临界点理论在时滞微分方程周期解的应用

【作者】 郭志明；

【导师】 徐远通；

【作者基本信息】 中山大学 ， 基础数学， 2001， 博士

【摘要】 本篇博士论文主要应用和推广临界点理论来研究含有偏差变元非线性微分方程的周期解、多重周期解与次调和解的存在性问题.本文共分四章.第一章简述问题产生的历史背景、发展现状及本文的主要工作.第二章建立一种新的几何指标理论（称为广义Z_p几何指标理论）,它是一个具有维数性质的规范指标,推广了Z₂指标与文献中的一些其它Z_p指标.应用这一指标理论,讨论Hamilton系统与二阶Hamilton系统次调和解的存在性.在相同的假设条件下,我们得到了更多的次调和解,并对R.Michalek、刘嘉荃等提出的猜测给出一个更精确的回答,改进了某些文献中的结果.本章还对位势变号的二阶Hamilton系统进行了讨论,证明了其非平凡周期解存在性的一个结论,从而在一定条件下解决了Antonacci[5]在Nonlinear Anal.T.M.A.上提出的一个公开问题.第三章首先给出微分方程具有变分框架的几个充分条件,根据这些结论,分别对几类微分差分方程与中立型泛函微分方程建立了适当的变分框架.借助于第二章建立的广义Z_p指标理论,应用最新的方法,我们证明了次线性耦合型时滞微分方程多重周期解存在性定理;并且研究了时滞倍周期的一阶与二阶微分差分方程的次调和解.这些结论即使对于不含时滞的常微分方程组也是最新的.本章还首次应用临界点理论研究中立型泛函微分方程的次调和解,在变分泛函中允许含有时滞变量,从而给出了研究这类方程周期解的新途径.第四章应用鞍点约化方法与Morse理论,讨论一类时滞微分方程的非平凡周期解,得到了这类方程的一个三周期解存在性定理;利用最近发展的E⁺-Morse理论,对一类微分差分方程非常数周期解的个数给出一个下界,将关于自治Hamilton系统与二阶保守系统的有关结论推广到微分差分方程中去.在本章中,我们还给出广义Z_p几何指标理论在含有偏差变元的波动方程中的应用,得到周期解与次调和解存在性的两个结果,即使对正常变元的波动方程,这些结果也是新的.由于对Z_p几何指标理论作了实质性的推广,成功地将临界点理论应用于研究时滞微分方程的周期解,并解决了若干重要问题.所以,本篇博士论文对泛函微分方程定性理论的发展有重要的价值.更多还原

【Abstract】 This PH.D.Thesis mainly concerns the applications and generalizations of critical point theory to investigating periodic solutions,multiple periodic solutions and subharmonic solutions for nonlinear differential equations with deviating arguments.It is composed of four chapters.In ChapterⅠ,we introduce the historical background and the recent development of problems to be studied,and main results of this paper are also outlined.In ChapterⅡ,a new geometrical index is developed,which is a normal index with dimensional property and is a generalization of the Z₂ index and Z_p index in the literature. By using this newly developed index theory,we investigate the multiplicity of subharmonic solutions for nonlinear Hamiltonian systems and second order Hamiltonian systems. Under the same hypotheses,more subharmonic solutions are obtained than those given in the literature,and by these results,a new answer to a conjecture posed by R.Michalek and Liu Jiaquan is given.In this chapter,we also establish an existence theorem of nontrivial periodic solutions to a class of second order Hamiltonian systems with potential changing its sign,and give a solution under centain hypotheses to one of the open problems posed by Antonacci.In ChapterⅢ,some sufficient conditions verifying variational structure for a given differential system are introduced.According to these results,we develop the variational structures for some first and second order differential difference equations,and study the existence of multiple periodic solutions and subharmonic solutions for these equations by newly developed Z_p geometrical index theory.Some new existence and multiplicity results are obtained for these delay differential systems and even for related ordinary differential systems.We also apply critical point theory to investigate the subharmonic solutions for a class of neutral functional differential equations,and therefore provide a new approach to deal with such problems.In ChapterⅣ,nontrivial periodic solutions for a delay differential equations has been studied.By using saddle point reduction and Morse theory,we obtain an existence theorem of three periodic solutions for this equation.By means of E⁺-Morse theory established by A.Abbondandolo,we also give a lower bound of the number of nontrivial periodic solutions for a differential difference equation.These conclusions generalize the same results on autonomous Hamiltonian systems and second order conservative systems to differential difference equations.In the last two sections of this chapter,we give two applications of the new Z_p geometrical index theory to the existence of periodic solutions and subharmonic solutions for a class of wave equations with deviating arguments.Some new existence and multiplicity theorems are given,which improve the related results on wave equations with normal variables. Due to a substantial generalization of Z_p geometrical index theory,we succeed in applying critical point theory to investigate the periodic solutions of delay differential equations,and solve some important problems.Therefore,this Ph.D.thesis will play an important role in the development of qualitative theory of functional differential equations.更多还原