概率度量空间中不动点定理及Banach空间中Lipschitz对偶算子数值域的研究

发布时间：2018-04-02 15:38

本文选题：概率度量空间　切入点：压缩映射　出处：《天津理工大学》2017年硕士论文

【摘要】：本文一共有两个部分,前半部分包含两章内容,研究了概率度量空间中几个不动点定理.后半部分研究了Banach空间中算子的数值域问题.第一章介绍了概率度量空间的历史背景和发展现状、基本概念、拓扑结构和相关性质.第二章改进了概率度量空间中的一类映射下的不动点定理,在辅助函数(37)与?只在零点连续的情况下证明了在G完备和M完备的概率度量空间下的不动点的存在性和唯一性,更深入研究了推广的?映射下的一个公共不动点定理.最后介绍了广义的概率度量空间,并对其中的性质做了初步研究,将之前的不动点定理推广到了新的空间中.第三章主要研究Lipschitz对偶算子的数值域和Lipschitz对偶空间.重点是研究跟Lipschitz算子及其对偶算子有关的数值半径问题.研究对比了Lipschitz算子T的Lipschitz对偶算子l*T与线性对偶算子*T之间的区别.更进一步证明了结论:如果Lipschitz算子T可以达范,那么它的对偶算子l*T的数值指标等于它的范数,并举例说明达范并不是l*T的数值指标等于范数的必要条件.
[Abstract]:In this paper, there are two parts, the first half contains two chapters, and we study some fixed point theorems in probabilistic metric spaces.In the second half of this paper, the problem of numerical range of operators in Banach spaces is studied.The first chapter introduces the historical background and development of probabilistic metric space, basic concepts, topological structure and related properties.In the second chapter, we improve the fixed point theorem for a class of maps in probabilistic metric spaces.The existence and uniqueness of fixed points in G complete and M complete probabilistic metric spaces are proved only in the case of zero continuity.A common fixed point theorem under a mapping.Finally, the generalized probabilistic metric space is introduced, and its properties are preliminarily studied, and the previous fixed point theorem is extended to a new space.In chapter 3, the numerical range and Lipschitz duality space of Lipschitz duality operator are studied.The emphasis is on the numerical radius problem related to Lipschitz operator and its dual operator.The difference between the Lipschitz dual operator LT of Lipschitz operator T and the linear dual operator T is studied and compared.It is further proved that if Lipschitz operator T can reach the norm, then the numerical index of its dual operator LT is equal to its norm, and the example shows that the numerical index of Lipschitz operator T is not the necessary condition for the numerical index of LT to be equal to the norm.
【学位授予单位】：天津理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177.2

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