Brouwer's fixed-point theorem: Under a continuous mapping of an -dimensional simplex into itself there exists at least one point such that ; this theorem was proved by L.E.J. Brouwer . An equivalent theorem had been proved by P.G. Bohl  at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an -dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.
|||L.E.J. Brouwer, "Ueber eineindeutige, stetige Transformationen von Flächen in sich" Math. Ann. , 69 (1910) pp. 176–180|
|||P. Bohl, "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage" J. Reine Angew. Math. , 127 (1904) pp. 179–276|
There are many different proofs of the Brouwer fixed-point theorem. The shortest and conceptually easiest, however, use algebraic topology. Completely-elementary proofs also exist. Cf. e.g. [a1], Chapt. 4. In 1886, H. Poincaré proved a fixed-point result on continuous mappings which is now known to be equivalent to the Brouwer fixed-point theorem, [a2]. There are effective ways to calculate (approximate) Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria, [a1]. The first such algorithm was proposed by H. Scarf, [a3]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [a4], [a5].
|[a1]||V.I. Istrăţescu, "Fixed point theory" , Reidel (1981)|
|[a2]||H. Poincaré, "Sur les courbes definies par les équations différentielles" J. de Math. , 2 (1886)|
|[a3]||H. Scarf, "The approximation of fixed points of continuous mappings" SIAM J. Appl. Math. , 15 (1967) pp. 1328–1343|
|[a4]||S. Karamadian (ed.) , Fixed points. Algorithms and applications , Acad. Press (1977)|
|[a5]||E. Allgower, K. Georg, "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations" SIAM Rev. , 22 (1980) pp. 28–85|
Brouwer's theorem on the invariance of domain: Under any homeomorphic mapping of a subset of a Euclidean space into a subset of that space any interior point of (with respect to ) is mapped to an interior point of (with respect to ), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer .
|||L.E.J. Brouwer, "Ueber Abbildungen von Mannigfaltigkeiten" Math. Ann. , 71 (1912) pp. 97–115|
For a modern account of the Brouwer invariance-of-domain theorem cf. [a1], Chapt. 7, Sect. 3. The result is important for the idea of the topological dimension ().
|[a1]||J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4)|
Brouwer theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brouwer_theorem&oldid=17262