# Brouwer fixed-point theorem

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Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function $f$ mapping a compact convex set to itself there is a point $x_{0}$ such that $f(x_{0})=x_{0}$ . 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 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. 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 into 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 (thus, in particular, bounded and closed) and convex (or homeomorphic to convex). The following examples show why the pre-conditions are important.

### Boundedness

Consider the function

$f(x)=x+1,$ which is a continuous function from $\mathbb {R}$ to itself. As it shifts every point to the right, it cannot have a fixed point. The space $\mathbb {R}$ is convex and closed, but not bounded.

### Closedness

Consider the function

$f(x)={\frac {x+1}{2}},$ 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. 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 f(1) = 1.

### Convexity

Convexity is not strictly necessary for BFPT. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball $D^{n}$ . For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also closed, bounded, connected, without holes, etc.).

The following example shows that BFPT doesn't work for domains with holes. Consider the function $f(x)=-x$ , 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 any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it has a hole (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.