Banach fixed-point theorem: Difference between revisions

Revision as of 18:57, 25 August 2020

Revision as of 18:57, 25 August 2020

Line 58:

==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> be an open set of a Banach space <math>E</math>; let <math>I\,\colon\Omega\to E</math> denote the identity (inclusion) map and let <math>g\,\colon\Omega\to E</math> be a Lipschitz map of constant <math>k<1</math>. Then

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

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

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