测地度量空间中若干广义非扩张型映射的不动点性质

发布时间：2018-06-28 18:05

本文选题：测地度量空间 + 不动点　；参考：《哈尔滨工业大学》2017年博士论文

【摘要】：不动点理论是目前蓬勃发展的非线性泛函分析的重要组成部分,特别是在解决各类方程解的存在性问题中起着关键作用。自20世纪初期,Brouwer和Banach分别提出“Brouwer不动点定理”和“Banach压缩映像原理”之后,国内外数学工作者们纷纷投身到不动点理论的研究中来,使得不动点理论成为重要的数学分支。传统上,不动点理论主要是利用Banach空间理论和拓扑度理论来研究不动点性质。近几十年来,关于不动点理论的研究逐步延展到各类度量空间,例如广义度量空间、概率度量空间等。测地度量空间是一类结合了微分几何、Banach空间性质以及度量空间性质的空间框架,主要包括CAT(0)空间(字母C,A,T分别代表Cartan,Alexandrov和Toponogov)、W-双曲空间、Busemann空间等。然而,与丰富的Banach空间不动点理论的研究成果相比,测地度量空间不动点性质的研究仍处于萌芽阶段,大量问题等待深入探讨。测地度量空间的不动点理论对变分不等式的求解以及计算机图论等方面均有着重要应用,从而在测地度量空间中研究非线性算子的不动点性质具有极大的理论价值与实际意义。本文围绕测地度量空间中若干广义非扩张型映射的不动点问题展开探讨,主要包括以下四个方面的内容:首先,研究CAT(0)空间平均非扩张映射的不动点性质。得到CAT(0)空间中有界闭凸子集上平均非扩张单值映射具有不动点性质的若干定理,包括存在性定理、收敛性定理及半闭原理。同时,给出CAT(0)空间中有界闭凸子集上平均非扩张集值映射存在稳定点的判别准则。其次,研究测地度量空间C-型集值映射的不动点性质。证明CAT(0)空间中可交换的满足条件(C)的单值与集值映射的公共不动点的存在性并给出满足条件(C)的集值映射的两类收敛性定理。得到W-双曲空间上C-型集值映射强收敛的充分必要条件。再次,研究CAT(0)空间新型成对映射的公共不动点性质。在度量空间中定义两类新型的成对映射,分别称为满足条件(PCλ)和满足条件(PEμ)的成对映射,并通过例子说明它们是比非扩张映射更广的映射类型。给出CAT(0)空间中满足条件(PCλ)的成对映射公共不动点存在的等价条件并得到CAT(0)空间中满足条件(PEμ)的成对映射的半闭原理。同时,利用S-迭代证明满足条件(PCλ)的成对映射的收敛性定理。最后,研究CAT(0)空间L-型映射的不动点性质。在CAT(0)空间中讨论L-型映射与其他非扩张型映射的关系。给出CAT(0)空间中L-型映射的不动点存在性定理。此外,证明L-型映射的公共不动点的存在性定理并利用新型的三步迭代得到L-型映射的逼近定理。
[Abstract]:Fixed point theory is an important part of the dynamic nonlinear functional analysis, especially plays a key role in solving the problem of the existence of solutions to various equations. Since Brouwer's fixed point theorem and Banach's principle of Banach contractive mapping were put forward in the early 20th century, many mathematics workers at home and abroad have devoted themselves to the study of fixed point theory, which makes fixed point theory become an important branch of mathematics. Traditionally, fixed point theory mainly uses Banach space theory and topological degree theory to study fixed point properties. In recent decades, the research on fixed point theory has been extended to various kinds of metric spaces, such as generalized metric spaces, probabilistic metric spaces and so on. Geodesic metric space is a kind of space frame which combines the properties of differential geometry Banach spaces and metric spaces. It mainly includes cat (0) spaces (the letters C _ (0) represent Cartanan Alexandrov and Toponogov) and Busemann spaces. However, compared with the abundant research results of fixed point theory in Banach spaces, the study of the properties of fixed points in geodesic metric spaces is still in its infancy, and a large number of problems are waiting to be deeply discussed. The fixed point theory of geodesic metric space has important applications in solving variational inequality and computer graph theory, so it is of great theoretical value and practical significance to study the fixed point properties of nonlinear operators in geodesic metric space. In this paper, the fixed point problem of some generalized nonexpansive type mappings in geodesic metric spaces is discussed, including the following four aspects: firstly, the fixed point properties of cat (0) space mean nonexpansive mappings are studied. In this paper, we obtain some theorems that mean nonexpansive single-valued mappings on a bounded closed convex subset in cat (0) spaces have fixed point properties, including existence theorem, convergence theorem and semi-closed principle. At the same time, a criterion for the existence of stable points for mean nonexpansive set-valued mappings on a bounded closed convex subset in cat (0) space is given. Secondly, the fixed point properties of C-type set-valued mappings in geodesic metric spaces are studied. This paper proves the existence of common fixed points of commutative satisfying condition (C) and set-valued mapping in cat (0) space, and gives two kinds of convergence theorems for set-valued mappings satisfying condition (C). A necessary and sufficient condition for the strong convergence of C-type set-valued mappings on W-hyperbolic spaces is obtained. Thirdly, the common fixed point properties of new pair mapping in cat (0) space are studied. Two new types of pairwise mappings are defined in metric spaces, which are called pairwise mappings of satisfying conditions (PC 位) and satisfying conditions (PE 渭), respectively. Examples show that these mappings are more extensive than nonexpansive mappings. The equivalent conditions for the existence of common fixed points of pairwise mappings in cat (0) spaces satisfying conditions (PC 位) are given, and the semi-closed principle of pairwise mappings satisfying conditions (PE 渭) in cat (0) spaces is obtained. At the same time, the convergence theorem of pairwise mappings satisfying the condition (PC 位) is proved by means of S- iteration. Finally, the fixed point properties of L-type mapping in cat (0) space are studied. In cat (0) space, the relation between L- type mapping and other nonexpansive type mappings is discussed. The existence theorem of fixed points for L- type mappings in cat (0) space is given. In addition, the existence theorem of common fixed points for L- type mappings is proved and the approximation theorem of L- type mappings is obtained by using a new three-step iteration.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O177.91

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