發(fā)布時(shí)間：2018-06-28 18:05

  本文選題：測(cè)地度量空間 + 不動(dòng)點(diǎn)��； 參考：《哈爾濱工業(yè)大學(xué)》2017年博士論文

【摘要】：不動(dòng)點(diǎn)理論是目前蓬勃發(fā)展的非線性泛函分析的重要組成部分,特別是在解決各類(lèi)方程解的存在性問(wèn)題中起著關(guān)鍵作用。自20世紀(jì)初期,Brouwer和Banach分別提出“Brouwer不動(dòng)點(diǎn)定理”和“Banach壓縮映像原理”之后,國(guó)內(nèi)外數(shù)學(xué)工作者們紛紛投身到不動(dòng)點(diǎn)理論的研究中來(lái),使得不動(dòng)點(diǎn)理論成為重要的數(shù)學(xué)分支。傳統(tǒng)上,不動(dòng)點(diǎn)理論主要是利用Banach空間理論和拓?fù)涠壤碚搧?lái)研究不動(dòng)點(diǎn)性質(zhì)。近幾十年來(lái),關(guān)于不動(dòng)點(diǎn)理論的研究逐步延展到各類(lèi)度量空間,例如廣義度量空間、概率度量空間等。測(cè)地度量空間是一類(lèi)結(jié)合了微分幾何、Banach空間性質(zhì)以及度量空間性質(zhì)的空間框架,主要包括CAT(0)空間(字母C,A,T分別代表Cartan,Alexandrov和Toponogov)、W-雙曲空間、Busemann空間等。然而,與豐富的Banach空間不動(dòng)點(diǎn)理論的研究成果相比,測(cè)地度量空間不動(dòng)點(diǎn)性質(zhì)的研究仍處于萌芽階段,大量問(wèn)題等待深入探討。測(cè)地度量空間的不動(dòng)點(diǎn)理論對(duì)變分不等式的求解以及計(jì)算機(jī)圖論等方面均有著重要應(yīng)用,從而在測(cè)地度量空間中研究非線性算子的不動(dòng)點(diǎn)性質(zhì)具有極大的理論價(jià)值與實(shí)際意義。本文圍繞測(cè)地度量空間中若干廣義非擴(kuò)張型映射的不動(dòng)點(diǎn)問(wèn)題展開(kāi)探討,主要包括以下四個(gè)方面的內(nèi)容:首先,研究CAT(0)空間平均非擴(kuò)張映射的不動(dòng)點(diǎn)性質(zhì)。得到CAT(0)空間中有界閉凸子集上平均非擴(kuò)張單值映射具有不動(dòng)點(diǎn)性質(zhì)的若干定理,包括存在性定理、收斂性定理及半閉原理。同時(shí),給出CAT(0)空間中有界閉凸子集上平均非擴(kuò)張集值映射存在穩(wěn)定點(diǎn)的判別準(zhǔn)則。其次,研究測(cè)地度量空間C-型集值映射的不動(dòng)點(diǎn)性質(zhì)。證明CAT(0)空間中可交換的滿足條件(C)的單值與集值映射的公共不動(dòng)點(diǎn)的存在性并給出滿足條件(C)的集值映射的兩類(lèi)收斂性定理。得到W-雙曲空間上C-型集值映射強(qiáng)收斂的充分必要條件。再次,研究CAT(0)空間新型成對(duì)映射的公共不動(dòng)點(diǎn)性質(zhì)。在度量空間中定義兩類(lèi)新型的成對(duì)映射,分別稱(chēng)為滿足條件(PCλ)和滿足條件(PEμ)的成對(duì)映射,并通過(guò)例子說(shuō)明它們是比非擴(kuò)張映射更廣的映射類(lèi)型。給出CAT(0)空間中滿足條件(PCλ)的成對(duì)映射公共不動(dòng)點(diǎn)存在的等價(jià)條件并得到CAT(0)空間中滿足條件(PEμ)的成對(duì)映射的半閉原理。同時(shí),利用S-迭代證明滿足條件(PCλ)的成對(duì)映射的收斂性定理。最后,研究CAT(0)空間L-型映射的不動(dòng)點(diǎn)性質(zhì)。在CAT(0)空間中討論L-型映射與其他非擴(kuò)張型映射的關(guān)系。給出CAT(0)空間中L-型映射的不動(dòng)點(diǎn)存在性定理。此外,證明L-型映射的公共不動(dòng)點(diǎn)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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O177.91

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