Sign-Changing Solutions for Discrete Second-Order Three-Point Boundary Value Problems

Tieshan He,Fengjian Yang,Wei Yang

doi:10.1155/2010/705387

Abstract

We consider the second-order three-point discrete boundary value problem. By using the topological degree theory and the fixed point index theory, we provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions. As an application, an example is given to demonstrate our main results.

Highlights

In this paper, we consider the following second-order three-point discrete boundary value problem BVP : Δ2u t − 1 f t, u t 0, t ∈ 1, n, 1.1 u 0 0, u n 1 αu m, where n ∈ {2, 3, . . .}, 1, n is the discrete interval {1, 2, . . . , n}, m ∈ 1, n, 0 ≤ α ≤ 1, Δu t u t 1 − u t, Δ2u t Δ Δu t, and f : 1, n × R → R is a continuous function.Boundary value problems for difference equations arise in different areas of applied mathematics and physics
By using the topological degree theory and the fixed point index theory, we provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions
Existence and multiplicity of positive solutions or nontrivial solutions for boundary value problems of difference equations have been extensively studied in the literature; see 1–9 and the references therein

Summary

Introduction

We consider the following second-order three-point discrete boundary value problem BVP : Δ2u t − 1 f t, u t 0, t ∈ 1, n , 1.1 u 0 0, u n 1 αu m , where n ∈ {2, 3, . . .}, 1, n is the discrete interval {1, 2, . . . , n}, m ∈ 1, n , 0 ≤ α ≤ 1, Δu t u t 1 − u t , Δ2u t Δ Δu t , and f : 1, n × R → R is a continuous function.Boundary value problems for difference equations arise in different areas of applied mathematics and physics. By using the topological degree theory and the fixed point index theory, we provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions.

Objectives

Results

Conclusion