Periodic Solutions Of Subquadratic Difference Equations

In this paper, we study the existence of periodic solutions of a class of higher-order difference equations or higher-dimensional difference systems, we obtain some important results, some of which extend some theorems in relevant papers.The paper is composed of there chapters.In the first chapter, we introduce the background of problem-searching, the recent developments of the research in this field and what we have done in this paper. At the same time, some basic definitions are given. Furthmore, we illustrate the importance of our work in theory and practice.In the second chapter, firstly, we discuss the existence of periodic solutions and subharmonic solutions of a class of higher-oder difference equations being under the conditions of subquadratic and half positive-definited matrix by using critical theory. We gain some new results and extend two results of Guo and Yu in paper [ 29 ]. Secondly, we discuss the existence of periodic solutions and sub-harmonic solutions of the above class of subquadratic higher-order difference equations being under the condition of positive-definited or negative-definited matrix by using minimax method. We gain some new results.In the third chapter, we discuss the existence of nontrivial periodic solutions of a class of second-order difference equations, which being with the condition of subquadratic linearity. We gain some important results.In the fourth chapter, we discuss the existence of periodic solutions of a class of higher-order difference equations under the general condition. We gain an important results.