I know the question sounds weird but I didn't know any other way of phrasing it. Suppose we have a function $f:X\times T \rightarrow X$, where $X \subseteq \mathbb{R}^{n}$ is an open metric space, and $T\subseteq\mathbb{R}$ is open.
Suppose at some value $t \in T$, $f(\cdot, t)$ is a contraction mapping, so for all $x,y \in X$,
$$ d(f(x,t), f(y,t))\leq cd(x,y),$$
where $c<1$.
Under what conditions can I guarantee that for some open set $\mathcal{O}$ around $t$, we have that each $t' \in \mathcal{O}$, $f(\cdot, t')$ remains a contraction? It seems like I should be able to say something about bounding a derivative, but I can't quite work it out.
Is there something known out there?