Sign-changing solution for a third-order boundary-value problem in ordered Banach space with lattice structure

Xiuli Lin,Zengqin Zhao

doi:10.1186/1687-2770-2014-132

Abstract

In this paper, the sign-changing solution of a third-order two-point boundary-value problem is considered. By calculating the eigenvalues and the algebraic multiplicity of the linear problem and using a new fixed point theorem in an ordered Banach space with lattice structure, we give some conditions to guarantee the existence for a sign-changing solution.

Highlights

In this paper, we consider the following nonlinear third-order two-point boundary-value problem–u (t) = f (t, u(t)), t ∈ [, ], ( . )u( ) = u ( ) = u ( ) =, where f ∈ C([, ] × R, R)
There has been much attention focused on the problem, especially to the two-point or multi-point boundary-value problem
For the second-order two-point or multi-point boundary-value problem, many beautiful results have been given on the existence and multiplicity of the sign-changing solutions

Summary

Introduction

We consider the following nonlinear third-order two-point boundary-value problem. For the second-order two-point or multi-point boundary-value problem, many beautiful results have been given on the existence and multiplicity of the sign-changing solutions (see [ – ] and the references therein). Xu and Sun [ ] obtained an existence result of the sign-changing solutions for the second-order threepoint boundary-value problem. Xu [ ] considered the sign-changing solutions for the second-order multi-point boundary-value problem. In [ ], Zhang and Sun obtained the existence and multiplicity of the sign-changing solutions for the integral boundary-value problem a(s)u(s) ds,. ), which is one of the key points that we can use to prove our main result; (c) some conditions are given to guarantee the existence for a sign-changing solution of the problem

Next we prove that the algebraic multiplicity of the eigenvalue

Proof Noticing