Existence And Multiplicity Of Periodic Solutions For Some Second Order Differential Equations

The existence and multiplicity of periodic solutions have always been an important part of the qualitative theory of differential equations.Because periodic phenomena are very common in life,and they are widely used in medicine,physics,and astronomy,periodic solutions has been affected attention.This paper mainly studies the existence and multiplicity of periodic solutions which is about several types of second-order differential equations with damping terms.The article is divided into six chapters for discussion.In chapter 1,we mainly explain the research background and the current research status at home and abroad of periodic solutions of second-order differential equations,in the meanwhile we give the main research content of this paper.In chapter 2,we provide a method for determining the Green's function of a second-order non-homogeneous differential equation as positive.In chapter 3,we describe the Liebau-type differential equation and the periodic solution of the equation which is under more general conditions.First of all,the operators defined separately,and we can obtain that the operators are completely continuous.Secondly,we assume that the coefficient function satisfies the condition that the Green function is positive in Chapter 2.Thirdly,the fixed point theorem in the cone and fixed point exponential theorem are applied respectively to obtain the existence and multiplicity of periodic solutions of Liebau-type differential equations.In chapter 4,we study the properties of the periodic solutions that is about a class of nonlinear second-order differential equations.The periodic solutions of the equations are converted into solutions of the periodic boundary value problems.Compared with the previous research,the damping term is added,in the meanwhile we also consider the singularities of the function and the fact that it can be negative and also variable.First,linear and non-linear operators are defined.According to Arscoli-Arzele theorem,we obtain that the operators are completely continuous.Next,it is also assumed that the coefficient function satisfies the condition that the Green function is positive in Chapter 2.Then,compare the relationship between the nonlinear term and the first eigenvalue.Finally,we can draw conclusions.Three examples aregiven at the end of this chapter to verify the correctness of our results.In chapter 5,we consider a class of functional differential equations with damping terms.First,the completely continuous operator is defined,and the relationship between the operator and the parameters is obtained.And then it is also assumed that the coefficient function satisfies the condition that the Green function is positive in Chapter 2.Next,according to apply the fixed point theorem in the cone,it is obtained that when the parameter meets certain conditions,the equation has one,two or no periodic solution.Finally,two examples are given to verify the correctness of the results,and it is found that if only the delay function is changed,the number of periodic solutions will change,and the situation of one solution is verified by numerical simulation.In chapter 6,we give a summary of the research content in this article,and prospect of research is given.