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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 263785, 6 pages
http://dx.doi.org/10.1155/2008/263785
Research Article

Uniqueness and Multiplicity of Solutions for a Second-Order Discrete Boundary Value Problem with a Parameter

1College of Mathematics and Information Science, Shaanxi Normal University, Xian, Shaanxi 710062, China
2Department of Mathematics, Qinghai University for Nationalities, Xining, Qinghai 810007, China

Received 15 November 2007; Accepted 10 February 2008

Academic Editor: Bing-Gen Zhang

Copyright © 2008 Xi-Lan Liu and Jian-Hua Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.

1. Introduction and Preliminaries

In this paper, we consider the existence, uniqueness, and multiplicity of solutions for a second-order discrete boundary value problem where is a parameter. Our technique is based on critical point theory, which is successfully used to deal with the existence of solutions for discrete problems (see [19]), especially in [7, 9]. Similarly to [7], we denote by , and the sets of all natural numbers, integers, and real numbers, respectively. For , , when . We assume that is nonzero and real-valued for each , is real-valued for each , and is real-valued for each and continuous in . Let be the real Euclidean space with dimension . For any , and , denote the usual norm and inner product in , respectively.

Consider the functional defined on ,where (⋅)T is the transpose of a vector in ,

It is easy to see that is Fréchet differentiable with Fréchet derivativewhere , and there is a one-to-one correspondence between the critical point of functional and the solution of BVP (1.1). Furthermore, is a critical point of if and only if is a solution of (1.1), where [7].

Recently, Yu and Guo [7] studied the BVP,They obtained some existence results for (1.5). One of the main results is as follows.

Theorem (). Suppose that satisfies the following assumption:
(f3) there exist constants , , , and such thatThen BVP (1.5) has at least one solution.

Equation (1.6) shows that for large enough. Since may be negative, then the conclusion of Theorem 1.1 cannot be drawn, which motivates us to consider (1.1). Note that if we take in (1.1), then . Under the similar condition to (1.6) when , we not only obtain the existence of solutions for (1.1), but also the multiplicity.

Let be a Banach space with a direct sum decomposition The functional has a local linking at if for some , and , The functional is said to satisfy the (PS) condition if any sequence for which is bounded, and possesses a convergent subsequence in .

Theorem 1 A (see[10]). Let be a Banach space. Suppose that satisfies the (PS) condition and has a local linking at Assume that is bounded below and Then has at least two nontrivial critical points.

Theorem 1 B (see[6, 11]). Let be a real Banach space, with even, bounded from below, and satisfying (PS) condition. Suppose . There is a set such that is homeomorphic to a unit sphere in () by an odd map, and Then possesses at least distinct pairs of critical points.

2. Main Results

Following conditions will be useful to prove our main results.

(H1)There exist numbers and such that(H2)

Theorem 2.1. Suppose that is positive definite, , for and , and that . Then (1.1) has only trivial solution for .

Proof. Note thatfor , where is the least eigenvalue of , which means that the Nahari manifold is empty. Thus (1.1) has only trivial solution.

Theorem 2.2. Suppose that (H1) and (H2) hold, and is neither positive definite nor negative definite. Then (1.1) has at least two nontrivial solutions for .

Proof. We will prove that the functional satisfies all conditions of Theorem A by two steps.
Step 1. is bounded from below and satisfies (PS) condition. Let denote all the eigenvalues of , where and . For any , set to be an eigenvector of corresponding to the eigenvalue , , such that
Let and be subspaces of defined byrespectively. Then has the direct sum decomposition . In view of (H1), we haveThe second inequality follows from the elementary inequality where , and which can be easily obtained by the fact that the function attains its minimum at Thus is bounded from below.
Equation (2.5) shows that is coercive, so we can obtain that any (PS) sequence must be bounded in and, by a standard argument, has a convergent subsequence.
Step 2. has a local linking at . Indeed, by (H2) for given and sufficiently small such that , there exists small enough such that for ,holds. Then for such that , we have
On the other hand, for with , we haveThe application of Theorem A finishes our proof.

Remark 2.3. By the above proof, we see that replacing (H1) with (f3), and by adding the condition that for , Theorem 2.2 still holds, where (f3) is the same as in Theorem 1.1. Indeed, we havewhich means that is bounded from below. The fact that has local linking at may be verified similarly.

If we further impose some condition on and matrix , then the following result can be derived.

Theorem 2.4. Suppose (H1) and (H2) hold, is odd in , that is, for , and that is neither positive definite nor negative definite and has distinct negative eigenvalues. Then (1.1) has at least distinct pairs of solutions for .

Proof. By the proof of Theorem 2.2, is bounded from below and satisfies (PS) condition. In addition, , is even. Consider the subset of :where is a positive number small enough to be determined later, is defined by Theorem 2.2 similarly. Define the mapping bywhere is a unit sphere in . Then is a homeomorphism between and , and is a subset of the finite dimensional space equipped with the Euclidian norm. We can choose and small enough such that for , and then we haveFor above , we havewhich together with Theorem B concludes the proof.

Remark 2.5. The condition that is bounded from below is crucial to prove both Theorems 2.2 and 2.4. As in Remark 2.3, if we replace (H1) by (f3) and the condition that for , then Theorem 2.4 is also true.

Acknowledgments

The authors would like to thank Professor B. G. Zhang and referee's valuable suggestions and comments. This research was partially supported by the NNSF (10571115) of China, China Postdoctoral Science Foundation (20070421106), the Key Project of Chinese Ministry of Education, and the Foundation of Chinese Nationalities Committee (07QH04).

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