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

Revision as of 07:56, 30 June 2008

Revision as of 23:29, 3 March 2009

Line 5:

The [[Schauder fixed point theorem]] states, in one version, that if ''C'' is a [[nonempty]] [[Closed set|closed]] [[Convex set|convex]] subset of a [[Banach space]] ''V'' and ''f'' is a [[continuous function|continuous map]] from ''C'' to ''C'' whose image is [[compact set|compact]], then ''f'' has a fixed point.

The '''Tikhonov (Tychonoff) fixed point theorem''' is applied to any [[locally convex topological vector space]] ''V''. It states that for any non-empty compact convex set ''X'' in ''V'', andany continuous function

:''fƒ'':''X'' → ''X'',

there ishas a fixed point for ''f''.

Other results are the [[Shizuo Kakutani|Kakutani]] and Markov fixed point theorems, as well as the [[Ryll-Nardzewski fixed point theorem]] (1967).