Banach fixed-point theorem: Difference between revisions

Revision as of 18:58, 25 August 2020

Revision as of 18:58, 25 August 2020

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==Applications==

*A standard application is the proof of the [[Picard–Lindelöf theorem]] about the existence and uniqueness of solutions to certain [[ordinary differential equation]]s. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.

*One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are [[Lipschitz continuity#Definitions|bi-lipschitz]] homeomorphisms. Let <math>\Omega</math>Ω be an open set of a Banach space <math>''E</math>''; let <math>''I\colon\Omega\to'' : Ω → ''E</math>'' denote the identity (inclusion) map and let <math>''g\colon\Omega\to'' : Ω → ''E</math>'' be a Lipschitz map of constant <math>''k'' < 1</math>. Then

#<math>\Omega'Ω′ := (''I''+''g'')(\OmegaΩ)</math> is an open subset of <math>''E</math>'': precisely, for any <math>''x\'' in\Omega</math> Ω such that <math>''B''(''x'', ''r'')\subset\Omega</math> ⊂ Ω one has <math>''B''((''I''+''g'')(''x''), ''r''(1-1−''k''))\subset\Omega'</math> ⊂ Ω′;

#<math>''I''+''g \colon\Omega\to\Omega'</math>' : Ω → Ω′ is a bi-lipschitz homeomorphism;

::precisely, <math>(''I''+''g'')^{-1}<sup>−1</mathsup> is still of the form <math>''I'' + ''h\colon\Omega\to\Omega'</math>' : Ω → Ω′ with <math>''h</math>'' a Lipschitz map of constant <math>''k''/(1-1−''k'')</math>. 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.

*It can be used to prove existence and uniqueness of solutions to integral equations.