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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 839474, 19 pages
http://dx.doi.org/10.1155/2010/839474
Research Article

Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 4 December 2009; Revised 30 January 2010; Accepted 25 March 2010

Academic Editor: Leonid Shaikhet

Copyright © 2010 Ruyun Ma et al. 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

Let be two integers with and let . We show the existence of solutions for nonlinear fourth-order discrete boundary value problem , , , under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.

1. Introduction

The deformations of an elastic beam whose both ends are simply supported are described by a fourth-order two-point boundary value problem See studies by Aftabizadeh [1] and Gupta in [2]. The existence of solutions of nonlinear boundary value problems of fourth-order differential equations has been studied by many authors; see [112] and the references therein. For example, Aftabizadeh [1] proved an existence theorem for nonlinear boundary value problems under several conditions that is a bounded function. Yang [3] obtained existence results of (1.2) under the following assumption.(A) There are constants with such that

Del Pino and Manásevich [4] extended Yang's result and proved the following.

Theorem A. Assume that the pair satisfies for all and that there are positive constants , and such that for all , , then (1.2) possesses at least one solution.

Of course, the natural question is whether or not the similar existence can be established for the corresponding discrete analog of (1.2) of the form where , for .

The purpose of this paper is to show that the answer is yes. To this end, we state and prove a spectrum result of two-parameter linear eigenvalue problem This result is a slightly generalized version of Shi and Wang [13, Theorem ]. In Section 3, we use Leray-Schauder principle to study the existence of solutions of (1.6), (1.7) under some nonresonant conditions involving the spectrum of (1.8), (1.9). Section 4 is considered with some perturbations of resonant linear problems. We established some a priori bounds and used these together with bifurcation arguments to prove the existence and multiplicity of solutions.

Finally, we note that the existence of solutions of second-order discrete boundary value problems has also received much attention; see studies by Agarwal and Wong in [14], Henderson in [15], and the references therein. However, relatively little is known about the existence of solutions of fourth-order discrete boundary value problems. The likely reason may be that the structure of spectrum of the corresponding linear eigenvalue problem is not very clear. To our best knowledge, only He and Yu [16] as well as Zhang et al. [17] dealt with the discrete problem of the form As we will see in Section 2, (1.9) has more advantage than (1.11) in the study of the spectrum of two-parameter linear eigenvalue problems.

2. Spectrum of Two-Parameter Linear Eigenvalue Problem

Let be two integers with . Recall Let Let be the Banach space under the norm Let be the Banach space equipped with the norm Let be the Banach space equipped with the norm

Remark 2.1. For any with it determines a unique element by and a unique element by Hence, the Banach spaces , , and are homomorphic with each other. Denote the natural homomorphism from to by .

Now, we define a linear operator by For , let be the th-eigenvalue of the second-order linear eigenvalue problem It is well known that is simple, and the corresponding eigenfunction See the study by Kelly and Peterson in [18].

The following result is considered with the spectrum of two-parameter eigenvalue problem: It is a slightly generalized version of Shi and Wang [13, Theorem ].

Proposition 2.2. is an eigenvalue pair of (2.15), (2.16) if and only if for some

Proof. Let such that Define two second-order difference operators by Then, for and , We claim that if (2.15), (2.16) possess a nontrivial solution , then either or for some . In either case, ), is a nontrivial solution of (2.15), (2.16).
In fact, if for all , then (2.20) implies that This is Thus, for some , and
If for some , then (2.20) implies that for some . This is Since , it follows that This implies that (2.25), (2.26) have a unique solution
We show that
In fact, from (2.25) we have which implies that , and, subsequently, .
Therefore, the claim is true.
Now, (2.17) follows by substituting this solution into (2.15), (2.16). Reciprocally, if (2.17) holds, then, clearly, , is a nontrivial solution of (2.15), (2.16).

Remark 2.3. From the proof of Proposition 2.2, we see that if (2.15) subjects to (1.9), then we can factor as follows: However, this cannot be done if (2.15) subjects to (1.11). So, (1.9) has more advantage than (1.11) in the study of the spectrum of two-parameter linear eigenvalue problems.

Next, for , let us set In view of the Proposition 2.2, we call an eigenline of (2.15), (2.16). We note that an eigenvalue pair can belong to at most two eigenlines. If belongs to just one , then the corresponding eigenspace is spanned by . If belongs to , then the corresponding eigenspace is spanned by and .

Suppose that the pair is not an eigenvalue pair of (2.15), (2.16), that is, for all , and that : From the Fredholm Alternative, it follows that the boundary value problem has a unique solution for each . Moreover, this solution admits a Fourier series expansion of the form Also, we have

From (2.36) and (2.37), we can easily see that the operators , defined by are compact linear operators. In (2.38), is the solution of (2.35), (2.16) corresponding to . The norms of and are, respectively, given by

Finally, as an immediate consequence of Proposition 2.2, we have the following.

Proposition 2.4. Let and be two constants with and . Then the generalized eigenvalues of problem are given by where The generalized eigenfunction corresponding to is

3. Existence Results for Nonresonant Problems

Theorem 3.1. Assume that the pair satisfies for all and that there are positive constants , and such that for all , , then (1.6), (1.7) possess at least one solution.

Remark 3.2. It is not difficult to see that (3.1), (3.2) imply that for . It turns out that (3.4) is equivalent to the fact that the square does not intersect any of the eigenlines of (2.15), (2.16). From this point of view, (3.1), (3.2) can be thought of as a two-parameter nonresonance condition relative to the eigenlines .

Proof of Theorem 3.1. It is easy to check that the problem has a unique solution . Set Then (1.6),(1.7) can be rewritten as Since with it follows that (3.2) and (3.3) still hold except that is replaced by . So, we may suppose that in (1.7).
Let us define by where and are the operators defined in (2.38). The growth condition (3.3) together with the compactness and implies that is a completely continuous operator. By Remark 2.1, the problem is equivalent to the fixed point problem in : We will study this fixed point problem by means of the well-known Leray-Schauder principle [18]. To do this, we show that there is a uniform bound independent of for the solutions of the equation Thus, let be a solution of (3.13). From the definition of and (3.3), we obtain the result that Combining (3.14) and (3.15) and using (3.2) and (2.38), we obtain the existence of a constant such that By the Leray-Schauder principle [19], we conclude the existence of at least one solution of (3.12), and the theorem follows.

4. Existence and Multiplicity Results for Perturbations of Resonant Linear Problems

In this section, we consider the perturbations of resonant linear problems of the form where with , , and and satisfy the following.(H1) (Sublinear growth condition) is continuous, and there exist , such that (H2) There exists such that (H3) satisfies

We will establish some a priori bounds and use these together with Leray-Schauder continuation and bifurcation arguments to reduce results which say that there are multiple solutions of , (4.1) for on one side of zero and guarantee the existence of at least one solution for and on the other side of zero. To wit, we have the following.

Theorem 4.1. Let (H1), (H2), and (H3) hold. Then there exist such that , (4.1) have(1)at least one solution if ,(2)at least three solutions if .

We have the following “dual” theorem if (H2) is replaced by the assumption(H) that there exists such that

Theorem 4.2. Let (H1), (H2), and (H3) hold. Then there exist such that , (4.1) have(1)at least one solution if ,(2)at least three solutions if .

Define by Define by It is easy to check that is continuous. Obviously , (4.1) are equivalent to

Define an operator by where

It is easy to show the following.

Lemma 4.3. is a projection and .

Define an operator by Obviously, we have the following.

Lemma 4.4. is a projection and .

It is clear that where represents the identity operator and , and are the images of , and , respectively.

It is obvious that the restriction of to is a bijection from onto , the image of . We define by Since , we see that each can be uniquely decomposed into for some , and . For , we also have the decomposition with and .

Lemma 4.5. Equations , (4.1) are equivalent to the system

Lemma 4.6. Let (H1) and (H2) hold. Then there exists such that any solution of , (4.1) satisfies as long as where is defined by

Proof. Obviously is invertible for . Moreover, by (4.17), Let be any solution of , (4.1). Then we have that, if , and hence where If then we have
If we assume that the conclusion of the lemma is false, we obtain a sequence with and , and a sequence of corresponding solutions of , (4.1) such that . From (4.24), we conclude that it is necessary that . We may assume that since the other case can be treated by the same way. Thus (4.24) yields that with .
Now from (4.15), we get that By (4.17) and (4.27), it follows that
Let It is easy to see that Combining (4.30) and (4.26), we conclude that there exists a positive constant such that, for , which implies that Applying (4.32), (4.30), and (H2), we conclude that which contracts (4.28).

Using the similar arguments, we may establish the following lemma.

Lemma 4.7. Let (H1) and (H2) hold. Then there exists such that any solution of , (4.1) satisfies as long as where is given in (4.17).

Lemma 4.8. Let (H1) and (H2) hold. Then there exists such that, for and , one has where is the natural homomorphism, , and “deg” denotes Leray-Schauder degree when and coincidence degree when (see the study by Gaines and Mawhin in [20]). Therefore has a solution in for .

Proof. By Lemma 4.6 and the definition of , the degree is well defined for and is a constant with respect to .
Now if is a solution of then we have Hence there exists such that . Thus if , then we have, whenever , that which completes the proof.

By a similar manner we may establish the following.

Lemma 4.9. Let (H1) and (H) hold. Then there exists such that, for and , one has Therefore has a solution in for .

Lemma 4.10. Let (H1) and (H2) hold. Then there exists such that, for , one has

Proof. Let Then it is not difficult to check that . Hence if we take so small that , then for ,

Lemma 4.11. Let (H1) and (H) hold. Then there exists such that, for , one has

Proof of Theorem 4.1. By the study of Massabò and Pejsachowicz in [21, Theorem ], has a continuum of solutions with and . On the other hand, since is -completely continuous and satisfies (H1) and since is a simple eigenvalue, it follows from the study by Rabinowitz in [22, Theorem ] that is a bifurcation point from infinity for . Moreover, there exist two continua of solutions of , bifurcating from infinity at , that is, there exists , such that for all there exist two continua and with and connects to . Notice that implies that , and implies that . So, Now, Lemma 4.6 implies that This completes the proof.

Proof of Theorem 4.2. Using similar arguments, we may get the desired results.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (no. 10671158), NWNU-KJCXGC-03-47, the Spring-sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006 []).

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