Research On Variational And Averaging Methods In Several Kinds Of Differential Equations

In this paper,we study the boundary value problems of 2nth-order differen-tial equations and second-order impulsive differential equations by variational method.The existence and multiplicity of solutions are obtained.In addition,we generalize the theory of averaging method and establish the higher-order zero point theorem.We obtain the existence of periodic and anti-periodic so-lutions by applying the averaging method to perturbed differential system.Ac-cording to the research methods and problems,this paper is divided into the following chapters:The first chapter is the introduction.It introduces the background of the research topic,research methods,the basic ideas of the variational and aver-aging methods,the status quo of the research topic,the research contents,the difficulties and innovations.The second chapter is the basic knowledge.It introduces the Holder in-equality,the critical point theorems,the maximum principle,the implicit func-tion theorem,and the existence and uniqueness theorem.The third chapter is the application of variational method to 2nth-order dif-ferential equation with Stunn-Liouville operator.By using critical point theo-rems due to Bonanno and Candito,the existence of three classical solutions for the boundary value problem is obtained.The fourth chapter is the applications of variational method to two im-pulsive problems.The first problem is the boundary value problem with non-instantaneous impulses.The existence of solution is obtained by Lax-Milgram theorem.The second problem is the boundary value problem with instanta-neous impulses and non-instantaneous impulses.The existence of solution is obtained by Ekeland variational principle.The fifth chapter is the generalization of averaging method and its ap-plication in higher-order perturbed differential system.We firstly establish a higher-order zero point theorem to prove the existence of zeros such as g(z,?)=g0(z)+?i=1m?igi(z)+O(?m+1).Secondly,the higher-order zero point theorem is applied to the higher-order perturbed differential system by the averaging method and the existence of periodic and anti-periodic solutions of the per-turbed system is obtained.The sixth chapter is the summary of current work and the prospect of future work.