The original question states
Let $g$ be a continuously differentiable function such that $g(0)=0$. Show that the solution $u \equiv 0$ of $u''+g(u)=0$ is not stable when $g'(0)<0$, and stable but not asymptotically stable when $g'(0)>0$
The first part is easy enough with Lyapunov linearization. However I am not able to prove the second part. I tried using Lyapunov functions but didn't find one.
I do think that the solutions whose value at 0 is close enough to 0 are all periodic. for example, $u''+u=0$ yield $\sin(x), \cos(x)$, while $u''+e^u-1=0$ and $u''+\sin(u)=0$ lead these and these solutions respectively. If this is true, it would solve the exercise, but I'm not sure if I can prove it.