Discrete Dynamics in Nature and Society

Volume 2008, Article ID 263785, 6 pages

http://dx.doi.org/10.1155/2008/263785

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

^{1}College of Mathematics and Information Science, Shaanxi Normal University, Xian, Shaanxi 710062, China^{2}Department 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 [1–9]), 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
that**Then 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).

#### References

- Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,”
*Science in China Series A*, vol. 46, no. 4, pp. 506–515, 2003. View at Google Scholar · View at MathSciNet - Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations,”
*Journal of the London Mathematical Society*, vol. 68, no. 2, pp. 419–430, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yu, Z. Guo, and X. Zou, “Periodic solutions of second order self-adjoint difference equations,”
*Journal of the London Mathematical Society*, vol. 71, no. 1, pp. 146–160, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Guo and J. Yu, “Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 55, no. 7-8, pp. 969–983, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Jiang and Z. Zhou, “Existence of nontrivial solutions for discrete nonlinear two point boundary value problems,”
*Applied Mathematics and Computation*, vol. 180, no. 1, pp. 318–329, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. H. Rabinowitz, in
*Minimax Methods in Critical Point Theory with Applications for Differential Equations*, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1984. View at Zentralblatt MATH - J. Yu and Z. Guo, “On boundary value problems for a discrete generalized Emden–Fowler equation,”
*Journal of Differential Equations*, vol. 231, no. 1, pp. 18–31, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yu and Z. Guo, “Boundary value problems of discrete generalized Emden–Fowler equation,”
*Science in China Series A*, vol. 49, no. 10, pp. 1303–1314, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Zhang and S. S. Cheng, “Existence of solutions for a nonlinear system with a parameter,”
*Journal of Mathematical Analysis and Applications*, vol. 314, no. 1, pp. 311–319, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Brezis and L. Nirenberg, “Remarks on finding critical points,”
*Communications on Pure and Applied Mathematics*, vol. 44, no. 8-9, pp. 939–963, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Gyulov and S. Tersian, “Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation,”
*Electronic Journal of Differential Equations*, no. 41, pp. 1–14, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet