非紧性测度与非线性分析若干问题的研究

The study about measures of noncompactness of Banach spaces is one of the central topics in functional analysis and it involves many branches of mathematics.In this project, measures of noncompactness are studied from a different perspective. We combine the theory of measures of noncompactness and that of Banach spaces and focus on the problems about representation of measures of noncompactness and their inequivalence by researching the intrinsic propertise of several measures. Some new ideas and serviceable tools in theory of Banach spaces and convex analysis will be imported to solve the following problems:.(1).Construction and representation of measures of noncompactness and measures of non-weak compactness by some method connecting space of continuous functions;.(2).Existence of inequivalent regular or homogeneous measures of noncompactness of all Banach spaces and existence of inequivalent measures of non-weak compactness of classical Banach spaces;.(3).Characterization or sufficient conditions for the existence of inequivalent homogeneous measures of noncompactness of operators.

Banach空间的非紧性测度是泛函分析的一个重要研究课题,它的研究联系着诸多数学分支.本项目从一个新的视角来研究非紧性测度,将非紧性测度理论和Banach空间理论相结合,通过对非紧性测度作为函数本身的性质研究,来讨论非紧性测度的表示和等价性等问题.本项目将Banach空间理论和凸分析的思想方法引入到非紧性测度理论的研究中,旨在研究并解决如下问题:.(1)公理化的非紧性测度和非弱紧性测度在连续函数空间下的构造和表示问题;.(2)不等价正则(或齐次)非紧性测度存在性和经典Banach空间中不等价非弱紧性测度的存在性问题;.(3)寻找使得算子的不等价齐次非紧性测度存在的Banach空间的特征或充分条件.

Banach空间的非紧性测度是泛函分析的一个重要研究课题,它的研究联系着诸多数学分支,本项目从一个新的视角研究非紧性测度,将Banach空间理论和凸分析的方法引入到非紧性测度理论的研究中,项目组作了如下工作:.(1)将公理化的非紧性测度表示成连续函数空间的范数;.(2)得到了齐次非紧性测度的一个简便的构造定理;.(3)在一些Banach空间中构造不等价的齐次测度,肯定回答了Mallet-Paret和Nussbaum问题.