脉冲随机泛函微分方程周期解的存在性

A fundamental problem in the study of the theory of differential equations is to study the behavior of the solution. One important aspect is to determine whether the system has periodic solutions, and under what conditions the periodic solutions exist. There are many ways to study the existence of periodic solutions for deterministic differential equations, such as: fixed point theory method, topological degree method, Liapunov second method, and so on. However, so far, there are very few methods to study the existence of periodic solutions for stochastic functional differential equations. In particular, there are less ways to study the existence of periodic solutions for impulsive stochastic functional differential equations. We plan to study the existence of periodic solutions of impulsive stochastic functional differential equations. Their research is more difficult due to the common perturbation of delay, random and pulse interference factors. To carry out research in this area can provide a rigorous mathematical foundation for practical applications, so it is very meaningful work.

微分方程理论研究中的一个基本问题是研究解的性态，其中一个重要的方面是确定系统是否存在周期解，什么条件下存在周期解。研究确定性微分方程周期解存在性的方法有很多种，比如：不动点理论方法、拓扑度方法、Liapunov第二方法等。然而，目前为止，研究随机泛函微分方程周期解存在性的方法还很少。特别是对随机泛函微分方程受脉冲干扰后周期解的存在性问题的研究方法更是缺少。我们计划研究脉冲随机泛函微分方程周期解的存在性。由于时滞、随机和脉冲三种干扰因素的共同介入，对其研究的难度更大。开展这方面的研究, 为实际应用提供严格的数学基础，是十分有意义的工作。

本项目计划研究随机泛函微分方程受脉冲干扰后周期解的存在性问题。基本上按项目计划完成了对微分方程周期解存在性问题的研究。主要工作有：（1）利用随机分析技巧，获得了一类受脉冲干扰的随机Logistic方程周期解存在的充分条件。（2）通过对方程解的精细估计，获得了一类受脉冲干扰的中立型微分方程周期解存在的充分条件。