Editing Brouwer fixed-point theorem (section)

==History==
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>

===Before 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.]]
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>

He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant [[flow (mathematics)|flow]]?<ref>This question was studied in: {{cite journal |first=H. |last=Poincaré |title=Sur les courbes définies par les équations différentielles |journal=[[Journal de Mathématiques Pures et Appliquées]] |volume=2 |issue=4 |pages=167–244 |year=1886 }}</ref> Poincaré discovered that the answer can be found in what we now call the [[topology|topological]] properties in the area containing the trajectory. If this area is [[compact space|compact]], i.e. both [[closed set|closed]] and [[bounded set|bounded]], then the trajectory either becomes stationary, or it approaches a [[limit cycle]].<ref>This follows from the [[Poincaré–Bendixson theorem]].</ref> Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval&nbsp;''t''. If the area is a circular band, or if it is not closed,<ref>Multiplication by {{sfrac|1|2}} on ]0,&nbsp;1[<sup>2</sup> has no fixed point.</ref> then this is not necessarily the case.

To understand differential equations better, a new branch of mathematics was born. Poincaré called it ''analysis situs''. The French [[Encyclopædia Universalis]] defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".<ref>"concerne les propriétés invariantes d'une figure lorsqu'on la déforme de manière continue quelconque, sans déchirure (par exemple, dans le cas de la déformation de la sphère, les propriétés corrélatives des objets tracés sur sa surface". From C. Houzel M. Paty ''[http://www.scientiaestudia.org.br/associac/paty/pdf/Paty,M_1997g-PoincareEU.pdf Poincaré, Henri (1854–1912)] {{webarchive|url=https://web.archive.org/web/20101008232932/http://www.scientiaestudia.org.br/associac/paty/pdf/Paty%2CM_1997g-PoincareEU.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.scientiaestudia.org.br/associac/paty/pdf/Paty%2CM_1997g-PoincareEU.pdf |archive-date=2022-10-09 |url-status=live |date=2010-10-08 }}'' Encyclopædia Universalis Albin Michel, Paris, 1999, p.&nbsp;696–706</ref> In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,<ref>Poincaré's theorem is stated in: V. I. Istratescu ''Fixed Point Theory an Introduction'' Kluwer Academic Publishers (réédition de 2001) p 113 {{isbn|1-4020-0301-3}}</ref> although the connection with the subject of this article was not yet apparent.<ref>{{SpringerEOM|title=Brouwer theorem |first=M.I. |last=Voitsekhovskii |isbn=1-4020-0609-8}}</ref> A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the [[fundamental group]] or sometimes the Poincaré group.<ref>{{cite book |first=Jean |last=Dieudonné |author-link=Jean Dieudonné |title=A History of Algebraic and Differential Topology, 1900–1960 |location=Boston |publisher=Birkhäuser |year=1989 |isbn=978-0-8176-3388-2 |pages=[https://archive.org/details/historyofalgebra0000dieu_g9a3/page/17 17–24] |url=https://archive.org/details/historyofalgebra0000dieu_g9a3/page/17 }}</ref> This method can be used for a very compact proof of the theorem under discussion.<!-- fr.wikipedia has it in its article on the fundamental group, we don't -->

Poincaré's method was analogous to that of [[Charles Émile Picard|Émile Picard]], a contemporary mathematician who generalized the [[Cauchy–Lipschitz theorem]].<ref>See for example: [[Charles Émile Picard|Émile Picard]] ''[http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires] {{Webarchive|url=https://web.archive.org/web/20110716055143/http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf# |archive-url=https://web.archive.org/web/20110716055143/http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf |archive-date=2011-07-16 |url-status=live |date=2011-07-16 }}'' Journal de Mathématiques p 217 (1893)</ref> Picard's approach is based on a result that would later be formalised by [[Banach fixed-point theorem|another fixed-point theorem]], named after [[Stefan Banach|Banach]]. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a [[contraction mapping|contraction]].

===First proofs===
At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. [[Piers Bohl]], a [[Latvia]]n mathematician, applied topological methods to the study of differential equations.<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bohl.html Piers Bohl]''</ref> In 1904 he proved the three-dimensional case of our theorem,<ref name="Bohl1904" /> but his publication was not noticed.<ref>{{cite journal |first1=A. D. |last1=Myskis |first2=I. M. |last2=Rabinovic |title=Первое доказательство теоремы о неподвижной точке при непрерывном отображении шара в себя, данное латышским математиком П.Г.Болем |trans-title=The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P. G. Bohl |language=ru |journal=Успехи математических наук |volume=10 |issue=3 |year=1955 |pages=188–192 |url=http://mi.mathnet.ru/eng/umn/v10/i3/p179 }}</ref>

It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially [[mathematical logic]] and [[topology]]. His initial interest lay in an attempt to solve [[Hilbert's fifth problem]].<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html Luitzen Egbertus Jan Brouwer]''</ref> In 1909, during a voyage to Paris, he met [[Henri Poincaré]], [[Jacques Hadamard]], and [[Émile Borel]]. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the [[hairy ball theorem]] for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.<ref>{{cite journal |first=Hans  |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 495] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |doi-access=free }}</ref> These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem.<ref>{{cite journal |first=Hans |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 495] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |quote=... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré |doi-access=free }}</ref> The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as [[homotopy]], the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. [[Hans Freudenthal]] comments on the respective roles as follows: <!-- NON-LITERAL QUOTATION! translated back from French -->"Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."<ref>{{cite journal |first=Hans |last=Freudenthal |author-link=Hans Freudenthal | title=The cradle of modern topology, according to Brouwer's inedita |journal=[[Historia Mathematica]] |volume=2 |issue=4 |pages=495–502 [p. 501] |year=1975 |doi=10.1016/0315-0860(75)90111-1 |doi-access=free }}</ref>

Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension,<ref name="brouwer-1910" /> as well as other key theorems such as the invariance of dimension.<ref>If an open subset of a [[manifold]] is [[homeomorphism|homeomorphic]] to an open subset of a Euclidean space of dimension ''n'', and if ''p'' is a positive integer other than ''n'', then the open set is never homeomorphic to an open subset of a Euclidean space of dimension ''p''.</ref> In the context of this work, Brouwer also generalized the [[Jordan curve theorem]] to arbitrary dimension and established the properties connected with the [[degree of a continuous mapping]].<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html Luitzen Egbertus Jan Brouwer]''.</ref> This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became [[algebraic topology]].<ref>The term ''algebraic topology'' first appeared 1931 under the pen of David van Dantzig: J. Miller ''[http://jeff560.tripod.com/t.html Topological algebra]'' on the site Earliest Known Uses of Some of the Words of Mathematics (2007)</ref>

===Reception===
[[Image:John f nash 20061102 2.jpg|thumb|220px|left|[[John Forbes Nash|John Nash]] used the theorem in [[game theory]] to prove the existence of an equilibrium strategy profile.]]
The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called [[fixed-point theory]].<ref>V. I. Istratescu ''Fixed Point Theory. An Introduction'' Kluwer Academic Publishers (new edition 2001) {{isbn|1-4020-0301-3}}.</ref>
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'''<sup>n</sup> 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 [[Set-valued function|set-valued functions]].<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>

Brouwer's celebrity is not exclusively due to his topological work. The proofs of his great topological theorems are [[constructive proof|not constructive]],<ref>For a long explanation, see: {{cite journal |first=J. P. |last=Dubucs |url=http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1988_num_41_2_4094 |title=L. J. E. Brouwer : Topologie et constructivisme |journal=Revue d'Histoire des Sciences |volume=41 |issue=2 |pages=133–155 |year=1988 |doi=10.3406/rhs.1988.4094 }}</ref> and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of [[constructivism (mathematics)|constructivity]]. He became the originator and zealous defender of a way of formalising mathematics that is known as [[intuitionistic logic|intuitionism]], which at the time made a stand against [[set theory]].<ref>Later it would be shown that the formalism that was combatted by Brouwer can also serve to formalise intuitionism, with some modifications. For further details see [[constructive set theory]].</ref> Brouwer disavowed his original proof of the fixed-point theorem.