Brouwer theorem

Brouwer's fixed-point theorem

Under a continuous mapping $f : S \rightarrow S$ of an $n$-dimensional simplex $S$ into itself there exists at least one point $x \in S$ such that $f(x) = x$; this theorem was proved by L.E.J. Brouwer [1]. An equivalent theorem had been proved by P.G. Bohl [2] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an $n$-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.

References

[1]	L.E.J. Brouwer, "Ueber eineindeutige, stetige Transformationen von Flächen in sich" Math. Ann. , 69 (1910) pp. 176–180
[2]	P. Bohl, "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage" J. Reine Angew. Math. , 127 (1904) pp. 179–276

Comments

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 $f : \mathbf{E}^n \rightarrow \mathbf{E}^n$ 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].

References

[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 $A$ of a Euclidean space $\mathbf{E}^n$ into a subset $B$ of that space any interior point of $A$ (with respect to $\mathbf{E}^n$) is mapped to an interior point of $B$ (with respect to $\mathbf{E}^n$), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer [1].

References

[1]	L.E.J. Brouwer, "Ueber Abbildungen von Mannigfaltigkeiten" Math. Ann. , 71 (1912) pp. 97–115

M.I. Voitsekhovskii

Comments

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 ($\dim \mathbf{E}^n = n$).

References

[a1]	J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4)