Positive solutions of discrete Neumann boundary value problems with signchanging nonlinearities
 Dingyong Bai^{1}Email author,
 Johnny Henderson^{2} and
 Yunxia Zeng^{1}
Received: 7 October 2015
Accepted: 29 November 2015
Published: 9 December 2015
Abstract
Keywords
difference equation Neumann boundary value problem positive solution fixed pointMSC
39A12 39A10 34B091 Introduction
In recent years, positive solutions of boundary value problems for difference equations have been widely studied. See [1–14] and the references therein. However, little work has been done that has referred to the existence of positive solutions for discrete boundary value problems with signchanging nonlinearities (see [15]).
 (C1):

\(f: [1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\to\mathbb{R}\) is continuous;
 (C2):

there exists a function \(h:[1,T]_{\mathbb{Z}}\to\mathbb {R}^{+}\) with \(h(t)\not\equiv0\) on \([1,T]_{\mathbb{Z}}\), and a constant number \(L>0\), such that$$\begin{aligned} f(t,z)+Lz+h(t)\geq0,\quad (t,z)\in[1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}. \end{aligned}$$(1.2)
 (1)
\(f(t,z)\) may be unbounded below and even be nonpositive for all \((t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\) (see the first parts of Theorems 3.1 and 3.3, in which the existence of at least one positive solution is presented);
 (2)
\(f(t,z)\) is ultimately nonpositive, i.e., \(f(t,z)\leq 0\) for all \(t\in[1,T]_{\mathbb{Z}}\) and \(z>0\) sufficiently large (see the second part of Theorem 3.1, in which the existence of at least two positive solutions is presented);
 (3)
\(f(t,z)\) is ultimately nonnegative, i.e., \(f(t,z)\geq 0\) for all \(t\in[1,T]_{\mathbb{Z}}\) and \(z>0\) sufficiently large, which implies that \(f(t,z)\) is bounded below (see the second part of Theorem 3.3, in which the existence of at least two positive solutions is presented);
 (4)
\(\lim_{z\rightarrow \infty}\frac{f(t,z)}{z}=0\) uniformly for \(t\in[1,T]_{\mathbb{Z}}\), which implies that \(f(t,z)\) may be either bounded or unbounded below, and that \(f(t,z)\) may be ultimately nonpositive or nonnegative or oscillating (see Corollary 3.2, in which the existence of at least two positive solutions is presented);
 (5)
\(\lim_{z\rightarrow \infty}\frac{f(t,z)}{z}=\infty\) uniformly for \(t\in[1,T]_{\mathbb {Z}}\), which is a special case of (3) and implies that \(f(t,z)\) is bounded below (see Corollary 3.4, in which the existence of at least two positive solutions is presented).
The remaining part of this paper is organized as follows. In Section 2, we provide some preliminary results for later use. Then, in Section 3, we show and prove the existence and multiplicity of positive solutions for boundary value problem (1.1).
2 Preliminaries
Lemma 2.1
Lemma 2.2
\(q(t)G(t,s)\leq G(t,s)\leq G(s,s)\), \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\).
Proof
Lemma 2.3
\(q(t)\geq\mu u_{0}(t)\), \(t\in[1,T]_{\mathbb{Z}}\), where \(\mu=\frac{(A^{2}1)(A^{T}A^{T})}{(A^{T}+A^{T+1})^{2}\sum_{s=1}^{T}h(s)}\).
Proof
Lemma 2.4
u is a positive solution of the boundary value problem (1.1) if and only if \(v=u+u_{0}\) is a solution of the boundary value problem (2.3) with \(v(t)>u_{0}(t)\) in \([1,T]_{\mathbb{Z}}\).
The proofs of our main results are based on the GuoKrasnosel’skiĭ fixed point theorem [28].
Lemma 2.5
 (i)
\(\Tu\\leq\u\\), \(u\in K\cap\partial\Omega_{1}\), and \(\Tu\\geq\u\\), \(u\in K\cap\partial\Omega_{2}\); or
 (ii)
\(\Tu\\geq\u\\), \(u\in K\cap\partial\Omega_{1}\), and \(\Tu\\leq\u\\), \(u\in K\cap\partial\Omega_{2}\).
3 Main results
Theorem 3.1
 (C3):

\(\phi(r)\leq\frac{r}{\max_{t\in[1,T]_{\mathbb {Z}}}\sum_{s=1}^{T}G(t,s)}\) and \(\psi(R)\geq \frac{R}{\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\).
 (C4):

\(f(t,z)\leq0\) for all \(t\in[1,T]_{\mathbb{Z}}\) and \(z>0\) sufficiently large, and
 (C5):

\(L<\frac{1}{\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\),
Proof
Therefore, by Lemma 2.5, F has a fixed point \(v_{1}\in P\) satisfying \(r\leq\v_{1}\\leq R\), which is a positive solution of problem (2.3). By Lemma 2.3, \(u_{1}(t)=v_{1}(t)u_{0}(t)\geq q(t)\v_{1}\u_{0}(t)\geq(\mu r1)u_{0}(t)>0\), \(t\in[1,T]_{\mathbb{Z}}\). Therefore, by Lemma 2.4, \(u_{1}\) is a positive solution of problem (1.1).
Therefore, by Lemma 2.5, F has a fixed point \(v_{2}\in P\) such that \(R\leq\v_{2}\\leq R_{\infty}\). By Lemma 2.4, \(u_{2}(t)=v_{2}(t)u_{0}(t)\) is a second positive solution of problem (1.1). The proof is complete. □
Corollary 3.2
Let (C1) and (C2) hold. Assume that there exist \(\frac{1}{\mu}< r< R\) such that (C3) hold. Then, if \(\lim_{z\to\infty}\frac {f(t,z)}{z}=0\) uniformly for \(t\in[1,T]_{\mathbb{Z}}\) and \(L<\frac{1}{\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\), problem (1.1) has at least two positive solutions.
Proof
Let F, P, and \(\Omega_{1}\), \(\Omega_{2}\) be defined as (3.1), (3.2), and (3.3), respectively. From the proof of Theorem 3.1, we know by (C3) that F has a fixed point \(v_{1}\) such that \(r\leq\v_{1}\\leq R\) and \(u_{1}(t)=v_{1}(t)u_{0}(t)\) is a positive solution of (1.1). Now, we prove that F has a second fixed point \(v_{2}\in P\).
Theorem 3.3
 (C3)^{∗} :

\(\phi(R)\leq\frac{R}{\max_{t\in[1,T]_{\mathbb {Z}}}\sum_{s=1}^{T}G(t,s)}\) and \(\psi(r)\geq \frac{r}{\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\).
 (C4)^{∗} :

\(f(t,z)\geq0\) for all \(t\in[1,T]_{\mathbb{Z}}\) and \(z>0\) sufficiently large, and
 (C5)^{∗} :

\(L> \frac{2}{q_{0}\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\),
Proof
Let F, P, and \(\Omega_{1}\), \(\Omega_{2}\) be defined as (3.1), (3.2), and (3.3), respectively. Consider the fixed point of operator F in the cone P. Similar to the arguments in the proof of Theorem 3.1, if (C3)^{∗} holds, then we have \(\Fv\\geq\v\\) for \(v\in P\cap\partial\Omega_{1}\) and \(\Fv\\leq \v\\) for \(v\in P\cap\partial\Omega_{2}\). Thus, by Lemma 2.5, F has a fixed point \(v_{1}\in P\) satisfying \(r\leq\v_{1}\\leq R\), which is a positive solution of problem (2.3) and satisfies \(v_{1}(t)u_{0}(t)>0\), \(t\in[1,T]_{\mathbb{Z}}\). Therefore, by Lemma 2.4, \(u_{1}=v_{1}(t)u_{0}(t)\) is a positive solution of problem (1.1).
The following result can be obtained directly from Theorem 3.3.
Corollary 3.4
Let (C1) and (C2) hold. Assume that there exist \(\frac{1}{\mu}< r< R\) such that (C3)^{∗} hold. Then, if \(\lim_{z\to\infty}\frac {f(t,z)}{z}=\infty\) uniformly for \(t\in[1,T]_{\mathbb{Z}}\) and \(L> \frac{2}{q_{0}\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\), problem (1.1) has at least two positive solutions.
Declarations
Acknowledgements
Supported partially by PCSIRT of China (No. IRT1226) and NSF of China (No. 11171078).
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.
Authors’ Affiliations
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