Contraction mapping

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 nonnegative real number ${\displaystyle 0\leq k<1}$ 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 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, and ${\displaystyle f:M\rightarrow N}$, then there is a constant ${\displaystyle k<1}$ such that

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