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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 Orlando, FL · Zbl 0635.34001
[2] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
[3] Yashizaw, T., Asymptotic behavior of solutions of differential equations, (), 1141-1172
[4] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[5] Krasovskii, N.N., Stability of motion, (1963), 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)
[20] Huang, L.; Cheng, Y.; Wu, J., Boundedness of solutions for a class of nonlinear planar systems, Tôhoku math. J., 54, 393-419, (2002)
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