Several Existence Theorems of Multiple Positive Solutions of Nonlinear m-Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales

Wei Han,Shugui Kang

doi:10.1155/2009/584145

Abstract

By using fixed point theorems in cones, the existence of multiple positive solutions is considered for nonlinear -point boundary value problem for the following second-order boundary value problem on time scales , , , , where is an increasing homeomorphism and homomorphism and . Some new results are obtained for the existence of twin or an arbitrary odd number of positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems, respectively. In particular, our criteria generalize and improve some known results by Ma and Castaneda (2001). We must point out for readers that there is only the -Laplacian case for increasing homeomorphism and homomorphism. As an application, one example to demonstrate our results is given.

Highlights

In this paper, we will be concerned with the existence of positive solutions for the following boundary value problem on time scales:φ uΔ ∇ a t f t, u t 0, t ∈ 0, T, 1.1 m−2 m−2 φ uΔ 0 aiφ uΔ ξi, biu ξi, i1i1 where φ : R → R is an increasing homeomorphism and homomorphism and φ 0 0.Boundary Value ProblemsA time scale T is a nonempty closed subset of R
By an interval 0, T, we always mean the intersection of the real interval 0, T with the given time scale, that is, 0, T ∩ T
There has been much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales, for examples, see 1–6 and references therein

Summary

Introduction

We will be concerned with the existence of positive solutions for the following boundary value problem on time scales:. There has been much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales, for examples, see 1–6 and references therein. For the existence problems of positive solutions of boundary value problems on time scales, some authors have obtained many results in the recent years, especially 6, 7, 9, 10, 14, 15 and the references therein. To date few papers have appeared in the literature concerning multipoint boundary value problems for an increasing homeomorphism and homomorphism on time scales. Sang et al 6 investigated the nonlinear m-point BVP on time scales 1.1 and 1.2

Preliminaries and Some Lemmas

Existence Theorems of Twin Positive Solutions

Existence Theorems of Triple Positive Solutions

14 Here we used the inequality

Application