Existence of Positive Solutions for Generalized p-Laplacian BVPs

Existence of Positive Solutions for Generalized p-Laplacian BVPs

Wei-Cheng Lian (National Kaohsiung Marine University, Taiwan), Fu-Hsiang Wong (National Taipei University of Education, Taiwan), Jen-Chieh Lo (Tamkang University, Taiwan) and Cheh-Chih Yeh (Lunghwa University of Science and Technology, Taiwan)
Copyright: © 2011 |Pages: 11
DOI: 10.4018/jalr.2011010105

Abstract

Using Kransnoskii’s fixed point theorem, the authors obtain the existence of multiple solutions of the following boundary value problem
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1. Introduction

In this paper, we concern with the existence of multiple solutions for higher order boundary value problem

jalr.2011010105.m03
where jalr.2011010105.m04 is a positive integer, jalr.2011010105.m05 are given and jalr.2011010105.m06 is the p-Laplacian operator, that is, jalr.2011010105.m07 for jalr.2011010105.m08. Clearly, jalr.2011010105.m09 is invertible with inverse jalr.2011010105.m10. Here jalr.2011010105.m11

In recent years, the existence of positive solutions for nonlinear boundary value problems with p-Laplacian operator received wide attention. As we know, two point boundary value problems are used to describe a number of physical, biological and chemical phenomena. Recently, some authors have obtained some existence results of positive solutions of multi-points boundary value problems for second order ordinary differential equations (Wang & Ge, 2007; Yu, Wong, Yeh, & Lin, 2007; Zhao, Wang, & Ge, 2007; Zhou, & Su, 2007). In this paper, we establish the existence of positive solutions of general multi-points boundary value problem (BVP) and related results (Bai, Gui, & Ge, 2004; Guo & Lakshmikantham, 1988; Guo, Lakshmikantham, & Liu, 1996; He & Ge, 2004; Lian & Wong, 2000; Liu, 2002; Ma, 1999; Ma & Cataneda, 2001; Sun, Ge, & Zhao, 2007; Wang, 1997).

In order to abbreviate our discussion, throughout this paper, we assume

jalr.2011010105.m13are both nondecreasing continuous and odd functions defined on jalr.2011010105.m14 and at least one of them satisfies the condition that there exists jalr.2011010105.m15 such that jalr.2011010105.m16 for all jalr.2011010105.m17jalr.2011010105.m18jalr.2011010105.m19

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2. Preliminaries And Lemmas

Let

Then, B is a Banach space with norm jalr.2011010105.m21 And let

Obviously, K is a cone in B.

In order to discuss our results, we need the following some lemmas:

Lemma 2.0

Assume that jalr.2011010105.m23 is a Banach space and jalr.2011010105.m24 is a cone in jalr.2011010105.m25jalr.2011010105.m26 are open subsets of jalr.2011010105.m27and jalr.2011010105.m28 Furthermore, let jalr.2011010105.m29 be a completely continuous operator satisfying one of the following conditions:

  • jalr.2011010105.m30jalr.2011010105.m31jalr.2011010105.m32jalr.2011010105.m33

  • jalr.2011010105.m34jalr.2011010105.m35jalr.2011010105.m36jalr.2011010105.m37

Then jalr.2011010105.m38 has a fixed point in jalr.2011010105.m39

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