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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) \(\dot x=(1/a(x))[c(y)-b(x)]\), \(\dot y=-a(x)[h(y)-e(t)]\) 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 \(t\to\infty\). Applying these results to the generalized Liénard equation \(\ddot x+(f(x)+g(x)\dot x)\dot x+h(x)=e(t)\), 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)].
Reviewer: W.Müller (Berlin)

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
34D20 Stability of solutions to ordinary differential equations
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