Concept

# Contraction mapping

Summary
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number 0 \leq k < 1 such that for all x and y in M, :d(f(x),f(y)) \leq k,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 instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive 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, then f:M \rightarrow N is a contractive mapping if there is a constant 0 \leq k < 1 such that :d'(f(x),f(y)) \leq k,d(x,y) for all x and y in M. Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, th
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