拟概自守性与泛函微分方程拟概自守解研究

The periodic, almost periodic and almost automorphic solutions to differential equations can profoundly characterize the asymptotical behavior of the corresponding dynamic systems, which have received great attention by many scholars. The almost automorphic phenomenon is the most common one among these three kinds of phenomenon. This project is mainly focused upon new quasi-almost automorphic solutions to functional differential equations with their applications by integrated applications of theories of nonlinear analysis, stochastic analysis and quasi-almost automorphic functions et al. First, we shall introduce some new concepts of quasi-almost automorphic functions and establish completeness and ergodicity theorems in the space of such functions; Second, we shall make these new quasi-almost automorphic function spaces "functional change" and obtain their ergodicity theorems, then we investigate properties of quasi-almost automorphic solutions to some functional differential equations with typical application backgrounds, and get some new meaningful results. The results of this study will further improve and perfect the basic theory of almost automorphic functions, and also provide new ideas and methods for the resolvement of some practical problems.

微分方程的周期解、概周期解以及概自守解可以深刻地刻画相应动力系统的渐近行为,一直受到众多学者的高度重视.其中, 概自守现象则是三种现象中最普遍的一种. 本项目拟综合应用非线性分析、随机分析和拟概自守函数等理论来研究泛函微分方程新拟概自守解及其应用. 首先提出一些新的拟概自守函数的概念, 建立其函数空间的完备性和遍历性; 其次将这些新拟概自守函数空间进行"泛函化", 得到其相应"泛函化"的遍历性结果, 进而研究一些具有典型应用背景的泛函微分方程拟概自守解的性质, 得到一些新的有意义的结果; 这些研究成果将进一步改进和完善概自守函数的基本理论, 也为某些实际问题的解决提供新的思路和方法.

微分方程的周期解、概周期解以及概自守解可以深刻地刻画相应动力系统的渐近行为，一直受到众多学者的高度重视。其中，概自守现象则是三种现象中最普遍的一种。本项目综合应用了非线性分析、随机分析和拟概自守函数等理论来研究泛函微分方程新拟概自守解及其应用。首先提出了一些新的拟概自守函数的概念, 建立了其函数空间的完备性和遍历性; 其次将这些新拟概自守函数空间进行“泛函化”, 得到其相应“泛函化”的遍历性结果, 进而获得一些具有典型应用背景的泛函微分方程拟概自守解的存在性；分数阶微积分被视为刻画长时记忆过程最有效率的工具之一，本项目针对一类(1,2)阶退化分数阶发展方程，引入了一类分数阶预解算子族，讨论了此类算子族的连续性和紧性，进而获得了此类退化分数阶发展方程驱动系统的近似可控性、最优控制问题及集值问题的可解性。