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

Revision as of 08:36, 6 May 2004

Revision as of 08:37, 6 May 2004

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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 [[Julius 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 map from ''C'' to ''C'' whose image is [[compact|countably compact]], then ''f'' has a fixed point.

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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-NardzeswkiNardzewski fixed point theorem (1967).