集值压缩型映射的不动点定理

发布时间：2018-01-07 09:40

本文关键词：集值压缩型映射的不动点定理　出处：《广东工业大学》2016年硕士论文　论文类型：学位论文

【摘要】：不动点理论是现代数学理论研究的一个很重要的组成部分,它在数学的许多分支及实际应用中均有着十分重要的地位.到目前为止,研究不动点理论的国内外学者有许多,取得了大量的研究结果.本文的内容是分别在序度量空间、仿拟度量空间以及拓扑空间中探讨与研究一些集值映射的不动点理论,给出了完备偏序度量空间中的若干个集值压缩型映射不动点定理,推广了Ismat Beg和Aama Rashid Butt在完备偏序空间中的不动点定理;将Erdal Karapinar、Salvador Romaguera和J.Marin在仿拟度量空间中所得到的不动点定理进行了推广,并列举了若干个例子来对所得的结果进行了详细的说明；最后是将拓扑空间中的单值型不动点定理推广到了集值型映射情形.全文内容总共可分为五章.第一章的内容为本文的绪论,我们首先介绍了有关于不动点理论的历史背景、发展历程以及在此过程中数学家们所取得的一些伟大成果,其次,我们介绍了有关于非对称度量空间的一些概念和有关于它的部分研究现状.并对论文中所需要用到的一些符号定义作简要的说明.第二章的主要内容是研究了在完备度量空间中部分集值型压缩映射的不动点定理并给出了具体详细的定理证明过程.第三章的内容主要是通过引入偏序的概念从而将本文第二章中探讨所得到的完备度量空间中集值型不动点定理相应推广到完备的偏序度量空间中.第四章主要的内容是对前人在仿拟度量空间中所得到的部分集值型不动点定理作一个更为广泛的推广以及深入的探究,即将他们证明所得的定理中关于函数的约束条件推广到更为广泛的一类函数中从而使得定理的应用范围变得更加的广泛了;同时也给出了具体的定理证明过程以及对应的例子说明.第五章的主要内容是通过引入有关上方下半连续的概念并利用紧拓扑空间上的任一上方下半连续函数必能达到其下界这一事实来将拓扑空间中的一些单值型不动点理论推广到集值型映射中去,从而得到了拓扑空间中的集值型不动点定理,并给出了定理的具体证明过程.
[Abstract]:Fixed point theory is an important part of modern mathematical theory research. It plays an important role in many branches and practical applications of mathematics. There are many scholars at home and abroad who study the fixed point theory, and a lot of research results have been obtained. The content of this paper is respectively in the ordered metric space. The fixed point theory of some set-valued mappings is discussed and studied in para-metric spaces and topological spaces. Some fixed point theorems of set-valued contractive mappings in complete partially ordered metric spaces are given. The fixed point theorems of Ismat Beg and Aama Rashid Butt in complete partially ordered spaces are generalized. The fixed point theorem obtained by Erdal Karapinarn Salvador Romaguera and J. Marin in para-metric spaces is generalized. Several examples are given to explain the results in detail. Finally, the single-valued fixed point theorem in topological space is extended to the set-valued mapping case. The whole paper can be divided into five chapters. The first chapter is the introduction of this paper. We first introduce the historical background of fixed point theory, the course of development and some great achievements made by mathematicians in this process. Secondly. In this paper, we introduce some concepts about asymmetric metric space and some research status about it, and give a brief explanation of some symbol definitions used in this paper. The main content of chapter 2 is to study. The fixed point theorems of partial set-valued contractive mappings in complete metric spaces are given and the process of proving the theorems in detail is given. In the third chapter, the concept of partial ordering is introduced to discuss the results obtained in chapter two. The fixed point theorem of set-valued type in complete metric space is extended to complete partial ordered metric space accordingly. The main content of Chapter 4th is to make a further study of the partial set-valued fixed point theorem obtained by predecessors in quasi-metric space. For a wide range of promotion and in-depth exploration. In this paper, the constraint conditions of the function in the theorem they prove are extended to a more extensive class of functions, which makes the application of the theorem more extensive. The main content of Chapter 5th is to introduce the concept of upper and lower semicontinuity and to use any upper lower semicontinuous function on compact topological space. The fact that the lower bound can be reached is used to extend some single-valued fixed point theory in topological space to set-valued mapping. Thus, the set-valued fixed point theorem in topological space is obtained, and the concrete proof process of the theorem is given.
【学位授予单位】：广东工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O177.91

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