About: Fixed-point theorems in infinite-dimensional spaces

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In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : C → C is continuous with a compact image, then f has a fixed point. Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point.

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rdfs:label	Fixed-point theorems in infinite-dimensional spaces (en)
rdfs:comment	In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : C → C is continuous with a compact image, then f has a fixed point. Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point. (en)
sameAs	Fixed-point theorems in infinite-dimensional spaces
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Subject	Fixed-point theorems
Link from a Wikipa... related subject.	dbpedia-fr:Théorème_du_point_fixe_de_Schauder
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Wikipage revision ID	1034813255 (xsd:integer)
Link from a Wikipage to another Wikipage	Locally convex topological vector space Uniformly convex space Algebraic topology Banach fixed-point theorem Banach space Brouwer fixed-point theorem James Dugundji Juliusz Schauder Fixed-point theorems Jean Leray Metric space Sheaf (mathematics) Closed set Compact space Continuous function Contraction mapping Convex set Earle–Hamilton fixed-point theorem Partial differential equation Empty set Fixed-point subgroup Ryll-Nardzewski fixed-point theorem Schauder fixed-point theorem Existence theorem Topological degree theory Kakutani fixed-point theorem Markov–Kakutani fixed-point theorem Mathematics Simplicial complex
has abstract	In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder's work. Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : C → C is continuous with a compact image, then f has a fixed point. Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point. Browder fixed-point theorem: Let K be a nonempty closed bounded convex set in a uniformly convex Banach space. Then any non-expansive function f : K → K has a fixed point. (A function is called non-expansive if for each and .) Other results include the Markov–Kakutani fixed-point theorem (1936-1938) and the Ryll-Nardzewski fixed-point theorem (1967) for continuous affine self-mappings of compact convex sets, as well as the Earle–Hamilton fixed-point theorem (1968) for holomorphic self-mappings of open domains. Kakutani fixed-point theorem: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces
is Wikipage redirect of	Tikhonov fixed point theorem Tikhonov's fixed point theorem Tikhonov fixed-point theorem Fixed point theorems in infinite-dimensional spaces Tychonoff fixed-point theorem Tychonoff fixed point theorem
is Link from a Wikipage to another Wikipage of	List of theorems Nonlinear functional analysis Banach fixed-point theorem Brouwer fixed-point theorem List of functional analysis topics List of convexity topics Fixed-point theorems Kakutani fixed-point theorem Fixed point theorems in infinite-dimensional spaces
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces

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