Periodic Solutions Of Difference Equations With Continuous Variables

In this paper, we study the existence of periodic solutions of several classes of difference equations with continuous variables and obtain a series of new results about the existence of periodic solutions, which generalize the conclusions of the corresponding difference equations with discrete variables. Moreover, we study carefully the continuity of periodic solutions of difference equations with continuous variables, which does not exist in the difference equations with discrete variables.The paper is composed of four chapters.In the first chapter, we introduce the background of problem-researching, the recent development of the research in this field and what we have done in this paper. Furthermore, we illustrate the importance of our work in theory and practice.In the second chapter, we discuss the existence of periodic solutions, asymptotically periodic solutions and continuous periodic solutions of first order linear difference equation with continuous variables and obtain a series of sufficient conditions for the existence of unique periodic solution, infinitely many periodic solutions and conditions for having no periodic solution, respectively.In the third chapter, we discuss the existence of periodic solutions, asymptotically periodic solutions and continuous periodic solutions of first order linear difference equations with continuous variables and obtain some new results, which expand the results of the second chapter, but there are still some distinction because of the difference between the matrix and the number.In the fourth chapter, we discuss the existence of periodic solutions of first order nonlinear difference equation by applying contraction mapping theorem and Leray-Schauder fixed point theorem and the implicit function theorem. Finally, we compare the sufficient conditions of the existence of continuous periodic solutions for first order linear difference equation with continuous variables with that for first order nonlinear difference equation.