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Boundedness and convergence of solutions of a second-order nonlinear differential system. (English) Zbl 0983.34021
The author considers the second-order nonlinear differential system \[ \dot{x}= \frac{1}{a(x)}[h(y)-F(x)],\qquad \dot{y}= -a(x)[g(x)-e(t)]. \] Sufficient and necessary conditions for all solutions to be bounded and to converge to zero are presented. The obtained results are applied to the differential equation \[ \ddot{x}+f_1(x)\dot{x}+f_2(x)\dot{x}^2+g(x)=e(t). \]

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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[1] Antosiewicz, H.A., On non-linear differential equations of second order with integrable forcing term, J. London math. soc., 30, 64-67, (1955) · Zbl 0064.08404
[2] Qian, Chuanxi, Boundedness and asymptotic behaviour of solutions of a second-order nonlinear system, Bull. London math. soc., 24, 281-287, (1992) · Zbl 0763.34021
[3] Freedman, H.I.; Kuang, Y., Uniqueness of limit cycles in Liénard-type equations, Nonlinear anal., 15, 333-338, (1990) · Zbl 0705.34038
[4] Guidorizzi, H.L., Oscillating and periodic solution of equations of the type +f1(x)ẋ+f2(x)ẋ2+g(x)=0, J. math. anal. appl., 176, 11-23, (1993) · Zbl 0778.34019
[5] Jin, Zhou, On the existence and uniqueness of periodic solution for Liénard-type equations, Nonlinear anal., 12, 1463-1470, (1996) · Zbl 0864.34031
[6] Qian, Chuanxi, On global asymptotic stability of second order nonlinear differential systems, Nonlinear anal., 7, 823-833, (1994) · Zbl 0801.34055
[7] Ji-Fa, Jian, On the qualitative behavior of solutions of equation +f1(x)ẋ+f2(x)ẋ2+g(x)=0, J. math. anal. appl., 194, 597-611, (1995)
[8] Ji-Fa, Jian, The global stability of a class of second order differential equations, Nonlinear anal., 5, 855-870, (1997)
[9] Burton, T.A., On the equation +f(x)h(ẋ)ẋ+g(x)=e(t), Ann. mat. pura. appl., 85, 277-286, (1970) · Zbl 0194.40203
[10] Graef, J.R., On the generalized Liénard equation with negative damping, J. differential equations, 12, 34-62, (1972) · Zbl 0254.34038
[11] 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
[12] Villari, G., On the qualitative behavior of solutions of the Liénard equation, J. differential equations, 67, 267-277, (1987) · Zbl 0613.34031
[13] D. W. Bushaw, The differential equation +g(x,ẋ)+h(x)=e(t), terminal report on contract, AF 29 (600)-1003, Holloman Air Force Base, 1958.
[14] Lasalle, J.P.; Lefschetz, S., Stability by Liapunov’s direct method with application, (1961), Academic Press New York
[15] Huiqing, Li, On the sufficient and necessary conditions of the global asymptotic stability of zero solution for Liénard equation, Acta math. sinica, 31, 209-214, (1988) · Zbl 0704.34059
[16] Zhigang, Pan; Ji-fa, Jiang, On the global asymptotic behavior of the generalized Liénard equation, J. sys. sci. math. sci., 12, 376-380, (1992) · Zbl 0767.34032
[17] Zhang, B., On the retarded Liénard equation, Proc. amer. math. soc., 115, 779-785, (1992) · Zbl 0756.34075
[18] Zhang, B., Boundedness and stability of solutions of the retarded Liénard equation with negative damping, Nonlinear anal., 20, 303-313, (1993) · Zbl 0773.34056
[19] 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
[20] Huang, Lihong; Zou, Xingfu, On the boundedness of solutions of the generalized Liénard system without the signum condition, Math. japon., 2, 283-292, (1995) · Zbl 0834.34037
[21] Lasalle, J.P., Stability of nonautonomous systems, Nonlinear anal., 1, 83-91, (1976) · Zbl 0355.34037
[22] Yoshizawa, T., Asymptotic behavior of solutions of a system of differential equations, Contrib. differential equations, 1, 371-387, (1963) · Zbl 0127.30802
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