Bifurcation techniques for Lidstone boundary value problems

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Abstract

By employing bifurcation techniques, this paper investigates the existence of nontrivial solutions (single and multiple) for Lidstone boundary value problems depending on higher order derivatives. Our results improve on those in the literature.

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) {(1)nu(2n)(t)=f(t,u(t),u(t),,u(2(n1))(t)),t(0,1);u(2i)(0)=u(2i)(1)=0,i=0,1,,n1, where n1, f is a given sign-changing function satisfying some assumptions that will be specified later. We first give some notation.

Let R+=[0,+), R=(,0], U=(u0,u1,,un1)Rn, |U|=max{|u0|,|u1|,,|un1|}, Rin=i=0n1(1)iR+, R+n=i=0n1R+, where (1)iR+={R+,i is even ;R,i is odd .

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 n=2, 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 f does not depend on any derivatives of u (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) f:[0,1]×RinR is continuous and there exist A=(a0,a1,,an1) , B=(b0,b1,,bn1)R+n{(0,0,,0)} such thatf(t,U)=i=0n1(1)iaiui+o(|U|),|U|0,f(t,U)=i=0n1(1)ibiui+o(|U|),|U|,uniformly in t[0,1] .

 (A2) f(t,U)>0 for any t[0,1] and U0 .

 (A3) There exists C=(c0,c1,,cn1)R+n{(0,0,,0)} such thatf(t,U)i=0n1(1)iciui,(t,U)[0,1]×Rin.

 (A4) λ1(B)<1<λ1(A) or λ1(A)<1<λ1(B) , whereλ1(A)=π2ni=0n1aiπ(2i),λ1(B)π2ni=0n1biπ(2i).

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 E be a real Banach space and L:EE be a compact linear map. If there exists μR and 0vE such that v=μLv, μ is said to be a real characteristic value of L. The set of real characteristic values of L will be denoted by r(L). The multiplicity of μr(L) is dimj=1N((IμL)j) where N(A) denotes the null space of A. Suppose that H:R×EE is compact and H(λ,u)=o(u) at u=0 uniformly on bounded λ intervals. Then u=λLu+H(λ,u) possesses the line of trivial solutions Θ={(λ,0)λR}. It is well known that if μR, a necessary condition for (μ,0) to be a bifurcation point of (1.2) with respect to Θ is that μr(L). If μ is a simple characteristic value of L, let v denote the eigenvector of L corresponding to μ normalized so that v=1. By Σ we denote the closure of the set of nontrivial solutions of (1.2). A component of Σ is a maximal closed connected subset. It was shown by Rabinowitz, [10, Theorem 1.3, 1.25, 1.27], that

Lemma 1.1

If μr(L) is simple, then Σ contains a component Cμ which can be decomposed into two subcontinua Cμ+ , Cμ such that for some neighborhood B of (μ,0) ,(λ,u)Cμ+(Cμ)B,and(λ,u)(μ,0)implies (λ,u)=(λ,αv+w) where α>0(α<0) and |λμ|=o(1) , w=o(|α|) at α=0 .

Moreover, each of Cμ+ , Cμ either

  • (i)

    meets infinity in Σ , or

  • (ii)

    meets (μˆ,0) where μμˆr(L) , or

  • (iii)

    contains a pair of points (λ,u) , (λ,u) , u0 .

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 L is compact and linear, H(λ,u) is continuous on R×E , H(λ,u)=o(u) at u= uniformly on bounded λ intervals, and u2H(λ,uu2) is compact. If μr(L) is of odd multiplicity, then Σ possesses an unbounded component Dμ which meets (μ,) . Moreover if ΛR is an interval such that Λr(L)={μ} and is a neighborhood of (μ,) whose projection on R lies in Λ and whose projection on E is bounded away from 0, then either

  • (i)

    Dμ is bounded in R×E in which case Dμ meets Θ={(λ,0)λR} or

  • (ii)

    Dμ is unbounded.

If (ii) occurs and Dμ has a bounded projection on R , then Dμ meets (μˆ,) where μμˆr(L) .

Lemma 1.3

Suppose the assumptions ofLemma 1.2hold. If μr(L) is simple, then Dμ can be decomposed into two subcontinua Dμ+ , Dμ and there exists a neighborhood of (μ,) such that (λ,u)Dμ+(Dμ) and (λ,u)(μ,) implies (λ,u)=(λ,αv+w) where α>0(α<0) and |λμ|=o(1) , w=o(|α|) at |α|= .

Section snippets

Preliminaries

Throughout this paper we assume f:I×RnR is continuous, where I=[0,1].

We first convert BVP (1.1) into another form. Suppose u(t) is a solution of BVP (1.1). Let v(t)=(1)n1u(2n2)(t). Notice that {[u(2n4)](t)=(1)n2v(t),tIu(2n4)(0)=u(2n4)(1)=0.

Thus u(2n4)(t) can be written as u(2n4)(t)=(1)n2A1v(t), where A1v(t)=01G(t,s)v(s)ds,tI,G(t,s)={s(1t),0st1;t(1s),0ts1.

Similarly, notice that (1)iu(2i)(t)=01G(t,s)(1)i+1u(2i+2)(s)ds,i=0,1,,n1. Thus we can obtain u(2i)(t)=(1)iAni

Main results

We now list the following hypotheses for convenience.

  • (H1)

    There exists a=(a0,a1,,an1)R+n{(0,0,,0)} such that f(t,U)=i=0n1(1)iaiui+o(|U|),as |U|0 uniformly in t[0,1].

  • (H2)

    There exists b=(b0,b1,,bn1)R+n{(0,0,,0)} satisfying f(t,U)=i=0n1(1)ibiui+o(|U|),as |U|+ uniformly in t[0,1].

  • (H3)

    There exists R>0 such that f(t,U)<813R,fortI,|U|R.

Let {λi}1 and {ηi}1 be the eigenvalues of the operator La and Lb (replacing d in (2.10) with a and b 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).

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