Definition:Contraction Mapping

From ProofWiki
Jump to: navigation, search


Let $\left({X, d_1}\right)$ and $\left({Y, d_2}\right)$ be metric spaces.

Let $f : X \to Y$ be a mapping.

Then $f$ is a contraction if and only if there exists $\kappa \in \R: 0 \le \kappa < 1$ such that:

$\forall x, y \in X: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le \kappa d_1 \left({x, y}\right)$

That is, $f$ is Lipschitz continuous for a Lipschitz constant less than $1$.

Also see