, a contraction mapping
, or contraction
, on a metric space M
is a function f
to itself, with the property that there is some real number k
< 1 such that, for all
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.