Brouwer fixed-point theorem: Difference between revisions

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The Brouwer fixed point theorem was one of the early achievements of [[algebraic topology]], and is the basis of more general [[fixed point theorem]]s which are important in [[functional analysis]]. The case ''n'' = 3 first was proved by [[Piers Bohl]] in 1904 (published in ''[[Journal für die reine und angewandte Mathematik]]'').<ref name=Bohl1904>{{cite journal |first=P. |last=Bohl |title= Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage |journal=J. Reine Angew. Math. |volume=127 |issue=3/4 |pages=179–276 |year=1904 }}</ref> It was later proved by [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] in 1909. [[Jacques Hadamard]] proved the general case in 1910,<ref name="hadamard-1910" /> and Brouwer found a different proof in the same year.<ref name="brouwer-1910" /> Since these early proofs were all [[Constructive proof|non-constructive]] [[indirect proof]]s, they ran contrary to Brouwer's [[intuitionist]] ideals. Although the existence of a fixed point is not constructive in the sense of [[Constructivism (mathematics)|constructivism in mathematics]], methods to [[Approximation theory|approximate]] fixed points guaranteed by Brouwer's theorem are now known.<ref name=Karamardian1977>{{cite book|last1=Karamardian|first1=Stephan|title=Fixed points: algorithms and applications|date=1977|publisher=Academic Press|___location=New York|isbn=978-0-12-398050-2}}</ref><ref name=Istratescu1981>{{cite book|last1=Istrăţescu|first1=Vasile|title=Fixed point theory|date=1981|publisher=D. Reidel Publishing Co.|___location=Dordrecht-Boston, Mass.|isbn=978-90-277-1224-0}}</ref>

===PrehistoryBefore discovery===

[[File:Théorème-de-Brouwer-(cond-1).jpg|thumb|right|For flows in an unbounded area, or in an area with a "hole", the theorem is not applicable.]]

[[File:Théorème-de-Brouwer-(cond-2).jpg|thumb|left|The theorem applies to any disk-shaped area, where it guarantees the existence of a fixed point.]]

To understand the prehistory of Brouwer's fixed point theorem one needs to pass through [[differential equation]]s. At the end of the 19th century, the old problem<ref>See F. Brechenmacher ''[https://arxiv.org/abs/0704.2931 L'identité algébrique d'une pratique portée par la discussion sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des planètes]'' CNRS Fédération de Recherche Mathématique du Nord-Pas-de-Calais</ref> of the [[stability of the solar system]] returned into the focus of the mathematical community.<ref>[[Henri Poincaré]] won the [[Oscar II, King of Sweden|King of Sweden]]'s mathematical competition in 1889 for his work on the related [[three-body problem]]: [[Jacques Tits]] ''[http://www.culture.gouv.fr/culture/actualites/celebrations2004/poincare.htm Célébrations nationales 2004]'' Site du Ministère Culture et Communication</ref>

Its solution required new methods. As noted by [[Henri Poincaré]], who worked on the [[three-body problem]], there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."<ref name=methodes>[[Henri Poincaré]] ''Les méthodes nouvelles de la mécanique céleste'' T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987.</ref>

He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".<ref>Quotation from [[Henri Poincaré]] taken from: P. A. Miquel ''[http://www.arches.ro/revue/no03/no3art03.htm La catégorie de désordre] {{Webarchive|url=https://web.archive.org/web/20160303205947/http://www.arches.ro/revue/no03/no3art03.htm# |date=2016-03-03 }}'', on the website of l'Association roumaine des chercheurs francophones en sciences humaines</ref>