Contraction mapping
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In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number <math>0 < k < 1</math> such that, for all x and y in M,
- <math>d(f(x),f(y))\leq k\,d(x,y).</math>
The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is satisfied for <math>0 < k \leq 1</math>, then the mapping is said to be non-expansive.
More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M,d) and (N,g) are two metric spaces, and <math>f:M\rightarrow N</math>, then one looks for the constant k such that <math>g(f(x),f(y))\leq k\,d(x,y)</math> for all x and y in M.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous.
A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for Iterated function systems where contraction mappings are often used. Banach fixed point theorem is also applied in proving the existence of solutions of ordinary differrential equations.
[edit] Operator Theory
In operator theory, as a special case of the above definition, a bounded operator T: X → Y between Banach spaces X and Y is a said to be a contraction if the operator norm ||T|| ≤ 1. Contractions on Hilbert spaces are the operator analogs of cos θ and are called operator angles in some contexts. Explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.
[edit] See also
[edit] References
- Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 90-277-1224-7 provides an undergraduate level introduction.
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5
- William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0-7923-7073-2de:Kontraktion (Mathematik)
