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

Revision as of 23:30, 19 June 2017

Revision as of 16:17, 8 June 2018

Line 3:

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 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]], first proved for finite [[simplicial complex]]es, 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.

<blockquote>'''[[Schauder fixed-point theorem]]:''' Let ''C'' be a [[nonempty]] [[Closed set|closed]] [[Convex set|convex]] subset of a [[Banach space]] ''V'',. ifIf ''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]],. forFor any non-emptynonempty 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]],. thenThen 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.