【作者】 肖嘉慧；

【导师】 姚慧丽；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2011， 硕士

【摘要】 众所周知,在讨论具有逐段常变量微分方程的概周期型解时,往往要用到相关差分方程的概周期型序列解。特别是近几年来,越来越多的数学工作者开始研究差分方程的各种解的存在性和唯一性问题,并把差分方程和微分方程结合起来,从而进一步来研究微分方程概周期型解的存在性和唯一性问题。还有一些是利用不动点理论来讨论某些方程的各种解的存在性的,并取得了很大的进展。本文主要讨论了两类微分方程的渐近概周期解的存在性和唯一性。具体包括以下内容:第一部分是通过微分方程解的连续性构造一个差分方程,随后结合指数二分并利用差分方程的渐近概周期序列解,讨论了一类具有逐段常变量微分方程的渐近概周期解的存在性。第二部分是利用指数型二分性定理和不动点定理,建立了一些保证一类具有有限时滞的微分方程有渐近概周期解的充分条件。本文所得的结果或是对已有结果的推广,从而使得相关结论应用更加广泛,或是对已有问题所给的条件作出调整和变化。这可以为其他问题的研究提供一些方法和借鉴。更多还原

【Abstract】 As well known, when the almost periodic type solutions of differential equations with piecewise constant argument are discussed, the almost periodic type sequence solutions of relevant difference equations are usually used. Especially in recent years, more and more mathematical workers began to study the existence and uniqueness of almost periodic type sequence solutions for difference equations, and the existence and uniqueness of almost periodic type solutions for some differential equations by joining difference equations and differential equations together. Moreover, the existence of various solutions for certain equations is discussed by using fixed point theory, and great progress is made.The existence and uniqueness of asympototically almost periodic solutions for two types differential equations are mainly discussed in this paper.Specific contents are as follows:A difference equation is constructed through the continuity of solutions of differential equations in the first part after this, the existence of asympototically almost periodic solutions for a class of differential equation with piecewise constant is discussed using asympototically almost periodic sequence solutions of relevant difference equations combinating exponential dichotomy.The exponential dichotomy theory and fixed point theory are used in the second part, and established the full conditions that assured a class of differential equations with limited delay exist the asympototically almost periodic solutionsSome of the results in the paper are promotion of the well-known results, thus the relevant conclusions will be more extensively applied. The others are obtained by changing the conditions of previous results, these methods can provide some reference for other problems.更多还原