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Boundedness and stability of solutions of a certain nonlinear differential equation of the second order. (English) Zbl 0791.34049

Using appropriate Lyapunov functions and combining up to 27 different assumptions, the author presents a series of sufficient conditions for uniform boundedness or uniform asymptotic stability of solutions of the nonlinear second-order differential equation \[ a(t) x''+ b(t) f(x,x') +g (t,x,x') + (1+c(t)) h(t,x) l(x') = e(t,x,x'), \tag{1} \] some of them generalizing or completing certain results of A. Malysheva [Sov. Math. 25, No. 7, 59–69 (1981); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1981, No. 7(230), 54–61 (1981; Zbl 0504.34030)].
In a second part he presents certain necessary conditions for existence and boundedness of solutions of (1), generalizing results of T. A. Burton and R. C. Grimmer [Monatsh. Math. 74, 211–222 (1970; Zbl 0195.09804)].

MSC:

34D40 Ultimate boundedness (MSC2000)
34D20 Stability of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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