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The generalized Liénard systems

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  • We consider the generalized Liénard system

    $\frac{dx}{dt} = \frac{1}{a(x)}[h(y)-F(x)],$

    $\frac{dy}{dt}= -a(x)g(x),\qquad\qquad\qquad\qquad\qquad$ (0.1)

    where $a$ is a positive and continuous function on $R=(-\infty, \infty)$, and $F$, $g$ and $h$ are continuous functions on $R$. Under the assumption that the origin is a unique equilibrium, we obtain necessary and sufficient conditions for the origin of system (0.1) to be globally asymptotically stable by using a nonlinear integral inequality. Our results substantially extend and improve several known results in the literature.

    Mathematics Subject Classification: 34D05, 34C05.


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