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{{Short description|Theorem about metric spaces}}
In [[mathematics]], the '''Banach fixed-point theorem''' (also known as the '''contraction mapping theorem''' or '''contractive mapping theorem''' or '''
==Statement==
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* One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are [[Lipschitz continuity#Definitions|bi-lipschitz]] homeomorphisms. Let Ω be an open set of a Banach space ''E''; let {{nobr|''I'' : Ω → ''E''}} denote the identity (inclusion) map and let ''g'' : Ω → ''E'' be a Lipschitz map of constant ''k'' < 1. Then
# Ω′ := (''I'' + ''g'')(Ω) is an open subset of ''E'': precisely, for any ''x'' in Ω such that {{nobr|''B''(''x'', ''r'') ⊂ Ω}} one has {{nobr|''B''((''I'' + ''g'')(''x''), ''r''(1 − ''k'')) ⊂ Ω′;}}
# ''I'' + ''g'' : Ω → Ω′ is a bi-
: precisely, (''I'' + ''g'')<sup>−1</sup> is still of the form {{nobr|''I'' + ''h'' : Ω → Ω′}} with ''h'' a Lipschitz map of constant ''k''/(1 − ''k''). A direct consequence of this result yields the proof of the [[inverse function theorem]].
* It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third-order method.
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Let ''T'' : ''X'' → ''X'' be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are:
*Assume that some iterate ''T<sup>n</sup>'' of ''T'' is a contraction. Then ''T'' has a unique fixed point.
*Assume that for each ''n'', there exist ''c<sub>n</sub>'' such that ''d''(''T''<sup>''n''</sup>(''x''), ''T''<sup>''n''</sup>(''y'')) ≤ ''c''<sub>''n''</sub>''d''(''x'', ''y
::<math>\sum\nolimits_n c_n <\infty.</math>
:Then ''T'' has a unique fixed point.
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i.e.
: <math>f(\pi)=\pi</math>
and also the function <math>f</math> is around {{pi}} the contraction mapping from the obvious reasons because its derivative in {{pi}} vanishes therefore {{pi}} can be obtained from the infinite superposition for example for the argument value 3:▼
▲<math>f</math> is around {{pi}} the contraction mapping from the obvious reasons because its derivative in {{pi}} vanishes therefore {{pi}} can be obtained from the infinite superposition for example for the argument value 3:
: <math>\pi=f(f(f(
Already the triple superposition of this function at <math>3</math> gives {{pi}} with accuracy to 33 digits:
: <math>f(f(f(3)))=3.141592653589793238462643383279502
==See also==
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