Category:Brouwer's Fixed Point Theorem
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This category contains pages concerning Brouwer's Fixed Point 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:
- $\forall f \in \map {C^0} { {B^n}^- \to {B^n}^-} : \exists x \in {B^n}^- : \map f x = x$
Pages in category "Brouwer's Fixed Point Theorem"
The following 8 pages are in this category, out of 8 total.
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- Brouwer's Fixed Point Theorem
- Brouwer's Fixed Point Theorem/Also known as
- Brouwer's Fixed Point Theorem/General Case
- Brouwer's Fixed Point Theorem/One-Dimensional Version
- Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof by Intermediate Value Theorem
- Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof Using Connectedness
- Brouwer's Fixed Point Theorem/Smooth Mapping
- Brouwer's Theorem