For intervals on a real line, I have found a result which states that for a continuous map, attracting fixed points are Lyapunov stable. However, I found no result about the converse.
So, is the converse also true (i.e. does Lyapunov stability imply atractivity for 1D systems on real line)? If so, what is the proof of this? Also, if not, what is a suitable counterexample for the same.
Consider the following example: $$ \dot x=\begin{cases}0,&x=0,\\ -x^3\sin \frac{1}{x}\,,&x\neq 0. \end{cases} $$ The point $\hat x=0$ is Lyapunov stable, but not asymptotically stable.