Existence Of Periodic Solutions For Some Classes Of Higher Order Difference System

As an important banch of modern mathematics, differential equation and difference equation have been widely applied in the area of computer science, economy, neutral net, ecology and control theory, so it is meaningful for the study of the solution of differential equations and difference equations. In the last decades, many researchers have deeply studied the existence and multiplicity of periodic solutions of differential equations with different approaches, such as critical point theory(which includes minimax theory, geometrical index theory and Morse theory), fixed point theory, coincidence degree theory, the Kaplan-Yorke method and so on. Among these approaches, critical point theory is a powerful tool to deal with such problem. However, results on periodic solutions of difference equations by using critical point theory are very scare in the literature because of the difficulty of finding the suitable variation structure. In this dissertation, the existence of periodic solutions for some class of higher order difference equation(system) and of solutions for an elliptic system is studied by using critical point theory, and a series of new results are obtained. The contents of the dissertation are introduced as follows:Firstly, we sketch the development of methods of variation. The historical background and the recent development of elliptic equations(systems) and Hamil-tonian systems related to our problems are introduced. At the same time, the main contents of the dissertation are outlined.Secondly, we construct some class of higher order difference system, and chang the existence of periodic solutions of some higher order difference equation (systems) and of solutions for an elliptic system into the existence of critical points of corresponding functional on suitable function space after finding the suitable variation structure. We have developed the second order models.In chapter 2, we study a higher-order difference system in the case that the nonlinearity is asyptotically linear and superlinear by combining Morse theory with computation of critical point groups at first. We conclude the following results: In the case that the nonlinearity is asymptotically linear, the existence of nontrivial periodic solutions are obtained under the conditions of resonance or nonresonance if the Morse index at infinity different from the one at zero; In the case that the nonlinearity is superlinear, at least three nontrivial periodic solutions are obtained. Then, multiple and infinite many periodic solutions for the higher order difference system are obtained by using linking theory and symmetric mountain pass lemma respectively, and some results improve or extend the related results in the literatures.At last, by combining Morse theory with Lyapunov-schmidt reduction method and three critical points theory, multiple and infinitely many periodic solutions for the higher order difference system are obtained. The method of studying differential equation has been developed to that of difference equation.In chapter 3, The existence of nontrivial periodic solutions of a higher order functional difference equation is considered by linking theorem. Nontrivial periodic solutions are obtained.The existence of nontrivial periodic solutions of a higher order difference equation with resonance is considered in chapter 4. Some sufficient conditions of the existence of at least three or four nontrivial periodic solutions are obtained by using local linking and abstract angle conditions at infinity.In the last chapter we consider an elliptic system by using the generalized Mountain Pass Lemma, and some results improve or extend the related resultes in the literatures.