In a handout I've found this definition of contractions:
Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $\varphi : X \to Y$ is called a contraction if there exists a positive number $c < 1$ such that $$ d_Y(\varphi(x), \varphi(y)) ≤ cd_X(x, y)\qquad\text{ for all }x, y \in X. $$
However, other sources always say that a contraction must map from a metric space into itself - not possibly into another one as happens here. So is this definition incorrect?
On
It seems an odd definition, as for one thing the CMT will no longer work. As an example, let $X$ be $\mathbb R$ with the usual distance function, and let $Y$ also be $\mathbb R$ but with the definition $d(y_1,y_2)=|y_1-y_2|/2$. Now consider the map $y=x+1$. It's easy to see that this is a contraction mapping (under the definition) but doesn't have a fixed point. Not convinced it is a useful construction.
Here they call it a 'contractive mapping', which I think makes no difference with the word 'contraction'. Well, you do not necessarily have fixed points.