The Existence Of Periodic And Subharmonic Solutions Of Several Classes Of Higher Order Difference Equations

In the last decade, nonlinear difference equation theory has been widely applied in the discrete models of computer science, economy, neutral net, ecology and control theory. There has been much progress on the qualitative properties of difference equations, which include results on stability and attractivity, results on oscillation and other topics. However, results on periodic solutions of difference equations are very scare in the literature because of lack of necessary methods. The main purpose of this paper is to study the existence of periodic solutions of some higher order difference equation (systems) by virtue of the critical theory.Based on the second order models, we consider some higher order difference models. We mainly make much investigation for the existence of periodic and sub-harmonic solutions of these new models. This thesis is composed of three chapters.In the first chapter, we state the existed related work and the origin of the problems we discuss. And the main works of this paper are also simply introduced.In the second chapter, at first we consider a higher-order difference system. By constructing suitable variational structure and using the linking theory and the maintain theory, we obtain new sufficient conditions for the existence of periodic solutions of this system. Later we still consider a higher order superlinear difference equation. The existence of periodic solutions of this equation is established by using linking theory. A series of new are acquaired in it. And some of them improve or extend the related resultes in the literatures.The purpose of the third chapter is to study the existence of periodic and sub-harmonic solutions of a higher order discrete system with delays. By constructing suitable variational structure and using the the maintain theory, we obtain some new criteria which are easily checked.