Boundedness of solutions for a class of retarded Liénard equation. (English) Zbl 1044.34023

The authors consider the retarded Liénard equation \[ x''+f_1(x) x+f_2(x) x^2+g(x(t-h))=e(t), \] where \(h\) is a nonnegative constant, \(f_1, f_2\), and \(g\) are continuous functions on \(\mathbb{R}=(-\infty, +\infty)\), and \(e(t)\) is a continuous function on \(\mathbb{R}^+=[0, +\infty)\). They obtain some new sufficient conditions, as well as some new necessary and sufficient conditions for all solutions and their derivatives to be bounded avoiding the following traditional conditions:
(H1) \(f_1(x)\geq hL\) for all \(x\in \mathbb{R}\) or when \(| x| \) is large;
(H2)  there exists a constant \(N>1\) such that \[ g(x)\left(\int^x_0 f_1(u)\,du- Nhg(x)\right)\geq 0\quad \text{for \( x\in \mathbb{R}\) or when \(x\) is large}; \] (H3) \(xg(x)\geq 0\) for all \(x\in \mathbb{R}\) or when \(| x| \) is large. Two illustrative examples are also included.


34K12 Growth, boundedness, comparison of solutions to functional-differential equations
Full Text: DOI


[1] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations (1985), Academic Press: Academic Press Orlando, FL · Zbl 0635.34001
[2] Hale, J. K., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0425.34048
[3] Yashizaw, T., Asymptotic behavior of solutions of differential equations, (Differential Equation: Qualitative Theory (Szeged, 1984). Differential Equation: Qualitative Theory (Szeged, 1984), Colloq. Math. Soc. János Bolyai, 47 (1987), North-Holland: North-Holland Amsterdam), 1141-1172 · Zbl 0616.34057
[4] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[5] Krasovskii, N. N., Stability of Motion (1963), Stanford Univ. Press: Stanford Univ. Press Stanford, CA · Zbl 0109.06001
[6] Murakami, S., Asymptotic behavior of solutions of some differential equations, J. Math. Anal. Appl., 109, 534-545 (1985) · Zbl 0594.34077
[7] Somolinos, A., Periodic solutions of the sunflower equation, Quart. Appl. Math., 35, 465-478 (1978) · Zbl 0385.34017
[8] Burton, T. A., On the generalized Liénard equation, SIAM J. Control Optim., 3, 223-230 (1965) · Zbl 0135.30201
[9] Burton, T. A.; Townsend, C. G., On the generalized Liénard equation with forcing function, J. Differential Equations, 4, 620-633 (1965) · Zbl 0174.13602
[10] Sugie, J., On the boundedness of solutions of the generalized Liénard equation without the signum condition, Nonlinear Anal., 11, 1391-1397 (1987) · Zbl 0648.34036
[11] Villari, G., On the qualitative behavior of solutions of the Liénard equation, J. Differential Equations, 67, 267-277 (1987) · Zbl 0613.34031
[12] Villari, G.; Zandin, F., On a dynamical system in the Liénard plane, necessary and sufficient conditions for the intersections with the vertical isocline and application, Funkcial. Ekvac., 33, 19-38 (1990) · Zbl 0731.34049
[13] Zhang, B., On the retarded Liénard equation, Proc. Amer. Math. Soc., 115, 779-785 (1992) · Zbl 0756.34075
[14] Zhang, B., Boundedness and stability of solutions of the retarded Liénard equation with negative damping, Nonlinear Anal., 20, 303-313 (1993) · Zbl 0773.34056
[15] Zhang, B., Necessary and sufficient conditions for boundedness and oscillation in the retarded Liénard equation, J. Math. Anal. Appl., 200, 453-473 (1996) · Zbl 0855.34090
[16] Burton, T. A.; Zhang, B., Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation, Ann. Mat. Pura Appl. (4), CLXV, 351-368 (1993) · Zbl 0803.34064
[17] Jin, Z., Boundedness and convergence of solutions of a second-order nonlinear differential system, J. Math. Anal. Appl., 256, 360-374 (2001) · Zbl 0983.34021
[18] Huang, L., On the necessary and sufficient conditions for the boundedness of the solutions of the nonlinear oscillating equation, Nonlinear Anal., 23, 1467-1475 (1994) · Zbl 0814.34022
[19] Huang, L., Boundedness of solutions for some nonlinear differential systems, Nonlinear Anal., 29, 839-847 (1997) · Zbl 0883.34042
[20] Huang, L.; Cheng, Y.; Wu, J., Boundedness of solutions for a class of nonlinear planar systems, Tôhoku Math. J., 54, 393-419 (2002) · Zbl 1048.34075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.