Brouwer's Fixed Point Theorem

Theorem

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.

Let $D \subseteq \R^2$ be the closed disk defined as:

$D = \set {\tuple {x, y} \in \R^2: x^2 + y^2 \le 1}$

Let $f: D \to D$ be a mapping which is continuous on $D$.

Then:

$\exists \xi \in D: \map f \xi = \xi$

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

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

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

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

is also known just as Brouwer's Theorem.

This entry was named for Luitzen Egbertus Jan Brouwer.

was published Luitzen Egbertus Jan Brouwer in $1912$.

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