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

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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.
 
The first result in the field was the '''Schauder fixed point theorem''', proved in 1930 by [[Juliusz Schauder]]. 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]] foundingwho founded [[sheaf theory]] came out of efforts to extend Schauder's work.
 
The '''Schauder fixed point theorem''' states, in one version, that if ''C'' is a [[nonempty]] [[closed set|closed]] [[convex]] subset of a [[Banach space]] ''V'' and ''f'' is a [[continuous function|continuous map]] from ''C'' to ''C'' whose image is [[compact set|countably compact]], then ''f'' has a fixed point.
 
The '''Tikhonov (Tychonoff) fixed point theorem''' is now applied to any [[locally convex topological vector space]] ''V''. ForIt states that for any non-empty compact convex set ''X'' in ''V'', and continuous function
 
:''f'':''X'' → ''X'',
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there is a fixed point for ''f''.
 
Other results are the Kakutani and Markov fixed point theorems, now subsumed in the [[Ryll-Nardzewski fixed point theorem]] (1967).
 
==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.
 
[[Category:Functional analysis]]