 Research
 Open Access
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 point
MSC
 39A12
 39A10
 34B09
1 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
References
 Agarwal, RP, Perera, K, O’Regan, D: Multiple positive solutions of singular discrete pLaplacian problems via variational methods. Adv. Differ. Equ. 2, 9399 (2005) MathSciNetGoogle Scholar
 Anderson, DR: Discrete thirdorder threepoint rightfocal boundary value problems. Comput. Math. Appl. 45, 861871 (2003) View ArticleMathSciNetMATHGoogle Scholar
 Anderson, D, Avery, RI: Multiple positive solutions to a thirdorder discrete focal boundary value problem. Comput. Math. Appl. 42, 333340 (2001) View ArticleMathSciNetMATHGoogle Scholar
 Aykut, N: Existence of positive solutions for boundary value problems of second order functional difference equations. Comput. Math. Appl. 48, 517527 (2004) View ArticleMathSciNetMATHGoogle Scholar
 Bai, D: A global result for discrete ϕLaplacian eigenvalue problems. Adv. Differ. Equ. 2013, 264 (2013) View ArticleGoogle Scholar
 Bai, D, Xu, Y: Nontrivial solutions of boundary value problems of second order difference equations. J. Math. Anal. Appl. 326(1), 297302 (2007) View ArticleMathSciNetMATHGoogle Scholar
 Bai, D, Xu, X: Existence and multiplicity of difference ϕLaplacian boundary value problems. Adv. Differ. Equ. 2013, 267 (2013) View ArticleMathSciNetGoogle Scholar
 Ji, D, Ge, W: Existence of multiple positive solutions for SturmLiouvillelike fourpoint boundary value problem with pLaplacian. Nonlinear Anal. 68, 26382646 (2008) View ArticleMathSciNetMATHGoogle Scholar
 Jiang, L, Zhou, Z: Existence of nontrivial solutions for discrete nonlinear two point boundary value problems. Appl. Math. Comput. 180, 318329 (2006) View ArticleMathSciNetMATHGoogle Scholar
 Karaca, IY: Discrete thirdorder threepoint boundary value problem. J. Comput. Appl. Math. 205, 458468 (2007) View ArticleMathSciNetMATHGoogle Scholar
 Liu, Y: Existence results for positive solutions of nonhomogeneous BVPs for second order difference equations with onedimensional pLaplacian. J. Korean Math. Soc. 47, 135163 (2010) View ArticleMathSciNetMATHGoogle Scholar
 Yang, Y, Meng, F: Eigenvalue problem for finite difference equations with pLaplacian. J. Appl. Math. Comput. 40, 319340 (2012) View ArticleMathSciNetMATHGoogle Scholar
 Yu, J, Guo, Z: On generalized discrete boundary value problems of EmdenFowler equation. Sci. China Ser. A 36, 721732 (2006) MathSciNetGoogle Scholar
 Yu, J, Guo, Z: On boundary value problems for a discrete generalized EmdenFowler equation. J. Differ. Equ. 231, 1831 (2006) View ArticleMathSciNetMATHGoogle Scholar
 Bai, D, Xu, Y: Positive solutions for semipositone BVPs of secondorder difference equations. Indian J. Pure Appl. Math. 39, 5968 (2008) MathSciNetMATHGoogle Scholar
 Anuradha, A, Hai, DD, Shivaji, R: Existence results for superlinear semipositone BVP’s. Proc. Am. Math. Soc. 124(3), 757763 (1996) View ArticleMathSciNetMATHGoogle Scholar
 Anuradha, V, Maya, C, Shivaji, R: Positive solutions for a class of nonlinear boundary value problems with NeumannRobin boundary conditions. J. Math. Anal. Appl. 236, 94124 (1999) View ArticleMathSciNetMATHGoogle Scholar
 Bai, D, Xu, Y: Existence of positive solutions for boundary value problems of secondorder delay differential equations. Appl. Math. Lett. 18, 621630 (2005) View ArticleMathSciNetMATHGoogle Scholar
 Bai, D, Xu, Y: Positive solutions of secondorder twodelay differential systems with twinparameter. Nonlinear Anal. 63, 601617 (2005) View ArticleMathSciNetMATHGoogle Scholar
 Bai, D, Yu, J: Semipositone problems of second order ordinary differential equations. Sciencepaper Online (10 October 2009); http://www.paper.edu.cn/
 Castro, A, Gadam, S, Shivaji, R: Evolution of positive solution curves in semipositone problems with concave nonlinearities. J. Math. Anal. Appl. 245, 282293 (2000) View ArticleMathSciNetMATHGoogle Scholar
 Hai, DD, Schmitt, K, Shivaji, R: Positive solutions of quasilinear boundary value problems. J. Math. Anal. Appl. 217, 672686 (1998) View ArticleMathSciNetMATHGoogle Scholar
 Hai, DD, Shivaji, R: An existence result on positive solutions for a class of pLaplacian systems. Nonlinear Anal. 56, 10071010 (2004) View ArticleMathSciNetMATHGoogle Scholar
 Ma, R: Multiple positive solutions for a semipositone fourthorder boundary value problem. Hiroshima Math. J. 33, 217227 (2003) MathSciNetMATHGoogle Scholar
 Maya, C, Shivaji, R: Multiple positive solutions for a class of semilinear elliptic boundary value problems. Nonlinear Anal. 38, 497504 (1999) View ArticleMathSciNetMATHGoogle Scholar
 Sun, J, Wei, J: Existence of positive solutions of positive solution for semi positone secondorder threepoint boundaryvalue problems. Electron. J. Differ. Equ. 2008, 41 (2008) MathSciNetGoogle Scholar
 Henderson, J, Kosmatov, N: Positive solutions of the semipositone Neumann boundary value problem. Math. Model. Anal. 20, 578584 (2015) View ArticleMathSciNetGoogle Scholar
 Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988) MATHGoogle Scholar