# American Institute of Mathematical Sciences

November  2002, 8(4): 1043-1057. doi: 10.3934/dcds.2002.8.1043

## The generalized Liénard systems

 1 Department of Mathematics, University of Turku, FIN-20014 Turku, Finland, Finland

Received  June 2001 Revised  May 2002 Published  July 2002

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.

Citation: Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
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