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When graph meets diagonal: an approximative access to fixed point theoryOkon, Thomas 30 August 2001 (has links)
The thesis deals with a general access to topological transversality in uniform spaces. / Die Arbeit behandelt einen allgemeinen Zugang zur Topologischen Transversalität in uniformen Räumen.
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On completeness of partial metric spaces, symmetric spaces and some fixed point resultsAphane, Maggie 12 1900 (has links)
The purpose of the thesis is to study completeness of abstract spaces. In particular,
we study completeness in partial metric spaces, partial metric type spaces, dislocated
metric spaces, dislocated metric type spaces and symmetric spaces that are
generalizations of metric spaces. It is well known that complete metric spaces have
a wide range of applications. For instance, the classical Banach contraction principle
is phrased in the context of complete metric spaces. Analogously, the Banach's
xed point theorem and xed point results for Lipschitzian maps are discussed in
this context, namely in, partial metric spaces and metric type spaces. Finally, xed
point results are presented for symmetric spaces. / Geography / Ph. D. (Mathematics)
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[en] ASPECTS OF TOPOLOGY AND FIXED POINT THEORY / [pt] ASPECTOS DA TOPOLOGIA E DA TEORIA DOS PONTOS FIXOSLEONARDO HENRIQUE CALDEIRA PIRES FERRARI 17 August 2017 (has links)
[pt] Esse trabalho tem como objetivo reunir os teoremas topológicos de ponto fixo clássicos e seus corolários, além de teoremas de ponto fixo provenientes da teoria do grau e algumas importantes aplicações desses teoremas a variadas áreas - desde as clássicas aplicações à teoria de EDOs e EDPs à uma aplicação à teoria dos jogos. Um exemplo é o Teorema do Ponto Fixo de Schauder-Tychonoff, para aplicações compactas em convexos de espaços localmente convexos, do qual segue como corolário que todo compacto convexo de
um espaço vetorial normado (não necessariamente de dimensão finita) possui a propriedade do ponto fixo. No que se refere à teoria dos jogos em particular, foi deduzido o Teorema de Nash, que determina condições sobre as quais certos jogos possuem equilíbrios nos seus espaços das estratégias. Toda a topologia geral necessária nas demonstrações foi desenvolvida extensiva e detalhadamente a partir de topologia elementar, seguindo algumas das referências bibliográficas. O Teorema de Extensão de Dugundji - uma extensão do Teorema de Extensão de Tietze a fechados de espaços métricos sobre espaços localmente convexos -, por exemplo, é demonstrado com detalhes e usado diversas vezes
ao longo da dissertação. / [en] The goal of the present work is to gather the classical fixed-point theorems and their corollaries, as well as other fixed-point theorems arising from degree theory, and some important applications to diverse fields -
from the classical applications to ODEs and PDEs to an application to the game theory. An example is the Schauder-Tychonoff Fixed-Point Theorem, 1 concerning compact mappings in convex subsets of locally convex spaces, from which it follows as a corollary that every compact convex subset of a normed
vector space is a fixed-point space. In regard to game theory in particular, we obtained Nash s theorem, 2 which ascertains conditions over which certain games have equilibria in their strategy spaces. All general topology necessary in the proofs was developed extensively and in details from a basic topology
starting point, following some of the bibliographic references. Dugundji s Extension Theorem 3 - an extension of Tietze s Extension Theorem 4 for closed subsets of metric spaces into locally convex spaces-, for instance, is obtained with detais and used throughout the dissertation.
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