# 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)
DOI: 10.4018/jalr.2011010105
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## 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

where is a positive integer, are given and is the p-Laplacian operator, that is, for . Clearly, is invertible with inverse . Here

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

are both nondecreasing continuous and odd functions defined on and at least one of them satisfies the condition that there exists such that for all

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## 2. Preliminaries And Lemmas

Let

Then, B is a Banach space with norm 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 is a Banach space and is a cone in are open subsets of and Furthermore, let be a completely continuous operator satisfying one of the following conditions:

Then has a fixed point in

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