泛函空间及其应用

Functional space and its application is an important branch in mathematics and closely connect with control theory, theoritic physic, economic system and so on. The researching direction of the project is some classes of concrete.functional spaces, such as K鰐he sequence space, Orlicz space, almost periodic.function space etc, fixed point theory, set-valued analysis and their application to the differential equation, integral equation and fuzzy analysis. The main achievements are as follows: (1) We get the characterization of the existence of continuous selection for a class of set-valued mappings and the approximate theorem for nonconvex set-valued mappings, which improve and generalize many previous results. We also define the.weak Radon-Nikodym derivative and obtain the weak Radon-Nikodym theorem, and.give some applications. (2) By using the ergodicity we prove the existence and uniqueness of pseudo almost periodic solution and asymptotically almost periodic solution for some classes.of nonlinear differential equations, and give the characterizations for the pseudoalmost periodic function space and the vector-valued weak almost periodic function space. (3) We introduce and characterize the AK-property of infinite matrix, and give the characterization of that the completely continuous infinite matrix operator ideal is minimal and closed in the strong topo logy sense..(4) We construct two concrete function spaces and obtain the existence theorem.of maximal and minimal solutions of the initial value problem for the implusive and discontinuous integro-differential equations of mixed type. (5) We establish the fixed point theorems for many kinds of operators and set-valued mappings which improve and generalize some previous results, and apply to the differential equations, integral equations and differential inclusions..(6) We construct some kinds of concrete Frechet spaces and Banach spaces and.solve the difficult problem for the embedding of multidimensional fuzzy numbers.

本项目是关于无穷维分析，包括对魁特型序列空间、奥尔里奇型空间及概周期函数空间等这些特殊空间的研究，以及它们的应用，并进一步刻划一般泛函空间（包括算子空间）的各种属性，反映了泛函分析中历来受到高度重视，而最近显示出格外突出的空间理论的研究的主流，且对非线性偏微分方程、动力系统、控制论及逼近论有广泛的应用。