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Boundedness and asymptotic behaviour of solutions of a second-order nonlinear system. (English) Zbl 0763.34021
The author considers the nonlinear second-order system (1) x ˙=(1/a(x))[c(y)-b(x)], y ˙=-a(x)[h(y)-e(t)] with a(x)>0. Using suitable Lyapunov functions he obtains sufficient conditions for all solutions of (1) to be bounded or to tend to zero as t. Applying these results to the generalized Liénard equation x ¨+(f(x)+g(x)x ˙)x ˙+h(x)=e(t), he is able to improve and extend, among others, certain results of H. A. Antosiewicz [J. London Math. Soc. 30, 64-67 (1955; Zbl 0064.084)] and T. Yoshizawa [Contrib. Differ. Equations 1, 371-387 (1963; Zbl 0127.308)].
34C11Qualitative theory of solutions of ODE: growth, boundedness
34D05Asymptotic stability of ODE
34D40Ultimate boundedness (MSC2000)
34D20Stability of ODE