Fixed-point theorems in infinite-dimensional spaces: Difference between revisions

Revision as of 16:17, 8 June 2018

Revision as of 00:04, 22 July 2021

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<blockquote>'''[[Schauder fixed-point theorem]]:''' Let ''C'' be a [[nonempty]] [[Closed set|closed]] [[Convex set|convex]] subset of a [[Banach space]] ''V''. If ''f'' : ''C'' → ''C'' is [[continuous function|continuous]] with a [[compact set|compact]] image, then ''f'' has a fixed point.</blockquote>

<blockquote>'''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.</blockquote>

<blockquote>'''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 <math>f</math> is called non-expansive if <math>\|f(x)-f(y)\|\leq \|x-y\| </math> for each <math>x</math> and <math>y</math>.)</blockquote>

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.

<blockquote>'''[[Kakutani fixed point theorem|Kakutani's 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.</blockquote>

==See also==

* [[Topological degree theory]]

==References==

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==External links==

* [http://planetmath.org/encyclopedia/TychonoffFixedPointTheorem.html PlanetMath article on the Tychonoff Fixed Point Theorem]

[[Category:Fixed-point theorems]]