On the limit cycle of the Liénard equation

summary:In the paper, we give an existence theorem of periodic solution for Liénard equation $\dot{x}=y-F(x)$, $\dot{y}=-g(x)$. As a result, we estimate the amplitude $\rho (\mu )$ (maximal $x$-value) of the limit cycle of the van der Pol equation $\dot{x}=y-\mu (x^3/3-x)$, $\dot{y}=-x$ from above by $\rho (\mu )<2.3439$ for every $\mu \ne 0$. The result is an improvement of the author’s previous estimation $\rho (\mu )