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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Oscillation of solutions of a generalized Liénard equation


Author: Donald C. Benson
Journal: Proc. Amer. Math. Soc. 33 (1972), 101-106
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1972-0291553-8
MathSciNet review: 0291553
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Abstract: The generalized Liénard equation, $ \ddot x + f(x,\dot x) + h(x) = 0$, with $ xh(x) > 0$ and $ yf(x,y) > 0$ for nonzero x and y, is considered here, subject to the additional condition that $ \vert f(x,y)\vert$ is not greater than $ k(x)\vert y{\vert^\alpha }$ where $ \alpha $ is a positive number and $ k(x)$ is a continuous function which is positive for nonzero x. In case $ \alpha \geqq 2$, all solutions of this Liénard equation are oscillatory. In case $ 0 < \alpha < 2$, sufficient conditions are given which insure that all solutions are oscillatory.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0291553-8
Keywords: Liénard equation, nonlinear vibrations, damping, oscillation
Article copyright: © Copyright 1972 American Mathematical Society

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