Contractive

Contractive describes a property of a function or operator in a metric space that brings points closer together. Formally, a map f: X → X on a metric space (X, d) is called contractive if there exists a constant c with 0 ≤ c < 1 such that for all x, y in X, d(f(x), f(y)) ≤ c d(x, y). This condition implies f is Lipschitz with constant c.

A related notion is a contraction, or contraction mapping. If X is complete, the Banach fixed-point theorem (also known as the contraction mapping theorem) asserts that every contractive self-map f has a unique fixed point x*, and for any initial x0 in X, the iterative sequence x_{n+1} = f(x_n) converges to x*. The theorem provides both existence and constructive convergence of solutions to equations f(x) = x.

There are variations: a strictly contractive map requires strict inequality d(f(x), f(y)) < d(x, y) for all x ≠ y, while nonexpansive maps satisfy d(f(x), f(y)) ≤ d(x, y). The common practical case in analysis is a contraction with a Lipschitz constant c ∈ [0, 1).

In practice, contractive mappings are used to prove existence and uniqueness of solutions to equations and integral or differential equations via fixed-point arguments, and in iterative numerical schemes where convergence to a stable solution is guaranteed for complete spaces.