The Existence of Countably Many Positive Solutions for Nonlinear nth-Order Three-Point Boundary Value Problems

Yude Ji,Yanping Guo

doi:10.1155/2009/572512

Abstract

We consider the existence of countably many positive solutions for nonlinear th-order three-point boundary value problem , , , , , where , for some and has countably many singularities in . The associated Green's function for the th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.

Highlights

The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see 1–12 and the references therein
There are a few papers dealing with the existence of positive solutions for the nth-order multipoint boundary value problems with infinitely many singularities
Hao et al 13 discussed the existence and multiplicity of positive solutions for the following nth-order nonlinear singular boundary value problems: untatft, u 0, t ∈ 0, 1, 1.1 u 0 0, u 0 · · · u n−2 0 0, u 1 αu η, where 0 < η < 1, 0 < αηn−1 < 1, a t may be singular at t 0 and/or t 1

Summary

Introduction

The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see 1–12 and the references therein. In 15 , Ji and Guo proved the existence of countably many positive solutions for the nth-order ordinary differential equation untatfut 0, t ∈ 0, 1 , 1.3 with one of the following m-point boundary conditions: m−2. Motivated by the result of 13–15 , in this paper we are interested in the existence of countably many positive solutions for nonlinear nth-order three-point boundary value problem untatfut 0, t ∈ 0, 1 , 1.5 u 0 αu η , u 0 · · · u n−2 0 0, u 1 βu η , where n ≥ 2, α ≥ 0, β ≥ 0, 0 < η < 1, α β − α ηn−1 < 1, f ∈ C 0, ∞ , 0, ∞ , a t ∈ Lp 0,1 for some p ≥ 1 and has countably many singularities in 0, 1/2.

Preliminary Results

Preliminary Lemmas

Main Results

Example