The

**Banach fixed point theorem**is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points.

Let (*X*, d) be a non-empty complete metric space. Let *T* : *X* `->` *X* be a *contraction mapping* on *X*, i.e: there is a real number *q* < 1 such that

*x*,

*y*in

*X*. Then the map

*T*admits one and only one fixed point

*x*

^{*}in

*X*(this means

*Tx*

^{*}=

*x*

^{*}). Furthermore, this fixed point can be found as follows: start with an arbitrary element

*x*

_{0}in

*X*and define a sequence by

*x*

_{n}=

*Tx*

_{n-1}for

*n*= 1, 2, 3, ... This sequence converges, and its limit is

*x*

^{*}. The following inequality describes the speed of convergence:

*Tx*,

*Ty*) < d(

*x*,

*y*) for all unequal

*x*and

*y*is in general not enough to ensure the existence of a fixed point, as is shown by the map

*T*: [1,∞) → [1,∞) with

*T*(

*x*) =

*x*+ 1/

*x*, which lacks a fixed point. However, if the space

*X*is compact, then this weaker assumption does imply all the statements of the theorem.

When using the theorem in practice, the most difficult part is typically to define *X* properly so that *T* actually maps elements from *X* to *X*, i.e. that *Tx* is always an element of *X*.

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.

An earlier version of this article was posted on Planet Math. This article is open content