Bifurcation techniques for Lidstone boundary value problems☆
Introduction
Bifurcation phenomena occur in many parts of physics and have been intensively studied. In particular, Rabinowitz (see [10], [11]) showed that bifurcation from both the trivial solution and infinity has global consequences. This theory has been successfully applied to Sturm–Liouville problems for ordinary differential equations, integral equations, and partial differential equations (see for example [4], [6], [8], [9], [10], [11], [12]).
Using Rabinowitz’s global bifurcation theorem, this paper examines the Lidstone boundary value problem (BVP, for short) where , is a given sign-changing function satisfying some assumptions that will be specified later. We first give some notation.
Let , , , , , , where
Recently many authors have studied the existence and multiplicity of positive solutions for the Lidstone boundary value problem (see [1], [2], [3], [5], [7], [9], [13], [14], [15] and the references therein) since it arises naturally in many different areas of applied mathematics and physics. In particular, if , such a problem describes the deformation of an elastic beam both ends of which are simply supported. The singular case has also been considered (see for example [1], [6], [7]). The approaches used in the literature are usually the monotone iterative method and the upper–lower solution method (see [3], [13]), the Leray–Schauder continuation theorem (see [1]) and topological degree approach (see [14], [6], [7]), or the Leggett–Williams theorem and the theorem of five functional fixed points (see [2], [5], [15]). Many of the results require that the nonlinearity does not depend on any derivatives of (see [14]). Recently in [9], Ma investigated the Lidstone BVP (1.1) by using a global bifurcation technique. The main result is the following:
Theorem A Suppose (A1) is continuous and there exist , such thatuniformly in . (A2) for any and . (A3) There exists such that (A4) or , where
Then BVP (1.1) has at least one positive solution.
To the best of our best knowledge, there is no paper which considers the case with sign-changing nonlinear terms. We try to fill this gap in the literature in this paper. Furthermore we extend Theorem A.
The paper is organized as follows. Section 2 gives some preliminaries. Section 3 is devoted to the existence of multiple solutions for BVP (1.1).
To conclude this section we give some notation and state three lemmas, which will be used in Section 3.
Following the notation of Rabinowitz, let be a real Banach space and be a compact linear map. If there exists and such that , is said to be a real characteristic value of . The set of real characteristic values of will be denoted by . The multiplicity of is where denotes the null space of . Suppose that is compact and at uniformly on bounded intervals. Then possesses the line of trivial solutions . It is well known that if , a necessary condition for to be a bifurcation point of (1.2) with respect to is that . If is a simple characteristic value of , let denote the eigenvector of corresponding to normalized so that . By we denote the closure of the set of nontrivial solutions of (1.2). A component of is a maximal closed connected subset. It was shown by Rabinowitz, [10, Theorem 1.3, 1.25, 1.27], that
Lemma 1.1 If is simple, then contains a component which can be decomposed into two subcontinua , such that for some neighborhood of ,implies where and , at . Moreover, each of , either meets infinity in , or meets where , or contains a pair of points , , .
The following are global results for (1.2) on bifurcation from infinity (see Rabinowitz, [11, Theorem 1.6 and Corollary 1.8]).
Lemma 1.2 Suppose is compact and linear, is continuous on , at uniformly on bounded intervals, and is compact. If is of odd multiplicity, then possesses an unbounded component which meets . Moreover if is an interval such that and is a neighborhood of whose projection on lies in and whose projection on is bounded away from 0, then either is bounded in in which case meets or is unbounded.
If (ii) occurs and has a bounded projection on , then meets where .
Lemma 1.3 Suppose the assumptions ofLemma 1.2hold. If is simple, then can be decomposed into two subcontinua , and there exists a neighborhood of such that and implies where and , at .
Section snippets
Preliminaries
Throughout this paper we assume is continuous, where .
We first convert BVP (1.1) into another form. Suppose is a solution of BVP (1.1). Let . Notice that
Thus can be written as , where
Similarly, notice that Thus we can obtain
Main results
We now list the following hypotheses for convenience.
- (H1)
There exists such that
- (H2)
There exists satisfying
- (H3)
There exists such that
Let and be the eigenvalues of the operator and (replacing in (2.10) with and of (H1) and (H2), respectively). Then by Lemma 2.3 we
Acknowledgements
The authors wish to thank the referees for their valuable suggestions. In addition, the first author would like to thank Professor Jianhong Wu and the Department of Mathematics and Statistics of York University for support and excellent facilities provided during his visit when this work was carried out.
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This project was supported by NNSF of PR China (10571111), China Scholarship Council and Natural Science Foundation of Shandong Province (Y2006A22).