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

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

(1.1)

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:

(1.2)

under the assumptions:

(H1) is a parameter, is given constant,

(H2) with on any subinterval of ,

(H3) satisfies for all , and

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

where

(1.7)

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

(1.8)

has a factorization of the form

(1.9)

on , where with on , and if and only if

(1.10)

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

(1.11)

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

(2.1)

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

(2.2)

on . A disconjugate operator can be written as

(2.3)

where and

(2.4)

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

(2.5)

satisfies

(2.6)

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

(2.7)

where with

(ii) if and only if

(2.8)

where

(2.9)

Proof.

We divide the proof into three cases.

Case 1.

. The case is obvious.

Case 2.

.

In the case, take

(2.10)

where , is a positive constant. Clearly, and then

(2.11)

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

(2.12)

Clearly, on

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

(2.13)

and accordingly

(2.14)

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

Case 3.

.

In the case, take

(2.15)

where .

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

(2.16)

From and , we have , so on

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

(2.17)

and accordingly

(2.18)

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

(2.19)

(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

(2.20)

with

(2.21)

To show (v), it is enough to prove

(2.22)

Clearly

(2.23)

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

(2.24)

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

(2.25)

which is a contradiction!

Theorem 2.6 (Maximum principle).

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

(2.26)

Then on .

Proof.

When , the homogeneous problem

(2.27)

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

(2.28)

where is Green's function.

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

(2.29)

that is,

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

Theorem 3.1.

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

(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 .

Let with the norm Let

(3.2)

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

Let be such that

(3.3)

here . Clearly

(3.4)

Let

(3.5)

then is nondecreasing and

(3.6)

Let us consider

(3.7)

as a bifurcation problem from the trivial solution .

Equation (3.7) can be converted to the equivalent equation

(3.8)

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

(3.9)

where

(3.10)

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

(3.11)

Suppose on the contrary that

(3.12)

where

(3.13)

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

(3.14)

we have that

(3.15)

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

Since

(3.16)

we have from (3.15) and (3.6) that

(3.17)

that is,

(3.18)

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

(3.19)

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.