Contraction Mapping and Fixed Point with two different distance metrics

Question

I have been looking at the fixed point theorems that use the contraction-mapping and all seem to use the same distance metric for the input and output spaces. If we have a differentiable mapping $f: X \rightarrow X$. I was wondering if we could use different distance metrics for the input $X$ and output $X$, say $d'$ and $d$. If we have $d'(f(x),f(y))\le k d(x,y)$ with $0<k<1$, would we still have a fixed point for $f$? Will it be unique?

Answer 1

Let $X = [0,1]$, $d'$ be the usual Euclidean metric and $d$ be a multiple of the discrete metric, that is $d(x,x') = 2$ unless $x = x'$. Is the map $$ f(x) = \begin{cases} 1,& \text{if }x< \frac12 \\ 0,& \text{otherwise} \end{cases} $$ contractive according to your definition?