Looking for a generalization of Lienard's theorem

Question

Let $g:\mathbb R\to\mathbb R$ be a $C^1-$function such that

1. $G(u)=\displaystyle\int_0^ug(r)dr$ is odd
2. $\displaystyle\lim_{u\to+\infty}G(u)=+\infty$ and there is a $\beta>0$ such that if $u>\beta$, $G$ is increasing.
3. There is a $\alpha>0$ such that $G(u)<0$ if $0<u<\alpha$.

Lienard's theorem.- Under the above conditions, the second order equation $u''+g(u)u'+u=0$ admits a non-constant periodic solution.

I found a natural generalization given by $u''+g(u)u'+f(u)=0$ with some conditions for f. But does anyone know other natural generalizations for this theorem? Please, it would be very useful for me if you could place a reference to read it carefully.