The domain of existence of a limit cycle of Liénard system (Q2357037)

Consider the Liénard system \[ {dx\over dt}= y,\quad {du\over dt}=- f(x)y- g(x),\tag{\(*\)} \] where \(f:\mathbb{R}\to \mathbb{R}\) is continuous and \(g:\mathbb{R}\to \mathbb{R}\) is locally Lipschitzian. The author derives conditions on \(g\), \(f\), \(F(x)= \int^x_0 f(\xi)\,d\xi\) such that \((*)\) has at least one limit cycle. Under the additional conditions \[ \phi(x):= \int^x_0 g(\xi) F(\xi)\,d\xi \] has one positive zero at \(x=\alpha_1\) and one negative zero at \(x=\alpha_2\), \[ \int^{\alpha_1}_{\alpha_2} [g(x)+ f(x) F(x)]\,dx= 0, \] he proves that these limit cycles are located in the strip \(\alpha_2\leq x\leq \alpha_1\).