Homoclinic Orbits, Periodic Solutions And Boundary Value Problems For Nonlinear Difference Equations

The existence of homoclinic orbits, periodic solutions and boundary value problems for nonlinear difference equations is studied by using critical point theory in this dissertation. It will motivate the development of qualitative theory of difference equations. This dissertation is composed of five chapters. The content of the dissertation is as follows.Chapter 1 gives a brief introduction to the historical background, status and the up-to-date progress for all the investigated problems together with preliminary tools and main results in this dissertation.The existence of homoclinic orbits for periodic discrete nonlinear Schrodinger equations is obtained in Chapter 2 by using Mountain Pass Lemma in combination with periodic approximations. Two open problems proposed by Alexander Pankov are partially solved under certain hypotheses.In Chaper 3, the existence of homoclinic orbits for higher order nonlinear difference equations is studied by using Mountain Pass Lemma. Some criteria for the existence of homoclinic orbits of these equations with periodic assumptions and without periodic assumptions are worked out, respectively. Moreover, some homoclinic orbits decaying exponentially at infinity are obtained. Our results extend some known results in the literature.The existence of periodic solutions to second order nonlinear difference equations is investigated in Chapter 4. The solutions to second order sublinear difference equations by using Saddle Point Theorem and to second order neither superlinear nor sublinear difference equations by using Linking Theorem are discussed. Some new results are obtained.In Chapter 5, boundary value problems to a class of second order nonlinear difference equations are studied. By establishing variational structure and applying critical point method, the existence of solutions of boundary value problems for second order nonlinear difference equations on a finite discrete segment with various boundary value conditions is considered. Some new sufficient conditions are obtained.