Consider the Liénard equation: $$u''+f(u, u')u'+g(u)=0$$ where $f, g$ are continuously differentiable, $f$ is definite positive and $g$ satisfies $ug(u)>0$ if $u\neq 0$. I want to find a strong Lyapunov function for this system, or rather its equivalent 2d system:
$$u_1' = u_2,\;\;\;\;\; u_2' = -u_2f+g$$
I have tried many functions, but they all seem to fail the $V'<0$ condition. For example, if one takes the $H = \dfrac{u_2^2}{2}+\displaystyle\int_0^{u_1}g$ as the Lyapunov function, then $H' = -u_2^2f$, which vanishes at $u_2=0$, even if $u_1\neq 0$.
Another function I tried was $V = H+\beta u_2g$, where $\beta$ is a sufficiently small constant that makes $V$ definite positive. In this case, $V' = (\beta g'-f)u_2^2-\beta u_2fg-\beta g^2$, but I don't know if this is indeed negative definite.
Here is an outline. I leave the details to you. Using the inequality $2|ab|\leq \frac{a^2}{\delta}+\delta b^2$, expand the term $-\beta u_2 f g\leq \beta f |u_2 g|$. You get an expression of the form $$V'\leq-f u_2^2(1-\beta( g'/f +\delta/2))-\beta(1-f/(2\delta))g^2$$ Restrict $u_1,u_2$ in a compact set to get bounds on $g'$ and $f$, and select the constants $\delta$, $\beta$ appropriately. Eventually you get some local results.
Edit: Unfortunately, this works only with $f>0$ for all $u_1,u_2$.