Sign-Changing Solutions for Discrete Dirichlet Boundary Value Problem

Yuhua Long,Baoling Zeng

doi:10.4236/jamp.2017.511182

Abstract

Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.

Highlights

Open AccessLet N, Z and R denote sets of all natural numbers, integers and real numbers, respectively
Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem
We consider the existence of sign-changing solutions, positive solutions and negative solutions for the following second-order nonlinear difference equation with Dirichlet boundary value problem (BVP for short)

Summary

Introduction

Let N, Z and R denote sets of all natural numbers, integers and real numbers, respectively. We consider the existence of sign-changing solutions, positive solutions and negative solutions for the following second-order nonlinear difference equation with Dirichlet boundary value problem (BVP for short). Agarwal [1] established the existence results of multiple positive solutions. By making using of a variety of methods and techniques, such as Leray-Schauder degree theory, fixed point index theory, topological degree theory, invariant sets of descending flow, critical point theory and etc. Invariant sets of descending flow play an important role, which was first used by Sun [10]. Making use of invariant sets of descending flow, [15] studied periodic boundary value problem. By using invariant sets of descending flow and Mountain pass lemma, we obtain sign-changing solutions, negative solutions and positive solutions for (1.1)

Preliminaries and Main Results

Applications

Conclusion