Displaying similar documents to “The Banach contraction mapping principle and cohomology”

Some cohomological aspects of the Banach fixed point principle

Ludvík Janoš (2011)

Mathematica Bohemica

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Let T : X X be a continuous selfmap of a compact metrizable space X . We prove the equivalence of the following two statements: (1) The mapping T is a Banach contraction relative to some compatible metric on X . (2) There is a countable point separating family 𝒞 ( X ) of non-negative functions f 𝒞 ( X ) such that for every f there is g 𝒞 ( X ) with f = g - g T .

Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

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We give a meaning to derivative of a function u X , where X is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space 𝒯 x X of x X . Let u , v [ 0 , 1 ) X , u ( 0 ) = v ( 0 ) be continuous at zero. Then by the definition u and v are in the same equivalence class if they are tangent at zero, that is if lim h 0 + d ( u ( h ) , v ( h ) ) h = 0 . By...

On the metric reflection of a pseudometric space in ZF

Horst Herrlich, Kyriakos Keremedis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show: (i) The countable axiom of choice 𝐂𝐀𝐂 is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom 𝐂𝐌𝐂 is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice 𝐀𝐂 is equivalent to each...

A generalization of boundedly compact metric spaces

Gerald Beer, Anna Di Concilio (1991)

Commentationes Mathematicae Universitatis Carolinae

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A metric space X , d is called a UC space provided each continuous function on X into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that UC spaces play relative to the compact metric spaces.