Banach fixed point theorem

Let (X,d) be a complete metric space. A function T:XX is said to be a contraction mapping if there is a constant q with 0q<1 such that


for all x,yX. Contractions have an important property.

Theorem 1 (Banach Theorem).

Every contraction has a unique pointPlanetmathPlanetmath.

There is an estimate to this fixed point that can be useful in applications. Let T be a contraction mapping on (X,d) with constant q and unique fixed point x*X. For any x0X, define recursively the following sequence

x1 := Tx0
x2 := Tx1
xn+1 := Txn.

The following inequalityMathworldPlanetmath then holds:


So the sequence (xn) converges to x*. This estimate is occasionally responsible for this result being known as the method of successive approximations.

Title Banach fixed point theorem
Canonical name BanachFixedPointTheorem
Date of creation 2013-03-22 12:31:10
Last modified on 2013-03-22 12:31:10
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 21
Author mathwizard (128)
Entry type Theorem
Classification msc 54A20
Classification msc 47H10
Classification msc 54H25
Synonym contraction principle
Synonym contraction mapping theorem
Synonym method of successive approximations
Synonym Banach-Caccioppoli fixed point theorem
Related topic FixedPoint
Defines contraction mapping
Defines contraction operator