Definition:Contraction Mapping (Metric Space)

Definition

Let $\struct {X, d_1}$ and $\struct {Y, d_2}$ be metric spaces.

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

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

$\forall x, y \in X: \map {d_2} {\map f x, \map f y} \le \kappa \map {d_1} {x, y}$

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

Also see

• Contraction Mapping Theorem

• Definition:Uniform Contraction Mapping

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): contraction mapping
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): contraction mapping
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): contraction mapping