- Research Article
- Open Access
Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems
© J. Xu and X. Han. 2010
- Received: 18 February 2010
- Accepted: 27 April 2010
- Published: 31 May 2010
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.
- Ordinary Differential Equation
- Eigenvalue Problem
- Fixed Point Theorem
- Positive Eigenvalue
- Elastic Beam
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  firstly investigated the existence of the solutions of the problem (1.1) by contraction mapping and iterative methods, subsequently, Ma and Wu  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  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:
This can be achieved under (H1) by using disconjugacy theory in .
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.
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 , Yang  and their references.
Definition 2.1 (see [7, Definition , page 2]).
Lemma 2.2 (see [7, Theorem , page 3]).
Lemma 2.3 (see [7, Theorem , page 9]).
Now using Lemmas 2.2 and 2.3, we will prove some preliminary results.
Let (H1) hold. Then
We divide the proof into three cases.
This completes the proof of the theorem.
(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 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 .
In what follows, we use the terminology of Rabinowitz .
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  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, .
Proof of Theorem 3.1.
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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