Positive Periodic Solutions Of Three Kinds Of Singular Differential Equations

Singular differential equations are widely used in many scientific fields,such as applied mathematics,applied physics,cybernetics and so on.Many physical phenomena can be transformed into singular differential equation models for study,so the existence of positive periodic solutions of singular differential equations play an indispensable role in many practical problems,which provides a framework for mathematical models in life.It is of great significance to predict the development of things in the future.In recent years,the problem of periodic positive solutions of singular differential equations has attracted the attention of many scholars at home and abroad.A large number of topics about the existence of positive periodic solutions of singular differential equations have emerged.These topics are mainly about the study of positive periodic solutions of repulsive singular differential equations and attractive singular differential equations.However,the study of repulsive-attracting singular differential equations,indefinite singular differential equations,higher-order singular differential equations and neutral singular differential equations remain to be further solved,the existence of positive periodic solutions are obtained by using different methods.The positive periodic solutions of the attracting-repulsive singular differential equation,Friedmann equation and the generalized Lazer-Solimini equation with indefinite weights are obtained respectively.An in-depth study has been carried out.In Chapter 1,we introduce the domestic and foreign trends of the existence of positive periodic solutions of singular differential equations at home and abroad,and put forward the main research content of this paper.In Chapter 2,some symbols and lemmas used are given.In Chapter 3,we study the existence of positive periodic solutions for a class of attractive-repulsive singular differential equations.This part mainly uses the fixed point theorem in the cone.Combined with the regularity of Green’s function,the existence of positive periodic solutions for a class of attractive-repulsive singular differential equation is discussed.In Chapter 4,the existence of positive periodic solutions of Friedmann’s equation are studied.In this chapter,we mainly use the Krasnoselski?i’s-Guo fixed point theorem,the Schauder’s fixed point theorem and the continuous theorem of Man′asevich-Mawhin to give a sufficient condition for the existence of at least one positive periodic solution of the Friedmann’s equation.In chapter 5,we study the existence of positive periodic solutions for a class of generalized Lazer-Solimini equations with indefinite weights.In this part,the proof is based on the continuous theorem of Man′asevich-Mawhin,and the results obtained are applicable to strong and weak singularities.In Chapter 6,we present a summary and prospect of this paper.