# Brouwer fixed-point theorem In 1886, Henri Poincaré (pictured) proved a result that is equivalent to Brouwer's fixed-point theorem. The three-dimensional case of the exact statement was proved in 1904 by Piers Bohl, and the general case in 1910 by Jacques Hadamard and Luitzen Egbertus Jan Brouwer.

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 x0 such that f(x0) = x0. 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, 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. This gives it a place among the fundamental theorems of topology. 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 Gérard Debreu and Kenneth Arrow.

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 and by Luitzen Egbertus Jan Brouwer.

## 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.

This can be generalized to an arbitrary finite dimension:

In Euclidean space
Every continuous function from a closed ball of a Euclidean space onto itself has a fixed point.

A slightly more general version is as follows:

Convex compact set
Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point.

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.

## Importance of the pre-conditions

The theorem holds only for sets that are compact, i.e. bounded and closed. The following examples show why these requirements are important.

### Boundedness

Consider the function

$f(x)=x+1$ which is a continuous function from R to itself. As it shifts every point to the right, it cannot have a fixed point.

### Closedness

Consider the function

$f(x)=(x+1)/2$ which is a continuous function from the open interval (-1,1) to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. It does have a fixed point for the closed interval [-1,1], namely f(x) = x = 1.