_{1}x(t)+d_{1}y(t))+k_{\mu _{3}} \sum_{t= \nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{1}x(t)+b_{1}y(t))-(k_{\mu _{4}}l_{1}+h _{\mu _{4}}l_{2})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix}. \end{aligned} $$

This implies that

$$\begin{aligned}& \begin{pmatrix} h_{\mu _{1}}b_{1}+k_{\mu _{1}}d_{1} & h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}- 1 \\ h_{\mu _{3}}d_{1}+k_{\mu _{3}}b_{1}-1 & h_{\mu _{3}}c_{1}+k_{\mu _{3}}a _{1} \end{pmatrix} \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \end{pmatrix}\\& \quad \le \begin{pmatrix} (h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\sum_{s=0}^{T-1}\rho (s) \\ (k_{\mu _{4}}l_{1}+h_{\mu _{4}}l_{2})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix}. \end{aligned}$$

Solving this matrix inequality, we have

$$\begin{aligned}& \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \end{pmatrix}\\& \quad \le \kappa _{1}^{-1} \begin{pmatrix} h_{\mu _{3}}c_{1}+k_{\mu _{3}}a_{1} & 1- h_{\mu _{1}}a_{1}-k_{\mu _{1}}c _{1} \\ 1-h_{\mu _{3}}d_{1}-k_{\mu _{3}}b_{1} & h_{\mu _{1}}b_{1}+k_{\mu _{1}}d _{1} \end{pmatrix} \begin{pmatrix} (h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\sum_{s=0}^{T-1}\rho (s) \\ (k_{\mu _{4}}l_{1}+h_{\mu _{4}}l_{2})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} & \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t)\\ &\quad \le \kappa _{1} ^{-1} \bigl[(h_{\mu _{3}}c_{1}+k_{\mu _{3}}a_{1}) (h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})+( 1- h_{\mu _{1}}a_{1}-k_{\mu _{1}}c_{1}) (k_{\mu _{4}}l_{1}+h _{\mu _{4}}l_{2})\bigr]\sum _{s=0}^{T-1}\rho (s), \\ & \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t)\\ &\quad \le \kappa _{1}^{-1}\bigl[(1-h _{\mu _{3}}d_{1}-k_{\mu _{3}}b_{1}) (h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2}) + (h_{\mu _{1}}b_{1}+k_{\mu _{1}}d_{1}) (k_{\mu _{4}}l_{1}+h_{\mu _{4}}l _{2})\bigr] \sum _{s=0}^{T-1}\rho (s). \end{aligned}$$

On the other hand, there exist \(t_{1},t_{2}\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\) such that

$$ \begin{aligned} &x(t_{1}) \rho ^{*}(t_{1})= \Vert x \Vert \rho ^{*}(t_{1})\le \sum _{t= \nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) ,\\ &y(t_{2}) \rho ^{*}(t_{2})= \Vert y \Vert \rho ^{*}(t_{2})\le \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t). \end{aligned} $$
(3.1)

Consequently, we have

$$\begin{aligned} \Vert x \Vert \le {}& \frac{1}{\kappa _{1}\rho ^{*}(t_{1})} \bigl[(1-h_{\mu _{3}}d_{1}-k _{\mu _{3}}b_{1}) (h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\\ &{} + (h_{\mu _{1}}b _{1}+k_{\mu _{1}}d_{1}) (k_{\mu _{4}}l_{1}+h_{\mu _{4}}l_{2})\bigr] \sum _{s=0}^{T-1}\rho (s), \\ \Vert y \Vert \le {}&\frac{1}{\kappa _{1}\rho ^{*}(t _{2})}\bigl[(h_{\mu _{3}}c_{1}+k_{\mu _{3}}a_{1}) (h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})\\ &{}+( 1- h_{\mu _{1}}a_{1}-k_{\mu _{1}}c_{1}) (k_{\mu _{4}}l_{1}+h _{\mu _{4}}l_{2})\bigr]\sum _{s=0}^{T-1}\rho (s). \end{aligned}$$

This proves that \(S_{1}\) is bounded in \(P\times P\). Then we can choose a positive number \(R_{1}>r_{1}\), \(R_{1}>\frac{1}{\kappa _{1}\rho ^{*}(t _{1})}[(1-h_{\mu _{3}}d_{1}-k_{\mu _{3}}b_{1})(h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2}) + (h_{\mu _{1}}b_{1}+k_{\mu _{1}}d_{1})(k_{\mu _{4}}l_{1}+h _{\mu _{4}}l_{2})] \sum_{s=0}^{T-1}\rho (s)\), and \(R_{1}> \frac{1}{ \kappa _{1}\rho ^{*}(t_{2})}[(h_{\mu _{3}}c_{1}+k_{\mu _{3}}a_{1})(h_{\mu _{2}} l_{1}+k_{\mu _{2}}l_{2})+( 1- h_{\mu _{1}}a_{1}-k_{\mu _{1}}c_{1})(k _{\mu _{4}}l_{1}+h_{\mu _{4}}l_{2})]\sum_{s=0}^{T-1}\rho (s)\) such that

$$ (x,y)\neq A(x,y)+\lambda (\varphi _{0},\varphi _{0}), \quad \text{for } (x,y) \in \partial B_{R_{1}}\cap (P\times P), \lambda \ge 0. $$
(3.2)

As a result, Lemma 2.8 implies

$$ i\bigl(A,B_{R_{1}} \cap (P \times P),P \times P\bigr)=0. $$
(3.3)

In what follows, we prove that

$$ (x,y)\neq \lambda A(x,y), \quad \text{for } (x,y)\in \partial B_{r_{1}} \cap (P\times P), \lambda \in [0,1], $$
(3.4)

where \(r_{1}\) is defined by (H2). Argument by contrary, there exist \((x,y)\in \partial B_{r_{1}}\cap (P\times P)\), \(\lambda _{0}\in [0,1]\) such that \((x,y)=\lambda _{0} A(x,y)\), and thus from (H2) we obtain

$$\footnotesize \begin{aligned} \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} &\le \begin{pmatrix} A_{1}(x,y)(t) \\ A_{2}(x,y)(t) \end{pmatrix} \\ & \le \begin{pmatrix} \sum_{s=0}^{T-1}H_{1}(t,s)(a_{2}x(s+\nu -1)+b_{2}y(s+\nu -1))+ \sum_{s=0}^{T-1}K_{1}(t,s)(c_{2}x(s+\nu -1)+d_{2}y(s+\nu -1)) \\ \sum_{s=0}^{T-1}H_{2}(t,s)(c_{2}x(s+\nu -1)+d_{2}y(s+ \nu -1))+\sum_{s=0}^{T-1}K_{2}(t,s)(a_{2}x(s+\nu -1)+b_{2}y(s+ \nu -1)) \end{pmatrix}. \end{aligned} $$

Multiplying both sides of the above inequality by \(\rho ^{*}(t)\) and summing from \(\nu -1\) to \(T+\nu -2\), together with (2.8)–(2.11), we obtain

$$\tiny \begin{aligned} & \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix} \\ &\quad \le \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) \sum_{s=0}^{T-1}H _{1}(t,s)(a_{2}x(s+\nu -1)+b_{2}y(s+\nu -1))+\sum_{t=\nu -1} ^{T+\nu -2} \rho ^{*}(t) \sum_{s=0}^{T-1}K_{1}(t,s)(c_{2}x(s+ \nu -1)+d_{2}y(s+\nu -1)) \\ \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) \sum_{s=0}^{T-1}H_{2}(t,s)(c_{2}x(s+\nu -1)+d_{2}y(s+ \nu -1))+\sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) \sum_{s=0}^{T-1}K_{2}(t,s)(a_{2}x(s+\nu -1)+b_{2}y(s+\nu -1)) \end{pmatrix} \\ &\quad \le \begin{pmatrix} h_{\mu _{2}}\sum_{s=0}^{T-1}\rho (s)(a_{2}x(s+\nu -1)+b_{2}y(s+ \nu -1))+k_{\mu _{2}} \sum_{s=0}^{T-1}\rho (s)(c_{2}x(s+\nu -1)+d _{2}y(s+\nu -1)) \\ h_{\mu _{4}}\sum_{s=0}^{T-1}\rho (s)(c _{2}x(s+\nu -1)+d_{2}y(s+\nu -1))+k_{\mu _{4}} \sum_{s=0}^{T-1} \rho (s)(a_{2}x(s+\nu -1)+b_{2}y(s+\nu -1)) \end{pmatrix} \\ &\quad = \begin{pmatrix} h_{\mu _{2}}\sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)(a_{2}x(s+\nu -1)+b _{2}y(s+\nu -1))+k_{\mu _{2}} \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)(c _{2}x(s+\nu -1)+d_{2}y(s+\nu -1)) \\ h_{\mu _{4}}\sum_{s=0} ^{T-1}\rho ^{*}(s+\nu -1)(c_{2}x(s+\nu -1)+d_{2}y(s+\nu -1))+k_{\mu _{4}} \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)(a_{2}x(s+\nu -1)+b _{2}y(s+\nu -1)) \end{pmatrix} \\ & \quad = \begin{pmatrix} h_{\mu _{2}} \sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{2}x(t)+b _{2}y(t))+k_{\mu _{2}} \sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(c _{2}x(t)+d_{2}y(t)) \\ h_{\mu _{4}} \sum_{t=\nu -1}^{T+ \nu -2}\rho ^{*}(t)(c_{2}x(t)+d_{2}y(t))+k_{\mu _{4}} \sum_{t= \nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{2}x(t)+b_{2}y(t)) \end{pmatrix} . \end{aligned} $$

Solving this matrix inequality, we have

$$ \begin{pmatrix} 1-h_{\mu _{2}}a_{2}-k_{\mu _{2}}c_{2} & -h_{\mu _{2}}b_{2}-k_{\mu _{2}}d _{2} \\ -h_{\mu _{4}}c_{2}-k_{\mu _{4}} a_{2} & 1-h_{\mu _{4}}d_{2}-k_{\mu _{4}}b _{2} \end{pmatrix} \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix}\le \begin{pmatrix} 0 \\ 0 \end{pmatrix}. $$

Consequently, we have

$$ \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix}\le \kappa _{2}^{-1} \begin{pmatrix} 1-h_{\mu _{4}}d_{2}-k_{\mu _{4}}b_{2} & h_{\mu _{2}}b_{2}+k_{\mu _{2}}d _{2} \\ h_{\mu _{4}}c_{2}+k_{\mu _{4}} a_{2} & 1-h_{\mu _{2}}a_{2}-k_{\mu _{2}}c _{2} \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \end{pmatrix}. $$

Note that \(\rho ^{*}(t)\not \equiv 0\) for \(t\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\), whence \(x(t)=y(t)\equiv 0\) for \(t\in [\nu -1,T+ \nu -2]_{\mathbf{N}_{\nu -1}}\), and this contradicts \((x,y)\in \partial B_{r_{1}}\cap (P\times P)\) with \(r_{1}>0\). As a result, (3.4) holds, and from Lemma 2.9 we have

$$ i\bigl(A,B_{r_{1}} \cap (P \times P),P \times P\bigr)=1. $$
(3.5)

Up to now, (3.3) and (3.5) enabled us to obtain \(i(A,(B_{R_{1}}\backslash \overline{B}_{r_{1}})\cap (P\times P), P \times P)=-1\neq 0\). Hence the operator A has at least one fixed point on \((B_{R_{1}}\backslash \overline{B}_{r_{1}})\cap (P\times P)\), and therefore (1.1) has at least one positive solution. This completes the proof. □

Theorem 3.2

Suppose that (H3)–(H4) hold. Then the fractional difference system (1.1) has at least one positive solution.

Proof

We first prove that

$$ (x,y)\neq A(x,y)+\lambda (\varphi _{1},\varphi _{1}), \quad \text{for } (x,y) \in \partial B_{r_{2}}\cap (P\times P), \lambda \ge 0, $$
(3.6)

where \(\varphi _{1}\in P\) is a given element, and \(r_{2}\) is defined by (H3). Suppose the contrary. Then there exist \((x,y)\in \partial B_{r _{2}}\cap (P\times P)\text{ and } \lambda _{0}\ge 0\) such that

$$(x,y) =A(x,y)+\lambda _{0} (\varphi _{1},\varphi _{1}). $$

Associated with condition (H3), this means that

$$\footnotesize \begin{aligned} \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} &\ge \begin{pmatrix} A_{1}(x,y)(t) \\ A_{2}(x,y)(t) \end{pmatrix} \\ & \ge \begin{pmatrix} \sum_{s=0}^{T-1}H_{1}(t,s)(a_{3}x(s+\nu -1)+b_{3}y(s+\nu -1))+ \sum_{s=0}^{T-1}K_{1}(t,s)(c_{3}x(s+\nu -1)+d_{3}y(s+\nu -1)) \\ \sum_{s=0}^{T-1}H_{2}(t,s)(c_{3}x(s+\nu -1)+d_{3}y(s+ \nu -1))+\sum_{s=0}^{T-1}K_{2}(t,s)(a_{3}x(s+\nu -1)+b_{3}y(s+ \nu -1)) \end{pmatrix}. \end{aligned} $$

Multiplying both sides of the above inequality by \(\rho ^{*}(t)\) and summing from \(\nu -1\) to \(T+\nu -2\), together with (2.8)–(2.11), we obtain

$$\tiny \begin{aligned} & \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix} \\ &\quad \ge \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) [ \sum_{s=0} ^{T-1}H_{1}(t,s)(a_{3}x(s+\nu -1)+b_{3}y(s+\nu -1))+\sum_{s=0} ^{T-1}K_{1}(t,s)(c_{3}x(s+\nu -1)+d_{3}y(s+\nu -1)) ] \\ \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) [ \sum_{s=0} ^{T-1}H_{2}(t,s)(c_{3}x(s+\nu -1)+d_{3}y(s+\nu -1))+\sum_{s=0} ^{T-1}K_{2}(t,s)(a_{3}x(s+\nu -1)+b_{3}y(s+\nu -1)) ] \end{pmatrix} \\ &\quad \ge \begin{pmatrix} h_{\mu _{1}}\sum_{s=0}^{T-1}\rho (s)(a_{3}x(s+\nu -1)+b_{3}y(s+ \nu -1))+k_{\mu _{1}} \sum_{s=0}^{T-1}\rho (s)(c_{3}x(s+\nu -1)+d _{3}y(s+\nu -1)) \\ h_{\mu _{3}}\sum_{s=0}^{T-1}\rho (s)(c _{3}x(s+\nu -1)+d_{3}y(s+\nu -1))+k_{\mu _{3}} \sum_{s=0}^{T-1} \rho (s)(a_{3}x(s+\nu -1)+b_{3}y(s+\nu -1)) \end{pmatrix} \\ &\quad = \begin{pmatrix} h_{\mu _{1}}\sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)(a_{3}x(s+\nu -1)+b _{3}y(s+\nu -1))+k_{\mu _{1}} \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)(c _{3}x(s+\nu -1)+d_{3}y(s+\nu -1)) \\ h_{\mu _{3}}\sum_{s=0} ^{T-1}\rho ^{*}(s+\nu -1)(c_{3}x(s+\nu -1)+d_{3}y(s+\nu -1))+k_{\mu _{3}} \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)(a_{3}x(s+\nu -1)+b _{3}y(s+\nu -1)) \end{pmatrix} \\ &\quad = \begin{pmatrix} h_{\mu _{1}} \sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{3}x(t)+b _{3}y(t))+k_{\mu _{1}} \sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(c _{3}x(t)+d_{3}y(t)) \\ h_{\mu _{3}} \sum_{t=\nu -1}^{T+ \nu -2}\rho ^{*}(t)(c_{3}x(t)+d_{3}y(t))+k_{\mu _{3}} \sum_{t= \nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{3}x(t)+b_{3}y(t)) \end{pmatrix} . \end{aligned} $$

This leads us to obtain

$$ \begin{pmatrix} h_{\mu _{1}}b_{3}+k_{\mu _{1}}d_{3} & h_{\mu _{1}}a_{3}+k_{\mu _{1}}c_{3}- 1 \\ h_{\mu _{3}}d_{3}+k_{\mu _{3}}b_{3}-1 & h_{\mu _{3}}c_{3}+k_{\mu _{3}}a _{3} \end{pmatrix} \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \end{pmatrix} \le \begin{pmatrix} 0 \\ 0 \end{pmatrix}. $$

Solving this matrix inequality, we have

$$ \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \end{pmatrix}\le \kappa _{3}^{-1} \begin{pmatrix} h_{\mu _{3}}c_{3}+k_{\mu _{3}}a_{3} & 1- h_{\mu _{1}}a_{3}-k_{\mu _{1}}c _{3} \\ 1-h_{\mu _{3}}d_{3}-k_{\mu _{3}}b_{3} & h_{\mu _{1}}b_{3}+k_{\mu _{1}}d _{3} \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix}. $$

Hence, we find

$$ \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \end{pmatrix}. $$

Note that \(\rho ^{*}(t)\not \equiv 0\) for \(t\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\), whence \(x(t)=y(t)\equiv 0\) for \(t\in [\nu -1,T+ \nu -2]_{\mathbf{N}_{\nu -1}}\), and this contradicts \((x,y)\in \partial B_{r_{2}}\cap (P\times P)\) with \(r_{2}>0\). Consequently, (3.6) is satisfied, and Lemma 2.8 implies that

$$ i\bigl(A,B_{r_{2}} \cap (P \times P),P \times P\bigr)=0. $$
(3.7)

On the other hand, we claim that the set

$$S_{2}=\bigl\{ (x,y)\in P\times P: (x,y) = \lambda A(x,y) \text{ for some } \lambda \in [0,1]\bigr\} $$

is bounded in \(P\times P\). If there exists \((x,y)\in S_{2}\), then from (H4) we have

$$\footnotesize \begin{aligned} & \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} \\ &\quad \le \begin{pmatrix} A_{1}(x,y)(t) \\ A_{2}(x,y)(t) \end{pmatrix} \\ & \quad \le \begin{pmatrix} \sum_{s=0}^{T-1}H_{1}(t,s)(a_{4}x(s+\nu -1)+b_{4}y(s+\nu -1)+l _{3})+\sum_{s=0}^{T-1}K_{1}(t,s)(c_{4}x(s+\nu -1)+d_{4}y(s+ \nu -1)+l_{4}) \\ \sum_{s=0}^{T-1}H_{2}(t,s)(c_{4}x(s+ \nu -1)+d_{4}y(s+\nu -1)+l_{4})+\sum_{s=0}^{T-1}K_{2}(t,s)(a _{4}x(s+\nu -1)+b_{4}y(s+\nu -1)+l_{3}) \end{pmatrix} \end{aligned} $$

for \(t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\). Multiplying both sides of the above inequality by \(\rho ^{*}(t)\) and summing from \(\nu -1\) to \(T+\nu -2\), together with (2.8)–(2.11), we obtain

$$\tiny \begin{aligned} & \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix} \\ &\quad \le \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) [\sum_{s=0} ^{T-1}H_{1}(t,s)(a_{4}x(s+\nu -1)+b_{4}y(s+\nu -1)+l_{3})+\sum_{s=0}^{T-1}K_{1}(t,s)(c_{4}x(s+\nu -1)+d_{4}y(s+\nu -1)+l_{4}) ] \\ \sum_{t=\nu -1}^{T+\nu -2} \rho ^{*}(t) [ \sum_{s=0}^{T-1}H_{2}(t,s)(c_{4}x(s+\nu -1)+d_{4}y(s+\nu -1)+l_{4})+ \sum_{s=0}^{T-1}K_{2}(t,s)(a_{4}x(s+\nu -1)+b_{4}y(s+\nu -1)+l _{3}) ] \end{pmatrix} \\ &\quad \le \begin{pmatrix} h_{\mu _{2}} \sum_{s=0}^{T-1}\rho (s)(a_{4}x(s+\nu -1)+b_{4}y(s+ \nu -1))+k_{\mu _{2}}\sum_{s=0}^{T-1}\rho (s)(c_{4}x(s+\nu -1)+d _{4}y(s+\nu -1))+(h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\sum_{s=0} ^{T-1}\rho (s) \\ h_{\mu _{4}} \sum_{s=0}^{T-1}\rho (s)(c _{4}x(s+\nu -1)+d_{4}y(s+\nu -1))+k_{\mu _{4}} \sum_{s=0}^{T-1} \rho (s)(a_{4}x(s+\nu -1)+b_{4}y(s+\nu -1))+(k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix} \\ &\quad = \begin{pmatrix} \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)[h_{\mu _{2}}(a_{4}x(s+\nu -1)+b _{4}y(s+\nu -1))+k_{\mu _{2}}(c_{4}x(s+\nu -1)+d_{4}y(s+\nu -1))]+(h _{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\sum_{s=0}^{T-1}\rho (s) \\ \sum_{s=0}^{T-1}\rho ^{*}(s+\nu -1)[h_{\mu _{4}}(c_{4}x(s+ \nu -1)+d_{4}y(s+\nu -1))+k_{\mu _{4}}(a_{4}x(s+\nu -1)+b_{4}y(s+ \nu -1))]+(k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})\sum_{s=0}^{T-1} \rho (s) \end{pmatrix} \\ &\quad = \begin{pmatrix} h_{\mu _{2}} \sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{4}x(t)+b _{4}y(t))+k_{\mu _{2}}\sum_{t=\nu -1}^{T+\nu -2}\rho ^{*}(t)(c _{4}x(t)+d_{4}y(t))+(h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\sum_{s=0}^{T-1}\rho (s) \\ h_{\mu _{4}} \sum_{t=\nu -1}^{T+ \nu -2}\rho ^{*}(t)(c_{4}x(t)+d_{4}y(t))+k_{\mu _{4}} \sum_{t= \nu -1}^{T+\nu -2}\rho ^{*}(t)(a_{4}x(t)+b_{4}y(t))+(k_{\mu _{4}}l_{3}+h _{\mu _{4}}l_{4})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix}. \end{aligned} $$

Solving this matrix inequality, we have

$$\begin{aligned}& \begin{pmatrix} 1-h_{\mu _{2}}a_{4}-k_{\mu _{2}}c_{4} & -h_{\mu _{2}}b_{4}-k_{\mu _{2}}d _{4} \\ -h_{\mu _{4}}c_{4}-k_{\mu _{4}} a_{4} & 1-h_{\mu _{4}}d_{4}-k_{\mu _{4}}b _{4} \end{pmatrix} \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix}\\& \quad \le \begin{pmatrix} (h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\sum_{s=0}^{T-1}\rho (s) \\ (k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix}. \end{aligned}$$

This indicates that

$$\begin{aligned}& \begin{pmatrix} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t) \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \end{pmatrix}\\& \quad \le \kappa _{4}^{-1} \begin{pmatrix} 1-h_{\mu _{4}}d_{4}-k_{\mu _{4}}b_{4} & h_{\mu _{2}}b_{4}+k_{\mu _{2}}d _{4} \\ h_{\mu _{4}}c_{4}+k_{\mu _{4}} a_{4} & 1-h_{\mu _{2}}a_{4}-k_{\mu _{2}}c _{4} \end{pmatrix} \begin{pmatrix} (h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\sum_{s=0}^{T-1}\rho (s) \\ (k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})\sum_{s=0}^{T-1}\rho (s) \end{pmatrix}. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \sum_{t=\nu -1}^{T+\nu -2} x(t) \rho ^{*}(t)\le{}& \kappa _{4} ^{-1} \bigl[(1-h_{\mu _{4}}d_{4}-k_{\mu _{4}}b_{4}) (h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\\ &{}+(h_{\mu _{2}}b_{4}+k_{\mu _{2}}d_{4}) (k_{\mu _{4}}l_{3}+h _{\mu _{4}}l_{4})\bigr] \sum _{s=0}^{T-1}\rho (s) , \\ \sum_{t=\nu -1}^{T+\nu -2} y(t) \rho ^{*}(t) \le{}& \kappa _{4}^{-1} \bigl[(h _{\mu _{4}}c_{4}+k_{\mu _{4}} a_{4}) (h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\\ &{}+(1-h _{\mu _{2}}a_{4}-k_{\mu _{2}}c_{4}) (k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})\bigr] \sum _{s=0}^{T-1}\rho (s). \end{aligned} $$

Similarly, using (3.1) we have

$$\begin{aligned} \Vert x \Vert \le{}& \frac{1}{\kappa _{4}\rho ^{*}(t_{1})} \bigl[(1-h_{\mu _{4}}d_{4}-k _{\mu _{4}}b_{4}) (h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\\ &{}+(h_{\mu _{2}}b _{4}+k_{\mu _{2}}d_{4}) (k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4}) \bigr] \sum_{s=0}^{T-1}\rho (s), \\ \Vert y \Vert \le {}&\frac{1}{\kappa _{4}\rho ^{*}(t _{2})}\bigl[(h_{\mu _{4}}c_{4}+k_{\mu _{4}} a_{4}) (h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})\\ &{}+(1-h_{\mu _{2}}a_{4}-k_{\mu _{2}}c_{4}) (k_{\mu _{4}}l_{3}+h _{\mu _{4}}l_{4})\bigr] \sum _{s=0}^{T-1}\rho (s). \end{aligned} $$

Then we can choose a positive number \(R_{2}>r_{2}\), \(R_{2}>\frac{1}{ \kappa _{4}\rho ^{*}(t_{1})}[(1-h_{\mu _{4}}d_{4}-k_{\mu _{4}}b_{4})(h _{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})+(h_{\mu _{2}}b_{4}+k_{\mu _{2}}d_{4}) (k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})] \sum_{s=0}^{T-1}\rho (s)\), and \(R_{2}> \frac{1}{\kappa _{4}\rho ^{*}(t_{2})}[(h_{\mu _{4}}c_{4}+k _{\mu _{4}} a_{4})(h_{\mu _{2}} l_{3}+k_{\mu _{2}}l_{4})+(1-h_{\mu _{2}}a _{4}-k_{\mu _{2}}c_{4})(k_{\mu _{4}}l_{3}+h_{\mu _{4}}l_{4})] \sum_{s=0}^{T-1}\rho (s)\) such that

$$ (x,y)\neq\lambda A(x,y), \quad \text{for } (x,y)\in \partial B_{R_{2}} \cap (P\times P), \lambda \in [0,1]. $$
(3.8)

As a result, Lemma 2.9 implies

$$ i\bigl(A,B_{R_{2}} \cap (P \times P),P \times P\bigr)=1. $$
(3.9)

Now, (3.7) and (3.9) enable us to obtain \(i(A,(B_{R_{2}}\backslash \overline{B}_{r_{2}})\cap (P\times P), P \times P)=1\neq 0\). Hence the operator A has at least one fixed point on \((B_{R_{2}}\backslash \overline{B}_{r_{2}})\cap (P\times P)\), and therefore (1.1) has at least one positive solution. This completes the proof. □

Example 3.3

Consider equation (1.1) with \(\nu = \frac{5}{2}\), \(T=4\), \(\alpha =\frac{1}{3}\), \(\xi =1\), \(\eta =2\), \(a=\frac{2}{3}\), \(b=\frac{4}{3}\). Then we need to calculate the following values: \(L= (\frac{\varGamma (T+\nu )}{T!} )^{2}- \frac{ab \varGamma (\xi +\nu +1)\varGamma (\eta +\nu +1)}{(\xi +1)!(\eta +1)!}= (\frac{ \varGamma (\frac{13}{2})}{24} ) ^{2}-\frac{\frac{8}{9}\varGamma ( \frac{9}{2})\varGamma (\frac{11}{2})}{12}\approx 98.9>0\), \(L_{1}=\frac{ \nu -1}{T(T+\nu -1)^{\underline{\nu -1}}(T+\nu -2)}\approx 0.007\), \(\varGamma (\nu )\approx 1.33\), \((\xi +\nu )^{\underline{\nu -1}}\approx 5.82, (\eta +\nu )^{\underline{\nu -1}}= (T+\nu -2)^{\underline{ \nu -1}}\approx 8.72,(T+\nu -1)^{\underline{\nu -1}}\approx 12\), \(\sum_{t=0}^{3} \rho (t)=\sum_{t=0}^{3} (T+\nu -t-2)^{\underline{ \nu -1}}\approx 19.19\), \(\sum_{t=0}^{3} {(t+\nu -1)}^{\underline{ \nu -1}} \rho (t)=\sum_{t=0}^{3} {(t+\nu -1)}^{\underline{ \nu -1}} (T+\nu -t-2)^{\underline{\nu -1}}\approx 61.84\). Then we have

$$\begin{aligned} & \begin{pmatrix} h_{\mu _{1}} & h_{\mu _{2}} \\ k_{\mu _{1}} & k_{\mu _{2}} \\ h_{\mu _{3}} & h_{\mu _{4}} \\ k_{\mu _{3}} & k_{\mu _{4}} \end{pmatrix} \\ &\quad = \begin{pmatrix} \sum_{t=0}^{T-1} \frac{abL_{1}(\xi +\nu )^{\underline{\nu -1}}( \eta +\nu )^{\underline{\nu -1}}{(t+\nu -1)}^{\underline{\nu -1}} \rho (t)}{L\varGamma (\nu )} & \sum_{t=0}^{T-1} \frac{[L+ab( \xi +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{\nu -1}}]\rho (t)}{L \varGamma (\nu )} \\ \sum_{t=0}^{T-1} \frac{aL_{1}(\xi +\nu )^{\underline{ \nu -1}}(\eta +\nu )^{\underline{\nu -1}}(t+\nu -1)^{\underline{ \nu -1}}\rho (t)}{L\varGamma (\nu )} & \sum_{t=0}^{T-1}\frac{a(T+ \nu -1)^{\underline{\nu -1}}(T+\nu -2)^{\underline{\nu -1}}\rho (t)}{L \varGamma (\nu )} \\ \sum_{t=0}^{T-1} \frac{abL_{1}(\xi +\nu )^{\underline{ \nu -1}}(\eta +\nu )^{\underline{\nu -1}}(t+\nu -1)^{\underline{ \nu -1}}\rho (t)}{L\varGamma (\nu )} & \sum_{t=0}^{T-1} \frac{[L+ab( \eta +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{\nu -1}}] \rho (t)}{L\varGamma (\nu )} \\ \sum_{t=0}^{T-1} \frac{bL_{1}( \xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{\nu -1}}(t+ \nu -1)^{\underline{\nu -1}}\rho (t)}{L\varGamma (\nu )} & \sum_{t=0}^{T-1}\frac{b(T+\nu -1)^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}\rho (t)}{L\varGamma (\nu )} \end{pmatrix} \\ &\quad \approx \begin{pmatrix} 0.15 & 21 \\ 0.11 & 10.35 \\ 0.15 & 24.23 \\ 0.23 & 20.7 \end{pmatrix}. \end{aligned} $$

Let \(a_{1}=a_{3}=3\), \(b_{1}=b_{3}=2\), \(c_{1}=c_{3}=4.5\), \(d_{1}=d_{3}=3\), \(a_{2}=a_{4}=\frac{1}{500}\), \(b_{2}=b_{4}=\frac{1}{420}\), \(c_{2}=c_{4}= \frac{1}{210}\), \(d_{2}=d_{4}=\frac{1}{500}\) and \(f_{1}(t,x,y)=(3x+2y)^{ \gamma _{1}}\), \(f_{2}(t,x,y)=(4.5x+3y)^{\gamma _{2}}\), for \((t,x,y) \in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\times \mathbf{R}^{+} \times \mathbf{R}^{+}\). Then we can calculate:

$$\begin{aligned}& h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}=h_{\mu _{1}}a_{3}+k_{\mu _{1}}c_{3}= 0.15 \times 3+0.11\times 4.5< 1,\\& h_{\mu _{3}}d_{1}+k_{\mu _{3}}b_{1}=h_{\mu _{3}}d_{3}+k_{\mu _{3}}b_{3}=0.15 \times 3+0.23\times 2< 1, \end{aligned}$$

and

$$\kappa _{1}=\kappa _{3}= \begin{vmatrix} h_{\mu _{1}}b_{1}+k_{\mu _{1}}d_{1} & h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}- 1 \\ h_{\mu _{3}}d_{1}+k_{\mu _{3}}b_{1}-1 & h_{\mu _{3}}c_{1}+k_{\mu _{3}}a _{1} \end{vmatrix}= \begin{vmatrix} 0.63 & -0.055 \\ -0.09 & 1.365 \end{vmatrix}=0.86>0. $$

Moreover,

$$\begin{aligned}& h_{\mu _{2}}a_{2}+k_{\mu _{2}}c_{2}=h_{\mu _{2}}a_{4}+k_{\mu _{2}}c_{4}=21 \times \frac{1}{500}+10.35\times \frac{1}{210} < 1,\\& h_{\mu _{4}}d_{2}+k _{\mu _{4}}b_{2}=h_{\mu _{4}}d_{4}+k_{\mu _{4}}b_{4}= 24.23\times \frac{1}{500}+20.7\times \frac{1}{420} < 1, \end{aligned}$$

and

$$\kappa _{2}=\kappa _{4}= \begin{vmatrix} 1-h_{\mu _{2}}a_{2}-k_{\mu _{2}}c_{2} & -h_{\mu _{2}}b_{2}-k_{\mu _{2}}d _{2} \\ -h_{\mu _{4}}c_{2}-k_{\mu _{4}} a_{2} & 1-h_{\mu _{4}}d_{2}-k_{\mu _{4}}b _{2} \end{vmatrix}= \begin{vmatrix} 0.91 & -0.071 \\ -0.16 & 0.90 \end{vmatrix}=0.81>0. $$

Case 1. When \(\gamma _{i}>1\), \(i=1,2\). Then we have

$$\begin{aligned}& \liminf_{a_{1}x+b_{1}y\to +\infty } \frac{f_{1}(t,x,y)}{a_{1}x+b_{1}y}=\liminf _{a_{1}x+b_{1}y\to +\infty } \frac{(3x+2y)^{\gamma _{1}} }{3x+2y}=+\infty , \\& \quad \text{uniformly on } t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}, \end{aligned}$$

and

$$\begin{aligned}& \liminf_{c_{1}x+d_{1}y\to +\infty } \frac{f_{2}(t,x,y)}{c_{1}x+d_{1}y}=\liminf _{c_{1}x+d_{1}y\to +\infty } \frac{(4.5x+3y)^{\gamma _{2}} }{4.5x+3y}=+\infty , \\& \quad \text{uniformly on } t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}. \end{aligned}$$

On the other hand, we also have

$$\begin{aligned}& \limsup_{a_{2}x+b_{2}y\to 0^{+}} \frac{f_{1}(t,x,y)}{a_{2}x+b_{2}y}= \limsup _{a_{2}x+b_{2}y\to 0^{+}} \frac{(3x+2y)^{\gamma _{1}} }{ \frac{x}{500}+\frac{y}{420}}=0, \\& \quad \text{uniformly on } t\in [\nu -1,T+ \nu -2]_{\mathbf{N}_{\nu -1}}, \end{aligned}$$

and

$$\begin{aligned}& \limsup_{c_{2}x+d_{2}y\to 0^{+}} \frac{f_{2}(t,x,y)}{c_{2}x+d_{2}y}= \limsup _{c_{2}x+d_{2}y\to 0^{+}} \frac{(4.5x+3y)^{\gamma _{2}} }{ \frac{x}{210}+\frac{y}{500}}=0, \\& \quad \text{uniformly on } t\in [\nu -1,T+ \nu -2]_{\mathbf{N}_{\nu -1}}. \end{aligned}$$

As a result, (H1)–(H2) hold.

Case 2. When \(\gamma _{i}\in (0,1)\), \(i=1,2\). Then we have

$$\begin{aligned}& \liminf_{a_{3}x+b_{3}y\to 0^{+}} \frac{f_{1}(t,x,y)}{a_{3}x+b_{3}y}= \liminf _{a_{3}x+b_{3}y\to 0^{+}} \frac{(3x+2y)^{\gamma _{1}} }{3x+2y}=+ \infty , \\& \quad \text{uniformly on } t \in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}, \end{aligned}$$

and

$$\begin{aligned}& \liminf_{c_{3}x+d_{3}y\to 0^{+}} \frac{f_{2}(t,x,y)}{c_{3}x+d_{3}y}= \liminf _{c_{3}x+d_{3}y\to 0^{+}} \frac{(4.5x+3y)^{\gamma _{2}} }{4.5x+3y}=+\infty ,\\& \quad \text{uniformly on } t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}. \end{aligned}$$

On the other hand, we also have

$$\begin{aligned}& \limsup_{a_{4}x+b_{4}y\to +\infty } \frac{f_{1}(t,x,y)}{a_{4}x+b_{4}y}=\limsup _{a_{4}x+b_{4}y\to +\infty } \frac{(3x+2y)^{\gamma _{1}} }{\frac{x}{500}+\frac{y}{420}}=0, \\& \quad \text{uniformly on } t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}, \end{aligned}$$

and

$$\begin{aligned}& \limsup_{c_{4}x+d_{4}y\to +\infty } \frac{f_{2}(t,x,y)}{c_{4}x+d_{4}y}=\limsup _{c_{4}x+d_{4}y\to +\infty } \frac{(4.5x+3y)^{\gamma _{2}} }{\frac{x}{210}+\frac{y}{500}}=0, \\& \quad \text{uniformly on } t\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}. \end{aligned}$$

As a result, (H3)–(H4) hold.

References

  1. 1.

    Fu, Z., Bai, S., O’Regan, D., Xu, J.: Nontrivial solutions for an integral boundary value problem involving Riemann–Liouville fractional derivatives. J. Inequal. Appl. 2019, Article ID 104 (2019)

  2. 2.

    He, J., Zhang, X., Liu, L., Wu, Y., Cui, Y.: Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions. Bound. Value Probl. 2018, Article ID 189 (2018)

  3. 3.

    Guo, L., Liu, L., Wu, Y.: Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters. Nonlinear Anal., Model. Control 23(2), 182–203 (2018)

  4. 4.

    Henderson, J., Luca, R.: Existence of positive solutions for a system of semipositone fractional boundary value problems. Electron. J. Qual. Theory Differ. Equ. 22, 1 (2016)

  5. 5.

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

  6. 6.

    Zou, Y., He, G.: The existence of solutions to integral boundary value problems of fractional differential equations at resonance. J. Funct. Spaces 2017, Article ID 2785937 (2017)

  7. 7.

    Qi, T., Liu, Y., Zou, Y.: Existence result for a class of coupled fractional differential systems with integral boundary value conditions. J. Nonlinear Sci. Appl. 10, 4034–4045 (2017)

  8. 8.

    Qi, T., Liu, Y., Cui, Y.: Existence of solutions for a class of coupled fractional differential systems with nonlocal boundary conditions. J. Funct. Spaces 2017, Article ID 6703860 (2017)

  9. 9.

    Meng, S., Cui, Y.: Multiplicity results to a conformable fractional differential equations involving integral boundary condition. Complexity 2019, Article ID 8402347 (2019)

  10. 10.

    Zhang, K., Xu, J., O’Regan, D.: Positive solutions for a coupled system of nonlinear fractional differential equations. Math. Methods Appl. Sci. 38(8), 1662–1672 (2015)

  11. 11.

    Zhang, X., Liu, L., Wu, Y., Zou, Y.: Existence and uniqueness of solutions for systems of fractional differential equations with Riemann–Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, Article ID 204 (2018)

  12. 12.

    Zhao, Y., Hou, X., Sun, Y., Bai, Z.: Solvability for some class of multi-order nonlinear fractional systems. Adv. Differ. Equ. 2019, Article ID 23 (2019)

  13. 13.

    Zhang, X., Liu, L., Zou, Y.: Fixed-point theorems for systems of operator equations and their applications to the fractional differential equations. J. Funct. Spaces 2018, Article ID 7469868 (2018)

  14. 14.

    Li, H., Zhang, J.: Positive solutions for a system of fractional differential equations with two parameters. J. Funct. Spaces 2018, Article ID 1462505 (2018)

  15. 15.

    Zhang, Y.: Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance. J. Inequal. Appl. 2018, Article ID 198 (2018)

  16. 16.

    Song, Q., Bai, Z.: Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, Article ID 183 (2018)

  17. 17.

    Zhai, C., Li, P., Li, H.: Single upper-solution or lower-solution method for Langevin equations with two fractional orders. Adv. Differ. Equ. 2018, Article ID 360 (2018)

  18. 18.

    Zhai, C., Wang, W., Li, H.: A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions. J. Inequal. Appl. 2018, Article ID 207 (2018)

  19. 19.

    Zou, Y.: Positive solutions for a fractional boundary value problem with a perturbation term. J. Funct. Spaces 2018, Article ID 9070247 (2018)

  20. 20.

    Sun, Q., Meng, S., Cui, Y.: Existence results for fractional order differential equation with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions at resonance. Adv. Differ. Equ. 2018, Article ID 243 (2018)

  21. 21.

    Wu, J., Zhang, X., Liu, L., Wu, Y., Cui, Y.: The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Bound. Value Probl. 2018, Article ID 82 (2018)

  22. 22.

    Yue, Z., Zou, Y.: New uniqueness results for fractional differential equation with dependence on the first order derivative. Adv. Differ. Equ. 2019, Article ID 38 (2019)

  23. 23.

    Zhang, K., Fu, Z.: Solutions for a class of Hadamard fractional boundary value problems with sign-changing nonlinearity. J. Funct. Spaces 2019, Article ID 9046472 (2019)

  24. 24.

    Zhang, K., Wang, J., Ma, W.: Solutions for integral boundary value problems of nonlinear Hadamard fractional differential equations. J. Funct. Spaces 2018, Article ID 2193234 (2018)

  25. 25.

    Zhang, X., Wu, J., Liu, L., Wu, Y., Cui, Y.: Convergence analysis of iterative scheme and error estimation of positive solution for a fractional differential equation. Math. Model. Anal. 23(4), 611–626 (2018)

  26. 26.

    Zuo, M., Hao, X., Liu, L., Cui, Y.: Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 2017, Article ID 161 (2017)

  27. 27.

    Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68–73 (2017)

  28. 28.

    Goodrich, C.S.: Solutions to a discrete right-focal fractional boundary value problem. Int. J. Difference Equ. 5(2), 195–216 (2010)

  29. 29.

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

  30. 30.

    Atici, F., Eloe, P.: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 17(4), 445–456 (2011)

  31. 31.

    Goodrich, C.S.: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 385(1), 111–124 (2012)

  32. 32.

    Goodrich, C.S.: On a first-order semipositone discrete fractional boundary value problem. Arch. Math. (Basel) 99(6), 509–518 (2012)

  33. 33.

    Lv, Z., Gong, Y., Chen, Y.: Multiplicity and uniqueness for a class of discrete fractional boundary value problems. Appl. Math. 59(6), 673–695 (2014)

  34. 34.

    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)

  35. 35.

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

  36. 36.

    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)

  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)

  38. 38.

    Chen, C., Xu, J., O’Regan, D., Fu, Z.: Positive solutions for a system of semipositone fractional difference boundary value problems. J. Funct. Spaces 2018, Article ID 6835028 (2018)

  39. 39.

    Xu, J., Goodrich, C.S., Cui, Y.: Positive solutions for a system of first-order discrete fractional boundary value problems with semipositone nonlinearities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(2), 1343–1358 (2019)

  40. 40.

    Ai, S., Lu, Y., Gao, P., Ge, Q.: Positive solutions for a class of singular semipositione fractional difference system with coupled boundary conditions. J. Northeast Petroleum University 38(4), 103–118 (2014)

  41. 41.

    Cheng, W., Xu, J., Cui, Y.: Positive solutions for a system of nonlinear semipositone fractional q-difference equations with q-integral boundary conditions. J. Nonlinear Sci. Appl. 10, 4430–4440 (2017)

  42. 42.

    Yang, W.: Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions. Appl. Math. Comput. 244, 702–725 (2014)

  43. 43.

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

  44. 44.

    Goodrich, C.S.: On a fractional boundary value problem with fractional boundary conditions. Appl. Math. Lett. 25(8), 1101–1105 (2012)

  45. 45.

    Ding, Y., Xu, J., Wei, Z.: Positive solutions for a system of discrete boundary value problem. Bull. Malays. Math. Sci. Soc. 38, 1207–1221 (2015)

  46. 46.

    Jiang, J., Henderson, J., Xu, J., Fu, Z.: Positive solutions for a system of Neumann boundary value problems of second order difference equations involving sign-changing nonlinearities. J. Funct. Spaces 2019, Article ID 3203401 (2019)

  47. 47.

    Zhang, K., O’Regan, D., Fu, Z.: Nontrivial solutions for boundary value problems of a fourth order difference equation with sign-changing nonlinearity. Adv. Differ. Equ. 2018, Article ID 370 (2018)

  48. 48.

    Cui, Y., Sun, J.: On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system. Electron. J. Qual. Theory Differ. Equ. 2012(41), 1 (2012)

  49. 49.

    Cui, Y., Zou, Y.: Monotone iterative method for differential systems with coupled integral boundary value problems. Bound. Value Probl. 2013, Article ID 245 (2013)

  50. 50.

    Cui, Y., Zou, Y.: An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions. Appl. Math. Comput. 256, 438–444 (2015)

  51. 51.

    Wang, F., Cui, Y.: Positive solutions for an infinite system of fractional order boundary value problems. Adv. Differ. Equ. 2019, Article ID 169 (2019)

  52. 52.

    Wang, G., Ren, X., Bai, Z., Hou, W.: Radial symmetry of standing waves for nonlinear fractional Hardy–Schrödinger equation. Appl. Math. Lett. 96, 131–137 (2019)

  53. 53.

    Yue, Y., Tian, Y., Bai, Z.: Infinitely many nonnegative solutions for a fractional differential inclusion with oscillatory potential. Appl. Math. Lett. 88, 64–72 (2019)

  54. 54.

    Xu, J., O’Regan, D., Zhang, K.: Positive solutions for a system of p-Laplacian boundary value problems. Fixed Point Theory 19(2), 823–836 (2018)

  55. 55.

    Yang, Z., Zhang, Z.: Positive solutions for a system of nonlinear singular Hammerstein integral equations via nonnegative matrices and applications. Positivity 16(4), 783–800 (2012)

  56. 56.

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

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Research supported by the National Natural Science Foundation of China(Grant No. 11601048), the Natural Science Foundation of Chongqing (Grant No. cstc2016jcyjA0181), the Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No. KJQN201800533), and the Natural Science Foundation of Chongqing Normal University (Grant No. 16XYY24).

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Correspondence to Yujun Cui.

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Keywords

  • Fractional difference systems
  • Positive solutions
  • Fixed point index
  • Nonnegative matrices