Geodesic
On Necessary Conditions for Global Asymptotic Stability of Equilibrium for the Li\'enard Equation
Matematičeskie zametki, Tome 93 (2013) no. 1, pp. 63-71

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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.
Keywords: Lienard differential equation, global asymptotic stability. 
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A. O. Ignatyev; V. V. Kirichenko. On Necessary Conditions for Global Asymptotic Stability of Equilibrium for the Li\'enard Equation. Matematičeskie zametki, Tome 93 (2013) no. 1, pp. 63-71. http://geodesic.mathdoc.fr/item/MZM_2013_93_1_a5/