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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 map from ''C'' to ''C'' whose image is [[compact|countably compact]], then ''f'' has a fixed point.
One variant states that if ''U'' is an open subset of ''C'' containing the origin (zero), then any bounded, 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]]
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