脉冲随机发展方程的均方逐段概自守型解及相关问题研究

Impulsive stochastic evolution equations have been applied successfully in several areas of Chemistry, Physics, Medicine and Engineering. Existence and stability of solutions, periodic solutions, almost periodic solutions to impulsive stochastic evolution equations have elicited a great deal of attention from many mathematicians. Almost automorphic functions are generalizations of periodic functions and almost periodic functions and describe the real phenomena more appropriately, the study on square-mean piecewise almost automorphic type solutions to impulsive stochastic evolution equations will be of great importance. In this project, we will set up two concrete function spaces, including the space of square-mean piecewise almost automorphic functions and the space of square-mean piecewise pseudo almost automorphic functions. We will investigate the existence and stability of square-mean piecewise almost automorphic type solutions to impulsive stochastic evolution equations in this project. The available literature deals mainly with the existence of almost periodic type solutions to impulsive stochastic evolution equations by virtue of the Lipschitz continuity conditions of perturbation functions and Contraction theorem. In this project, we will establish a criterion of the relatively compact square-mean piecewise continuous function set and investigate the existence of square-mean piecewise almost automorphic type solutions by Schauder’s fixed point theorem, which overcome the limitations of the Lipschitz continuity conditions of perturbation functions. The expected results will be of great significance for almost periodic function theory and its applications in practical problems.

脉冲随机发展方程已经成功应用于生物学、物理学、医学、工程学等越来越多的领域，关于脉冲随机发展方程的解、周期解、概周期解的存在性和稳定性引起了很多学者的关注。概自守函数是周期函数和概周期函数的推广，能在更广范围内较好地刻画客观现象，因此，对脉冲随机发展方程的均方逐段概自守型解的研究具有重要意义。本项目将建立两类具体的函数空间理论：均方逐段概自守函数空间和均方逐段伪概自守函数空间，并分析脉冲随机发展方程的均方逐段概自守型解的存在性及稳定性。目前已有文献研究脉冲随机发展方程的概周期型解的存在性大多是基于扰动函数的Lipschitz连续条件，利用压缩映射定理证明的。本项目将给出均方逐段连续函数集相对紧的一个等价条件，利用Schauder不动点定理证明存在性结果，这种方法克服了对扰动函数的Lipschitz连续条件的限制，预期结果对概周期函数理论及其在很多实际问题中的应用都具有重要意义。

概自守函数是概周期函数的一个重要推广，自提出后引起了许多学者的关注，在很多实际问题中有重要应用。本项目在脉冲随机过程中引入概自守函数，定义了均方逐段概自守函数和均方逐段伪概自守函数，分析了其值域性质、完备性和复合性质等；对一类脉冲随机发展方程，给出了此类方程的均方逐段概自守解和均方逐段伪概自守解的存在性定理和稳定性定理。在本项目中，分别在扰动函数满足Lipschitz连续条件和一致连续条件下，使用压缩映射定理和Schauder不动点定理证明了均方逐段概自守型解的存在性。这些结果无论是对基础理论的研究，还是对应用技术的研究都具有十分重要的意义。