Banach Fixed-point Theorem - Applications

Applications

• A standard application is the proof of the Picard–Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. 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 perturbation of the identity are bi-lipschitz homeomorphisms. Let be an open set of a Banach space ; let denote the identity (inclusion) map and let be a Lipschitz map of constant k<1. Then (i) is an open subset of :precisely, for any such that one has

; (ii)

is a bi-lipschitz homeomorphism; precisely, is still of the form

,

with a Lipschitz map of constant

A direct consequence of this result yields the proof of the inverse function theorem.