关于广义度量空间中算子方程解的问题研究

发布时间：2018-12-07 11:42

【摘要】：1992年,为了解决计算机理论中的相关问题,Matthews在度量空间的基础上建立了偏度量空间的理论,并提出了偏度量空间中的Banach压缩映射原理.这使得偏度量空间的不动点理论及其应用成为了目前非线性分析问题的重要内容.从而,对该空间不动点定理的探讨对非线性算子理论的发展有着深远影响.当然,对于度量空间中的不动点理论进行深入挖掘也将推动整个方向的发展.本文主要研究了包括多值映射在内的多种压缩映射在偏度量空间和度量空间中的不动点定理.全文分为四章.第一章,主要介绍了度量空间与偏度量空间中的不动点理论的研究背景、现状以及发展前景和一些相关的概念.第二章,引入关于F-函数的新型压缩映射的概念,用以研究多值映射在偏度量空间中不动点理论,并举例说明了结果的实用性.第三章,引入I-函数的同时加入图论作为辅助工具构建新型压缩映射,得到了单值映射在度量空间中的新不动点定理,并将其推广到了偏度量空间中.而这些结果推广了度量空间中的一些已有的定理.第四章,将F-函数的新型压缩映射推广到类度量空间之中,证明了其中的多值映射的不动点定理.
[Abstract]:In 1992, in order to solve the related problems in computer theory, Matthews established the theory of partial metric space on the basis of metric space, and put forward the principle of Banach contraction mapping in partial metric space. This makes the fixed point theory of partial metric space and its application become an important content of nonlinear analysis. Therefore, the discussion of the fixed point theorem in this space has a profound influence on the development of nonlinear operator theory. Of course, mining the fixed point theory in metric space will promote the development of the whole direction. In this paper, we study the fixed point theorems of many kinds of contractive mappings in partial metric spaces and metric spaces, including multivalued mappings. The full text is divided into four chapters. In the first chapter, the research background, present situation, development prospect and some related concepts of fixed point theory in metric space and partial metric space are introduced. In chapter 2, we introduce the concept of new contractive mappings about F- functions to study the fixed point theory of multivalued mappings in partial metric spaces, and illustrate the practicability of the results. In chapter 3, we introduce I- function and add graph theory as auxiliary tool to construct new contractive mapping. We obtain a new fixed point theorem of single valued mapping in metric space, and extend it to partial metric space. These results generalize some existing theorems in metric spaces. In chapter 4, the new contractive mapping of F- function is extended to the class metric space, and the fixed point theorem of multivalued mapping is proved.
【学位授予单位】：南昌大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189.11

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