Abstract
We consider the existence of at least two positive solutions for a system of Caputo fractional difference equations \(\Delta _{{\mathrm{C}}}^{\nu _{j}}y_{j}(t)=-\lambda_{j}f_{j}(y_{1}(t+\nu_{1}-1), \ldots,y_{n}(t+\nu_{n}-1))\), subject to boundary conditions \(y_{j}(\nu_{j}-3)=\Delta y_{j}(\nu_{j}+b)=\Delta ^{2} y_{j}(\nu_{j}-3)=0\), where \(2<\nu_{j}\leqslant3\), \(j=1,\ldots,n\). We use the Krasnosel’skiĭ fixed point theorem to obtain the sufficient conditions of the existence of two positive solutions for this boundary value problem of Caputo fractional difference equations depending on parameters.
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1 Introduction
In this paper we consider a system of Caputo fractional difference boundary value problem (FBVP) of the form:
where \(t\in[0,b+1]_{{\mathbb{N}}_{0}}:=\{0,1,\ldots,b+1\}\), \(b>3\), \(\lambda_{j}>0\), \(2<\nu_{j}\leqslant3\), \(f_{j}:[0,+\infty)\times\cdots\times[0,+\infty)\rightarrow [0,+\infty)\) are continuous functions for each j (\(j=1,2,\ldots, n\)). \(\Delta _{{\mathrm{C}}}^{\nu}y(t)\) is the standard Caputo difference.
Fractional difference equations have been of great interest recently. It is caused by intensive development of the theory of discrete fractional calculus itself, see [1–19] and the references therein. Abdeljawad [1] defined left and right Caputo fractional sums and differences, studied some of their properties. Holm [2] introduced the fractional sum and difference operators. He developed and presented a complete and precise theory for composing fractional sums and differences. Atici and Sengül [3] provided some analysis of discrete fractional variational problems, their paper also provided some initial attempts at using the discrete fractional calculus to model biological processes. Abdeljawad and Baleanu [4] defined the right fractional sum and difference operators and obtained many of their properties. Then by using those properties they obtained a by-part formula analogous to that in the usual fractional calculus. In [5] the authors studied the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference by using the Lyapunov direct method. They discussed the conditions for uniform stability, uniform asymptotic stability, and uniform global stability. Mohammadi and Rezapour [6] discussed the existence and uniqueness of solutions for some nonlinear fractional differential equations via some boundary value problems by using fixed point results on ordered complete gauge spaces. Recently, Wu and Baleanu introduced some applications of the Caputo fractional difference to discrete chaotic maps in [7, 8].
In particular, the authors [9–19] developed some of the basic theory of fractional difference both IVPs and BVPs with delta derivative on the time scale \(\mathbb{Z}\). In [10], we obtained some results on the existence of one or more positive solutions for the Caputo fractional boundary value problems by means of cone theoretic fixed point theorems. Thus, the fractional difference equation has recently attracted increasing attention from a growing number of researchers. However, systems of discrete fractional boundary value problems are limited (see [14–19]). Among them, Atici and Eloe [14] studied a linear system of fractional nabla difference equation with constant coefficients of the form
where \(0<\nu<1\), A is an \(n\times n\) matrix with constant entries, and f are n-vector valued functions. The operator \(\nabla_{0}^{\nu}\) is a Riemann-Liouville fractional difference. They constructed the fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system.
In [15], the authors investigated the existence of solutions for a k-dimensional system of fractional finite difference equations:
where \(b\in{{\mathbb{N}}_{0}}\), \(1<\nu_{i}\leqslant2\), \(f_{i}:{{\mathbb{R}}^{k}}\rightarrow{\mathbb{R}}\) are continuous functions for \(k = 1, 2,\ldots \) . They investigated the existence of solutions for this k-dimensional system of fractional finite difference equations by using the Krasnosel’skiĭ fixed point theorem.
In [16], Goodrich studied the following pair of discrete fractional boundary value problems:
where \(t\in[0,b]_{{\mathbb{N}}_{0}}:=\{0,1,\ldots,b\}\), \(\lambda_{1},\lambda_{2}>0\), \(\nu_{1},\nu_{2}\in(1,2]\). Goodrich obtained the existence of at least one positive solution to this problem by means of the Krasnosel’skiĭ theorem for cons.
In [18], the authors considered the existence of at least one positive solution to the discrete fractional system:
where \(\nu_{1}, \nu_{2}\in(1,2]\).
Following this trend, in [19], we discussed the boundary value problems of fractional difference system of the form
where \(t\in[0,b]_{{\mathbb{N}}_{0}}:=\{0,1,\ldots,b\}\), \(\lambda_{j}>0\), \(1<\nu_{j}\leqslant2\), \(f_{j}:[0,+\infty)\times\cdots\times[0,+\infty)\rightarrow [0,+\infty)\) are continuous functions. For each j we have that \(\psi_{j}, \phi_{j}:{\mathbb{R}}^{b+3}\rightarrow\mathbb{R}\) (\(j=1,2,\ldots,n\)) are given functions. We obtained the sufficient conditions for the existence of two positive solutions to the boundary value problem of a fractional difference system. In this paper we open our studies in this field. We establish some conditions on parameters \(\lambda_{i}\) which are able to guarantee that FBVP (1.1)-(1.2) has at least two positive solutions and one positive solution, respectively, based on the Krasnosel’skiĭ theorem.
This paper is organized as follows. In Section 2, we provide basic definitions and demonstrate some lemmas in order to prove our main results. In Section 3, we establish some results for the existence of at least two positive solutions to FBVP (1.1)-(1.2), and we conclude with an example explicating our main result.
2 Preliminaries
In this section, we present some basic definitions in the discrete fractional calculus and establish some lemmas.
Definition 2.1
[1]
We define
for any t and ν for which the right-hand side is defined. We also appeal to the convention that if \(t+1-\nu\) is a pole of the gamma function and \(t+1\) is not a pole, then \(t^{(\nu)}=0\).
Definition 2.2
[1]
The νth fractional sum of a function f is defined by
for \(\nu>0\) and \(t\in \{a+\nu,a+\nu+1,\ldots \}={\mathbb{N}}_{a+\nu}\). We also define the νth Caputo fractional difference for \(\nu>0\) by
where \(n-1<\nu\leqslant n \).
Lemma 2.3
Assume that \(\nu>0\) and f is defined on domains \({\mathbb{N}}_{a}\), then
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\); \(n-1<\nu\leqslant n\).
Lemma 2.4
[2]
Let \(f:{\mathbb{N}}_{a+\nu}\times{\mathbb{N}}_{a}\to{\mathbb{R}}\) be given. Then
In order to get our main results, we now state an important lemma. This lemma gives a representation for the solution of (1.1)-(1.2), provided that the solution exists.
Lemma 2.5
[10]
Let \(2<\nu\leqslant3\) and \(g:[\nu-2, \nu-1, \ldots, \nu+b]_{{\mathbb{N}}_{\nu-2}}\to{\mathbb{R}}\) be given. Then the solution of the FBVP
is given by
where the Green’s function \(G:[\nu-2,\nu-1,\ldots,\nu+b]_{{\mathbb{N}}_{\nu-2}}\times[0,b+1]_{{\mathbb{N}}_{0}}\rightarrow{\mathbb{R}}\) is defined by
Remark
Notice that \(G(\nu-3,s)=0\), \(G(t,b+2)=0\). G could be extended to \([\nu-3,\nu+b]_{{\mathbb{N}}_{\nu-3}}\times [0,b+2]_{{\mathbb{N}}_{0}}\), so we only discuss on \((t,s)\in [\nu-2,\nu+b]_{{\mathbb{N}}_{\nu-2}}\times[0,b+1]_{{\mathbb{N}}_{0}}\).
Lemma 2.6
[10]
The Green’s function G satisfies the following conditions:
-
(i)
\(G(t,s)>0\), \((t,s)\in[\nu-2,\nu+b]_{{\mathbb{N}}_{\nu-2}}\times[0,b+1]_{{\mathbb{N}}_{0}}\).
-
(ii)
\(\max_{t\in[\nu-2,\nu+b]_{{\mathbb{N}}_{\nu-2}}}G(t,s)=G(\nu+b, s)\), \(s\in[0, b+1]_{{\mathbb{N}}_{0}}\).
-
(iii)
\(\min_{\frac{\nu+b}{4}\leqslant t\leqslant\frac{3(\nu+b)}{4}} G(t,s)\geqslant \frac{1}{4} \max_{t\in[\nu-2,\nu+b]_{{\mathbb{N}}_{\nu-2}}}G(t,s)=\frac{1}{4}G(\nu+b, s)\), \(s\in[0, b+1]_{{\mathbb{N}}_{0}}\).
The proofs of Lemma 2.5 and Lemma 2.6 can be found in [10], so we omit their proofs.
Let
be equipped with the usual maximum norm \(\|\cdot\|\), it is easy to verify that \(\mathcal{B}_{j}\) is the Banach space. Then we put \(\mathcal{K}:={\mathcal{B}}_{1}\times {\mathcal{B}}_{2}\times\cdots\times{\mathcal{B}}_{n}\). By equipping \(\mathcal{K}\) with the norm
it follows that \((\mathcal{K},\|\cdot\|)\) is a Banach space.
Now consider the operator \(T:\mathcal{K}\rightarrow\mathcal{K}\) defined by
where we define \(T_{j}:\mathcal{K}\rightarrow{\mathcal{B}}_{j}\) by
Let \(I:=[\frac{\nu_{1}+b}{4},\frac{3(\nu_{1}+b)}{4}]\times\cdots\times[\frac{\nu_{n}+b}{4},\frac{3(\nu_{n}+b)}{4}]\). In the sequel, we shall also make use of the cone
Lemma 2.7
Let T be the operator defined as in (2.3). Then \(T: \Lambda\rightarrow\Lambda\).
Proof
We show first that for \((y_{1},\ldots,y_{n})\in\mathcal{K}\), by the definitions \(T_{j}\) (\(j=1,2,\ldots,n\)), it is clear that
On the other hand, we show that
for \((y_{1},\ldots,y_{n})\in\mathcal{K}\). In fact, by Lemma 2.6(iii), we have
for \((y_{1},\ldots,y_{n})\in\mathcal{K}\) and \(j=1,2,\ldots,n\). Then we obtain
for \((y_{1},\ldots,y_{n})\in\mathcal{K}\). So, we conclude that \(T: \Lambda\rightarrow\Lambda\). This completes the proof. □
Theorem 2.8
Let \(f_{j}:[0,+\infty)\times\cdots\times[0,+\infty )\rightarrow[0,+\infty)\) be given for \(j=1,\ldots,n\). If \((y_{1},\ldots ,y_{n})\in\mathcal{K}\) is a fixed point of T, then \((y_{1},\ldots,y_{n})\in\mathcal{K}\) is a solution of FBVP (1.1)-(1.2).
Proof
Suppose that the operator T has a fixed point, say \((y_{1},\ldots,y_{n})\in\mathcal{K}\). Let \((t_{1},\ldots,t_{n})\in\mathbb{N}_{{\nu_{1}-2}}\times\cdots \times\mathbb{N}_{{\nu_{n}-2}}\), then we have
where \(T_{j}\) is defined as in (2.4). It is easy to check that
and
Finally, when \(0\leqslant t_{j}-\nu_{j}+1\leqslant s\leqslant b+1\),
then
Therefore, we can get
So the boundary conditions are satisfied, which completes the proof. □
Finally, to accomplish proof of our main results, we state cones theory. In particular, we require the following well-known fixed point theorem for cons in [20].
Theorem 2.9
[20]
Let \(\mathcal{B}\) a Banach space and let \(\mathcal{K}\subseteq\mathcal{B}\) be a cone. Assume that \(\Omega_{1}\) and \(\Omega_{2}\) are bounded open sets contained in \(\mathcal{B}\) such that \(0\in\Omega_{1}\) and \(\overline{\Omega}_{1}\subseteq\Omega_{2}\). Assume further that \(T:{\mathcal{K}}\cap (\overline{\Omega}_{2}\backslash\Omega_{1})\to\mathcal{K}\) is a completely continuous operator. If either
-
(i)
\(\|Ty\|\leqslant\|y\|\) for \(y\in{\mathcal{K}}\cap{\partial \Omega_{1}}\) and \(\|Ty\|\geqslant\|y\|\) for \(y\in{\mathcal{K}}\cap {\partial\Omega_{2}}\), or
-
(ii)
\(\|Ty\|\geqslant\|y\|\) for \(y\in{\mathcal{K}}\cap{\partial \Omega_{1}}\) and \(\|Ty\|\leqslant\|y\|\) for \(y\in{\mathcal{K}}\cap {\partial\Omega_{2}}\),
then the operator T has at least one fixed point in \({\mathcal{K}}\cap(\overline{\Omega}_{2}\backslash\Omega_{1})\).
3 Main results
In this section, we state and prove the existence of at least two positive solutions regarding FBVP (1.1)-(1.2). Then we conclude this section with examples to illustrate our main results. First, denote
For convenience, we now present the conditions that we presume in the sequel.
- (H1):
-
\(\lim_{(y_{1}+\cdots+y_{n})\rightarrow0^{+}} \frac{f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=\infty\), \({t_{j}\in[\nu _{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2,\ldots,n\),
- (H2):
-
\(\lim_{(y_{1}+\cdots+y_{n})\rightarrow\infty} \frac{f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=\infty\), \({t_{j}\in[\nu _{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2,\ldots,n\),
- (H3):
-
\(\lim_{(y_{1}+\cdots+y_{n})\rightarrow0^{+}} \frac{f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=0\), \({t_{j}\in[\nu_{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2,\ldots,n\),
- (H4):
-
\(\lim_{(y_{1}+\cdots+y_{n})\rightarrow\infty} \frac{f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=0\), \({t_{j}\in[\nu_{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2,\ldots,n\),
- (H5):
-
\(\lim_{(y_{1}+\cdots+y_{n})\rightarrow0^{+}} \frac{ f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=l_{j}\), \({t_{j}\in[\nu _{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2,\ldots,n\),
- (H6):
-
\(\lim_{(y_{1}+\cdots+y_{n})\rightarrow\infty} \frac {f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=L_{j}\), \({t_{j}\in[\nu _{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2,\ldots,n\),
where \(0< l_{j}, L_{j}<+\infty\).
Theorem 3.1
Suppose that there exist two different positive numbers \(r_{1}\) and \(r_{2}\) (\(r_{1}< r_{2}\)) such that
Then the operator T has a fixed point \((\overline{y}_{1},\ldots,\overline{y}_{n})\in\Lambda\) such that
Proof
For any \((y_{1},\ldots,y_{n})\in\Omega_{r_{1}}\) and \(\|(y_{1},\ldots,y_{n})\|=r_{1}\), we have
\(j=1,\ldots,n\). That is,
for \((y_{1},\ldots,y_{n})\in\partial\Omega_{r_{1}}\).
On the other hand, for any \((y_{1},\ldots,y_{n})\in\Omega_{r_{2}}\) and \(\frac{\nu_{j}+b}{4}\leqslant t_{j}\leqslant\frac{3(\nu_{j}+b)}{4}\), note that \({[\frac{b-\nu_{j}}{2}]+\nu_{j}}\in[\frac{\nu_{j}+b}{4}, \frac{3(\nu_{j}+b)}{4}]\), we have
Then
That is,
for \((y_{1},\ldots,y_{n})\in\partial\Omega_{r_{2}}\).
By the use of Theorem 2.9, there exists \((\overline{y}_{1},\ldots,\overline{y}_{n})\in\Lambda\) such that \(T(\overline{y}_{1},\ldots,\overline{y}_{n}) =(\overline{y}_{1},\ldots,\overline{y}_{n})\), the proof is complete. □
Theorem 3.2
Suppose that conditions (H1) and (H2) hold. Then, for every \(\lambda_{j}\in(0,\lambda_{j}^{*})\), FBVP (1.1)-(1.2) has at least two positive solutions, where
Proof
Define the function
It is easy to know that \(\varphi_{j}:(0,+\infty)\to(0,+\infty)\) is a continuous function. From (H1), we see that \(\lim_{r\rightarrow0}\frac{r}{f_{j}(r)}=0\), that is, \(\lim_{r\rightarrow0}\frac{r}{n\alpha_{j}f_{j}(r)}=0\), and
so \(\lim_{r\rightarrow0}\varphi_{j}(r)=0\).
From (H2), we see further that \(\lim_{r\rightarrow \infty}\varphi_{j}(r)=0\). Then there exists \(r_{0}>0\) such that \(\varphi_{j}(r_{0})=\max_{r>0}\varphi_{j}(r)=\lambda^{*}_{j}\), \(j=1,\ldots,n\). For any \(\lambda_{j}\in(0,\lambda^{*}_{j})\), by the intermediate value theorem, there exist two points \(d_{1}\in(0,r_{0})\), \(d_{2}\in(r_{0},\infty)\) such that \(\varphi_{j}(d_{1})=\varphi_{j}(d_{2})=\lambda_{j}\). Thus, we have
On the other hand, since (H1) and (H2) hold, there exist \(e_{1}\in(0,d_{1})\), \(e_{2}\in(d_{2},\infty)\) such that
Thus
Application of Theorem 3.1 and Theorem 2.8 leads to two distinct positive solutions of FBVP (1.1)-(1.2) which satisfy
The proof is complete. □
By the proof of Theorem 3.2, we obtain the following.
Corollary 3.3
If one of conditions (H1) and (H2) is satisfied, then for every \(0<\lambda_{j}<\lambda_{j}^{*}\), FBVP (1.1)-(1.2) has at least one positive solution.
Theorem 3.4
Suppose that (H3), (H4) hold. Then, for any \(\lambda_{j}\geqslant\lambda_{j}^{**}\), FBVP (1.1)-(1.2) has at least two positive solutions, where
Proof
Define the function
We know that \(\psi_{j}: (0,+\infty)\to (0,+\infty)\) is a continuous function. For \(\lambda_{j}>\lambda_{j}^{**}\), there exists \(0< e_{3}<+\infty \) such that
By condition (H3), there exists \(0< d_{3}<e_{3}\) such that
From condition (H4), there exists \(e_{3}< d_{0}<+\infty\) such that
Let \(M_{j}=\max_{0\leqslant y_{1}+\cdots+y_{n}\leqslant d_{0}}f_{j}(y_{1}, \ldots,y_{n})\). Choose \(d_{4}>d_{0}\) such that \(d_{4}\geqslant \lambda_{j}M_{j}\alpha_{j}\). Then
By Theorem 3.1 and Theorem 2.8, the proof is complete. □
From the proof of Theorem 3.4, we get the following.
Corollary 3.5
Suppose that one of conditions (H3) and (H4) holds. Then, for every \(\lambda_{j}>\lambda_{j}^{**}\), FBVP (1.1)-(1.2) has at least one positive solution.
Theorem 3.6
Suppose that one of the following cases is satisfied:
-
(1)
(H1), (H6) hold, and \(0<\lambda_{j}< \frac{1}{n\alpha_{j} L_{j}}\);
-
(2)
(H2), (H5) hold, and \(0<\lambda_{j}< \frac {1}{n\alpha_{j} l_{j}}\).
Then FBVP (1.1)-(1.2) has at least one positive solution.
Proof
(1) From (H6) (namely \(\lim_{(y_{1}+\cdots+y_{n})\rightarrow\infty} \frac{ f_{j}(y_{1},\ldots,y_{n})}{y_{1}+\cdots+y_{n}}=L_{j}\), \({t_{j}\in[\nu _{j}-2,\nu_{j}+b]_{{\mathbb{N}}_{\nu_{j}-2}}}\), \(j=1,2, \ldots,n\)), for any \(\epsilon _{j}>0\), there exists a number \(R_{0}>0\), for \(y_{1}+\cdots+y_{n}\in(R_{0},+\infty)\), we have
Let \(M_{j}=\max_{0\leqslant y_{1}+\cdots+y_{n}\leqslant R_{0}}f_{j}(y_{1},\ldots,y_{n})\). Choose \(R>\max\{R_{0}, \frac{M_{j}}{L_{j}+\epsilon _{j}}\}\), then
for \(y_{1}+\cdots+y_{n}\in[0,R]\). As arbitrarily of \(\epsilon _{j}\), so \(f_{j}(y_{1},\ldots,y_{n})\leqslant L_{j}R\). Note that \(0<\lambda_{j}< \frac{1}{n\alpha_{j} L_{j}}\), then
Namely \(0<\lambda_{j}< \lambda_{j}^{*}\). By means of Corollary 3.3, FBVP (1.1)-(1.2) has at least one positive solution.
The proof is similar to (2) and hence omitted. □
Similarly, we have the following.
Theorem 3.7
Suppose that one of the following cases is satisfied:
-
(1)
(H3), (H6) hold, and \(\frac{4}{nL_{j}\beta_{j}}<\lambda_{j}<+\infty\);
-
(2)
(H4), (H5) hold, and \(\frac{4}{nl_{j}\beta _{j}}<\lambda_{j}<+\infty\).
Then FBVP (1.1)-(1.2) has at least one positive solution.
Theorem 3.8
Suppose that conditions (H5) and (H6) hold. If \(\lambda_{j}\) satisfies \(\frac{4}{n\beta_{j}L_{j}}<\lambda_{j}<\frac{1}{n\alpha_{j}l_{j}}\) or \(\frac{4}{n\beta_{j}l_{j}}<\lambda_{j}<\frac{1}{n\alpha_{j}L_{j}}\), then FBVP (1.1)-(1.2) has at least one positive solution.
Proof
Suppose that \(\frac{4}{n\beta _{j}L_{j}}<\lambda_{j}<\frac{1}{n\alpha_{j}l_{j}}\) holds. Choose \(\epsilon_{j}>0\) such that \(\frac{4}{n\beta_{j}(L_{j}-\epsilon_{j})}\leqslant \lambda_{j}\leqslant\frac{1}{n\alpha_{j}(l_{j}+\epsilon_{j})}\). With condition (H5), there exists \(\tau>0\) such that \(f_{j}(y_{1},\ldots ,y_{n})\leqslant(l_{j}+\epsilon_{j})(y_{1}+\cdots+y_{n}) \) for \(y_{1}+\cdots+y_{n}\in(0,\tau)\). Thus, for \(y_{1}+\cdots+y_{n}\in\partial \Omega_{\tau}\),
\(j=1,\ldots,n\). That is,
for \((y_{1},\ldots,y_{n})\in \partial\Omega_{\tau}\).
By condition (H6), there exists \(R_{1}>0\) such that \(f_{j}(y_{1},\ldots,y_{n})\geqslant(L_{j}-\epsilon_{j})(y_{1}+\cdots+y_{n})\) for \(y_{1}+\cdots+y_{n}\geqslant\frac{1}{4}R_{1}\).
Let \(R_{2}=\max\{2\tau, R_{1}\}\), for \((y_{1},\ldots,y_{n})\in\partial\Omega_{R_{2}}\), we get
Then
That is,
for \((y_{1}, \ldots, y_{n})\in\partial\Omega_{R_{2}}\)
By using Theorem 2.9, we obtain the conclusion.
A similar proof holds when \(\frac{4}{n\beta_{j}l_{j}}<\lambda_{j}<\frac{1}{n\alpha_{j}L_{j}}\). The proof is complete. □
We now present an example illustrating the sorts of boundary conditions that can be treated by Theorem 3.2.
Example 3.1
Consider the following boundary value problems:
where \(b=19\), \(\nu_{1}= \frac{17}{8}\), \(\nu_{2}=\frac{33}{16}\), we take
\(f_{1}, f_{2}:[0,+\infty)\times[0,+\infty)\rightarrow[0,+\infty)\), and \(y_{1}\) is defined on the time scale \(\{- \frac{7}{8},\frac{1}{8}, \ldots, \frac{169}{8}\}\), \(y_{2}\) is defined on the time scale \(\{- \frac{15}{16},\frac{1}{16}, \ldots, \frac{337}{16}\}\). \(f_{1}\) and \(f_{2}\) satisfy conditions of Theorem 3.2. A computation shows that \(\lambda_{1}^{\ast}\approx0.01456\), \(\lambda_{2}^{\ast}\approx0.032845\), then, for every \(\lambda_{j}\in(0,\lambda_{j}^{*})\) (\(j=1,2\)), problem (3.1) has at least two positive solutions.
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Acknowledgements
The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper. Project was supported by the National Natural Science Foundation of China (Grant No. 11271235) and Shanxi Datong University Institute (2009-Y-15, 2010-B-01, 2013K5) and the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111020).
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SK conceived of the study and participated in its design. HC drafted the manuscript and participated in its design and coordination. JG participated in the sequence correction. All authors read and approved the final manuscript.
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Kang, S., Chen, H. & Guo, J. Existence of positive solutions for a system of Caputo fractional difference equations depending on parameters. Adv Differ Equ 2015, 138 (2015). https://doi.org/10.1186/s13662-015-0466-y
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DOI: https://doi.org/10.1186/s13662-015-0466-y