Existence of Positive Solutions for Second-Order Neumann Difference System

Abstract

This paper studies a class of Neumann difference system which is not cooperative but its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative solutions of the system in terms of the principal eigenvalue of the corresponding linearized system.

1. Introduction

There has been a long history of interest in difference equations and difference systems; for example, see [1–8] and the references therein. In particular, a wide variety of nonlinear difference systems have been studied because they model numerous real-life problems in biology, physics, population dynamics, economics, and so on.

One of the difference equations that has attracted some attention is where for all , . In 2003, Cabada and Otero-Espinar [4] studied the existence of solution of the problem (1) and obtained optimal existence results by lower and upper solutions methods.

Besides, we note that some systems of discrete boundary value problems are investigated by several authors in recent years; see [5, 7, 8] and the references therein. For example, Sun and Li [5] studied the following boundary value problem of discrete system: Under some assumptions on , they obtained some sufficient conditions for the existence of one or two positive solutions to the system by using nonlinear alternative of Leray-Schauder and the fixed point theorem in cones.

Henderson et al. [7] considered the following system of three-point discrete boundary value problem: where , , , , , , are nonnegative functions and , . They deduced the existence of the eigenvalues and yielding at least one positive solution to the system (3) under some assumptions on , , , and with weakly coupling behaviors. Their main tool is the fixed point theorem in cones.

However, very little work has been done for the existence of positive solutions of second-order Neumann difference systems. Inspired by the above works, we study the existence of positive solutions of the following second order Neumann difference system: where , the coefficients , , , and are positive functions on , and , are positive constants. Through careful analysis, we have found that (4) is not cooperative but its linear part is and this makes it possible to establish existence and nonexistence results for nonnegative solutions of (4) in terms of the principal eigenvalue of the corresponding linearized system. These conditions are different from those given in [5, 7].

Although system (4) is very simple, it contains an interesting mathematical feature. It is well known that cooperative systems can be analysed by using lower and upper solutions method and, in general, possesses many of the properties of scalar equations. Although system (4) is not cooperative, its linear part is and the cooperative nature of the linear system plays a key role in formulating and helping to prove our existence results for (4). For recent works in the literature on continuous Neumann boundary value problem as well as system, we refer to [9–15] and the references therein.

Sufficiently motivated, the paper is organized as follows: in Section 2, we discuss the properties of cooperative system which we will require. In Section 3, we establish necessary and sufficient conditions for the existence of a positive solution of (4) in terms of the principal eigenvalue of the associated linear system. Finally, an example is presented to illustrate the main result.

2. Cooperative Systems

The difference system is said to be cooperative if is a nondecreasing function for any fixed and is a nondecreasing function for any fixed . Because of maximum principle arguments cooperative systems possesses many of the properties of single difference equation, in this section, we will discuss the results that we require for linear cooperative systems. We are primarily interested in the existence of principal eigenvalues for such systems and on the monotone behavior of such eigenvalues with respect to coefficients of the system. Our proofs depend on the connection between the existence of positive upper solutions for the system and the maximum principle. The continuous case has been obtained by Sweers [13], López-Gómez and Molina-Meyer [14], and Brown and Zhang [15], but as far as we known, there is no result for the discrete case.

Throughout the rest of this section, we will consider the cooperative system where , , , and are functions on with , and , for .

Let and be a Banach space with the norm . In addition, we introduce the notation on that means that for .

In the following, we give some important Lemmas to show the main result.

Lemma 1. Assume that , and are constants. Then the boundary value problem has the solution that satisfies either or on .

Proof. Let be the solution of initial value problems respectively. It is easy to compute and show that(i) and is increasing on ;(ii) and is decreasing on .
It is not difficult to verify that the problem (8) has the solution where Clearly, it follows from the properties of , together with the above that either or on .

Theorem 2. Suppose that there exist functions , such that , and where equality does not hold in all of the equations in (12). Then (7) satisfies the maximum principle; that is, if such that then either (i) on or (ii) and on .

Proof. Suppose on the contrary that there exist , not both identically zero satisfying inequalities (13) but not satisfying (ii) in the conclusion of the theorem. For , define and . Then there exists , such that , for and either or has a zero in . Without loss of generality, we may assume that there exists such that ; then, Moreover, and so it follows from Lemma 1 that or on . Obviously, , so . Hence, by (14), it also follows that . From are not both identically zero for , this goether with and , implies that and equal the same negative multiples of and , respectively. This is impossible as , satisfy (12) and satisfy (13) and so the proof is complete.

Corollary 3. Suppose that and for all . Then the cooperative system (7) satisfies the maximum principle.

Proof. The result follows from Theorem 2 by choosing where is any positive number.

Theorem 4. The cooperative system (7) has a principal eigenvalue; that is, there exists and such that and

Proof. Let and . Define by and define the matrix by It is well know that if is sufficiently large, then is an invertible operator such that is compact. If, moreover, is chosen sufficiently large to ensure that and for all , it follows from Corollary 3 that is strongly positive.
Since is compact and strongly positive, has a positive principal eigenvalue . Thus there exists with such that . Hence and has a principal eigenvalue.

For convenience, we will denote the principal eigenvalue of by .

Corollary 5. Suppose that and are cooperative matrices (i.e., matrices with positive entries in the off-diagonal elements) such that (i.e., the th element of the th element of for all but ). Then .

Proof. There exists such that and . Then but and so by Theorem 2, the cooperative system with satisfies the maximum principle. Hence, if denotes the principal eigenvalue for the system , it follows that . Clearly, has principal eigenvalue and so .

3. Existence and Nonexistence of Positive Solutions

System (4) can be rewritten as where and such that Although is a cooperative matrix, (20) is not a cooperative system. The following general theorem describes how the method of lower and upper solutions must be modified to deal with general, possibly non-cooperative system such as (6). The proofs of the method of lower and upper solutions are based upon the connectivity properties of the solution sets of parameterized families of compact vector fields; see [16, Theorem 1.0].

Lemma 6. Let , and suppose that there exist functions such that and, for all , Then the problem has a solution that satisfies

Proof. Let us consider the auxiliary problem where and is defined by Clearly, to show that (23) has a solution satisfies , it is enough to prove that (25) has a solution that satisfies .
Define the operator By the standard of compact operator argument, it is easy to show that is a completely continuous operator. Obviously, is a bounded function; this combined with Lemma 1 and Brouwer fixed point theorem and we can get that has a fixed point ; that is, is a solution of the problem (25).
Next, we will show that .
We only have to deal with and the case can be proved by a similar argument. Suppose on the contrary that there exists a point such that . Let ; then, there exists such that and , . Subsequently, On the other hand, we have that which is a contradiction. So . The proof is complete.

Theorem 7. Suppose that there exist functions , , , such that and and, for all , Then there exists a solution of the system such that and for all .

Proof. From Lemma 6, it is easy to show that for any fixed , there exists the solution set Moreover, by the connectivity properties of the solution sets of parameterized families of compact vector fields [16, Theorem 1.0], the set contains a connected subset which joins and with .
For every , we have that and for any , we get that Therefore, by the connectivity of , there exist and such that and satisfies Thus is solution of the system (31).
We can proceed in the same way, proving that there exists , such that is a solution of the system (31), where The proof is complete.

We can give necessary and sufficient conditions for the existence of a positive solution of the system (4).

Theorem 8. There exists a positive solution of system (4) if and only if .

Proof. Suppose . Then there exist such that ; that is, Let and . We will show that and satisfy the hypotheses of Theorem 7 provided that is chosen sufficiently small and is chosen sufficiently large. Let Then for all , and so whenever . Similarly, whenever .
Let Then if , we get that and for all . Hence, when and , we have when is sufficiently small.
Similarly, if , and is sufficiently small, we have Hence, by Theorem 7, there exists a positive solution of system (4).
Suppose now that system (4) has a solution with . Then is a solution of the system where . Hence may be regarded as the principal eigenfunction corresponding to the principal eigenvalue of the system , where . Hence . As but , it follows from Corollary 5 that and the proof is complete.

Remark 9. From Theorem 8, it is easy to see that the system (4) has no positive solution if .

Example 10. Let us consider the Neumann difference system Obviously, ; here denote the unit matrix of order . By simple computing, . From Theorem 8, the problem (46) has at least one positive solution on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

Yanqiong Lu completed the main study, carried out the results of this article, and drafted the paper. Ruyun Ma checked the proofs and verified the calculation. All the authors read and approved the paper.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by NSFC (no. 11361054 and no. 11201378), SRFDP (no. 20126203110004), and Gansu provincial National Science Foundation of China (no. 1208RJZA258).