Showing that the fixed point of a contraction depends continuously on the contraction

Question

I want to show that the fixed point of a contraction depends continuously on the contraction with respect to the $C^1$ topology.

What does it mean that the point $\textbf{depends continuously}$ on the contraction?

Thank you!

Answer 1

Just apply the definition of continuity (for metric spaces). If we denote the fixpoint of a contraction $f$ as $\operatorname{fix}(f)$, then you want to show that for every contraction $f$ and every $\epsilon>0$ there exists a $\delta>0$ such that for every contraction with $d(f,g)<\delta$ we have $|\operatorname{fix}(f)-\operatorname{fix}(g)|<\epsilon$. Here, $d$ denotes the metric in $C^1$.