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Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function ''f'' mapping a compact convex set into itself there is a point ''x''0 such that ''f''(''x''0) = ''x''0. The simplest forms of Brouwer's theorem are for continuous functions ''f'' from a closed interval ''I'' in the real numbers to itself or from a closed disk ''D'' to itself. A more general form than the latter is for continuous functions from a convex compact subset ''K'' of Euclidean space to itself. Among hundreds of fixed-point theorems,〔E.g. F & V Bayart ''(Théorèmes du point fixe )'' on Bibm@th.net〕 Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.〔See page 15 of: D. Leborgne ''Calcul différentiel et géométrie'' Puf (1982) ISBN 2-13-037495-6〕 This gives it a place among the fundamental theorems of topology.〔More exactly, according to Encyclopédie Universalis: ''Il en a démontré l'un des plus beaux théorèmes, le théorème du point fixe, dont les applications et généralisations, de la théorie des jeux aux équations différentielles, se sont révélées fondamentales.'' (Luizen Brouwer ) by G. Sabbagh〕 The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu. The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard〔Jacques Hadamard: ''(Note sur quelques applications de l’indice de Kronecker )'' in Jules Tannery: ''Introduction à la théorie des fonctions d’une variable'' (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French)〕 and by Luitzen Egbertus Jan Brouwer.〔L. E. J. Brouwer ''(Über Abbildungen von Mannigfaltigkeiten )'' Mathematische Annalen 71, pp. 97–115, (German; published 25 July 1911, written July 1910)〕 ==Statement== The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: :;In the plane: Every continuous function from a closed disk to itself has at least one fixed point.〔D. Violette ''(Applications du lemme de Sperner pour les triangles )'' Bulletin AMQ, V. XLVI N° 4, (2006) p 17.〕 This can be generalized to an arbitrary finite dimension: :;In Euclidean space:Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.〔Page 15 of: D. Leborgne ''Calcul différentiel et géométrie'' Puf (1982) ISBN 2-13-037495-6.〕 A slightly more general version is as follows:〔This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see 〕 :;Convex compact set:Every continuous function from a convex compact subset ''K'' of a Euclidean space to ''K'' itself has a fixed point.〔V. & F. Bayart ''(Point fixe, et théorèmes du point fixe )'' on Bibmath.net.〕 An even more general form is better known under a different name: :;Schauder fixed point theorem:Every continuous function from a convex compact subset ''K'' of a Banach space to ''K'' itself has a fixed point.〔C. Minazzo K. Rider ''(Théorèmes du Point Fixe et Applications aux Equations Différentielles )'' Université de Nice-Sophia Antipolis.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brouwer fixed-point theorem」の詳細全文を読む スポンサード リンク
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