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Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems
Boundary Value Problems volume 2010, Article number: 570932 (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.
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

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:
(H1) is a parameter,
is given constant,
(H2) with
on any subinterval of
,
(H3) 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.
2. Preliminary Results
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
.
3. Statement of the Results
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.
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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.
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Xu, J., Han, X. Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems. Bound Value Probl 2010, 570932 (2010). https://doi.org/10.1155/2010/570932
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DOI: https://doi.org/10.1155/2010/570932
Keywords
- Ordinary Differential Equation
- Eigenvalue Problem
- Fixed Point Theorem
- Positive Eigenvalue
- Elastic Beam