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,

${\displaystyle d(f(x),f(y))\leq k\,d(x,y).}$

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 ${\displaystyle f:M\rightarrow N}$ is a contractive mapping if there is a constant ${\displaystyle k<1}$ such that

${\displaystyle d'(f(x),f(y))\leq k\,d(x,y)}$

for all x and y in M.

Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).

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.[1]

Contraction mapping plays an important role in dynamic programming problems.[2][3]

Firmly non-expansive mapping

A non-expansive mapping with ${\displaystyle k=1}$ 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:

${\displaystyle \|f(x)-f(y)\|^{2}\leq \,\langle x-y,f(x)-f(y)\rangle .}$

where

${\displaystyle d(x,y)=\|x-y\|}$

This is a special case of ${\displaystyle \alpha }$ averaged nonexpansive operators with ${\displaystyle \alpha =1/2}$.[4] A firmly non-expansive mapping is always non-expansive, via the Cauchy–Schwarz inequality.

Subcontraction map

A subcontraction map or subcontractor is a map f on a metric space (M,d) such that

${\displaystyle d(f(x),f(y))\leq d(x,y)\ ;}$
${\displaystyle d(f(f(x)),f(x))

If the image of a subcontractor f is compact, then f has a fixed point.[5]

Locally convex spaces

In a locally convex space (E,P) with topology given by a set P of seminorms, one can define for any pP 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 pP 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.[6]