Brouwer's Fixed Point Theorem/Two-Dimensional Version

Theorem

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:

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


Proof


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Source of Name

This entry was named for Luitzen Egbertus Jan Brouwer.


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Brouwer's Fixed Point Theorem
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