Periodic Solutions And Homoclinic Solutions For Several Types Of Differential Equations

In this dissertation,we mainly study the existence of periodic solutions and homoclinic solutions for several types of differential equations by using the dual least action principle,Mountain Pass lemma and Critical Point theory.The whole dissertation is divided into four chapters,as follows:Chapter 1: we mainly introduce the research background,research status and the main work of this dissertation.Chapter 2: we study the existence of periodic solutions for a class of convex Hamiltonian systems with a perturbation.Compared with related literatures,the main difficulty of the problem is due to the appearance of the perturbation.We prove the problem admits at least one non-zero periodic solution by employing the dual least action principle.In the fourth part of this chapter,a specific example is given to verify the main conclusion.Chapter 3: we study the existence of homoclinic solutions for a class of fourth-order impulsive differential equation.That is,on the basis of the existing literature,the potential and the impulse term are added.Under certain conditions,the nontrivial homoclinic solution of the system is also obtained by employing Mountain Pass theorem,which makes the existing literature popularized.Chapter 4: we study the existence of homoclinic solutions for a class of Hamiltonian systems with impulses,and establish the sufficient condition for the existence of existing at least one homoclinic solution for the system by employing Mountain Pass lemma,which extended the existing results in the literature.