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Banach fixed point theorem (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

$\displaystyle d(Tx,Ty)\leq q\cdot d(x,y) $
for all $ x,y\in X$. Contractions have an important property.
Theorem 1 (Banach Fixed Point Theorem)   Every contraction has a unique fixed 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:
$\displaystyle d(x^*,x_n)\leq \frac{q^n}{1-q}d(x_1,x_0). $
So the sequence $ (x_n)$ converges to $ x^*$. This recursive estimate is occasionally responsible for this result being known as the method of successive approximations.



"Banach fixed point theorem" is owned by mathwizard. [ full author list (2) | owner history (1) ]
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See Also: fixed point

Other names:  contraction principle, contraction mapping theorem, method of successive approximations, Banach-Caccioppoli fixed point theorem
Also defines:  contraction mapping, contraction operator

Attachments:
proof of Banach fixed point theorem (Proof) by asteroid
contractive maps are uniformly continuous (Theorem) by mathcam
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Cross-references: converges, inequality, sequence, fixed point, estimate, function, metric space, complete
There are 9 references to this entry.

This is version 18 of Banach fixed point theorem, born on 2002-03-07, modified 2007-10-01.
Object id is 2758, canonical name is BanachFixedPointTheorem.
Accessed 79191 times total.

Classification:
AMS MSC: 54A20 (General topology :: Generalities :: Convergence in general topology )
 47H10 (Operator theory :: Nonlinear operators and their properties :: Fixed-point theorems)
 54H25 (General topology :: Connections with other structures, applications :: Fixed-point and coincidence theorems)
Pending Errata and Addenda
None.
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Discussion
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a question regarding the radius of the attraction balls by ecatinas on 2005-02-28 05:41:40
 Provided that the metric comes from a norm, is anyone aware about results regarding the evaluation of the radius of the attraction balls?
 More precisely, given T : X -> X Lipschitz (with constant L), with the fixed point x*, q = ||T'(x*)|| < 1, estimate r > 0 in terms of q and L such that for any initial approximation from the ball B(x*,r), the successive approximations converge to x*.
 I have obtained such an estimation and sent for publication, but since the result is obtained in a rather elementary manner, I am still seeking for related results. The mathematicians I have contacted so far are not aware of such results.
 Thanks in advance.
 Emil Catinas
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Otherwise known as "Banach-Caccioppoli" by Oblomov on 2003-10-27 06:13:33
Hello,

I would just like to remark that (at least in Italy) this is known as the Banach-Caccioppoli fixed point theorem. Would it be possible to update the entry to add synonyms?
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