Contraction mapping

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>

Every contraction mapping is Lipschitz continuous and hence uniformly continuous, and has at most one fixed point.

An important property of contraction mappings is given by the Banach fixed point theorem. This states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that, for any x in M, the sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point.