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

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.

The deformations of an elastic beam in equilibrium state with fixed both endpoints can be described by the fourth-order ordinary differential equation boundary value problem

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.

In this paper, applying disconjugate operator theory and bifurcation techniques, we consider the existence of nodal solution of more general the problem:

under the assumptions:

() is a parameter, is given constant,

() with on any subinterval of ,

() satisfies for all , and

for some .

However, in order to use bifurcation technique to study the nodal solutions of the problem (1.2), we firstly need to prove that the generalized eigenvalue problem

(where satisfies (H2)) has an infinite number of positive eigenvalues

and each eigenvalue corresponding an essential unique eigenfunction which has exactly simple zeros in and is positive near 0. Fortunately, Elias [6] developed a theory on the eigenvalue problem

where

and with on . are called the 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

has a factorization of the form

on , where with on , and if and only if

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

The rest of paper is arranged as follows: in Section 2, we state some disconjugacy theory which can be used in this paper, and then show that (H1) implies the equation

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.

Let

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]).

Equation (2.1) is disconjugate on a compact interval if and only if there exists a basis of solutions such that

on . A disconjugate operator can be written as

where and

and

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

Green's function of the disconjugate Equation (2.3) and the two-point boundary value conditions

satisfies

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

Theorem 2.4.

Let (H1) hold. Then

(i) is disconjugate on , and has a factorization

where with

(ii) if and only if

where

Proof.

We divide the proof into three cases.

Case 1.

. The case is obvious.

Case 2.

.

In the case, take

where , is a positive constant. Clearly, and then

It is easy to check that ,,, form a basis of solutions of . By simple computation, we have

Clearly, on

By Lemma 2.2, is disconjugate on , and has a factorization

and accordingly

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

Case 3.

.

In the case, take

where .

It is easy to check that ,  ,  ,   form a basis of solutions of . By simple computation, we have

From and , we have , so on

By Lemma 2.2, is disconjugate on , and has a factorization

and accordingly

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

(i)Equation (1.4) has an infinite number of positive eigenvalues

(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).

Let

with

To show (v), it is enough to prove

Clearly

Suppose on the contrary that the algebraic multiplicity of is greater than 1. Then there exists , and subsequently

for some . Multiplying both sides of (2.24) by and integrating from 0 to 1, we deduce that

which is a contradiction!

Theorem 2.6 (Maximum principle).

Let (H1) hold. Let with on and in . If satisfies

Then on .

Proof.

When , the homogeneous problem

has only trivial solution. So the boundary value problem (2.26) has a unique solution which may be represented in the form

where is Green's function.

By Theorem 2.4 and Lemma 2.3 (take ), we have

that is,

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

Theorem 3.1.

Let (H1), (H2), and (H3) hold. Assume that for some ,

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 .

Let with the norm Let

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

Let be such that

here . Clearly

Let

then is nondecreasing and

Let us consider

as a bifurcation problem from the trivial solution .

Equation (3.7) can be converted to the equivalent equation

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).

In fact, (3.7) can be rewritten to

where

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

Proof of Theorem 3.1.

We only need to show that

Suppose on the contrary that

where

Since   joins to infinity in and is the unique solutions of (3.7) in , there exists a sequence such that and as . We may assume that as . Let . From the fact

we have that

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

Since

we have from (3.15) and (3.6) that

that is,

By (H2), (H3), and (3.17) and the fact that , we conclude that , and consequently

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.

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.

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