Open Access

Positive solutions of discrete Neumann boundary value problems with sign-changing nonlinearities

Boundary Value Problems20152015:231

https://doi.org/10.1186/s13661-015-0500-8

Received: 7 October 2015

Accepted: 29 November 2015

Published: 9 December 2015

Abstract

Our concern is the existence of positive solutions of the discrete Neumann boundary value problem
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} u(t-1)=f(t, u(t)), \quad t\in[1,T]_{\mathbb{Z}},\\ \Delta u(0)=\Delta u(T)=0, \end{array}\displaystyle \right . \end{aligned}$$
where \(f: [1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\to\mathbb{R}\) is a sign-changing function. By using the Guo-Krasnosel’skiĭ fixed point theorem, the existence and multiplicity of positive solutions are established. The nonlinear term \(f(t,z)\) may be unbounded below or nonpositive for all \((t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\).

Keywords

difference equation Neumann boundary value problem positive solution fixed point

MSC

39A12 39A10 34B09

1 Introduction

For \(a, b\in\mathbb{Z}\) with \(a< b\), let \([a,b]_{\mathbb{Z}}=\{a, a+1, a+2,\ldots,b-1, b\}\). We consider the following discrete Neumann boundary value problem:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} u(t-1)=f(t, u(t)), \quad t\in[1,T]_{\mathbb{Z}},\\ \Delta u(0)=\Delta u(T)=0, \end{array}\displaystyle \right . \end{aligned}$$
(1.1)
where \(T>1\) is a given positive integer, \(\Delta u(t)=u(t+1)-u(t)\). Our purpose is to establish existence results for positive solutions of (1.1) when the nonlinearity term \(f: [1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}\to\mathbb{R}\) is a sign-changing function.

In recent years, positive solutions of boundary value problems for difference equations have been widely studied. See [114] 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 sign-changing nonlinearities (see [15]).

Usually, in order to obtain positive solutions of semipositone problems for ordinary differential equations or difference equations by using fixed point methods, the nonlinearity terms need to be bounded below and ultimately positive. For example, Anuradha et al. [16] studied the following problem:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (p(t)u')'+\lambda g(t, u)=0, \quad t\in(a,b),\\ \gamma_{1}u(a)-\gamma_{2}p(a)u'(a)=0,\qquad \gamma_{3}u(b)+\gamma_{4}p(b) u'(b)=0, \end{array}\displaystyle \right . \end{aligned}$$
where \(g: [a,b]\times\mathbb{R}^{+}\rightarrow\mathbb{R}\) is continuous, bounded below (i.e., \(g(t,z)+M>0\) for some \(M>0\)), and \(\lim_{z\rightarrow \infty}\frac{g(t,z)}{z}=\infty\) uniformly for \(t\in[\alpha, \beta]\subset(a,b)\). Motivated by the method in [16], the first author and Xu [15] discussed its discrete analog and the nonlinearity term also required boundedness below, as well as a superlinear condition at ∞. We also refer to [1726] for some references.
In this paper, our interest is with the existence and multiplicity of positive solutions of (1.1), where \(f(t,z)\) may be nonpositive or unbounded below for all \((t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb {R}^{+}\). For this purpose, we make our basic assumptions as follows.
(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)
Under (C1) and (C2), we give some sufficient conditions such that (1.1) has at least one positive solution and two positive solutions, respectively. Under these conditions, \(f(t,z)\) is allowed to meet one of the following cases:
  1. (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. (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. (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. (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. (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 idea of this paper comes from the method in [27] by Henderson and Kosmatov, in which the Neumann boundary value problem of ordinary differential equation at resonance
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -u''(t)=f(t,u(t)), \quad 0< t< 1,\\ u'(0)=u'(1)=0, \end{array}\displaystyle \right . \end{aligned}$$
(1.3)
was first studied. The nonlinear term \(f(t,z)\) satisfied a similar inequality as (1.2). The problem (1.3) was transformed into a non-resonant positone problem and an existence result for at least one positive solution was obtained by means of the Guo-Krasnosel’skiĭ fixed point theorem. In our results, we present not only the existence, but also the multiplicity of positive solutions for the discrete Neumann boundary value problem (1.1).

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

For convenience, let
$$A=\frac{1}{2}\bigl(L+2+\sqrt{L^{2}+4L}\bigr), \qquad\rho= \bigl(A^{T}-A^{-T}\bigr) \bigl(A^{2}-1\bigr). $$
Consider the linear Neumann boundary value problem
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} u(t-1)+L u(t)=h(t), \quad t\in[1,T]_{\mathbb{Z}},\\ \Delta u(0)=\Delta u(T)=0. \end{array}\displaystyle \right . \end{aligned}$$
(2.1)
The following lemma is easy to check.

Lemma 2.1

Problem (2.1) has unique solution
$$\begin{aligned} u_{0}(t)=\sum_{s=1}^{T}G(t,s)h(s), \end{aligned}$$
(2.2)
where
$$\begin{aligned} G(t,s)=\frac{1}{\rho} \left \{ \textstyle\begin{array}{@{}l@{\quad}l} (A^{s}+A^{-s+1})(A^{t-T}+A^{T-t+1}),& 1\leq s\leq t\leq T+1,\\ (A^{t}+A^{-t+1})(A^{s-T}+A^{T-s+1}),& 0\leq t\leq s\leq T. \end{array}\displaystyle \right . \end{aligned}$$
In addition, \(G(t,s)>0\) for all \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\).
Let
$$\begin{aligned} q(t)=\frac{1}{A^{T}+A^{-T+1}}\min \bigl\{ A^{t-T}+A^{T-t+1}, A^{t}+A^{-t+1} \bigr\} ,\quad t\in[0,T+1]_{\mathbb{Z}}. \end{aligned}$$
It is easy to see that \(0< q(t)<1\), \(t\in[1,T]_{\mathbb{Z}}\).

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

If \(s\leq t\), we have, for \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\),
$$\begin{aligned} \frac{A^{t-T}+A^{T-t+1}}{A^{T}+A^{-T+1}}\leq\frac{G(t,s)}{G(s,s)}=\frac {A^{t-T}+A^{T-t+1}}{A^{s-T}+A^{T-s+1}}\leq1. \end{aligned}$$
If \(t\leq s\), we have, for \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\),
$$\begin{aligned} \frac{A^{t}+A^{-t+1}}{A^{T}+A^{-T+1}}\leq\frac{G(t,s)}{G(s,s)}=\frac {A^{t}+A^{-t+1}}{A^{s}+A^{-s+1}}\leq1. \end{aligned}$$
Therefore, \(q(t)G(t,s)\leq G(t,s)\leq G(s,s)\), \((t,s)\in[0,T+1]_{\mathbb{Z}}\times[1,T]_{\mathbb{Z}}\). The proof is complete. □

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

On the one hand, for \(t\in[1,T]_{\mathbb{Z}}\),
$$\begin{aligned} q(t) =&\frac{1}{A^{T}+A^{-T+1}}\min \bigl\{ A^{t-T}+A^{T-t+1}, A^{t}+A^{-t+1} \bigr\} \\ \geq& \min \biggl\{ \frac{A^{t-T}+A^{T-t+1}}{A^{T}+A^{-T+1}}, \frac {A^{t}+A^{-t+1}}{A^{T}+A^{-T+1}} \biggr\} \cdot\max \biggl\{ \frac {A^{t-T}+A^{T-t+1}}{A^{T}+A^{-T+1}}, \frac {A^{t}+A^{-t+1}}{A^{T}+A^{-T+1}} \biggr\} \\ =&\frac{(A^{t-T}+A^{T-t+1})(A^{t}+A^{-t+1})}{(A^{T}+A^{-T+1})^{2}}\\ =&\frac{(A^{2}-1)^{2}(A^{T}-A^{-T})}{(A^{T}+A^{-T+1})^{2}}G(t,t). \end{aligned}$$
On the other hand,
$$\begin{aligned} u_{0}(t)=\sum_{s=1}^{T}G(t,s)h(s) \leq G(t,t)\sum_{s=1}^{T}h(s),\quad t\in [1,T]_{\mathbb{Z}}. \end{aligned}$$
Thus, \(q(t)\geq\mu u_{0}(t)\), \(t\in[1,T]_{\mathbb{Z}}\). The proof is complete. □
Define
$$\begin{aligned} \tilde{f}(t,z)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} f(t,z)+Lz+h(t),& (t,z)\in[1,T]_{\mathbb{Z}}\times(0,\infty),\\ f(t,0)+h(t), & (t,z)\in[1,T]_{\mathbb{Z}}\times (-\infty,0), \end{array}\displaystyle \right . \end{aligned}$$
and consider
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} v(t-1)+L v(t)=\tilde{f}(t, v(t)-u_{0}(t)),\quad t\in [1,T]_{\mathbb{Z}},\\ \Delta v(0)=\Delta v(T)=0. \end{array}\displaystyle \right . \end{aligned}$$
(2.3)
It is easy to check that the following lemma holds.

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 Guo-Krasnosel’skiĭ fixed point theorem [28].

Lemma 2.5

Let X be a Banach space and \(K\subset X\) be a cone. Assume \(\Omega_{1}\), \(\Omega_{2}\) are open sets of X with \(\overline{\Omega}_{1}\subset\Omega_{2}\), and let \(T:K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\longrightarrow K\) be a completely continuous operator such that
  1. (i)

    \(\|Tu\|\leq\|u\|\), \(u\in K\cap\partial\Omega_{1}\), and \(\|Tu\|\geq\|u\|\), \(u\in K\cap\partial\Omega_{2}\); or

     
  2. (ii)

    \(\|Tu\|\geq\|u\|\), \(u\in K\cap\partial\Omega_{1}\), and \(\|Tu\|\leq\|u\|\), \(u\in K\cap\partial\Omega_{2}\).

     
Then T has a fixed point in \(K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).

3 Main results

In this section, we show the existence and multiplicity of positive solutions for (1.1). Let \(q_{0}=\min_{t\in[1, T]_{\mathbb{Z}}}q(t)\) and define
$$\begin{aligned} \phi(r)=\max \bigl\{ \tilde{f}(t,z): t\in[1,T]_{\mathbb{Z}}, z\in [0,r] \bigr\} , \quad\mbox{for } r>0, \end{aligned}$$
and
$$\begin{aligned} \psi(r)= \min \biggl\{ \tilde{f}(t,z): t\in[1, T]_{\mathbb{Z}}, z\in \biggl[ \biggl(r-\frac{1}{\mu} \biggr)q_{0}, r \biggr] \biggr\} , \quad\mbox{for }r>\frac {1}{\mu}. \end{aligned}$$

Theorem 3.1

Assume that (C1) and (C2) hold. Suppose that there exist \(r, R>0\) such that \(\frac{1}{\mu}< r< R\) and
(C3): 

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

Then problem (1.1) has at least one positive solution. In addition, if
(C4): 

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

(C5): 

\(L<\frac{1}{\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\),

then problem (1.1) has at least two positive solutions.

Proof

Assume (C3) holds. We first prove by Lemma 2.5 that problem (2.3) has at least one positive solution v. In the Banach space \(\mathbb{E}=\{v: [0,T+1]_{\mathbb{Z}}\to\mathbb {R}\}\) endowed with the norm \(\|v\|=\max_{[0,T+1]_{\mathbb{Z}}}|v(t)|\), define
$$\begin{aligned} Fv(t)=\sum_{s=1}^{T} G(t,s) \tilde{f}\bigl(s,v(s)-u_{0}(s)\bigr). \end{aligned}$$
(3.1)
Then by (C1), \(F: \mathbb{E}\to\mathbb{E}\) is completely continuous. Define the cone
$$\begin{aligned} P=\bigl\{ v\in\mathbb{E}: v(t)\geq q(t)\|v\|, t \in[1,T]_{\mathbb{Z}} \bigr\} . \end{aligned}$$
(3.2)
By Lemma 2.2, \(F(P)\subset P\). Thus, a fixed point of F in P is a positive solution of problem (2.3).
Let
$$\begin{aligned} \Omega_{1}=\bigl\{ v\in\mathbb{E}: \|v\|< r\bigr\} ,\qquad \Omega_{2}=\bigl\{ v\in\mathbb{E}: \|v\|< R\bigr\} . \end{aligned}$$
(3.3)
For \(v\in P\cap\partial\Omega_{1}\), we have \(v(s)-u_{0}(s)\geq q(s)\|v\|-u_{0}(s)\geq(\mu r-1)u_{0}(s)> 0\), \(s\in[1,T]_{\mathbb{Z}}\). It follows that \(\tilde{f}(s, v(s)-u_{0}(s))\leq\phi(r)\), \(s\in[1,T]_{\mathbb{Z}}\). Notice that \(G(0, s)=G(1,s)\) and \(G(T,s)=G(T+1,s)\), \(s\in[1,T]_{\mathbb {Z}}\). Then, by (C3), we have
$$\begin{aligned}[b] \|Fv\|&=\max_{t\in[0,T+1]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \tilde{f}\bigl(s,v(s)-u_{0}(s)\bigr) \\ &\leq \max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \phi (r) \\ &\leq r. \end{aligned} $$
That is, \(\|Fv\|\leq\|v\|\) for \(v\in P\cap\partial\Omega_{1}\).
For \(v\in P\cap\partial\Omega_{2}\), we have by Lemma 2.3, for \(s\in[1, T]\),
$$\begin{aligned} R\geq v(s)-u_{0}(s)\geq q(s)\|v\|-\frac{q(s)}{\mu} \geq \biggl(R-\frac {1}{\mu} \biggr)q_{0}. \end{aligned}$$
(3.4)
This implies \(\tilde{f}(s, v(s)-u_{0}(s))\geq\psi(R)\) for \(s\in[1, T]\), \(v\in P\cap\partial\Omega_{2}\). Then, by (C3), we have, for \(v\in P\cap \partial\Omega_{2}\),
$$\begin{aligned} \|Fv\| =&\max_{t\in[0,T+1]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \tilde{f}\bigl(s,v(s)-u_{0}(s)\bigr) \\ \geq& \max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \psi (R) \\ \geq& R. \end{aligned}$$
That is, \(\|Fv\|\geq\|v\|\) for \(v\in P\cap\partial\Omega_{2}\).

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, \(u_{1}(t)=v_{1}(t)-u_{0}(t)\geq q(t)\|v_{1}\|-u_{0}(t)\geq(\mu r-1)u_{0}(t)>0\), \(t\in[1,T]_{\mathbb{Z}}\). Therefore, by Lemma 2.4, \(u_{1}\) is a positive solution of problem (1.1).

Now, let (C4) and (C5) also hold. We prove that problem (1.1) has a distinct second positive solution \(u_{2}(t)\). By (C4), there exists \(D>0\) such that, for \(z>D\),
$$\tilde{f}(t,z)=f(t,z)+Lz+h(t)\leq Lz+h(t),\quad t\in[1,T]_{\mathbb{Z}}. $$
By (C5), we can choose \(R_{\infty}>\max \{R, \frac{D}{q_{0}}+\frac{1}{\mu} \}\) such that
$$\begin{aligned} L+\frac{\bar{h}}{R_{\infty}}\leq\frac{1}{\max_{t\in [1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}, \end{aligned}$$
(3.5)
where \(\bar{h}=\max_{t\in[1,T]_{\mathbb{Z}}}h(t)\). Let \(\Omega_{3}=\{ v\in\mathbb{E}: \|v\|< R_{\infty}\}\). For \(v\in P\cap\partial\Omega_{3}\), similar to (3.4), we have
$$\begin{aligned} v(t)-u_{0}(t)\geq \biggl(R_{\infty}- \frac{1}{\mu} \biggr)q_{0}>D,\quad t\in [1,T]_{\mathbb{Z}}, \end{aligned}$$
(3.6)
which implies that
$$\tilde{f}\bigl(t, v(t)-u_{0}(t)\bigr)\leq L\bigl(v(t)-u_{0}(t) \bigr)+h(t)\leq LR_{\infty}+\bar {h},\quad t\in[1,T]_{\mathbb{Z}}. $$
Thus, by (3.5), we have, for \(v\in P\cap\partial\Omega_{3}\),
$$\begin{aligned}[b] \|Fv\|&= \max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \tilde {f}\bigl(s,v(s)-u_{0}(s)\bigr) \\ &\leq (LR_{\infty}+\bar{h})\max_{t\in[1,T]_{\mathbb{Z}}}\sum _{s=1}^{T}G(t,s) \\ &\leq R_{\infty}. \end{aligned} $$
That is, \(\|Fv\|\leq\|v\|\) for \(v\in P\cap\partial\Omega_{3}\).

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 \(v_{2}\in P\).

Take \(\epsilon>0\) sufficiently small such that \(L+\epsilon\leq\frac {1}{\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\). Note that \(\lim_{z\to\infty}\frac{f(t,z)}{z}=0\) implies
$$\lim_{z\to\infty}\frac{\tilde{f}(t,z)}{z}=L $$
uniformly for \(t\in[1,T]_{\mathbb{Z}}\). Then there exists \(D>0\) such that, for \(z>D\), \(\tilde{f}(t,z)\leq(L+\epsilon)z\) holds for all \(t\in [1,T]_{\mathbb{Z}}\). Choose \(R_{\infty}=\max \{R, \frac{D}{q_{0}}+\frac{1}{\mu} \}+1\) and \(\Omega_{3}=\{v\in\mathbb{E}: \|v\|< R_{\infty}\}\). For \(v\in P\cap\partial\Omega_{3}\), we see that (3.6) holds and hence \(\tilde{f}(t, v(t)-u_{0}(t))\leq(L+\epsilon)(v(t)-u_{0}(t))\leq (L+\epsilon)R_{\infty}\) for \(t\in[1,T]_{\mathbb{Z}}\). Thus, for \(v\in P\cap\partial\Omega_{3}\),
$$\begin{aligned} \|Fv\| =& \max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \tilde {f}\bigl(s,v(s)-u_{0}(s)\bigr) \\ \leq& (L+\epsilon)R_{\infty}\max_{t\in[1,T]_{\mathbb{Z}}}\sum _{s=1}^{T}G(t,s) \\ \leq& R_{\infty}. \end{aligned}$$
That is, \(\|Fv\|\leq\|v\|\) for \(v\in P\cap\partial\Omega_{3}\). Therefore, by Lemma 2.5, F has a fixed point \(v_{2}\in P\) such that \(R\leq\|v_{2}\|\leq R_{\infty}\), and hence \(u_{2}(t)=v_{2}(t)-u_{0}(t)\) is a second positive solution of problem (1.1). The proof is complete. □

Theorem 3.3

Assume that (C1) and (C2) hold. Suppose that there exist \(r, R>0\) such that \(\frac{1}{\mu}< r< R\) and
(C3)

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

Then problem (1.1) has at least one positive solution. In addition, if
(C4)

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

(C5)

\(L> \frac{2}{q_{0}\max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}\),

then 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. 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).

Now, let (C4) and (C5) hold. By (C4), there exists \(D>0\) such that, for \(z>D\),
$$\tilde{f}(t,z)=f(t,z)+Lz+h(t)\geq Lz+h(t),\quad t\in[1,T]_{\mathbb{Z}}. $$
By (C5), one can choose \(R_{\infty}>R+\max \{\frac{2D}{q_{0}}, \frac{2}{\mu} \}\) such that
$$\begin{aligned} \frac{1}{2}q_{0}L+\frac{h_{0}}{R_{\infty}}\geq \frac{1}{\max_{t\in [1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s)}, \end{aligned}$$
(3.7)
where \(h_{0}=\min_{t\in[1,T]_{\mathbb{Z}}}h(t)\). Let \(\Omega_{3}=\{v\in\mathbb{E}: \|v\|< R_{\infty}\}\). Then, for \(v\in P\cap\partial\Omega_{3}\), we have
$$\begin{aligned} v(t)-u_{0}(t)\geq \biggl(R_{\infty}-\frac{1}{\mu} \biggr)q_{0}\geq\frac{1}{2} q_{0}R_{\infty}>D,\quad t \in[1,T]_{\mathbb{Z}}, \end{aligned}$$
which implies that
$$\begin{aligned} \tilde{f}\bigl(t, v(t)-u_{0}(t)\bigr)\geq L\bigl(v(t)-u_{0}(t) \bigr)+h(t)\geq\frac {1}{2}Lq_{0}R_{\infty}+h_{0},\quad t\in[1,T]_{\mathbb{Z}}. \end{aligned}$$
Thus, by (3.7), we have, for \(v\in P\cap\partial\Omega_{3}\),
$$\begin{aligned} \|Fv\| =& \max_{t\in[1,T]_{\mathbb{Z}}}\sum_{s=1}^{T}G(t,s) \tilde {f}\bigl(s,v(s)-u_{0}(s)\bigr) \\ \geq& \biggl(\frac{1}{2}Lq_{0}R_{\infty}+h_{0} \biggr)\max_{t\in[1,T]_{\mathbb {Z}}}\sum_{s=1}^{T}G(t,s) \\ \geq& R_{\infty}. \end{aligned}$$
That is, \(\|Fv\|\geq R_{\infty}=\|v\|\) for \(v\in P\cap\partial\Omega_{3}\). Therefore, F has another fixed point \(v_{2}\in P\) such that \(r\leq \|v_{1}\|\leq R\leq\|v_{2}\|\leq R_{\infty}\). By Lemma 2.4, problem (1.1) has two positive solutions \(u_{1}(t)=v_{1}(t)-u_{0}(t)\) and \(u_{2}(t)=v_{2}(t)-u_{0}(t)\). The proof is complete. □

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

(1)
School of Mathematics and Information Science, Guangzhou University
(2)
Department of Mathematics, Baylor University

References

  1. Agarwal, RP, Perera, K, O’Regan, D: Multiple positive solutions of singular discrete p-Laplacian problems via variational methods. Adv. Differ. Equ. 2, 93-99 (2005) MathSciNetGoogle Scholar
  2. Anderson, DR: Discrete third-order three-point right-focal boundary value problems. Comput. Math. Appl. 45, 861-871 (2003) View ArticleMathSciNetMATHGoogle Scholar
  3. Anderson, D, Avery, RI: Multiple positive solutions to a third-order discrete focal boundary value problem. Comput. Math. Appl. 42, 333-340 (2001) View ArticleMathSciNetMATHGoogle Scholar
  4. Aykut, N: Existence of positive solutions for boundary value problems of second order functional difference equations. Comput. Math. Appl. 48, 517-527 (2004) View ArticleMathSciNetMATHGoogle Scholar
  5. Bai, D: A global result for discrete ϕ-Laplacian eigenvalue problems. Adv. Differ. Equ. 2013, 264 (2013) View ArticleGoogle Scholar
  6. Bai, D, Xu, Y: Nontrivial solutions of boundary value problems of second order difference equations. J. Math. Anal. Appl. 326(1), 297-302 (2007) View ArticleMathSciNetMATHGoogle Scholar
  7. Bai, D, Xu, X: Existence and multiplicity of difference ϕ-Laplacian boundary value problems. Adv. Differ. Equ. 2013, 267 (2013) View ArticleMathSciNetGoogle Scholar
  8. Ji, D, Ge, W: Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with p-Laplacian. Nonlinear Anal. 68, 2638-2646 (2008) View ArticleMathSciNetMATHGoogle Scholar
  9. Jiang, L, Zhou, Z: Existence of nontrivial solutions for discrete nonlinear two point boundary value problems. Appl. Math. Comput. 180, 318-329 (2006) View ArticleMathSciNetMATHGoogle Scholar
  10. Karaca, IY: Discrete third-order three-point boundary value problem. J. Comput. Appl. Math. 205, 458-468 (2007) View ArticleMathSciNetMATHGoogle Scholar
  11. Liu, Y: Existence results for positive solutions of non-homogeneous BVPs for second order difference equations with one-dimensional p-Laplacian. J. Korean Math. Soc. 47, 135-163 (2010) View ArticleMathSciNetMATHGoogle Scholar
  12. Yang, Y, Meng, F: Eigenvalue problem for finite difference equations with p-Laplacian. J. Appl. Math. Comput. 40, 319-340 (2012) View ArticleMathSciNetMATHGoogle Scholar
  13. Yu, J, Guo, Z: On generalized discrete boundary value problems of Emden-Fowler equation. Sci. China Ser. A 36, 721-732 (2006) MathSciNetGoogle Scholar
  14. Yu, J, Guo, Z: On boundary value problems for a discrete generalized Emden-Fowler equation. J. Differ. Equ. 231, 18-31 (2006) View ArticleMathSciNetMATHGoogle Scholar
  15. Bai, D, Xu, Y: Positive solutions for semipositone BVPs of second-order difference equations. Indian J. Pure Appl. Math. 39, 59-68 (2008) MathSciNetMATHGoogle Scholar
  16. Anuradha, A, Hai, DD, Shivaji, R: Existence results for superlinear semipositone BVP’s. Proc. Am. Math. Soc. 124(3), 757-763 (1996) View ArticleMathSciNetMATHGoogle Scholar
  17. Anuradha, V, Maya, C, Shivaji, R: Positive solutions for a class of nonlinear boundary value problems with Neumann-Robin boundary conditions. J. Math. Anal. Appl. 236, 94-124 (1999) View ArticleMathSciNetMATHGoogle Scholar
  18. Bai, D, Xu, Y: Existence of positive solutions for boundary value problems of second-order delay differential equations. Appl. Math. Lett. 18, 621-630 (2005) View ArticleMathSciNetMATHGoogle Scholar
  19. Bai, D, Xu, Y: Positive solutions of second-order two-delay differential systems with twin-parameter. Nonlinear Anal. 63, 601-617 (2005) View ArticleMathSciNetMATHGoogle Scholar
  20. Bai, D, Yu, J: Semipositone problems of second order ordinary differential equations. Sciencepaper Online (10 October 2009); http://www.paper.edu.cn/
  21. Castro, A, Gadam, S, Shivaji, R: Evolution of positive solution curves in semipositone problems with concave nonlinearities. J. Math. Anal. Appl. 245, 282-293 (2000) View ArticleMathSciNetMATHGoogle Scholar
  22. Hai, DD, Schmitt, K, Shivaji, R: Positive solutions of quasilinear boundary value problems. J. Math. Anal. Appl. 217, 672-686 (1998) View ArticleMathSciNetMATHGoogle Scholar
  23. Hai, DD, Shivaji, R: An existence result on positive solutions for a class of p-Laplacian systems. Nonlinear Anal. 56, 1007-1010 (2004) View ArticleMathSciNetMATHGoogle Scholar
  24. Ma, R: Multiple positive solutions for a semipositone fourth-order boundary value problem. Hiroshima Math. J. 33, 217-227 (2003) MathSciNetMATHGoogle Scholar
  25. Maya, C, Shivaji, R: Multiple positive solutions for a class of semilinear elliptic boundary value problems. Nonlinear Anal. 38, 497-504 (1999) View ArticleMathSciNetMATHGoogle Scholar
  26. Sun, J, Wei, J: Existence of positive solutions of positive solution for semi- positone second-order three-point boundary-value problems. Electron. J. Differ. Equ. 2008, 41 (2008) MathSciNetGoogle Scholar
  27. Henderson, J, Kosmatov, N: Positive solutions of the semipositone Neumann boundary value problem. Math. Model. Anal. 20, 578-584 (2015) View ArticleMathSciNetGoogle Scholar
  28. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988) MATHGoogle Scholar

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