【作者】 刘鹏；

【导师】 郭飞；

【作者基本信息】 天津大学 ， 基础数学， 2020， 硕士

【摘要】 本文主要研究了两类问题:一类是带渐近二次条件的二阶Hamilton系统周期解的多重性问题,另一类是在原点处带有局部条件的二阶Hamilton系统同宿解的多重性问题.关于问题一,我们弱化了文献[1]的条件,利用极大极小方法证明了周期奇解的多重性结果(具体见第二章).关于问题二,我们弱化了文献[2,3]的条件,且构造了一个不同于文献[2,3]的全新截断泛函,之后利用新的对称山路引理(由数学家Kajikiya确立)证明了同宿解的多重性结果(具体见第三章).本文总共分为四章:第一章绪论介绍了 Hamilton系统的研究背景、过往历史、研究方法和本文的研究内容、方法、创新性;第二章介绍了二阶Hamilton系统周期解的研究历史、方法以及相关的预备知识,证明了该系统周期奇解的多重性结果,并给出了具体实例.此结果已发表在数学类SCI 期刊《Acta Mathematica Sinica,English Series》上;第三章介绍 了二阶Hamilton系统同宿解的研究方法、主要定理以及相关的预备知识,证明了该系统同宿解的多重性结果,并给出了具体实例.此结果已投稿在数学类SCI期刊《Chinese Annals of Mathematics,Series B》上,目前处于审稿中.第四章总结了本文已得到的结果,并指出尚未解决的问题和之后的研究方向。更多还原

【Abstract】 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.更多还原