I have a general question about the properties of contractive/non-contractive maps.
Assuming I have a map $F$ which I know is not a contraction. Can it map a ball back to itself, i.e. for some $g\in\mathcal{B}_{r}(g_{0})$ (that is $||g-g_{0}||<r$ for a given norm), is it possible that $||F(g)-g_{0}||<r$?
Thank you
Yes, take as an example $f : (0,1) \rightarrow (0,1), x \mapsto x^2$. Then $f'(x) = 2x > 1$ for $x > 1/2$, so it $f$ is no contraction (show this!).
But nevertheless, $f$ maps $(0,1) = B_{1/2}(\frac{1}{2})$ into itself.