【作者】 张志信；

【导师】 蒋威；

【作者基本信息】 安徽大学 ， 应用数学， 2006， 硕士

【摘要】 退化时滞微分方程作为模拟现实世界中相关模型的有效工具，很早就引起了数学家的注意，在对诸如工业工程系统，电力系统，生态系统，金融系统等等实际系统的建模、设计、分析和应用研究中，人们发现实际系统中普遍存在退化、时滞现象，因此学者们对退化时滞系统进行了系统的研究并取得了全面的进展，特别是对于退化时滞微分方程的周期解和概周期解的研究得到了较为系统的研究成果[1-30]。然而，随着对比较复杂的数学模型(如：生物数学人工神经网络)的研究表明，现在的周期系统理论还需进一步完善。 本文就退化时滞微分方程的周期解及概周期解等问题作了一些讨论，并得出了一些结论。 本文共分四章，下面将四章内容作简要概述： 第一章，给出本文所必需的预备知识，介绍了线性系统指数型二分性、矩阵测度的基本知识。 第二章，利用线性系统指数型二分性理论和泛函分析方法，并通过技巧性代换获得了保证一类中立型系统周期解存在性和唯一性的充分性条件。 第三章，讨论混合型退化时滞微分方程周期解存在的充分必要条件，并对二维的混合型退化时滞微分方程的周期解存在性给出代数判据。 第四章，考虑具有无穷时滞的非线性积分微分方程概周期解的存在性和唯一性问题，利用矩阵测度和不动点方法获得了一类方程的概周期解的存在性与唯一性的充分条件。更多还原

【Abstract】 Degenerate differential system with delay as an effective method of emulating models in realistic world have aroused the notice of mathematicians. In the course of practical systems emulation, design, analysis and application research, people found that there were many degenerate and time delay phenomena existing in many practical fields such as industrial project system, power system, ecosystem, financial system and so on, so that many scholars have research the system with degenerate and delay systematically and made comprehensive progress, Particularly, many systematic results[l-30] have been obtained on the studying of periodic solutions and almost periodic solutions of the degenerate differential system with delay. However, many more complex mathematics models( as: Biological mathematics artificial nerve network) research show that Current period systematic theory need to be improved further. And in the paper, the periodic solution and almost periodic solution of the degenerate differential systems with delay are discussed and many important results are also given in it.There are four chapters in this paper.In the first chapter, the preliminary knowledge which is necessary in the paper is given. The basic concept of exponential dichotomies of linear system and matrix measure are given.In chapter 2, By combining the theory of exponential dichotomies of linear system and the method of functional analysis, some sufficient conditions that guarantee the existence and uniqueness of periodic solution of the systems are obtained.更多还原