【作者】 汪凯；

【导师】 鲁世平；

【作者基本信息】 安徽师范大学 ， 应用数学， 2007， 硕士

【摘要】 本文通过运用Mawhin拓展定理与Lyapunov第二方法研究了几类中立型泛函微分方程周期解的存在性及全局吸引性.并探讨了具有相互干扰的Volterra种群模型周期正解的存在性与唯一性.第一章介绍了泛函微分方程的发展历史及本文的主要工作.第二章主要介绍了几个必备的引理并建立了几个重要的不等式.第三章利用Mawhin拓展定理研究了两类具有分布时滞的高阶中立型泛函微分方程周期解的存在性,与已有工作不同的是我们着重考虑的是连续时滞的影响,这样能更好的反映真实情况.并且把结果推广到了高阶的情形.第四章主要研究中立型时滞微分系统周期解的存在性与全局吸引性.对于中立型时滞微分系统已有一些文献探讨了其周期解的存在性,但据作者所知他们大多数是利用指数型二分性的方法与Krasnoselskii不动点定理加以研究并建立了在差分算子D稳定(即|c|<1)的前提下存在周期解的充分条件,而对此类系统周期解的全局吸引性还很少有学者进行研究过.原因是研究吸引性常用的方法往往是通过构造恰当的Lyapunov函数,然而遗憾的是对于Lyapunov函数的构造方法还没有一个一般性的准则,这就给构造适合具体方程的Lyapunov函数带来了困难.通过运用Mawhin拓展定理我们把周期解存在的条件放宽到允许差分算子D不稳定的情形(即|c|>1).并在算子D稳定的条件下通过巧妙地构造恰当的Lyapunov函数,获得了其存在唯一全局吸引的周期解的充分条件.本章的最后,探讨了具有多个偏差变元的中立型泛函微分系统存在全局吸引的周期解的充分条件.目前对此类方程周期解的吸引性研究还不多见.我们的工作改进了已有文献中的一些结果.第五章着重研究了具有相互干扰的Volterra种群模型周期正解的存在性与唯一性.更多还原

【Abstract】 By applying Mawhin’s continuation theorem(MCT) and Lyapunov’s second method, this thesis mainly studies the existence and global attractivity of periodic solution for several types of neutral functional differential equations(NFDEs), and investigates the existence and uniqueness of positive periodic solution for a Volterra model with mutual interference.In the first Chapter, we introduce the history of functional differential equation and the work of this paper. Chapter 2 introduces some necessary lemmas and establishes several inequalities. In Chapter 3, by utilizing MCT we analysis two kinds of higher order NFDEs with distributed delay. Differing from the previous results we mainly study the influence of distributed delay which is more realistic than discrete delays, and we generate the results to higher order equations. In Chapter 4, we study the existence of globally attractive periodic solution for neutral differential system(NDS) with delays. As far as we know there are some papers on the existence of periodic solutions for NDS, but the methods they used are the exponential dichotomy and Krasnoselakii fixed point theorem, and the sufficient conditions for the existence of periodic solutions which they obtained are under the condition that the difference operator D is stable(i.e.|c| < 1). The main reason why few scholars have studied the global attractivity of periodic solution for NDS is that the general way is to construct a suitable Lyapunov function. It is regrettable that there is no general methods to construct the suitable Lyapunov function, so how to construct a suitable Lyapunov function is a difficult work. By using MCT we relax the conditions which guarantee the existence of periodic solutions to that the operator D is unstable (i.e.|c| > 1), under the stability of the operator D and by constructing a suitable Lyapunov function we obtain some sufficient conditions for insuring the existence of only one globally attractive periodic solution. Lastly, we discuss the existence and global attractivity of periodic solution to a kind of NFDE with multiple deviating arguments. There are few works on the global attractivity of periodic solution for NFDEs. Our work improves the previous results. We mainly investigate the existence and uniqueness of positive periodic solution for a Volterra model with mutual interference in the last Chapter.更多还原