On Necessary Conditions for Global Asymptotic Stability of Equilibrium for the Liénard Equation

In [1], necessary and sufficient conditions for the global asymptotic stability of the trivial solution of the Liénard equation $\ddot x+f(x)\dot x+g(x)=0$, $g(0)=0$, were obtained under the condition
\begin{equation} x\int_0^xf(s)\,ds\geqslant 0 \tag{A} \end{equation}
In [1], the following problem was also posed: To determine whether condition (A) is a necessary condition for the global asymptotic stability of the trivial solution of the Liénard equation. The present paper answers this question, and the answer is negative, i.e., condition (A) is not a necessary condition.