Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

On the retarded Liénard equation


Author: Bo Zhang
Journal: Proc. Amer. Math. Soc. 115 (1992), 779-785
MSC: Primary 34K20; Secondary 34D20, 34D40, 34K15
DOI: https://doi.org/10.1090/S0002-9939-1992-1094508-1
MathSciNet review: 1094508
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $ \vert x\vert \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.


References [Enhancements On Off] (What's this?)

  • [1] T. A. Burton, On the Equation $ x''(t) + f(x)h(x')x' + g(x) = e(t)$, Ann. Mat. Pure Appl. 85 (1970), 277-286. MR 0262595 (41:7201)
  • [2] -, Stability and periodic solutions of ordinary and functional differential equations, Academic Press, Orlando, FL, 1985. MR 837654 (87f:34001)
  • [3] -, The generalized Liénard equation, SIAM J. Control Optim. 3 (1965), 223-230. MR 0190462 (32:7874)
  • [4] J. R. Graef, On the generalized Liénard equation with negative damping, J. Differential Equations 12 (1972), 34-62. MR 0328200 (48:6542)
  • [5] J. K. Hale, Sufficient conditions for stability and instability of autonomous functional differential equations, J. Differential Equations 1 (1965), 452-482. MR 0183938 (32:1414)
  • [6] T. Hara and T. Yoneyama, On the global center of generalized Liénard equation and its application to stability problems, Funkcial. Ekvac. 28 (1985), 171-192. MR 816825 (87b:34055)
  • [7] Q. C. Huang and X. F. Shi, Global asymptotic behavior of solutions of second order differential equations, Acta Math. Sinica 27 (1984), 449-457. MR 767319 (86f:34095)
  • [8] J. F. Jiang, The asymptotic behavior of a class of second order differential equations with application to electrical circuit equations, J. Math. Anal. Appl. 149 (1990), 26-37. MR 1054791 (91f:34072)
  • [9] N. N. Krasovskii, Stability of motion, Stanford Univ. Press, Stanford, CA, 1963. MR 0147744 (26:5258)
  • [10] N. Minorsky, Nonlinear oscillation, Van Nostrand, New York, 1962. MR 0137891 (25:1339)
  • [11] A. Somolinos, Periodic solutions of the sunflower equation, Quart. Appl. Math. 35 (1978), 465-478. MR 0465265 (57:5170)
  • [12] J. Sugie, On the boundedness of solutions of the generalized Liénard equation without the Signum condition, Nonlinear Anal. 11 (1987), 1391-1397. MR 917860 (89b:34082)
  • [13] -, On the generalized Liénard equation without the Signum condition, J. Math. Anal. Appl. 128 (1987), 80-91. MR 915968 (88k:34037)
  • [14] G. Villari, On the qualitative behaviour of solutions of Liénard equation, J. Differential Equations 67 (1987), 269-277. MR 879697 (88i:34086)
  • [15] G. Villari and F. Zanolin, On a dynamical system in the Liénard plane. Necessary and sufficient conditions for the intersection with the vertical isocline and application, Funkcial. Ekvac. 33 (1990), 19-38. MR 1065466 (91h:34055)
  • [16] T. Yoshizawa, Asymptotic behavior of solutions of differential equations, Differential Equations: Qualitative Theory (Szeged, 1984), Colloq. Math. Soc. János Bolyai, vol. 47, North-Holland, Amsterdam, pp. 1141-1172. MR 890596 (89a:34085)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34K20, 34D20, 34D40, 34K15

Retrieve articles in all journals with MSC: 34K20, 34D20, 34D40, 34K15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1094508-1
Keywords: Necessary and sufficient conditions, boundedness and global asymptotic stability
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society