The Study On Periodic Solutions Of Second Order Differential Equations With Singularity

In the thesis,we study existence of positive periodic solutions for some singular second order differential equations by using Mawhin continuation Theorem of coincidence degree theory.The paper is divided into four chapters,the main contents organized as follows:The first chapter introduces the research status and development trend of periodic solutions of differential equations,summarizes the main work of this paper,and briefly introduces some basic theories.In the second chapter,we are concerned with the existence of positive periodic solu-tions for a singular Rayleigh equation.The outline of the proof is that we find a priori estimate for all positive periodic solutions to the Rayleigh equation,then we verify that the priori estimate satisfies all the assumptions in the Mawhin continuation theorem of coincidence degree theory,so we can obtain some new results on the existence of periodic solutions to the equation.The innovation of this chapter is that we allow restoring force term changes its sign.The idea how to obtain the main results is similar to the previous chapter.Different from the second chapter,the third chapter intends to find positive periodic solutions on subset of CT1 rather than on whole space.In this chapter we define an open subset and estimate priori bounds of all possible positive periodic solutions on the subset.The method of proof relies on a continuation theorem of coincidence degree principle.In the fourth chapter,we study the existence of positive periodic solutions of relativis-tic singular differential equations.First,based on the continuation theorem of coincidence degree theory,we study the existence of positive periodic solutions.Secondly,we define a set and estimate priori bounds of all possible positive periodic solutions on the set.In the last chapter,we summarize the article and put forword the prospect of the future work.