Brouwer fixed-point theorem: Difference between revisions

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Brouwer's theorem is probably the most important.<ref>"... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu ''Fixed Point Theory an Introduction'' Kluwer Academic Publishers (new edition 2001) {{isbn|1-4020-0301-3}}.</ref> It is also among the foundational theorems on the topology of [[topological manifold]]s and is often used to prove other important results such as the [[Jordan curve theorem]].<ref>E.g.: S. Greenwood J. Cao'' [http://www.math.auckland.ac.nz/class750/section5.pdf Brouwer’s Fixed Point Theorem and the Jordan Curve Theorem]'' University of Auckland, New Zealand.</ref>

Besides the fixed-point theorems for more or less [[contraction mapping|contracting]] functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the [[Borsuk–Ulam theorem]] says that a continuous map from the ''n''-dimensional sphere to '''R'''ⁿ has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the [[Lefschetz fixed-point theorem]] provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to [[Banach space]]s.<ref>{{cite journal |first=J. |last=Schauder |title=Der Fixpunktsatz in Funktionsräumen |journal=[[Studia Mathematica]] |volume=2 |year=1930 |pages=171–180 |doi= 10.4064/sm-2-1-171-180|doi-access=free }}</ref> This generalization is known as [[Fixed-point theorems in infinite-dimensional spaces|Schauder's fixed-point theorem]], a result generalized further by S. Kakutani to [[multivaluedSet-valued function|set-valued functions]]s.<ref>{{cite journal |first=S. |last=Kakutani |title=A generalization of Brouwer's Fixed Point Theorem |journal= Duke Mathematical Journal|volume=8 |year=1941 |issue=3 |pages=457–459 |doi=10.1215/S0012-7094-41-00838-4 }}</ref> One also meets the theorem and its variants outside topology. It can be used to prove the [[Hartman-Grobman theorem]], which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the [[Central Limit Theorem]]. The theorem can also be found in existence proofs for the solutions of certain [[partial differential equation]]s.<ref>These examples are taken from: F. Boyer ''[http://www.cmi.univ-mrs.fr/~fboyer/ter_fboyer2.pdf Théorèmes de point fixe et applications]'' CMI Université Paul Cézanne (2008–2009) [https://www.webcitation.org/5refXIDvI?url=http://www.cmi.univ-mrs.fr/%7Efboyer/ter_fboyer2.pdf Archived copy] at [[WebCite]] (August 1, 2010).</ref>

Other areas are also touched. In [[game theory]], [[John Forbes Nash|John Nash]] used the theorem to prove that in the game of [[Hex (board game)|Hex]] there is a winning strategy for white.<ref>For context and references see the article [[Hex (board game)]].</ref> In economics, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria ([[Hotelling's law]]), financial equilibria and incomplete markets.<ref>P. Bich ''[http://www.ann.jussieu.fr/~plc/code2007/bich.pdf Une extension discontinue du théorème du point fixe de Schauder, et quelques applications en économie] {{webarchive |url=https://web.archive.org/web/20110611140634/http://www.ann.jussieu.fr/~plc/code2007/bich.pdf |date=June 11, 2011 }}'' Institut Henri Poincaré, Paris (2007)</ref>