%0 Journal Article
%T On the Nonhomogeneous Fourth-Order -Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems
%A Jian Liu
%A Zengqin Zhao
%J Discrete Dynamics in Nature and Society
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/831960
%X We study the nonlinear nonhomogeneous -point generalized Sturm-Liouville fourth-order -Laplacian boundary value problem by using Leray-Schauder nonlinear alternative and Leggett-Williams fixed-point theorem. 1. Introduction In this paper, we prove the existence of one and multiple positive solutions of the following differential equations: where is -Laplacian operator, that is, , , , , , , , , , , . Recently, much attention has been paid to the existence of positive solutions for nonlocal nonlinear boundary value problems (BVPs for short), see [1¨C4] and references therein. Such problems have potential applications in physics, biology, chemistry, and so forth. For example, a second-order three-point is used as a model for the membrane response of a spherical cap in nonlinear diffusion generated by nonlinear sources and in chemical reactor theory. At the same time, the boundary value problems with -Laplacian operator have been discussed extensively, for example, see [1¨C3, 5¨C7]. In [1], Feng et al. researched the boundary value problem they obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of by applying Krasnoselskii fixed-point theorem. Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with -Laplacian operator in [3]: they established a corresponding iterative scheme for (1.4) by employing the monotone iterative technique. We would also like to mention the work of Zhang and Liu in [7], in which they considered the existence of positive solutions for by virtue of monotone iterative techniques, and they established a necessary and sufficient condition of positive solutions for their problem. However, to the best of our knowledge, there are not many results concerning about the existence and multiple solutions of fourth-order -Laplacian generalized Sturm-Liouville -point boundary value problems. In this paper, motivated by the study of [4, 8], we committed to consider the fourth-order -Laplacian generalized Sturm-Liouville nonlocal boundary value problem without assuming any monotonicity condition on the nonlinearity . The rest of the paper is arranged as follows. We state some definitions and several preliminary results in Section 2 that we will use in the sequel. Then in Section 3 we present the existence of one positive solution of BVP (1.1) by Leray-Schauder nonlinear alternative. In Section 4 we get three solutions by Leggett-Williams fixed-point theorem. 2. Preliminaries and Some Lemmas The basic space used
%U http://www.hindawi.com/journals/ddns/2012/831960/