Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

Haiping Shi,Tao Zhou,Xia Liu

doi:10.1155/2018/6703503

Abstract

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence ofndistinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

Highlights

Throughout this paper, we denote by N, Z, R the sets of all natural numbers, integers, and real numbers, respectively
Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence of n distinct pairs of nontrivial solutions, and nonexistence of solutions
We are concerned with the existence and nonexistence solutions to the fourth-order nonlinear difference equation

Summary

Introduction

Throughout this paper, we denote by N, Z, R the sets of all natural numbers, integers, and real numbers, respectively. We are concerned with the existence and nonexistence solutions to the fourth-order nonlinear difference equation. We may think of boundary value problem (BVP) (1), (2) as being a discrete analogue of the following fourth-order nonlinear differential equation:. Using the critical point theory, Yang [27] studied the following higher order nonlinear difference equation: n. Some sufficient conditions for the existence of the solution to the boundary value problem (7), (8) are obtained. We shall study the boundary value problem for a fourth-order nonlinear difference equation (1), (2). Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence of n distinct pairs of nontrivial solutions, and nonexistence of solutions. For the basic knowledge of variational methods, the reader is referred to [30,31,32]

Preliminary Lemmas

Main Results

Proofs of the Main Results

Some Examples