Brouwer's Fixed Point Theorem

Theorem

One-Dimensional Version

Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.

Then:

$\exists \xi \in \closedint a b: \map f \xi = \xi$

That is, a continuous real function from a closed real interval to itself fixes some point of that interval.

Smooth Mapping

A smooth mapping $f$ of the closed unit ball $B^n \subset \R^n$ into itself has a fixed point:

$\forall f \in \map {C^\infty} {B^n \to B^n}: \exists x \in B^n: \map f x = x$

General Case

A continuous mapping $f$ of the closed unit ball ${B^n}^- \subset \R^n$ into itself has a fixed point:

$\forall f \in \map {C^0} { {B^n}^- \to {B^n}^-} : \exists x \in {B^n}^- : \map f x = x$

Also known as

Brouwer's Fixed Point Theorem is also known just as Brouwer's Theorem.

Also see

• Definition:Fixed Point Theorem

Source of Name

This entry was named for Luitzen Egbertus Jan Brouwer.

Historical Note

Brouwer's fixed point theorem was published Luitzen Egbertus Jan Brouwer in $1912$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Brouwer's theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Brouwer's theorem