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 k < 1 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 Lipschitz constant is equal to one, 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.
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 0-7923-7073-2sv:Kontraktionsavbildning pl:Kontrakcja (matematyka)