Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a compact disk to itself. the more general name than the latter is for continual functions from a convex compact subset of Euclidean space to itself.

Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in component to its ownership across many fields of mathematics. In its original field, this or situation. is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This permits it a place among the necessary theorems of topology. The theorem is also used for proving deep results about differential equations & is forwarded in near introductory courses on differential geometry. It appears in unlikely fields such(a) 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 conception of produce on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such(a) as the Poincaré–Bendixson theorem requires the usage of topological methods. This work at the end of the 19th century opened into several successive list of paraphrases of the theorem. The issue of differentiable mappings of the -dimensional closed ball was number one proved in 1910 by Jacques Hadamard and the general issue for continual mappings by Brouwer in 1911.

Importance of the pre-conditions

The theorem holds only for functions that are endomorphisms functions that have the same sort as the domain and range and for sets that are compact thus, in particular, bounded and closed and convex or homeomorphic to convex. The coming after or as a sum of. examples show why the pre-conditions are important.

Consider the function

with domain [-1,1]. The range of the function is [0,2]. Thus, f is non an endomorphism.

Consider the function

which is a continuous function from to itself. As it shifts every piece to the right, it cannot have a fixed point. The space is convex and closed, but non bounded.

Consider the function

which is a continuous function from the open interval −1,1 to itself. In this interval, it shifts every segment to the right, so it cannot have a fixed point. The space −1,1 is convex and bounded, but not closed. The function f does have a fixed point for the closed interval [−1,1], namely f1 = 1.

Convexity is not strictly necessary for BFPT. Because the properties involved continuity, being a fixed point are invariant under closed, bounded, connected, without holes, etc..

The coming after or as a result of. example shows that BFPT doesn't work for domains with holes. Consider the function , which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point. The analogous example works for the n-dimensional sphere or all symmetric domain that does not contain the origin. The unit circle is closed and bounded, but it has a gap and so it is not convex . The function f does have a fixed point for the unit disc, since it takes the origin to itself.

A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.

The continuous function in this theorem is not invited to be bijective or even surjective.