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 such that, for all x and y in M,

$d(f(x),f(y))\backslash leq\; k\backslash ,d(x,y).$

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 , 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 $f:M\backslash rightarrow\; N$, then one looks for the constant k such that $g(f(x),f(y))\backslash leq\; k\backslash ,d(x,y)$ 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.

See also

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• Short map

References

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• 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 0-7923-7073-2

Metric geometry | Fixed points

Kontraktion (Mathematik) | Kontrakcja (matematyka) | Kontraktionsavbildning