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.
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1 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:
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
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
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
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
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:
and
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.
2 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
We also define the νth fractional difference for \(\nu >0\) by
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
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
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
has the unique solution, which takes the form
where
Proof
From Lemma 2.3 we have
and
Substituting \(x(\nu -3)=y(\nu -3)=0\) into (2.6), (2.7), we obtain \(C_{3}=\overline{C}_{3}=0\). Because of
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
and
Note that
so we have
As a result, we have
Similarly, we can obtain
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
-
(i)
\(G(t,s)>0\), for \((t,s)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{ \nu -1}}\times [0,T-1]_{\mathbf{N}_{0}}\);
-
(ii)
for all \((t,s)\in [\nu -1,T+\nu -2]_{\mathbf{N}_{\nu -1}}\times [0,T-1]_{\mathbf{N}_{0}}\), there holds
$$\begin{aligned}& \frac{abL_{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{ \nu -1}}t^{\underline{\nu -1}} \rho (s)}{L\varGamma (\nu )}\\& \quad \le H_{1}(t,s) \le \frac{[L+ab(\xi +\nu )^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}]\rho (s)}{L\varGamma (\nu )}, \\& \frac{aL_{1}(\xi +\nu )^{\underline{\nu -1}}(\eta +\nu )^{\underline{ \nu -1}}t^{\underline{\nu -1}}\rho (s)}{L\varGamma (\nu )}\\& \quad \le K_{1}(t,s) \le \frac{a(T+\nu -1)^{\underline{\nu -1}}(T+\nu -2)^{\underline{ \nu -1}}\rho (s)}{L\varGamma (\nu )}; \end{aligned}$$ -
(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:
and
and
and
where
Proof
We only prove (2.8). Indeed, for all \(s\in [0,T-1]_{\mathbf{N}_{0}}\), from Lemma 2.6(ii) we have
On the other hand, we obtain
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:
Then we define an operator \(A:P\times P\to P\times P\) as follows:
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
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
then \(i(A,\varOmega \cap P,P)=1\).
3 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 , we mean that \(x_{1}(t)\ge (\text{or } \le ) y(t)\), \(x_{2}(t) \ge (\text{or } \le ) y_{2}(t)\) for \(t\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\).
-
(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
$$\begin{aligned}& h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}< 1,\qquad h_{\mu _{3}}d_{1}+k_{\mu _{3}}b _{1}< 1,\\& \det \begin{pmatrix} h_{\mu _{1}}b_{1}+k_{\mu _{1}}d_{1} & h_{\mu _{1}}a_{1}+k_{\mu _{1}}c_{1}- 1 \\ h_{\mu _{3}}d_{1}+k_{\mu _{3}}b_{1}-1 & h_{\mu _{3}}c_{1}+k_{\mu _{3}}a _{1} \end{pmatrix}:=\kappa _{1}>0. \end{aligned}$$ -
(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
$$\begin{aligned}& h_{\mu _{2}}a_{2}+k_{\mu _{2}}c_{2}< 1,\qquad h_{\mu _{4}}d_{2}+k_{\mu _{4}}b _{2}< 1, \\& \det \begin{pmatrix} 1-h_{\mu _{2}}a_{2}-k_{\mu _{2}}c_{2} & -h_{\mu _{2}}b_{2}-k_{\mu _{2}}d _{2} \\ -h_{\mu _{4}}c_{2}-k_{\mu _{4}} a_{2} & 1-h_{\mu _{4}}d_{2}-k_{\mu _{4}}b _{2} \end{pmatrix}:=\kappa _{2}>0. \end{aligned}$$ -
(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
$$\begin{aligned}& h_{\mu _{1}}a_{3}+k_{\mu _{1}}c_{3}< 1,\qquad h_{\mu _{3}}d_{3}+k_{\mu _{3}}b _{3}< 1,\\& \det \begin{pmatrix} h_{\mu _{1}}b_{3}+k_{\mu _{1}}d_{3} & h_{\mu _{1}}a_{3}+k_{\mu _{1}}c_{3}- 1 \\ h_{\mu _{3}}d_{3}+k_{\mu _{3}}b_{3}-1 & h_{\mu _{3}}c_{3}+k_{\mu _{3}}a _{3} \end{pmatrix}:=\kappa _{3}>0. \end{aligned}$$ -
(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
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
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
This implies that
Solving this matrix inequality, we have
Therefore, we have
On the other hand, there exist \(t_{1},t_{2}\in [\nu -1,T+\nu -2]_{ \mathbf{N}_{\nu -1}}\) such that
Consequently, we have
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
As a result, Lemma 2.8 implies
In what follows, we prove that
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
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
Solving this matrix inequality, we have
Consequently, we have
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
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
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
Associated with condition (H3), this means that
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
This leads us to obtain
Solving this matrix inequality, we have
Hence, we find
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
On the other hand, we claim that the set
is bounded in \(P\times P\). If there exists \((x,y)\in S_{2}\), then from (H4) we have
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
Solving this matrix inequality, we have
This indicates that
Consequently, we have
Similarly, using (3.1) we have
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
As a result, Lemma 2.9 implies
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
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:
and
Moreover,
and
Case 1. When \(\gamma _{i}>1\), \(i=1,2\). Then we have
and
On the other hand, we also have
and
As a result, (H1)–(H2) hold.
Case 2. When \(\gamma _{i}\in (0,1)\), \(i=1,2\). Then we have
and
On the other hand, we also have
and
As a result, (H3)–(H4) hold.
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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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Cheng, W., Xu, J., Cui, Y. et al. Positive solutions for a class of fractional difference systems with coupled boundary conditions. Adv Differ Equ 2019, 249 (2019). https://doi.org/10.1186/s13662-019-2184-3
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DOI: https://doi.org/10.1186/s13662-019-2184-3