On the retarded Liénard equation

Zhang, Bo

doi:10.1090/S0002-9939-1992-1094508-1

On the retarded Liénard equation
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Proc. Amer. Math. Soc. 115 (1992), 779-785 Request permission

Abstract:

We consider the equation $x'' + f(x)x’ + g(x(t - h)) = 0$ in which $f,g$ are continuous with $f(x) > 0$ for $x \in R,h$ is a nonnegative constant, and $xg(x) > 0$ if $|x| \geq k$ for some $k \geq 0$. Necessary and sufficient conditions are given for boundedness of all solutions and their derivatives. When $k = 0$ we give necessary and sufficient conditions for all solutions and their derivatives to converge to zero.

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