Banach fixed point theorem

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

d(Tx,Ty)\leq q\cdot d(x,y)

for all $x,y\in X$ . Contractions have an important property.

Theorem 1 (Banach Theorem).

Every contraction has a unique http://planetmath.org/node/2777fixed point.

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 $x^{*}\in X$ . For any $x_{0}\in X$ , define recursively the following sequence

$\displaystyle x_{1}$	$\displaystyle:=$	$\displaystyle Tx_{0}$
$\displaystyle x_{2}$	$\displaystyle:=$	$\displaystyle Tx_{1}$
	$\displaystyle\vdots$
$\displaystyle x_{n+1}$	$\displaystyle:=$	$\displaystyle Tx_{n}.$

The following inequality then holds:

d(x^{*},x_{n})\leq\frac{q^{n}}{1-q}d(x_{1},x_{0}).

So the sequence $(x_{n})$ 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