Triple Positive Solutions of Fourth-Order Four-Point Boundary Value Problems for -Laplacian Dynamic Equations on Time Scales

Mei-Qiang Feng,Xiang-Gui Li,Wei-Gao Ge

doi:10.1155/2008/496078

Abstract

Recently, there have beenmany papersworking on the existence of positive solutions to boundary value problems for differential equations on time scales see, e.g., 1–4 and the references therein . This has been mainly due to their unification of the theory of differential and difference equations. An introduction to this unification is given in 5 . Now, this study is still a new area of fairly theoretical exploration in mathematics. However, it has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models see, e.g., 1, 5 . We let T be any time scale nonempty closed subset of R and let a, b be subset of T such that a, b {t ∈ T : a ≤ t ≤ b}. Thus, R, Z, N, No, that is, the real numbers, the integers, the natural numbers, and the nonnegative integers, are examples of time scales. In this paper, we study the existence of multiple positive solutions for the fourth-order four-point nonlinear dynamic equation on time scales with p-Laplacian:

Highlights

There have been many papers working on the existence of positive solutions to boundary value problems for differential equations on time scales see, e.g., 1–4 and the references therein
We study the existence of multiple positive solutions for the fourth-order four-point nonlinear dynamic equation on time scales with p-Laplacian: φp xΔ∇ t ∇Δ − wtfxt 0, t ∈ 0, 1, 1.1
By using a new triple fixed-point theorem due to Avery 8 in a cone, we prove that there exist at least triple positive solutions of problem 1.1 - 1.2

Summary

Introduction

There have been many papers working on the existence of positive solutions to boundary value problems for differential equations on time scales see, e.g., 1–4 and the references therein. This has been mainly due to their unification of the theory of differential and difference equations. We study the existence of multiple positive solutions for the fourth-order four-point nonlinear dynamic equation on time scales with p-Laplacian: φp xΔ∇ t ∇Δ − wtfxt 0, t ∈ 0, 1 ,. P-Laplacian problems with two-, three-, m-point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively see 6, 7 and the references therein. Throughout this paper, we assume that T is a closed subset of R with 0 ∈ Tk2 , 1 ∈ Tk2

Preliminaries

Main results