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单词 BanachFixedPointTheorem
释义

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

d(Tx,Ty)qd(x,y)

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

Theorem 1 (Banach Theorem).

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

d(x*,xn)qn1-qd(x1,x0).

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

TitleBanach fixed point theorem
Canonical nameBanachFixedPointTheorem
Date of creation2013-03-22 12:31:10
Last modified on2013-03-22 12:31:10
Ownermathwizard (128)
Last modified bymathwizard (128)
Numerical id21
Authormathwizard (128)
Entry typeTheorem
Classificationmsc 54A20
Classificationmsc 47H10
Classificationmsc 54H25
Synonymcontraction principle
Synonymcontraction mapping theorem
Synonymmethod of successive approximations
SynonymBanach-Caccioppoli fixed point theorem
Related topicFixedPoint
Definescontraction mapping
Definescontraction operator
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