View source for Fixed-point theorems in infinite-dimensional spaces

In [[Mathematics|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|existence theorem]]s for [[Partial differential equation|partial differential equation]]s. 
The first result in the field was the '''[[Schauder fixed-point theorem]]''', proved in 1930 by [[Biography:Juliusz Schauder|Juliusz Schauder]] (a previous result in a different vein, the [[Banach fixed-point theorem]] for [[Contraction mapping|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|algebraic topology]], first proved for finite [[Simplicial complex|simplicial complex]]es, to spaces of infinite dimension. For example, the research of [[Biography:Jean Leray|Jean Leray]] who founded sheaf theory came out of efforts to extend Schauder's work.

<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 image, then ''f'' has a fixed point.</blockquote>

<blockquote>'''Tikhonov (Tychonoff) fixed-point theorem:''' Let ''V'' be a [[Locally convex topological vector space|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]]:''' 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 [[Biography:James Dugundji|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}}.

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

[[Category:Fixed-point theorems]]

fr:Théorème du point fixe de Schauder

{{Sourceattribution|Fixed-point theorems in infinite-dimensional spaces}}