Study On Periodic And Homoclinic Solutions Of Second Order Hamiltonian Systems

This paper mainly considers two kinds of problems:the first category is the multiplicity of periodic solutions for second order Hamiltonian systems with asymptotically quadratic conditions,the second category is the multiplicity of homoclinic solutions for second order Hamiltonian systems with local conditions at the origin.For the first problem,this paper has weakened the assumptions of paper[1].The multiplicity result of odd period solutions is proved via minimax principle(see chapter 2).For the second problem,this paper has weakened the assumptions of paper[2,3].By constructing a new truncated functional different from papers[2,3],the multiplicity result of homoclinic solutions is proved via a new Symmetric Mountain Pass Lemma established by Kajikiya(see chapter 3).This paper is divided into four parts.In the first chapter,which is the pref-ace,we introduce the research background,history and methods of Hamiltonian systems and the research content,methods,innovation of this paper.In the second chapter,we introduce the research history,methods and related basic knowledge of periodic solutions for second order Hamiltonian systems,and give the proof of the multiplicity of periodic solutions for the Hamiltonian systems as well as an example.This result has been published in the mathematical SCI journal Ac-ta Mathematica Sinica,English Series.In the third chapter,we introduce the research methods,main theorem and related basic knowledge of homoclinic solu-tions for second order Hamiltonian systems,and give the proof of the multiplicity of homoclinic solutions for the Hamiltonian systems as well as an example.This result which has been submitted to the mathematical SCI journal Chinese Annals of Mathematics,Series B is under review at present.In the fourth chapter,we summarize the results obtained in this paper,and point out unresolved problems and future research directions.