Existence Of Sign-changing Solutions Of Boundary Value Problems For Nonlinear Difference Equations

The purpose of this thesis is to study the existence of multiple solutions for several types of nonlinear difference equations with different boundary value conditions.By establishing suitable variational framework,and using invariant set of descending flow together with mountain pass lemma,we obtain a series of results on the existence of multiple solutions and sign-changing solutions for some second-order and fourth-order nonlinear difference equations.Furthermore,some examples are provided to illustrate the validity of our results.The main details of this thesis read as follows:In chapter 1,at first we introduce the research background of this topic.Then we state the research progress and our main results of this paper.Besides,some relevant preliminaries are given.The existence of sign-changing solutions of Neumann boundary value problems for two classes of second-order nonlinear difference equations are considered in Chapter 2.Applying invariant set of descending flow,we obtain that these difference equations admit at least three nontrivial solutions,which contain one positive,one negative and one sign-changing solution.Chapter 3 concerns a class of second-order nonlinear difference equation with Robin boundary condition.Similar to Chapter 2,some sufficient conditions on the existence of multiple solutions and sign-changing solutions are established.In addition,making use of mountain pass lemma,we get some results on the existence of two nontrivial solutions(one positive and one negative)under suitable conditions.On the basis of the above techniques and results,we discuss a class of fourthorder nonlinear difference equation in Chapter 4.We achieve that there are at least three nontrivial solutions which include one positive,one negative and one sign-changing solution for the periodic boundary value problem.