SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research Article

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

Jia Xu12* and XiaoLing Han1

Author Affiliations

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

2 College of Physical Education, Northwest Normal University, Lanzhou 730070, China

For all author emails, please log on.

Boundary Value Problems 2010, 2010:570932  doi:10.1155/2010/570932


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2010/1/570932


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

© 2010 The Author(s)

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.

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.

1. Introduction

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

(11)

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:

(12)

under the assumptions:

() is a parameter, is given constant,

() with on any subinterval of ,

() satisfies for all , and

(13)

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

(14)

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

(15)

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

(16)

where

(17)

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

(18)

has a factorization of the form

(19)

on , where with on , and if and only if

(110)

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

(111)

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 [812], An and Ma [13], Yang [14] and their references.

2. Preliminary Results

Let

(21)

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

(22)

on . A disconjugate operator can be written as

(23)

where and

(24)

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

(25)

satisfies

(26)

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

(27)

where with

(ii) if and only if

(28)

where

(29)

Proof.

We divide the proof into three cases.

Case 1.

. The case is obvious.

Case 2.

.

In the case, take

(210)

where , is a positive constant. Clearly, and then

(211)

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

(212)

Clearly, on

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

(213)

and accordingly

(214)

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

Case 3.

.

In the case, take

(215)

where .

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

(216)

From and , we have , so on

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

(217)

and accordingly

(218)

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

(219)

(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

(220)

with

(221)

To show (v), it is enough to prove

(222)

Clearly

(223)

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

(224)

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

(225)

which is a contradiction!

Theorem 2.6 (Maximum principle).

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

(226)

Then on .

Proof.

When , the homogeneous problem

(227)

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

(228)

where is Green's function.

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

(229)

that is,

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

3. Statement of the Results

Theorem 3.1.

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

(31)

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

(32)

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

Let be such that

(33)

here . Clearly

(34)

Let

(35)

then is nondecreasing and

(36)

Let us consider

(37)

as a bifurcation problem from the trivial solution .

Equation (3.7) can be converted to the equivalent equation

(38)

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

(39)

where

(310)

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

(311)

Suppose on the contrary that

(312)

where

(313)

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

(314)

we have that

(315)

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

Since

(316)

we have from (3.15) and (3.6) that

(317)

that is,

(318)

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

(319)

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.

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.

References

  1. Agarwal, RP, Chow, YM: Iterative methods for a fourth order boundary value problem. Journal of Computational and Applied Mathematics. 10(2), 203–217 (1984). Publisher Full Text OpenURL

  2. Ma, R, Wu, HP: Positive solutions of a fourth-order two-point boundary value problem. Acta Mathematica Scientia A. 22(2), 244–249 (2002)

  3. Yao, Q: Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Applied Mathematics Letters. 17(2), 237–243 (2004). Publisher Full Text OpenURL

  4. Yao, Q: Solvability of an elastic beam equation with Caratheodory function. Mathematica Applicata. 17(3), 389–392 (2004)

  5. Korman, P: Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proceedings of the Royal Society of Edinburgh A. 134(1), 179–190 (2004). Publisher Full Text OpenURL

  6. Elias, U: Eigenvalue problems for the equations . Journal of Differential Equations. 29(1), 28–57 (1978). Publisher Full Text OpenURL

  7. Elias, U: Oscillation Theory of Two-Term Differential Equations, Mathematics and Its Applications,p. viii+217. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997)

  8. Ma, R: Existence of positive solutions of a fourth-order boundary value problem. Applied Mathematics and Computation. 168(2), 1219–1231 (2005). Publisher Full Text OpenURL

  9. Ma, R: Nodal solutions for a fourth-order two-point boundary value problem. Journal of Mathematical Analysis and Applications. 314(1), 254–265 (2006)

  10. Ma, R: Nodal solutions of boundary value problems of fourth-order ordinary differential equations. Journal of Mathematical Analysis and Applications. 319(2), 424–434 (2006). Publisher Full Text OpenURL

  11. Ma, R, Xu, J: Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. Nonlinear Analysis: Theory, Methods & Applications. 72(1), 113–122 (2010). PubMed Abstract | Publisher Full Text OpenURL

  12. Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis. 59(1–4), 225–231 (1995)

  13. An, Y, Ma, R: Global behavior of the components for the second order -point boundary value problems. Boundary Value Problems. 2008, (2008)

  14. Yang, Z: Existence and uniqueness of positive solutions for higher order boundary value problem. Computers & Mathematics with Applications. 54(2), 220–228 (2007). PubMed Abstract | Publisher Full Text OpenURL

  15. Rabinowitz, PH: Some global results for nonlinear eigenvalue problems. Journal of Functional Analysis. 7(3), 487–513 (1971). Publisher Full Text OpenURL