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 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, then is a contractive mapping if there is a constant such that
for all x and y in M.
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, and is used in one proof of the inverse function theorem.
Firmly non-expansive mapping
A non-expansive mapping with can be strengthened to a firmly non-expansive mapping in a Hilbert space H if the following holds for all x and y in H:
A subcontraction map or subcontractor is a map f on a metric space (M,d) such that
Locally convex spaces
In a locally convex space (E,P) with topology given by a set P of seminorms, one can define for any p ∈ P a p-contraction as a map f such that there is some kp < 1 such that p(f(x) - f(y)) ≤ kp p(x - y). If f is a p-contraction for all p ∈ P and (E,P) is sequentially complete, then f has a fixed point, given as limit of any sequence xn+1 = f(xn), and if (E,P) is Hausdorff, then the fixed point is unique.
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