Abstract
We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions, under some assumptions on the nonlinearities of the system which contains concave functions. In the proofs of our main results we use some theorems from the fixed point index theory.
Similar content being viewed by others
1 Introduction
In this paper, we consider the system of nonlinear second-order difference equations
subject to the multi-point boundary conditions
where \(N\in \mathbb{N}\), \(N\ge 2\), \(p, q, r, l\in \mathbb{N}\), Δ is the forward difference operator with stepsize 1, \(\Delta u_{n}=u_{n+1}-u_{n}\), \(\Delta ^{2}u_{n-1}=u_{n+1}-2u_{n}+u_{n-1}\), and \(n=\overline{k,m}\) means that \(n=k, k+1,\ldots ,m\) for \(k, m\in \mathbb{N}\), \(\xi _{i}\in \mathbb{N}\) for all \(i=\overline{1,p}\), \(\eta _{i}\in \mathbb{N}\) for all \(i=\overline{1,q}\), \(\zeta _{i}\in \mathbb{N}\) for all \(i= \overline{1,r}\), \(\rho _{i}\in \mathbb{N}\) for all \(i=\overline{1,l}\), \(1\le \xi _{1}<\cdots <\xi _{p}\le N-1\), \(1\le \eta _{1}<\cdots <\eta _{q}\le N-1\), \(1\le \zeta _{1}<\cdots <\zeta _{r}\le N-1\), and \(1\le \rho _{1}<\cdots <\rho _{l}\le N-1\).
Under sufficient conditions on the nonnegative nonlinearities f and g which contain some concave functions, we investigate the existence and multiplicity of positive solutions of problem (S)–(BC) by using the fixed point index theory. By a positive solution of (S)–(BC), we mean a pair of sequences \((u,v)=((u_{n})_{n=\overline{0,N}}, (v_{n})_{n= \overline{0,N}})\) satisfying (S) and (BC) with \(u_{n}\ge 0\) and \(v_{n}\ge 0\) for all \(n=\overline{0,N}\), and \(u_{n}>0\) for all \(n=\overline{1,N}\) or \(v_{n}>0\) for all \(n=\overline{1,N}\). The existence and nonexistence of nonnegative and nontrivial solutions \((u,v)\) (\(u_{n}\ge 0\), \(v_{n}\ge 0\) for all \(n=\overline{0,N}\) and \((u,v)\neq(0,0)\)) of problem (S)–(BC) with some positive parameters in system (S) were studied in the papers [14] and [15] by using the Guo–Krasnosel’skii fixed point theorem. We also mention the paper [19], where the authors investigated the existence and multiplicity of positive solutions for problem (S)–(BC) under some assumptions on the functions f and g which are different than those we use in this paper. The existence, nonexistence, and multiplicity of positive solutions for system (S) with parameters or without parameters, subject to the multi-point coupled boundary conditions
were studied in the papers [16] and [18].
The mathematical modeling of many nonlinear problems from computer science, economics, mechanical engineering, control systems, biological neural networks, and others leads to the consideration of nonlinear difference equations (see [2, 4, 21, 23]). In the last decades, many authors have investigated such problems by using various methods, such as fixed point theorems, the critical point theory, upper and lower solutions, the fixed point index theory, and the topological degree theory (see, for example, [1, 6,7,8,9,10,11,12, 17, 20, 24,25,26,27,28]).
The paper is organized as follows. In Sect. 2, we investigate a system of second-order linear difference equations subject to the boundary conditions (BC), and we present the properties of the corresponding Green functions. In Sect. 3, we prove the main theorems for the existence and multiplicity of positive solutions of problem (S)–(BC) which are based on some theorems from the fixed point index theory, and we present two examples to support our results.
2 Preliminary results
We begin this section with a result from [14] related to the following system of second-order difference equations:
subject to the multi-point boundary conditions
where \(p, q\in \mathbb{N}\), \(\xi _{i}\in \mathbb{N}\) for all \(i=\overline{1,p}\), \(\eta _{i}\in \mathbb{N}\) for all \(i= \overline{1,q}\), \(1\le \xi _{1}<\cdots <\xi _{p}\le N-1\), \(1\le \eta _{1}<\cdots <\eta _{q}\le N-1\), and \(y_{n}\in \mathbb{R}\) for all \(n=\overline{1,N-1}\).
We denote \(\Delta _{1}= (1- \sum_{i=1}^{q}b_{i} ) \sum_{i=1}^{p}a_{i}\xi _{i}+ (1- \sum_{i=1}^{p}a_{i} ) (N- \sum_{i=1}^{q}b_{i}\eta _{i} )\).
Lemma 2.1
([14])
If \(\Delta _{1}\neq0\), then the solution \((u_{n})_{n= \overline{0,N}}\) of problem (1)–(2) is given by \(u_{n}=\sum_{j=1}^{N-1}G_{1}(n,j)y_{j}\) for all \(n=\overline{0,N}\), where the Green function \(G_{1}\) is defined by
and
Next we will present some properties of the function \(g_{0}\) and the Green function \(G_{1}\).
Lemma 2.2
The function \(g_{0}\) given by (4) has the following properties:
-
(a)
\(g_{0}(n,j)\ge 0\) for all \(n=\overline{0,N}\), \(j= \overline{1,N-1}\);
-
(b)
\(g_{0}(n,j)\le h(j)\) for all \(n=\overline{0,N}\), \(j= \overline{1,N-1}\), where \(h(j)=g_{0}(j,j)=\frac{1}{N}j(N-j)\) for all \(j=\overline{1,N-1}\);
-
(c)
\(g_{0}(n,j)\ge k(n)h(j)\) for all \(n=\overline{0,N}\), \(j= \overline{1,N-1}\), where \(k(n)=\frac{1}{N(N-1)}n(N-n)\) for all \(n=\overline{0,N}\).
Proof
For the proofs of (a) and (b), see [7].
For (c), if \(1\le j\le n\le N\), then we have
which is satisfied for all \(j=\overline{1,N-1}\) and \(n=\overline{0,N}\).
If \(0\le n\le j\le N-1\), then we obtain
which is satisfied for all \(j=\overline{1,N-1}\) and \(n=\overline{0,N}\). □
Lemma 2.3
If \(a_{i}\ge 0\) for all \(i=\overline{1,p}\), \(\sum_{i=1}^{p}a_{i}<1\), \(b_{i}\ge 0\) for all \(i=\overline{1,q}\), \(\sum_{i=1}^{q}b_{i}<1\), then the Green function \(G_{1}\) of problem (1)–(2) given by (3) satisfies the inequalities
-
(a)
\(G_{1}(n,j)\le Ah(j)\) for all \(n=\overline{0,N}\), \(j= \overline{1,N-1}\), where
$$ A=1+ \frac{1}{\Delta _{1}} \Biggl(N- \sum_{i=1}^{q} b_{i}\eta _{i} \Biggr) \Biggl( \sum _{i=1}^{p}a_{i} \Biggr) + \frac{1}{\Delta _{1}} \Biggl(N- \sum_{i=1}^{p} a_{i}(N- \xi _{i}) \Biggr) \Biggl( \sum_{i=1}^{q} b_{i} \Biggr)>0. $$ -
(b)
\(G_{1}(n,j)\ge k(n)h(j)\) for all \(n=\overline{0,N}\), \(j= \overline{1,N-1}\).
Proof
By the assumptions on the coefficients \(a_{i}\), \(i= \overline{1,p}\) and \(b_{j}\), \(j=\overline{1,q}\), we can easily see that \(\Delta _{1}>0\) and \(A>0\). By using Lemma 2.2, for all \(n=\overline{0,N}\) and \(j=\overline{1,N-1}\), we deduce
and
that is, we obtain inequalities (a) and (b). □
Lemma 2.4
Assume that \(a_{i}\ge 0\) for all \(i=\overline{1,p}\), \(\sum_{i=1}^{p}a _{i}<1\), \(b_{i}\ge 0\) for all \(i=\overline{1,q}\), \(\sum_{i=1}^{q}b _{i}<1\), and \(y_{n}\ge 0\) for all \(n=\overline{1,N-1}\). Then the solution \((u_{n})_{n=\overline{0,N}}\) of problem (1)–(2) satisfies the inequality \(u_{n}\ge \frac{1}{A}k(n)u_{m}\) for all \(n, m=\overline{0,N}\).
Proof
By using Lemmas 2.1–2.3, we deduce
□
We can also formulate similar results as Lemmas 2.1–2.4 for the discrete boundary value problem
where \(r, l\in \mathbb{N}\), \(c_{i}\ge 0\) for all \(i=\overline{1,r}\), \(\sum_{i=1}^{r}c_{i}<1\), \(\zeta _{i}\in \mathbb{N}\) for all \(i= \overline{1,r}\), \(d_{i}\ge 0\) for all \(i=\overline{1,l}\), \(\sum_{i=1} ^{l} d_{i}<1\), \(\rho _{i}\in \mathbb{N}\) for all \(i=\overline{1,l}\), \(1\le \zeta _{1}<\cdots <\zeta _{r}\le N-1\), \(1\le \rho _{1}<\cdots <\rho _{l}\le N-1\), and \(\widetilde{y}_{n}\ge 0\) for all \(n= \overline{1,N-1}\).
We denote by
Then we deduce the inequalities \(G_{2}(n,j)\le B h(j)\) and \(G_{2}(n,j) \ge k(n)h(j)\) for all \(n=\overline{0,N}\), \(j=\overline{1,N-1}\). In addition the solution \((v_{n})_{n=\overline{0,N}}\) of problem (5)–(6) satisfies the inequality \(v_{n}\ge \frac{1}{B}k(n)v_{m}\) for all \(n, m=\overline{0,N}\).
We recall now some theorems concerning the fixed point index theory. Let E be a real Banach space with the norm \(\|\cdot \|\), \(P\subset E\) be a cone, “≤” be the partial ordering defined by P, and 0 be the zero element in E. For \(\varrho >0\), let \(B_{\varrho }=\{u \in E, \|u\|<\varrho \}\) be the open ball of radius ϱ centered at 0, and its boundary \(\partial B_{\varrho }=\{u\in E, \|u\|=\varrho \}\). The proofs of our results are based on the following fixed point index theorems (see [3, 5, 13, 22]).
Theorem 2.1
Let \(A:\overline{B}_{\varrho }\cap P\to P\) be a completely continuous operator which has no fixed points on \(\partial B_{\varrho }\cap P\). If \(\|Au\|\le \|u\|\) for all \(u\in \partial B_{\varrho }\cap P\), then \(i(A,B_{\varrho }\cap P,P)=1\).
Theorem 2.2
Let \(A:\overline{B}_{\varrho }\cap P\to P\) be a completely continuous operator. If there exists \(u_{0}\in P\setminus \{0\}\) such that \(u-Au\neq \lambda u_{0}\) for all \(\lambda \ge 0\) and \(u\in \partial B_{\varrho }\cap P\), then \(i(A,B_{\varrho }\cap P,P)=0\).
Theorem 2.3
Let \(\varOmega \subset E\) be a bounded open set with \(0\in \varOmega \). Assume that \(A:\overline{\varOmega }\cap P\to P\) is a completely continuous operator.
-
(a)
If \(u\not \le Au\) for all \(u\in \partial \varOmega \cap P\), then the fixed point index \(i(A,\varOmega \cap P,P)=1\).
-
(b)
If \(Au\not \le u\) for all \(u\in \partial \varOmega \cap P\), then the fixed point index \(i(A,\varOmega \cap P,P)=0\).
3 Existence and multiplicity of positive solutions
In this section we present sufficient conditions on the functions f and g such that problem (S)–(BC) has positive solutions with respect to a cone.
We present the assumptions that we shall use in the sequel.
- \((H1)\) :
-
\(a_{i}\ge 0\), \(\xi _{i}\in \mathbb{N}\) for all \(i=\overline{1,p}\), \(1\le \xi _{1}<\cdots <\xi _{p}\le N-1\),
\(b_{i}\ge 0\), \(\eta _{i}\in \mathbb{N}\) for all \(i=\overline{1,q}\), \(1\le \eta _{1}<\cdots <\eta _{q}\le N-1\),
\(c_{i}\ge 0\), \(\zeta _{i}\in \mathbb{N}\) for all \(i=\overline{1,r}\), \(1\le \zeta _{1}<\cdots <\zeta _{r}\le N-1\),
\(d_{i}\ge 0\), \(\rho _{i}\in \mathbb{N}\) for all \(i=\overline{1,l}\), \(1\le \rho _{1}<\cdots <\rho _{l}\le N-1\), and
\(\sum_{i=1}^{p}a_{i}<1\), \(\sum_{i=1}^{q}b_{i}<1\), \(\sum_{i=1}^{r}c_{i}<1\), \(\sum_{i=1}^{l}d_{i}<1\).
- \((H2)\) :
-
The functions \(f, g:\{1,\ldots ,N-1\}\times \mathbb{R} _{+}\times \mathbb{R}_{+}\to \mathbb{R}_{+}\) are continuous, (\(\mathbb{R}_{+}=[0,\infty )\)).
- \((H3)\) :
-
There exist functions \(a, b\in C(\mathbb{R}_{+}, \mathbb{R}_{+})\) such that
-
(a)
\(a(\cdot )\) is concave and strictly increasing on \(\mathbb{R}_{+}\) with \(a(0)=0\);
-
(b)
$$\textstyle\begin{cases} f_{0}^{i}= \liminf_{v\to 0+} \frac{f(n,u,v)}{a(v)}\in (0,\infty ], \\ \quad \text{uniformly with respect to } (n,u)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}, \quad \text{and} \\ g_{0}^{i}= \liminf_{u\to 0+} \frac{g(n,u,v)}{b(u)}\in (0,\infty ], \\ \quad \text{uniformly with respect to } (n,v)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}; \end{cases} $$
-
(c)
\(\lim_{u\to 0+} \frac{a(Cb(u))}{u}=\infty \) exists for any constant \(C>0\).
-
(a)
- \((H4)\) :
-
There exist \(\alpha _{1}, \alpha _{2}>0\) with \(\alpha _{1}\alpha _{2}\le 1\) such that
$$\textstyle\begin{cases} f_{\infty }^{s}= \limsup_{v\to \infty } \frac{f(n,u,v)}{v^{\alpha _{1}}}\in [0,\infty ), \\ \quad \text{uniformly with respect to } (n,u)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}, \quad \text{and} \\ g_{\infty }^{s}= \lim_{u\to \infty } \frac{g(n,u,v)}{u^{\alpha _{2}}}=0 \\ \quad \text{exists uniformly with respect to } (n,v)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}. \end{cases} $$ - \((H5)\) :
-
There exist the functions \(c, d\in C(\mathbb{R}_{+}, \mathbb{R}_{+})\) such that
-
(a)
\(c(\cdot )\) is concave and strictly increasing on \(\mathbb{R}_{+}\);
-
(b)
$$\textstyle\begin{cases} f_{\infty }^{i}= \liminf_{v\to \infty } \frac{f(n,u,v)}{c(v)}\in (0,\infty ], \\ \quad \text{uniformly with respect to } (n,u)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}, \quad \text{and} \\ g_{\infty }^{i}= \liminf_{u\to \infty } \frac{g(n,u,v)}{d(u)}\in (0,\infty ], \\ \quad \text{uniformly with respect to } (n,v)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}; \end{cases} $$
-
(c)
\(\lim_{u\to \infty } \frac{c(Cd(u))}{u}=\infty \) exists for any constant \(C>0\).
-
(a)
- \((H6)\) :
-
There exist \(\beta _{1}, \beta _{2}>0\) with \(\beta _{1} \beta _{2}\ge 1\) such that
$$\textstyle\begin{cases} f_{0}^{s}= \limsup_{v\to 0+} \frac{f(n,u,v)}{v^{\beta _{1}}}\in [0,\infty ), \\ \quad \text{uniformly with respect to } (n,u)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}, \quad \text{and} \\ g_{0}^{s}= \lim_{u\to 0+} \frac{g(n,u,v)}{u^{\beta _{2}}}=0 \\ \quad \text{exists uniformly with respect to } (n,v)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}. \end{cases} $$ - \((H7)\) :
-
The functions \(f(n,u,v)\) and \(g(n,u,v)\) are nondecreasing with respect to u and v, and there exists \(N_{0}>0\) such that
$$f(n,N_{0},N_{0})< \frac{3N_{0}}{(N^{2}-1)\max \{A,B\}} \quad \text{and} \quad g(n,N_{0},N _{0})< \frac{3N_{0}}{(N^{2}-1)\max \{A,B\}} $$for all \(n\in \{1,\ldots ,N-1\}\).
By using the Green functions \(G_{1}\) and \(G_{2}\) from Sect. 2, our problem (S)–(BC) can be written equivalently as the following system:
Then \((u,v)=((u_{n})_{n=\overline{0,N}},(v_{n})_{n=\overline{0,N}})\) is a solution of problem (S)–(BC) if and only if \((u,v)\) is a solution of system (7).
We consider the Banach space \(X=\mathbb{R}^{N+1}=\{u=(u_{n})_{n= \overline{0,N}}, u_{i}\in \mathbb{R}, i=\overline{0,N}\}\) with the maximum norm \(\|\cdot \|\), \(\|u\|=\max_{i=\overline{0,N}}|u_{i}|\), and the Banach space \(Y=X\times X\) with the norm \(\|(u,v)\|_{Y}=\|u\|+\|v \|\). We define the cones
and \(P=P_{1}\times P_{2}\subset Y\).
We introduce the operators \(Q_{1}, Q_{2}:Y\to X\) and \(Q:Y\to Y\) defined by
The pair \((u,v)\) is a solution of problem (S)–(BC) if and only if \((u,v)\) is a fixed point of operator Q in the space Y. So, we will investigate the existence of fixed points of operator Q. Under assumptions \((H1)\) and \((H2)\) and by using Lemma 2.4, we can easily prove that \(Q(P)\subset P\) and the operator \(Q:P\to P\) is completely continuous.
Theorem 3.1
Assume that \((H1)\), \((H2)\), \((H3)\), and \((H4)\) hold. Then problem (S)–(BC) has at least one positive solution.
Proof
By \((H3)\), there exist \(C_{1}>0\), \(C_{2}>0\), and a sufficiently small \(r_{1}>0\) such that
and
where \(C_{3}=\max \{\frac{(N-1)C_{2}}{N}h(j), j= \overline{1,N-1} \}\).
We will show that \((Q_{1}(u,v),Q_{2}(u,v))\not \le (u,v)\) for all \((u,v)\in \partial B_{r_{1}}\cap P\). We suppose that there exists \((u,v)\in \partial B_{r_{1}}\cap P\), that is, \(\|(u,v)\|_{Y}=r_{1}\), such that \((Q_{1}(u,v),Q_{2}(u,v))\le (u,v)\). Then \(u\ge Q_{1}(u,v)\) and \(v\ge Q_{2}(u,v)\). By using the monotonicity and concavity of \(a(\cdot )\), the Jensen inequality, Lemma 2.3, relations (8) and (9), we obtain
So, \(\|u\|\ge \max_{n=\overline{1,N-1}}u_{n}\ge 2\|u\|\), and then
In a similar manner, we deduce
Then we conclude that \(a(\|v\|)=a(\sup_{i=\overline{0,N}}v_{i})\ge a(v _{1})\ge 2 a(\|v\|)\), and hence \(a(\|v\|)=0\). By \((H3)\)(a), we obtain
Therefore, by (10) and (11), we deduce that \(\|(u,v)\|_{Y}=0\), which is a contradiction. Hence \((Q_{1}(u,v),Q_{2}(u,v)) \not \le (u,v)\) for all \((u,v)\in \partial B_{r_{1}}\cap P\). By Theorem 2.3(b), we conclude that the fixed point index
On the other hand, by \((H4)\) we deduce that there exist \(C_{4}>0\), \(C_{5}>0\), and \(C_{6}>0\) such that
with
Then, by (13), we have
We consider now the functions \(\widetilde{p}, \widetilde{q}: \mathbb{R}_{+}\to \mathbb{R}_{+}\) defined by
Because
we conclude that there exists \(R_{1}>r_{1}\) such that
We will show that \((u,v)\not \le (Q_{1}(u,v),Q_{2}(u,v))\) for all \((u,v)\in \partial B_{R_{1}}\cap P\). We suppose that there exists \((u,v)\in \partial B_{R_{1}}\cap P\), that is, \(\|(u,v)\|_{Y}=R_{1}\), such that \((u,v)\le (Q_{1}(u,v),Q_{2}(u,v))\). So, by (14), we obtain
Then, for all \(n=\overline{0,N}\), we deduce
and
By using (16), (17), and (15), we conclude that \(u_{n}\le \frac{1}{4}\|(u,v)\|_{Y}\) and \(v_{n}\le \frac{1}{4}\|(u,v) \|_{Y}\) for all \(n=\overline{0,N}\). Therefore we obtain that \(\|(u,v)\|_{Y}\le \frac{1}{2}\|(u,v)\|_{Y}\), and so \(\|(u,v)\|_{Y}=0\), which is a contradiction because \(\|(u,v)\|_{Y}=R_{1}>0\). So, \((u,v)\not \le (Q_{1}(u,v),Q_{2}(u,v))\) for all \((u,v)\in \partial B _{R_{1}}\cap P\). By Theorem 2.3(a), we deduce that the fixed point index
Because Q has no fixed points on \(\partial B_{r_{1}}\cup \partial B _{R_{1}}\), by (12) and (18), we conclude that
So the operator Q has at least one fixed point \((u^{1},v^{1})\in (B _{R_{1}}\setminus \overline{B}_{r_{1}})\cap P\), with \(r_{1}<\|(u^{1},v ^{1})\|_{Y}<R_{1}\), that is, \(\|u^{1}\|>0\) or \(\|v^{1}\|>0\). Because \(u^{1}\in P_{1}\) and \(v^{1}\in P_{2}\), we obtain \(u^{1}_{n}>0\) for all \(n=\overline{1,N}\) or \(v_{n}^{1}>0\) for all \(n=\overline{1,N}\). □
Theorem 3.2
Assume that \((H1)\), \((H2)\), \((H5)\), and \((H6)\) hold. Then problem (S)–(BC) has at least one positive solution.
Proof
By \((H5)\) there exist \(C_{i}>0\), \(i=7,\ldots ,11\), such that
and
where \(C_{12}=\max \{\frac{C_{9}(N-1)}{N}h(i), i=\overline{1,N-1} \}>0\). Then we obtain
We will prove that the set \(U=\{(u,v)\in P, (u,v)=Q(u,v)+\lambda (\varphi ^{1},\varphi ^{2}), \lambda \ge 0\}\) is bounded, where \((\varphi ^{1},\varphi ^{2})\in P\setminus \{(0,0)\}\). Indeed, \((u,v)\in U\) implies that \(u\ge Q_{1}(u,v)\), \(v\ge Q_{2}(u,v)\) for some \(\varphi ^{1}, \varphi ^{2}\ge 0\). By (21), we obtain
where \(C_{13}=C_{8}(N^{2}-1)/(6N)\), \(C_{14}=C_{10}(N^{2}-1)/(6N)\).
By the monotonicity and concavity of \(c(\cdot )\) and the Jensen inequality, inequality (23) implies that
Since \(c(v_{n})\ge c(v_{n}+C_{14})-c(C_{14})\), by relations (22), (23), and (24), we deduce
where \(C_{15}=\frac{C_{7}c(C_{14})(N^{2}-1)}{6N}+C_{13}\), \(C_{16}=\frac{C _{7}C_{9}C_{11}(N^{2}-1)^{2}}{36N^{2}C_{12}}+C_{15}\).
Therefore \(\|u\|\ge u_{1}\ge 2\|u\|-C_{16}\), and then
Since \(c(v_{n})\ge c (\frac{1}{B}k(n)\|v\| )\ge c (\frac{1}{BN} \|v\| )\ge \frac{1}{BN}c(\|v\|)\) for all \(n=\overline{1,N-1}\), then by relations (19), (22), (23), and (24), we obtain
where \(C_{17}=\frac{C_{9}C_{11}(N^{2}-1)}{6NC_{12}}+c(C_{14})\), \(C_{18}=\frac{12C_{13}N^{2}\max \{A,B\}}{C_{7}(N^{2}-1)}+C_{17}\).
Then \(c(\|v\|)\ge c(v_{1})\ge 2c(\|v\|)-C_{18}\), and so \(c(\|v\|) \le C_{18}\). By \((H5)\)(a) and (c), we deduce that \(\lim_{v\to \infty }c(v)=\infty \). Thus there exists \(C_{19}>0\) such that
By (25) and (26), we conclude that \(\|(u,v)\|_{Y} \le C_{16}+C_{19}\) for all \((u,v)\in U\). That is the set U is bounded. Then there exists a sufficiently large \(R_{2}>0\) such that \((u,v) \neq Q(u,v)+\lambda (\varphi ^{1},\varphi ^{2})\) for all \((u,v)\in \partial B_{R_{2}}\cap P\) and \(\lambda \ge 0\). By Theorem 2.2 we deduce that
On the other hand, by \((H6)\) there exist \(C_{20}>0\) and a sufficiently small \(r_{2}>0\), (\(r_{2}< R_{2}\), \(r_{2}\le 1\)) such that
where \(\varepsilon _{2}= (2AB^{\beta _{1}}C_{20} (\frac{N^{2}-1}{6} ) ^{\beta _{1}+1} )^{-1/\beta _{1}}>0\).
We will show that \((u,v)\not \le Q(u,v)\) for all \((u,v)\in \partial B _{r_{2}}\cap P\). We suppose that there exists \((u,v)\in \partial B _{r_{2}}\cap P\), that is, \(\|(u,v)\|_{Y}=r_{2}\le 1\), such that \((u,v)\le (Q_{1}(u,v),Q_{2}(u,v))\), or \(u\le Q_{1}(u,v)\) and \(v\le Q_{2}(u,v)\). Then by (28) we obtain
Therefore \(\|u\|\le \frac{1}{2}\|u\|\), so
In addition
By (29) and (30) we deduce that \(\|v\|=0\), and then \(\|(u,v)\|_{Y}=0\), which is a contradiction because \(\|(u,v)\|_{Y}=r _{2}>0\). Then \((u,v)\not \le Q(u,v)\) for all \((u,v)\in \partial B_{r _{2}}\cap P\). By Theorem 2.3(a), we conclude that
Because Q has no fixed points on \(\partial B_{r_{2}}\cup \partial B _{R_{2}}\), by (27) and (31), we deduce that
So the operator Q has at least one fixed point \((u^{2},v^{2})\in (B _{R_{2}}\setminus \overline{B}_{r_{2}})\cap P\), with \(r_{2}<\|(u^{2},v ^{2})\|_{Y}<R_{2}\), which is a positive solution for our problem (S)–(BC). □
Theorem 3.3
Assume that assumptions \((H1)\), \((H2)\), \((H3)\), \((H5)\), and \((H7)\) hold. Then problem (S)–(BC) has at least two positive solutions.
Proof
By using \((H7)\), for any \((u,v)\in \partial B_{N_{0}} \cap P\), we obtain
Then we deduce
Because Q has no fixed points on \(\partial B_{N_{0}}\), by Theorem 2.1 we conclude that
On the other hand, from \((H3)\) and \((H5)\), and the proofs of Theorems 3.1 and 3.2, we know that there exist a sufficiently \(r_{1}>0\) (\(r_{1}< N_{0}\)) and a sufficiently large \(R_{2}>N_{0}\) such that
Because Q has no fixed points on \(\partial B_{r_{1}}\cup \partial B _{R_{2}}\cup \partial B_{N_{0}}\), by relations (32) and (33), we obtain
Then \(\mathcal{Q}\) has at least one fixed point \((u^{1},v^{1})\in (B _{R_{2}}\setminus \bar{B}_{N_{0}})\cap P\) and has at least one fixed point \((u^{2},v^{2})\in (B_{N_{0}}\setminus \bar{B}_{r_{1}})\cap P\). Therefore, problem (S)–(BC) has two distinct positive solutions \((u^{1},v^{1})\), \((u^{2},v^{2})\). □
Remark 3.1
In \((H3)\), if \(a(v)=v^{p}\) with \(p\le 1\) and \(b(u)=u^{q}\) with \(q>0\), the condition from \((H3)\)(c) is satisfied if \(pq<1\). In \((H5)\), if \(c(v)=v^{p}\) with \(p\le 1\), and \(d(u)=u^{q}\) with \(q>0\), the condition from \((H5)\)(c) is satisfied if \(pq>1\).
Examples
-
(1)
We consider \(f(n,u,v)=\frac{n}{n+1}(1+e^{-(u+v)})\) and \(g(n,u,v)=(1+e ^{-n})u^{\theta }\) for \((n,u,v)\in \{1,\ldots ,N-1\}\times \mathbb{R} _{+}\times \mathbb{R}_{+}\). For \(a(v)=v^{p}\) with \(p\le 1\), and \(b(u)=u^{q}\) for \(q>0\) and \(pq<1\), then assumptions \((H3)\) and \((H4)\) are satisfied if \(q>\theta \) and \(\alpha _{2}>\theta \). For example, if \(\theta =\frac{5}{4}\), \(p=\frac{1}{3}\), \(q=\frac{4}{3}\), \(\alpha _{1}=\frac{1}{3}\), and \(\alpha _{2}=3\), we can apply Theorem 3.1, and we deduce that problem (S)–(BC) has at least one positive solution.
-
(2)
We consider \(f(n,u,v)=(1+e^{-u})v^{\theta _{1}}\) and \(g(n,u,v)=(1+e ^{-v})u^{\theta _{2}}\) for \((n,u,v)\in \{1,\ldots ,N-1\}\times \mathbb{R}_{+}\times \mathbb{R}_{+}\). For \(c(v)=v^{p}\) with \(p\le 1\), and \(d(u)=u^{q}\) for \(q>0\) and \(pq>1\), then assumptions \((H5)\) and \((H6)\) are satisfied if \(p<\theta _{1}\), \(q<\theta _{2}\), \(\beta _{1}<\theta _{1}\), and \(\beta _{2}<\theta _{2}\). For example, if \(\theta _{1}=4\), \(\theta _{2}=2\), \(p=\frac{3}{5}\), \(q=\frac{9}{5}\), \(\beta _{1}=3\), and \(\beta _{2}=\frac{1}{3}\), we can apply Theorem 3.2, and we conclude that problem (S)–(BC) has at least one positive solution.
References
Afrouzi, G.A., Hadjian, A.: Existence and multiplicity of solutions for a discrete nonlinear boundary value problem. Electron. J. Differ. Equ. 2014, 35 1–13 (2014)
Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Monographs and Textbooks in Pure and Applied Mathematics, vol. 228. Dekker, New York (2000)
Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)
Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (1999)
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)
Anderson, D.R.: Solutions to second-order three-point problems on time scales. J. Differ. Equ. Appl. 8(8), 673–688 (2002)
Atici, F., Peterson, A.C.: Inequality for a 2nth order difference equation. Panam. Math. J. 6, 41–49 (1996)
Avery, R.: Three positive solutions of a discrete second order conjugate problem. Panam. Math. J. 8, 79–96 (1998)
Cheung, W.S., Ren, J.: Positive solution for discrete three-point boundary value problems. Aust. J. Math. Anal. Appl. 1(2), Article 9, 1–7 (2004)
Goodrich, C.S.: Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale. Comment. Math. Univ. Carol. 54(4), 509–525 (2013)
Graef, J.R., Kong, L., Wang, M.: Multiple solutions to a periodic boundary value problem for a nonlinear discrete fourth order equation. Adv. Dyn. Syst. Appl. 8(2), 203–215 (2013)
Graef, J.R., Kong, L., Wang, M.: Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Discrete Contin. Dyn. Syst. 2013(Supplement), 291–299 (2013)
Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Henderson, J., Luca, R.: Existence of positive solutions for a system of second-order multi-point discrete boundary value problems. J. Differ. Equ. Appl. 19(11), 1889–1906 (2013)
Henderson, J., Luca, R.: On a second-order nonlinear discrete multi-point eigenvalue problem. J. Differ. Equ. Appl. 20(7), 1005–1018 (2014)
Henderson, J., Luca, R.: Positive solutions for a system of difference equations with coupled multi-point boundary conditions. J. Differ. Equ. Appl. 22(2), 188–216 (2016)
Henderson, J., Luca, R.: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations. Positive Solutions. Elsevier, Amsterdam (2016)
Henderson, J., Luca, R.: Existence and multiplicity of positive solutions for a system of difference equations with coupled boundary conditions. J. Appl. Anal. Comput. 7(1), 134–146 (2017)
Henderson, J., Luca, R., Tudorache, A.: Multiple positive solutions for a multi-point discrete boundary value problem. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 63(2), 59–70 (2014)
Iannizzotto, A., Tersian, S.A.: Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory. J. Math. Anal. Appl. 403(1), 173–182 (2013)
Kelley, W.G., Peterson, A.C.: Difference Equations. An Introduction with Applications, 2nd edn. Academic Press, San Diego (2001)
Krasnoselskii, M.A.: Positive Solution of Operator Equations. Noordhoff, Groningen (1964)
Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations. Numerical Methods and Applications. Mathematics in Science and Engineering, vol. 181. Academic Press, Boston (1988)
Li, W.T., Sun, H.R.: Positive solutions for second-order m-point boundary value problems on times scales. Acta Math. Sin. Engl. Ser. 22(6), 1797–1804 (2006)
Rodriguez, J.: Nonlinear discrete Sturm–Liouville problems. J. Math. Anal. Appl. 308(1), 380–391 (2005)
Sun, H.R., Li, W.T.: Positive solutions for nonlinear three-point boundary value problems on time scales. J. Math. Anal. Appl. 299(2), 508–524 (2004)
Wang, D.B., Guan, W.: Three positive solutions of boundary value problems for p-Laplacian difference equations. Comput. Math. Appl. 55(9), 1943–1949 (2008)
Wang, L., Chen, X.: Positive solutions for discrete boundary value problems to one-dimensional p-Laplacian with delay. J. Appl. Math. 2013, Article ID 157043 (2013)
Acknowledgements
The authors thank the referee for his/her valuable comments and suggestions.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this paper. The authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Additional information
Abbreviation
Not applicable.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Agarwal, R.P., Luca, R. Positive solutions for a system of second-order discrete boundary value problems. Adv Differ Equ 2018, 470 (2018). https://doi.org/10.1186/s13662-018-1929-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-018-1929-8