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Abstract
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In this paper we consider the generalized Li´enard system x˙ = h(z − F(x)), z˙ =
−g(x), where h is strictly increasing and h(±1) = ±1. We present some sufficient
and necessary conditions for this system to have a positive and a negative semiorbit
which starts at a point on the curve z = F(x) and approaches the origin without
intersecting the x-axes which are very important in the theory of oscillation and
global asymptotic stability of the solutions of this system.
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