Global structure of positive solutions for second-order discrete Neumann problems involving a superlinear nonlinearity with zeros
- Yanqiong Lu^{1}Email author
Received: 27 November 2015
Accepted: 24 February 2016
Published: 8 April 2016
Abstract
This paper studies the global structure of positive solutions of a class discrete Neumann problem which includes the superlinear nonlinearity with zeros. The main results are based on the method of lower and upper solutions, a priori estimates and Brouwer degree theory.
Keywords
positive solutions global structure degree difference equationMSC
34B15 39A121 Introduction
The difference equations play an important role in many fields such as science, biology, engineering, and technology where discrete phenomena abound, in addition, from the advent and the rise of computers, where differential equations are solved by employing their approximative difference equations formulations; e.g., see [1–12] and the references therein.
In 2003, Cabada and Otero-Espinar [3] studied the existence of solutions of (1.1) by the method of upper and lower solutions whenever \(p(\cdot )\equiv1\), \(q(\cdot)=0\). Anderson et al. [2] obtained the existence of solutions of (1.1) by a fixed point theorem whenever \(p(\cdot)\equiv1\) and Lu and Gao [10] obtained the existence of positive solutions of (1.1) by the fixed point theorem in cones. Jun Ji, Bo Yang, Candito, G. D’Aguì and Gao also studied the discrete Neumann problem by different methods, see [1, 4, 6–9] and references therein.
- (C1)
\(f\in C (I\times[0,\infty),[0,\infty) )\) and there exists a function \(m:I\to(0,\infty)\), satisfying \(m(k_{1})+m(k_{2})\leq m (\frac {k_{1}+k_{2}}{2} )\) with \(k_{1},k_{2}\in I\), such that \(f(k,0)=f(k,m(k))=0\) and \(f(k,y)>0\) if \(0< y< m(k)\).
- (C2)There exists a function \(h:I\to(0,\infty)\) such that$$ \lim_{y\to0^{+}}\frac{f(k,y)}{y}=h(k) \quad \text{uniformly in } k\in I. $$(1.3)
- (C3)There exists a subset \(I_{1}\subset I\) such that$$\lim_{y\to\infty}\frac{f(k,y)}{y}=\infty \quad \text{uniformly in } k\in I_{1}. $$
- (C4)The function \(f_{y}:=\frac{\partial f}{\partial y}\) exists and is continuous in the set \(\{(k,y) : k\in I, y\in[0, m(k)]\}\); further,$$ f_{y}(k,y)< y^{-1}f(k,y), \quad (k,y)\in\bigl\{ (k,y) : k\in I, y\in \bigl(0, m(k)\bigr)\bigr\} . $$(1.4)
Through careful analysis we have found that the nonlinearity has a zero at a variable positive value and has linear growth at zero and locally superlinear growth at infinity. The effect of the variable zero and the condition of superlinear growth in a small interval are the main differences when comparing to the results in [10, 11]. For the results concerning the existence of positive solutions for nonlinear differential equation boundary value problems involving nonlinearity with zero points, see, e.g. [13–15] and references therein.
Theorem 1.1
Let (C1)-(C3) hold. Then for every \(0<\lambda<\lambda_{1,ah}\), the problem (1.2) has at least one positive solution.
Theorem 1.2
Let (C1)-(C4) hold. Then for every \(\lambda >\lambda_{1,ah}\), the problem (1.2) has at least two ordered positive solutions.
Theorem 1.3
- (i)
one has \(\Vert y_{\lambda} \Vert \to\infty\) as \(\lambda\to0^{+}\);
- (ii)if \(f(k,y)>0\) for \(y\neq m(k)\), \(y\neq0\), then$$y_{\lambda}\to m \quad \textit{pointwise in } I \quad \textit{and}\quad \Vert y_{\lambda} \Vert \to \Vert m\Vert ,\quad \textit{as } \lambda\to\infty. $$
Remark 1.1
Remark 1.2
Notice that (C2) means that f has asymptotically linear growth at \(u=0\). From Figure 1, we see that (1.2) has a positive solution u satisfying \(\min_{k\in I}u(k)>\Vert m\Vert \) for any \(\lambda\in(0,\lambda_{ah})\) and (1.2) has two positive solutions \(u_{1}\) and \(u_{2}\) with \(\Vert u_{1}\Vert <\Vert m\Vert \) and \(\min_{k\in I}u_{2}(k)>\Vert m\Vert \) for any \(\lambda>\lambda_{ah}\).
If \(f_{0}=\infty\), then for any \(\lambda>0\) (1.2) has two positive solutions \(u_{1}\) and \(u_{2}\) with \(\Vert u_{1}\Vert <\Vert m\Vert \) and \(\min_{k\in I}u_{2}(k)>\Vert m\Vert \). Especially, under the conditions (C1)-(C3), we can show (1.2) has at least one positive solution u with \(\Vert u\Vert <\Vert m\Vert \) for any \(\lambda>0\) by using the fixed point theorem and Proposition 3.3. Compared with the main results of [5], the interval of λ is sharp. However, Theorem 1.1-Theorem 1.2 and [5], Theorem 3.3, Theorem 3.7, Theorem 3.9, with \(p=2\) complement each other but cannot contain each other.
The rest of this paper is organized as follows. In Section 2, we state some notations and preliminary results. Section 3 contains the proof of existence of one solution for λ small enough and the proof of the existence of two solutions for λ large enough. Finally, we study the asymptotic behavior of the solutions compact operator equation and prove Theorem 1.3 in Section 4.
2 Preliminaries
Let \(\hat{I}:=\{0, 1,\ldots, N, N+1\}\), and define \(E=\{y \mid y:\hat{I}\to\mathbb{R}\} \) be the space of all maps from Î into \(\mathbb{R}\). Then it is a Banach space with the norm \(\Vert y\Vert = \max_{k\in\hat{I}}\vert y(k)\vert \).
Let \(P:=\{y\in E \mid y(k)\geq0, k\in\hat{I}\}\). Then P is a cone which is normal and has a nonempty interior and \(E=\overline{P-P}\).
- (i)
\(\phi(k)=1+ \sum_{s=1}^{k-1} (\sum_{j=s}^{k-1}\frac{1}{p(j)} )q(s)\phi(s)>0\), and ϕ is increasing on Î;
- (ii)
\(\psi(k)=1+\sum_{s=k+1}^{N} (\sum_{j=k}^{s-1}\frac {1}{p(j)} )q(s)\psi(s)>0\), and ψ is decreasing on Î.
Lemma 2.1
([10], Lemma 2.4)
Lemma 2.2
- (i)
\(G(k,s)\geq0\), \(\forall (k,s)\in\hat{I}\times\hat{I}\); \(G(k,s)>0\), \(\forall (k,s)\in I\times I\).
- (ii)
\(G(k,s)\leq G(s,s)\), \(\forall (k,s)\in\hat{I}\times\hat{I}\).
- (iii)
\(G(k,s)\geq\min\{\frac{\phi(k)}{\phi(N+1)}, \frac{\psi(k)}{\psi (0)}\}G(s,s)\), \(\forall (k,s)\in\hat{I}\times\hat{I}\).
First, we introduce the following strict monotonicity property with respect to the weight function b for the problem (2.4) (see Gao [6], Lemma 2.5 and Ji and Yang [7], Theorem 3.1).
Lemma 2.3
Let b and b̃ be two bounded weights with \(b\leq\tilde{b}\), and let \(j \in\{-N,\ldots,-1,1, \ldots,N\}\). Then the eigenvalue \(\lambda_{j,b}>\lambda_{j,\tilde{b}}\).
3 The existence of one solution and two solutions for (1.2)
In this section, we will show the existence of a solution for \(\lambda \in(0,\lambda_{1,ah})\) and the existence of two solutions for \(\lambda \in(\lambda_{1,ah},\infty)\).
Lemma 3.1
Proof
Lemma 3.2
Proof
Consider the norm \(\Vert y\Vert _{*}=\inf\{\eta : \eta(\varphi_{1,ah}+M_{1})\geq y\}=\Vert \frac{y}{\varphi_{1,ah}+M_{1}}\Vert \), which is equivalent to \(\Vert \cdot \Vert \).
From (C1)-(C2), there exists a \(\rho=\rho(\varepsilon)>0\) such that \(0< y<\rho\) implies that \(f(k,y)<(1+\varepsilon)h(k)y(k)\) for all \(k\in I\). Let \(r>0\) be such that \(r(\Vert \varphi_{1,ah}\Vert +M_{1})<\rho\), so that \(\Vert y\Vert _{*}=r\) implies \(\Vert y\Vert <\rho\).
Lemma 3.1 and Lemma 3.2 may imply the existence of a positive solution of (1.2).
Proof of Theorem 1.1
The existence of a positive solution is a consequence of the fixed point theorem (see Deimling [16], Theorem 20.1) by virtue of Lemma 3.1 and Lemma 3.2. □
Now we shall show (1.2) has at least two positive solutions for \(\lambda >\lambda_{1,ah}\). First of all, we start with the existence of a first solution y which satisfies \(y(k)\leq m(k)\).
Proposition 3.3
Suppose (C1)-(C3) hold. Then for every \(\lambda>\lambda_{1,ah}\), the problem (1.2) has a positive solution y which satisfies \(y(k)\leq m(k)\).
Proof
By (C1)-(C3), we may use the method of lower and upper solutions (see [3]).
From (C1), m is positive and \(f(k,m(k))=0\), we see that \(m(k)\) is always an upper solution of (1.2). Clearly, it is a strict upper solution, because \(m(k)>0\) and the concave of m implies that it cannot satisfy \(\Delta y(0)=\Delta y(N)=0\). We recall that a strict lower solution of (1.2) is a lower solution which is not a solution of (1.2).
Remark 3.1
Obviously, the choice of τ, the values of ϱ and ϵ in the preceding proof depends on λ. However, once chosen τ for a given value of λ, the same choice works for any larger value of λ. Thus, for any given \(\hat{\lambda}>\lambda_{1,ah}\), it is possible to find a unique function \(\epsilon\varphi_{1,ah}\) which is a lower solution for any \(\lambda>\hat{\lambda}\).
In the following we will show the existence of a second solution for all \(\lambda>\lambda_{1,ah}\). From (C1)-(C3), we fix λ and denote by \(y_{1}=y_{1}(\lambda)\) the solution of (1.2) found above.
It is easy to see that \(w\geq0\) is a nontrivial solution of (3.4), then \(y_{1}+w\) is a second positive solution of (1.2), which satisfies \(y_{1}+w\geq y_{1}\).
Our purpose is to find a nontrivial fixed point of \(A_{1,1,0}\). We need the following lemmas.
Lemma 3.4
Suppose (C1), (C2), and (C4) hold. Then if w is a solution of (3.4) (or of (3.6)), then \(w\geq0\).
Proof
Lemma 3.5
Suppose (C1)-(C4) hold. Then there exists a constant \(M_{3}>0\), which does not depend on ρ, such that \(A_{1,1,\rho}w=w\) implies \(\Vert w\Vert \leq M_{3}\) for any \(\rho\geq0\).
Proof
Lemma 3.6
Proof
Obviously, \(y_{1}\) depends on λ, and \(\Vert y_{1}\Vert \leq \Vert m\Vert \), thus \(\tilde{\rho}(\tilde{R})\) can be chosen uniformly with respect to \(\lambda>\lambda_{1,ah}\). □
Lemma 3.7
Proof
Lemma 3.8
Proof
However, since \(\lambda>0\), it cannot coincide with any \(\lambda_{j,\hat {b}}\) with \(j<0\), and because \(\lambda=\lambda_{1, \tilde{b}}<\lambda_{1,\hat{b}}\), it cannot coincide with any \(\lambda_{j,\hat{b}}\) with \(j>0\) neither. This concludes that \(w\equiv0\). □
Proposition 3.9
Suppose (C1)-(C4) hold. Then the problem (3.4) has a nontrivial positive solution w, that is, (1.2) has a second positive solution.
Proof
4 Asymptotic behavior of solutions for (1.2)
Proposition 4.1
Suppose (C1) and (C2) hold. If \(\{ y_{\lambda}\}\) is a family of positive solutions of (1.2), then \(\Vert y_{\lambda} \Vert \to\infty\) as \(\lambda\to0^{+}\).
Proof
Next, we study the asymptotic behavior of the positive solutions of (1.2) as \(\lambda\to\infty\), we need to prove a uniform estimate for the positive solutions of (1.2), for λ large enough.
Lemma 4.2
Proof
Finally, we shall give the proof of Theorem 1.3, with the following result, which reveals the asymptotic behavior of the solutions of (1.2) as \(\lambda\to\infty\).
Proposition 4.3
Proof
Consequently, \(\Vert y_{\lambda} \Vert \to \Vert m\Vert \), as \(\lambda\to \infty\). □
Declarations
Acknowledgements
The author is very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11401479), Gansu provincial National Science Foundation of China (No. 1208RJZA258), SRFDP (No. 20126203110004).
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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