Some Research On Homoclinic And Periodic Solutions Of Second Order Differential Equations With Singularity

Since the formation of the Calculus Theory,we used the calculus equation to explain various natural phenomena,and have achieved remarkable results continuously.Differen-tial equations come from human social practice and are the most powerful mathematical method to solve practical problems.Singular differential equations are derived from many subject areas such as physics,biology and medicine,and have important application val-ue.In mathematics,singular conditions have an important influence on the dynamics of differential equations,which makes singular differential equations The research has re-ceived more extensive attention.In this paper,the existence of homoclinic or periodic solutions of four kinds of singular differential equations is studied by using coincidence degree extension theorem and critical point theory.This kind of problem has always been a hot topic in the study of differential equation theory.The main content is arranged as follows:The first chapter introduces the research status and development trend of periodic solutions and homoclinic solutions of differential equations,summarizes the main work of this paper,and briefly introduces some basic theories.In the second chapter,we study the existence of the same solution of a class of second order differential equations with singularity.The Mawhin coincidence degree extension theorem is used to obtain at least one homoclinic solution.In the third chapter,the existence of the same solution of a class of singular Rayleigh equations is studied.Using the coincidence degree extension theorem,the equation has one and only one solution.In the fourth chapter,we study the existence of periodic positive solutions for a class of singular neutral neutral Rayleigh equations.The Mawhin coincidence degree theorem is used to obtain at least one T-periodic solution.Finally,the fifth chapter studies the homoclinic-like solutions of a class of second-order Hamiltonian systems with singularities under locally quadratic potentials.