A D18: Map $f : X \to Y$ is a

**proper contraction**from $M_X$ to $M_Y$ if and only if \begin{equation} \exists \, C \in [0, 1) : \forall \, x, y \in X : d_Y ( f(x), f(y) ) \leq C d_X(x, y) \end{equation}

Definition D49

Proper contraction

Formulation 0

Let $M_X = (X, d_X)$ and $M_Y = (Y, d_Y)$ each be a D1107: Metric space.

A D18: Map $f : X \to Y$ is a**proper contraction** from $M_X$ to $M_Y$ if and only if
\begin{equation}
\exists \, C \in [0, 1) : \forall \, x, y \in X : d_Y ( f(x), f(y) ) \leq C d_X(x, y)
\end{equation}

A D18: Map $f : X \to Y$ is a

Results

▶ | Proper contraction has at most a single fixed point |

▶ | Proper contraction map is continuous |