# “Contraction mapping principle”

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.

• I have my doubts whether this question is off-topic: it does not ask about the standard contraction mapping principle but a generalization of it. It anything the question might have poor motivation, but it seems to be research level mathematics to me. – Jaap Eldering Nov 4 '13 at 10:12
• I'd say this is a typical case of a question that may have 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. – Pietro Majer Nov 4 '13 at 11:44
• See Meta: meta.mathoverflow.net/a/1142/12898 – András Bátkai Nov 4 '13 at 14:35
• cuckoo: is there a source for this fact? – Todd Trimble Nov 4 '13 at 17:28
• what if $d_\alpha(x,y)=0$ for all $x,y$? – Suvrit Nov 4 '13 at 19:29