On necessary conditions for global asymptotic stability of equilibrium for the Liénard equation (Q1947770)

The authors consider the Liénard equation \[ x''(t) + f({x}'(t)){x}'(t) + g(x(t)) = 0,\;g(0) = 0. \tag{1} \] In [\textit{C. Qian}, Nonlinear Anal., Theory Methods Appl. 22, No. 7, 823--833 (1994; Zbl 0801.34055)], necessary and sufficient conditions for the global asymptotic stability of the zero solution of equation (1) were obtained under the condition \[ x\int_0^x f(s)\,ds \geq 0. \tag{2} \] In [loc. cit.], also, the following problem was posed. Determine whether condition (2) is a necessary condition for the global asymptotic stability of the zero solution of the Liénard equation (1). The present paper answers this question, and the answer is negative, i.e., condition (2) is not necessary.