二阶和带p-Laplace算子的Hamilton系统周期解的存在性

【作者】 刘健；

【导师】 唐先华；

【作者基本信息】 中南大学 ， 应用数学， 2010， 硕士

【摘要】 本文主要利用变分方法研究了Hamilton系统周期解的存在性,全文共分为四章,其主要内容如下：第一章阐述了Hamilton系统的历史背景、应用以及它的研究方向,并介绍了本文的主要工作和创新点；第二章介绍了Hamilton系统的一些基础知识和本文所用到的一些定理；第三章是利用极小化原理和鞍点定理研究了在次线性条件和线性条件下n-维Duffing型系统周期解的存在性问题,并且得到了三个新的可解性结论,为判断Hamilton系统是否存在周期解提供了强有力的判断条件；第四章首先利用了鞍点定理研究了在次二次条件下带p-laplace算子的Hamilton系统周期解的存在性问题.之后又应用广义山路引理证明了在超二次条件下A=0时的周期解的存在性.更多还原

【Abstract】 The periodic solutions of Hamiltonian systems are studied by variational methods in this paper. This dissertation is divided into four chapters. The main contents are as follows:Firstly, the source, applications and its research direction of Hamiltonian systems are introduced in Chapter 1, and it also introduces main works and innovative contents;In Chapter 2, some basic knowledge and necessary theorems of Hamiltonian system are described.In Chapter 3, the existence of periodic solutions for n-dimension Duffing system are proved in sublinear potential condition and linear potential condition by using the least action principle and the saddle point theorem. And then three new solvable conclusions are obtained which are some powerful judge conditions to determine the Hamiltonian systems that there are some solutions or not.In Chapter 4, it is firstly considered that the existence of periodic solutions for the second order Hamiltonian system with a p-Laplacian operator under the "subquadratic" potential condition, and then it is also studied that the existence of periodic solutions of Hamiltonian system where A=0 by using the Generalized Mountain Pass Lemma.更多还原