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.

mayhave interesting answers. I would give the OP the opportunity to improve the question before closing, and in any case I would give an explanation for closing. $\endgroup$2more comments