Contraction mapping

Let $(X,d)$ be a metric space and let $T:X\to X$.

The map $T$ is a contraction mapping (or contraction) if there exists a constant $c\in[0,1)$ such that for all $x,y\in X$, $d\bigl(T(x),T(y)\bigr)\le c\,d(x,y).$ The constant $c$ is called a contraction constant.

A contraction is a special case of a Lipschitz map: $T$ is Lipschitz with constant $L$ if $d(Tx,Ty)\le L d(x,y)$ for all $x,y$. Contractions are exactly Lipschitz maps with $L<1$.

Contractions are important because on complete metric spaces they have unique fixed points and the fixed point can be found by iteration (Banach fixed point theorem ).

Examples:

• On $\mathbb{R}$ with $d(x,y)=|x-y|$, the affine map $T(x)=ax+b$ is a contraction iff $|a|<1$, with contraction constant $c=|a|$.
• On $\mathbb{R}^k$, $T(x)=\tfrac12 x$ is a contraction with $c=\tfrac12$.
• The map $T(x)=x+1$ is Lipschitz with constant $1$ but is not a contraction (and it has no fixed point).