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

Our concern is the existence of positive solutions of the discrete Neumann boundary value problem

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 \((t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\).

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:

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 [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 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:

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 [17–26] 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

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).

For convenience, let

Consider the linear Neumann boundary value problem

The following lemma is easy to check.

Lemma 2.1

Problem (2.1) has unique solution

where

In addition, \(G(t,s)>0\) for all \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\).

Let

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 \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\),

If \(t\leq s\), we have, for \((t,s)\in[0,T+1]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\),

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 \(t\in[1,T]_{\mathbb{Z}}\),

On the other hand,

Thus, \(q(t)\geq\mu u_{0}(t)\), \(t\in[1,T]_{\mathbb{Z}}\). The proof is complete. □

Define

and consider

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 \(K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).

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

and

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

(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

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

Then by (C1), \(F: \mathbb{E}\to\mathbb{E}\) is completely continuous. Define the cone

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

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

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 \(s\in[1, T]\),

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 \(v\in P\cap \partial\Omega_{2}\),

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 \(z>D\),

By (C5), we can choose \(R_{\infty}>\max \{R, \frac{D}{q_{0}}+\frac{1}{\mu} \}\) such that

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

which implies that

Thus, by (3.5), we have, for \(v\in P\cap\partial\Omega_{3}\),

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

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 \(v\in P\cap\partial\Omega_{3}\),

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

(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).

Now, let (C4)^∗ and (C5)^∗ hold. By (C4)^∗, there exists \(D>0\) such that, for \(z>D\),

By (C5)^∗, one can choose \(R_{\infty}>R+\max \{\frac{2D}{q_{0}}, \frac{2}{\mu} \}\) such that

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

which implies that

Thus, by (3.7), we have, for \(v\in P\cap\partial\Omega_{3}\),

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.

Supported partially by PCSIRT of China (No. IRT1226) and NSF of China (No. 11171078).

Affiliations

1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P.R. China
   • Dingyong Bai
   •  & Yunxia Zeng
2. Department of Mathematics, Baylor University, Waco, TX, 76798-7328, USA
   • Johnny Henderson

Corresponding author

Correspondence to Dingyong Bai.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results and they read and approved the final manuscript.

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