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 research of [[Jean Leray]] that proved influential for [[algebraic topology]] and [[sheaf theory]] was motivated by the need to go beyond the Schauder fixed point theorem, proved in 1930 by [[Juliusz Schauder]].

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.

Another variant of this theorem statessays that if ''U'' is an [[open subset]] of ''C'' containing the origin (zero), then any bounded, [[Contraction mapping|contractive map]] ''f'' on the closure of ''U'' has one, or both of the following properties: (1) ''f'' has a unique fixed point, or (2) there is a point ''x'' on the boundary of ''U'' such that ''f''(''x'') = ''a'' ''x'' for some 0 < a < 1.

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

:''f'':''X'' → ''X'',