The Existence Of Periodic Solutions To Liénard Equations With Singularities

In this paper,we investigate the existence of positive periodic solutions to singular differential equations with an indefinite attractive or a repulsive singularity.The paper is divided into five chapters,the main contents are organized as follows:The first chapter has three sections.At first,we introduce the research status of the periodic problems with singular differential equations.Secondly,we summary the main works of this paper.At last,we introduce Mawhin continuation theorem of coincidence degree theory.In second chapter,we use Mawhin continuation theorem of coincidence degree theory to study the existence of positive T-periodic solutions to a class of Liénard equations with a repulsive singularity.The interesting point is that we not only prove a sufficient condition for the equations to have a positive periodic solution,but also we present the necessity.The innovation of this chapter is that we allow the coefficient of the friction term f(x) has a singularity at the origin and the restoring force trem is superlinerly incerasing.In third chapter,we use Mawhin continuation theorem of coincidence degree theory to study the existence of positive T-periodic solutions to a class of Liénard equations with an indefinite attractive singularity.The innovation of this chapter is that we allow ?< 1 and ?(t) is not necessarily always positive,that is,we allow ?(t)? 0 holds in some subinterval with positive measure.The fourth chapter,we use Mawhin continuation theorem of coincidence degree theory to study the existence of positive T-periodic solutions for delay second order differential equations with a repulsive singularity.The innovation of this chapter is that we allow ?(t) changes its sign,and we have a time-delay in our problem.At the end of this paper,we summary the main results in this thesis,and give a further consideration on related problems.