Dynamical Behaviour Of Difference Equations With Resonance

In this paper,we study the dynamical behaviour of some difference equations with resonance.Undr some suitable assumptions,we obtain the existence and multiplicity of nontrivial solutions of the problems via variational methods.The results of this paper improve,extend,and complement some related results in the relevant literatures.The paper is divided into six chapters,the details are as follows:In Chapter 1,the research background and status of difference equations are briefly introduced.We present the main results of this paper.In Chapter 2,we study a boundary value problems of second order differ-ence equation with resonance at zero.There are two cases to be considered,first,when the nonlinearity is sublinear case,we obtain the existence and multiplicity of nontrivial solutions of the problem by means of Morse theory,second,when the nonlinearity is superlinear case,we define a contraction deformation,the ex-istence and multiplicity of nontrivial solutions of the problem are obtained by the deformation lemma and Morse theory.In addition,some examples are given to illustrate our results.In Chapter 3,we study the boundary value problems of-Laplacian difference equations containing both advance and retardation depending on a parameter.We make use of a suitable oscillating behaviour of the nonlinearity at infinity or at zero,establish a real continuous interval for appropriate parameters,the existence of infinitely many solutions are obtained.In particular,when the problem is resonant at infinity,we give that the parameterbelong to suitable interval whose left endpoint does not contain any variable.In Chapter 4,we consider the discrete nonlinear Schr¨odinger equations with resonance and unbounded potentials.The method is to find a bounded critical sequence by means of linking methods in critical point theory.Under suitable conditions,there is a subsequence of this critical sequence which converges to some element u inl~2,we prove thatis one nontrivial homoclinic solution of the problem.In Chapter 5,we consider a discrete nonlinear periodic Schr¨odinger equations with resonance.When the temporal frequency w?(?,?),we obtain a bounded critical sequence by linking theorem,moreover,there is a subsequence of this critical sequence which converges to some element u ?0 in l~2,we prove thatis one nontrivial homoclinic solution of the problem.In Chapter 6,a summary of this paper and the outlook for future research work are stated.