 Contraction mapping

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 nonnegative real number k < 1 such that for all x and y in M,
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 instead satisfied for k ≤ 1, then the mapping is said to be a nonexpansive map.
More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M,d) and (N,d') are two metric spaces, and , then there is a constant k such that
for all x and y in M.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).
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's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.^{[1]}
Contents
Firmly nonexpansive mapping
A nonexpansive mapping with k = 1 can be strengthened to a firmly nonexpansive mapping in a Hilbert space H if the following holds for all x and y in H:
where
This is a special case of α averaged nonexpansive operators with α = 1 / 2.^{[2]} A firmly nonexpansive mapping is always nonexpansive, via the Cauchy–Schwarz inequality.
See also
 Short map
 Contraction (operator theory)
Note
 ^ Theodore Shifrin, Multivariable Mathematics, Wiley, 2005, ISBN 047152638X, pp. 244–260.
 ^ Solving monotone inclusions via compositions of nonexpansive averaged operators, Patrick L. Combettes, 2004
References
 Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 9027712247 provides an undergraduate level introduction.
 Andrzej Granas and James Dugundji, Fixed Point Theory (2003) SpringerVerlag, New York, ISBN 0387001735
 William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0792370732
Categories: Metric geometry
 Fixed points
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