Revision as of 16:35, 5 February 2005 edit Oleg Alexandrov (talk \| contribs) Extended confirmed users 47,270 edits m small rewordings. In the third paragraph, the requirement ''U'' is an open subset of ''C'' is suspicious; there seems to be no reason for C to be here. ← Previous edit		Revision as of 16:04, 6 February 2005 edit undo Oleg Alexandrov (talk \| contribs) Extended confirmed users 47,270 edits Rm pargrph until I hear from contrib. This page needs a careful read from somebody knowing these things (eg, explain why the name of Leray was mentioned in the first paragraph), elaborate on Tikhonov. Next edit →
Line 2: 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 says 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

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