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

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{{Short description|Theorems generalizing the Brouwer fixed-point theorem}}

In [[mathematics]], a number of '''[[fixed point (mathematics)|fixed-point]] theorems in infinite-dimensional spaces''' generalise the [[Brouwer fixed-point theorem]]. They have applications, for example, to the proof of [[existence theorem]]s for [[partial differential equation]]s.

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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. Also, Aniki & Rauf (2019) presented some interesting results on the stability of partially ordered metric spaces for coupled fixed point iteration procedures for mixed monotone mappings.

<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==

* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D.Reidel, Holland (1981). {{isbn|90-277-1224-7}}.

* Andrzej Granas and [[James Dugundji]], ''Fixed Point Theory'' (2003) Springer-Verlag, New York, {{isbn|0-387-00173-5}}.

* William A. Kirk and [[Brailey Sims]], ''Handbook of Metric Fixed Point Theory'' (2001), Kluwer Academic, London {{isbn|0-7923-7073-2}}.

* Samuel A. Aniki and Kamilu Rauf, ''Some stability results in partially ordered metric spaces for coupled fixed point iteration of procedures for mixed monotone mappings'' (2019), Islamic University Multidisciplinary Journal, 6(3), 175-186 https://www.iuiu.ac.ug/journaladmin/iumj/ArticleFiles/49305.pdf

==External links==

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

[[Category:Fixed-point theorems]]