Existence Of Periodic Solutions For Some Classes Of Dynamic Equations With P-Laplacian

Differential equations and difference equations have played an important role in natural science, biology, medicine, economics and control theory, etc.. The theory of dynamic equations on time scales, as the unification of differential and difference equations, can explain deeply the essential difference between them, and provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. Hence, the studying of dynamic equations on time scales is worth with theoretical and practical values.This thesis aims to study the existence of periodic solutions for p-Laplacian differential equations and dynamic equations on time scales. By using Mawhin's continuation theorem, general Mawhin's continuation theorem and inequality technique, we give some new sufficient conditions for the existence of periodic solutions to second-order, third-order and fourth-order p-Laplacian equations. In this thesis, with the help of Mawhin's continuation theorem, we study the second-order and third-order dynamic equations on time scales for the first time, and obtain some significative results.This paper is divided into five chapters and organized as follows:In chapter 1, we present the history and development of this problem, basic definition of time scales and the development of this theory, the end, we give the brief statement of the main work.There are three sections in chapter 2, applying Mawhin's continuation theorem, general Mawhin's continuation theorem and inequality technique, we study the existence and uniqueness of periodic solutions of three classes of differential equations, including general Liénard type p-Laplacian equations, third-order p-Laplacian equations and fourth-order p-Laplacian equations, and obtain some preferable sufficient conditions, which extend the results of second-order to third-order and fourth-order differential equation. The results in this chapter embodied in Advances in Difference Equations, Journal of University of Jinan (Science and Technology) and Acta Scientiarum Naturalium Universitatis Sunyatseni, separately.There are four sections in chapter 3, we study the existence of periodic solutions of three classes of dynamic equations with p-Laplacian, where the Mawhin's continuation theorem is first time used to research the second-order and third-order dynamic equations which with p-Laplacian operator, and obtain some excellent results. It makes the thesis with lively innovation sense. The results of first section embodied in the SCI journal named as Advances in Difference Equations. In chapter 4, by using critical point theory, we consider the periodic solutions of a class of high-order difference equations, and give some sufficient conditions about the existence of periodic solutions for this class of equations.