contraction mapping

A contraction mapping has at most one fixed point.

Thus a contraction mapping is strictly metric, but not necessarily the other way around.

This fixed point iteration is a contraction mapping for x around P.

The functions are normally contraction mapping which means they bring points closer together and make shapes smaller.

It is easy to generalize the definition by using more than two contraction mappings.

This concept is very useful for iterated function systems where contraction mappings are often used.

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.

The contraction mappings and are then defined as complex functions in the complex plane by:

Since "T" is a contraction mapping, it is continuous, so we may take the limit inside:

The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed point theorem.

The image of the curve, i.e. the set of points , can be obtained by an Iterated function system using the set of contraction mappings .

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).

Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem.

P.J. Bushell, Hilbert's Metric and Positive Contraction Mappings in a Banach Space, Arch.

The open semigroup of the complexification G taking the closure of D into D acts by contraction mappings, so again the Banach fixed-point theorem can be applied.

A contraction mapping may then be thought of as a way of approximating the final result of a computation (which can be guaranteed to exist by the Banach fixed point theorem).

There exist set-valued extensions of the following concepts from point-valued analysis: continuity, differentiation, integration, implicit function theorem, contraction mappings, measure theory, fixed-point theorems, optimization, and topological degree theory.

But the result of an iterated function system with two contraction mappings is a de Rham curve if and only if the contraction mappings satisfy the continuity condition.

A simplified proof of the second Nash embedding theorem was obtained by who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be applied.

Such a generalization allows, for example, to produce the Sierpiński arrowhead curve (whose image is the Sierpiński triangle), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.

Geometric arithmetic was used by Muttalip Ozavsar and Adem C. Cevikel (both of Yildiz Technical University in Turkey) in an article on multiplicative metric spaces and multiplicative contraction mappings.

Krasnosel'skii also presented many new general principles on solvability of a large variety of nonlinear equations, including one-sided estimates, cone stretching and contractions, fixed-point theorems for monotone operators and a combination of the Schauder fixed point and contraction mapping theorems that was the genesis of condensing operators.