Brouwer fixed-point theorem
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 Henri Poincaré and Charles Émile Picard.
Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball
Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other.
The case n = 3 first was proved by Piers Bohl in 1904 (published in Journal für die reine und angewandte Mathematik).
[15][16] At the end of the 19th century, the old problem[17] of the stability of the solar system returned into the focus of the mathematical community.
[22] 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.
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".
[24] In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,[25] although the connection with the subject of this article was not yet apparent.
Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations.
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.
[33] 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.
[35] 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.
[40] Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion.
In the finite-dimensional case, the Lefschetz fixed-point theorem provided from 1926 a method for counting fixed points.
He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand against set theory.
Brouwer's original 1911 proof relied on the notion of the degree of a continuous mapping, stemming from ideas in differential topology.
are precisely the fixed points of the original function f. This requires some work to make fully general.
The more modern advent of homology theory simplifies the construction of the degree, and so has become a standard proof in the literature.
If there were a fixed-point-free continuous self-mapping f of the closed unit ball B of the n-dimensional Euclidean space V, set Since f has no fixed points, it follows that, for x in the interior of B, the vector w(x) is non-zero; and for x in S, the scalar product x ⋅ w(x) = 1 – x ⋅ f(x) is strictly positive.
Set By construction X is a continuous vector field on the unit sphere of W, satisfying the tangency condition y ⋅ X(y) = 0.
For n odd, one can apply the fixed point theorem to the closed unit ball B in n + 1 dimensions and the mapping F(x,y) = (f(x),0).
The advantage of this proof is that it uses only elementary techniques; more general results like the Borsuk-Ulam theorem require tools from algebraic topology.
The impossibility of a retraction can also be shown using the de Rham cohomology of open subsets of Euclidean space En.
It is an easy numerical task to follow such a path from q to the fixed point so the method is essentially computable.
[57] gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.
The basic theorem regarding Hex, first proven by John Nash, is that no game of Hex can end in a draw; the first player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater).
Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.
The Kakutani fixed point theorem generalizes the Brouwer fixed-point theorem in a different direction: it stays in Rn, but considers upper hemi-continuous set-valued functions (functions that assign to each point of the set a subset of the set).
