Are there any applications of the following fact?:

Let $X$ be a complete Hausdorff semi-metric space with a collection of semi-metrics  $\{d_\alpha(\cdot,\cdot)\}_{\alpha\in A}.$
Further let $f:X\to X$ be a continuous mapping such that
for any $\alpha \in A$ there is a $\gamma \in A$ and a number $c > 0$ such that
$$d_\gamma(f(x),f(y))\le d_\gamma(x,y)-cd_\alpha(x,y),\quad \forall x,y\in X.$$
Then $f$ has a unique fixed point.