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)
DOI: 10.4018/978-1-4666-3890-7.ch007

Abstract

where are given. The authors examine and discuss these solutions.
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1. Introduction

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

978-1-4666-3890-7.ch007.m05
where 978-1-4666-3890-7.ch007.m06 is a positive integer, 978-1-4666-3890-7.ch007.m07 are given and 978-1-4666-3890-7.ch007.m08 is the p-Laplacian operator, that is, 978-1-4666-3890-7.ch007.m09 for 978-1-4666-3890-7.ch007.m10. Clearly, 978-1-4666-3890-7.ch007.m11 is invertible with inverse 978-1-4666-3890-7.ch007.m12. Here 978-1-4666-3890-7.ch007.m13

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

978-1-4666-3890-7.ch007.m15are both nondecreasing continuous and odd functions defined on 978-1-4666-3890-7.ch007.m16 and at least one of them satisfies the condition that there exists 978-1-4666-3890-7.ch007.m17 such that 978-1-4666-3890-7.ch007.m18 for all 978-1-4666-3890-7.ch007.m19978-1-4666-3890-7.ch007.m20978-1-4666-3890-7.ch007.m21

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2. Preliminaries And Lemmas

Let

Then, B is a Banach space with norm 978-1-4666-3890-7.ch007.m23 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 978-1-4666-3890-7.ch007.m25 is a Banach space and 978-1-4666-3890-7.ch007.m26 is a cone in 978-1-4666-3890-7.ch007.m27978-1-4666-3890-7.ch007.m28 are open subsets of 978-1-4666-3890-7.ch007.m29and 978-1-4666-3890-7.ch007.m30 Furthermore, let 978-1-4666-3890-7.ch007.m31 be a completely continuous operator satisfying one of the following conditions:

  • 978-1-4666-3890-7.ch007.m32978-1-4666-3890-7.ch007.m33978-1-4666-3890-7.ch007.m34978-1-4666-3890-7.ch007.m35

  • 978-1-4666-3890-7.ch007.m36978-1-4666-3890-7.ch007.m37978-1-4666-3890-7.ch007.m38978-1-4666-3890-7.ch007.m39

Then 978-1-4666-3890-7.ch007.m40 has a fixed point in 978-1-4666-3890-7.ch007.m41

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