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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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  • [1] E. F. Beckenbach and R. Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 30, Springer-Verlag, Berlin, 1961. MR 28 #1266. MR 0158038 (28:1266)
  • [2] T. A. Burton, The generalized Liénard equation, J. Soc. Indust. Appl. Math. Ser. A Control 3 (1965), 223-230. MR 32 #7874. MR 0190462 (32:7874)
  • [3] J. K. Hale, Ordinary differential equations, Wiley, New York, 1969. MR 0419901 (54:7918)
  • [4] J. W. Heidel, Global asymptotic stability of a generalized Liénard equation, SIAM J. Appl. Math. 19 (1970), 629-636. MR 0269938 (42:4831)
  • [5] K. Magnus, Vibrations, Blackie and Son, London, 1965.
  • [6] D. W. Willet and J. S. W. Wong, The boundedness of solutions of the equation $ \ddot x + f(x,\dot x) + g(x) = 0$, SIAM J. Appl. Math. 14 (1966), 1084-1098. MR 34 #7901. MR 0208091 (34:7901)

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Additional Information

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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