Mathematics-Online lexicon: Banach Fixed-Point Theorem

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	Mathematics-Online lexicon:
	Banach Fixed-Point Theorem

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overview

is a contraction of a non-empty closed set $D\subset\mathbb{R}^n$ , i.e., if

$D=\overline{D}$ ,
$x \in D \implies g(x) \in D$ ,
$\Vert g(x)-g(y)\Vert\le c \Vert x-y\Vert\quad \forall x,y \in D$

with

, then

has a unique fixed point $x_*=g(x_*)\in D$ . Starting with any point $x_0\in D$ , $x_\ast$ can be approximated by the sequence

$\displaystyle x_0,\, x_1 = g(x_0),\, x_2=g(x_1),\,\ldots \,.$

The error satisfies

$\displaystyle \Vert x_*-x_k\Vert\le \frac{c^k}{1-c}\, \Vert x_1-x_0\Vert \,,$

i.e., the iteration converges linearly.

More generally, the fixed point theorem holds in complete metric spaces. Since the proof neither requires translation invariance nor homogeneity of the norm, $\Vert x-y\Vert$ can be replaced by a general distance function .

Annotation:

Proof

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automatically generated 9/22/2016