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
A contraction mapping has at most one fixed point.
Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty 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.
[1] Contraction mappings play an important role in dynamic programming problems.
can be generalized to a firmly non-expansive mapping in a Hilbert space
The class of firmly non-expansive maps is closed under convex combinations, but not compositions.
[5] This class includes proximal mappings of proper, convex, lower-semicontinuous functions, hence it also includes orthogonal projections onto non-empty closed convex sets.
The class of firmly nonexpansive operators is equal to the set of resolvents of maximally monotone operators.
[6] Surprisingly, while iterating non-expansive maps has no guarantee to find a fixed point (e.g. multiplication by -1), firm non-expansiveness is sufficient to guarantee global convergence to a fixed point, provided a fixed point exists.
[5] A subcontraction map or subcontractor is a map f on a metric space (M, d) such that If the image of a subcontractor f is compact, then f has a fixed point.
[7] 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.