导　　师： 庚建设

学科专业： 070104

授予学位： 博士

作　　者： ;

机构地区： 湖南大学

摘　　要： 本篇博士学位论文主要应用临界点理论/(包括直接变分法、对偶变分法、扰动技巧、对偶最小作用原理、极小极大方法和几何指标理论/)研究非线性离散Hamilton系统的具有固定极小周期的次调和解的存在性与多重性。全文由如下四部分组成。 第一章简述问题的产生和研究的意义。周期现象普遍存在于自然科学的各个研究领域。周期解问题一直是广大学者和专家关注的问题，其中具有极小周期的周期解问题更是吸引着国内外许多著名学者的注意力。我们对离散Hamilton系统与二阶差分方程的周期解的研究现状进行回顾，同时归纳总结关于研究微分方程极小周期解的几种方法。然后，对本文的主要工作做简要介绍。 对遵从“变分”原理的系统采用变分法，将原系统的极小周期解问题转化为相应变分泛函临界点的相关问题进行研究是解决离散Hamilton系统极小周期解的存在性问题的一种基本方法。第二章采用直接变分法研究离散单摆方程及其推广形式的具有固定极小周期的次调和解的存在性，得到一系列全新的结果。 第三章利用对偶方法、扰动技巧与对偶最小作用原理研究二阶次二次离散Hamilton系统。当二阶离散Hamilton系统在无穷远处满足次二次增长条件时，得到了系统具有任意大极小周期的次调和解的充分条件，并且考虑了所得次调和解的一些性质。当二阶离散Hamilton系统在原点及无穷远处满足次二次增长条件时，通过考虑其对偶变分泛函，得到具有极小周期的次调和解的存在性结果。 极大极小方法与几何指标理论是研究泛函临界点存在性与多重性的强有力的工具。第四章考虑了一阶和二阶次二次离散Hamilton系统。首先采用对偶变分法，将原系统的周期解问题转化为对偶变分泛函的临界点问题� This dissertation mainly deals with the existence and multiplicity of subharmonic solutions with prescribed minimal periods to discrete Hamiltonian systems by applying the critical point theory including variational methods, duality variational methods, perturbation argument, dual least action principle, minimax theory and geometrical index theory. It is organized as follows by four parts. In Chapter 1, the historical background of problems to be studied and the significance of this dissertation are introduced. Periodic phenomena occur in various fields of natural science. The problem on periodic solutions attracts important attentions of many famous mathematicians all around the world, especially on periodic solutions with prescribed minimal periods. In this chapter, recent development of periodic solutions to discrete Hamiltonian systems and second-order difference equations is also viewed. Several kinds of methods for studying the minimal period problems of differential equations are outlined. And main results of this dissertation are also summarized. Reducing the problem on periodic solutions to discrete Hamiltonian systems which conform the variational principle to the problem on critical points of the corresponding variational functionals is one of main approaches to study the periodic solution problem. In Chapter 2, the existence of subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation and a class of second order discrete Hamiltonian systems are studied. Some new results on the existence of subharmonic solutions with prescribed minimal periods are obtained. By making use of Clark duality, perturbation technique and dual least action principle, the existence of subharmonic solutions to second order subquadratic discrete Hamiltonian systems is studied in Chapter 3. When the Hamiltonian function satisfies subquadratic growth conditions at infinity, some results on the existence of subharmonics which have any large minimal period are obtained.

关 键 词： 离散 系统 变分结构 临界点理论 几何指标 对偶变分 周期解 次调和解 极小周期

领　　域： [理学] [理学]