- Research Article
- Open Access

# Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems

- Jia Xu
^{1, 2}Email author and - XiaoLing Han
^{1}

**Received:**18 February 2010**Accepted:**27 April 2010**Published:**31 May 2010

## Abstract

We consider the fourth-order two-point boundary value problem , , , where is a parameter, is given constant, with on any subinterval of , satisfies for all , and , , for some . By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.

## Keywords

- Ordinary Differential Equation
- Eigenvalue Problem
- Fixed Point Theorem
- Positive Eigenvalue
- Elastic Beam

## 1. Introduction

where is continuous, is a parameter. Since the problem (1.1) cannot transform into a system of second-order equation, the treatment method of second-order system does not apply to the problem (1.1). Thus, existing literature on the problem (1.1) is limited. In 1984, Agarwal and chow [1] firstly investigated the existence of the solutions of the problem (1.1) by contraction mapping and iterative methods, subsequently, Ma and Wu [2] and Yao [3, 4] studied the existence of positive solutions of this problem by the Krasnosel'skii fixed point theorem on cones and Leray-Schauder fixed point theorem. Especially, when , Korman [5] investigated the uniqueness of positive solutions of the problem (1.1) by techniques of bifurcation theory. However, the existence of sign-changing solution for this problem have not been discussed.

under the assumptions:

(H1) is a parameter, is given constant,

(H2) with on any subinterval of ,

for some .

*quasi-derivatives*of . To apply Elias's theory, we have to prove that (1.4) can be rewritten to the form of (1.6), that is, the linear operator

This can be achieved under (H1) by using *disconjugacy theory* in [7].

is disconjugacy on , moreover, we establish some preliminary properties on the eigenvalues and eigenfunctions of the generalized eigenvalue problem (1.4). Finally in Section 3, we state and prove our main result.

Remark 1.1.

For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of ordinary differential equations based on bifurcation techniques, see Ma [8–12], An and Ma [13], Yang [14] and their references.

## 2. Preliminary Results

be th-order linear differential equation whose coefficients are continuous on an interval .

Definition 2.1 (see [7, Definition , page 2]).

Equation (2.1) is said to be disconjugate on an interval if no nontrivial solution has zeros on , multiple zeros being counted according to their multiplicity.

Lemma 2.2 (see [7, Theorem , page 3]).

and

Lemma 2.3 (see [7, Theorem , page 9]).

Now using Lemmas 2.2 and 2.3, we will prove some preliminary results.

Theorem 2.4.

Let (H1) hold. Then

where with

Proof.

We divide the proof into three cases.

Case 1.

. The case is obvious.

Case 2.

.

Clearly, on

Using (2.14), we conclude that is equivalent to (2.8).

Case 3.

.

where .

From and , we have , so on

Using (2.18), we conclude that is equivalent to (2.8).

This completes the proof of the theorem.

Theorem 2.5.

Let (H1) hold and satisfy (H2). Then

(ii) as

(iii) To each eigenvalue there corresponding an essential unique eigenfunction which has exactly simple zeros in and is positive near 0.

(iv) Given an arbitrary subinterval of , then an eigenfunction which belongs to a sufficiently large eigenvalue change its sign in that subinterval.

(v) For each , the algebraic multiplicity of is 1.

Proof.

(i)–(iv) are immediate consequences of Elias [6, Theorems ] and Theorem 2.4. we only prove (v).

which is a contradiction!

Theorem 2.6 (Maximum principle).

Then on .

Proof.

where is Green's function.

that is,

Using (2.28), when on with in , then on .

## 3. Statement of the Results

Theorem 3.1.

Then there are at least nontrivial solutions of the problem (1.2). In fact, there exist solutions such that for , has exactly simple zeros on the open interval and and there exist solutions such that for , has exactly simple zeros on the open interval and .

with the norm Then is completely continuous, here is given as in (2.20).

as a bifurcation problem from the trivial solution .

Further we note that for near 0 in .

In what follows, we use the terminology of Rabinowitz [15].

Let under the product topology. Let denote the set of function in which have exactly interior nodal (i.e., nondegenerate) zeros in and are positive near , set , and . They are disjoint and open in . Finally, let and .

The results of Rabinowitz [13] for (3.8) can be stated as follows: for each integer , , there exists a continuum of solutions of (3.8), joining to infinity in . Moreover, .

Notice that we have used the fact that if is a nontrivial solution of (3.7), then all zeros of on are simply under (H1), (H2), and (H3).

clearly satisfies (H2). So Theorem 2.5(iii) yields that all zeros of on are simple.

Proof of Theorem 3.1.

Furthermore, since is completely continuous, we may assume that there exist with such that as .

By Theorem 2.6, we know that in . This means is the first eigenvalue of and is the corresponding eigenfunction. Hence . Since is open and , we have that for large. But this contradict the assumption that and , so (3.12) is wrong, which completes the proof.

## Declarations

### Acknowledgments

This work is supported by the NSFC (no. 10671158), the Spring-sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), the SRF for ROCS, SEM (2006[ ]), NWNU-KJCXGC-SK0303-23, and NWNU-KJCXGC-03-69.

## Authors’ Affiliations

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