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 0 < k < 1 such that, for all x and y in M, Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Partial plot of a function f. ... In mathematics, the real numbers may be described informally in several different ways. ...

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 , then one looks for the constant k such that for all x and y in M.

Every contraction mapping is Lipschitz continuous and hence uniformly continuous. In mathematics, a function f : M &#8594; N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) &#8804; K d(x, y) for all x and y... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...

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. In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ... The Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ... Menger sponge, created by using IFS. Iterated function systems or IFS, are a kind of fractal that was conceived in its present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ...

Operator Theory

Contraction Mapping is teh Suck. 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. In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, the operator norm is a means to measure the size of certain linear operators. ...

See also

• Short map

In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have . Here and denote metrics on and , respectively. ...

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 0-7923-7073-2