# 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