The Existence Of Positive Periodic Solutions To The Second Order Differential Equations With Singularities

In this paper,we investigate the existence of positive periodic solutions to some second order differential equations with an indefinite singularity of attractive type as well as a repulsive one by using Mawhin continuation Theorem of coincidence degree theory and the method of the upper and lower solutions.The paper is divided into four chapters,the main contents organized as follows:The first chapter has four sections.Firstly,we introduce the research status of the periodic problems of differential equations with singularities.Secondly,we summary the main works of this paper.Thirdly,we introduce Mawhin continuation Theorem of coincidence degree theory as well as the upper and lower solutions method.At last,we give some notations which are used throughout this paper.In second chapter,we study the existence of positive periodic solutions for the Lienard equation with a repulsive singularity by using Mawhin continuation theorem of coincidence degree theory.The interesting point is that we allow the coefficient of friction term has two singularities.The outline of the proof is that find a priori estimates for all positive periodic solutions to the Lienard equation with parameter ?,then we verify that the priori estimates satisfies all the assumptions in the Mawhin continuation theorem of coincidence degree theory,and so we can obtain some new results on the existence of periodic solutions to the equation.On the other hand,we get some corollaries about the non-existence of positive periodic solutions in the specified intervals.In third chapter,we study the existence of positive periodic solutions for Lienard equation with an indefinite singularity of attractive type by means of the upper and lower solutions method.Under the proper assumptions,we can construct the upper and lower functions to the Lienard equation,and then we can obtain a series of new results on the existence of positive periodic solutions.Our conclusions promote the existing research results.In the last chapter,we summarize the main results obtained in this thesis,and give a further consideration on related problems.