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Banach fixed point theorem
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(Theorem)
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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.
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 \begin{eqnarray*} x_1 &:=& Tx_0 \\ x_2 &:=& Tx_1 \\ &\vdots& \\ x_{n+1} &:=& Tx_n. \end{eqnarray*}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 recursive estimate is occasionally responsible for this result being known as the method of successive approximations.
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"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 |
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Cross-references: converges, inequality, sequence, applications, fixed point, estimate, theorem, 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 84631 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) |
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Pending Errata and Addenda
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