The Banach contraction mapping principle and cohomology

Ludvík Janoš

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 $T : X \to X$ 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 $ℱ \subset 𝒞 (X)$ of non-negative functions $f \in 𝒞 (X)$ such that for every $f \in ℱ$ there is $g \in 𝒞 (X)$ with $f = g - g \circ T$ .

The nonexistence of universal metric flows

Stefan Geschke (2018)

Commentationes Mathematicae Universitatis Carolinae

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We consider dynamical systems of the form $(X, f)$ where $X$ is a compact metric space and $f : X \to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $ω$ -limit sets, answering a question by Will Brian.

Differential equations in metric spaces

Jacek Tabor (2002)

Mathematica Bohemica

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We give a meaning to derivative of a function $u ℝ \to X$ , 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 $x \in X$ . Let $u, v [0, 1) \to X$ , $u (0) = v (0)$ 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 $lim_{h \to 0^{+}} \frac{d (u (h), v (h))}{h} = 0 .$ By...

Some approximate fixed point theorems without continuity of the operator using auxiliary functions

Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang (2019)

Mathematica Bohemica

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We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $α$ -complete metric spaces and $μ$ -complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained....