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On the Asymptotic Stability for Nonlinear Oscillators with Time-Dependent Damping

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Abstract

The equation

$$\begin{aligned} x''+h(t,x,x')x'+f(x)=0 \qquad (x\in \mathbb {R},\ xf(x)\ge 0,\ t\in [0,\infty )) \end{aligned}$$

is considered, where the damping coefficient h allows an estimate

$$\begin{aligned} a(t)|x'|^\alpha w(x,x')\le h(t,x,x')\le b(t) W(x,x'). \end{aligned}$$

Sufficient conditions on the lower and upper control functions a, b are given guaranteeing that along every motion the total mechanical energy tends to zero as \(t\rightarrow \infty \). The key condition in the main theorem is of the form

$$\begin{aligned} \int _0^\infty a(t)\psi (t;a,b)\,{\mathrm{d}}t=\infty , \end{aligned}$$

which is required for every member \(\psi \) of a properly defined family of test functions. In the second part of the paper corollaries are deduced from this general result formulated by explicit analytic conditions on a, b containing certain integral means. Some of the corollaries improve earlier theorems even for the linear case.

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Acknowledgements

The research was supported by the Hungarian National Foundation for Scientific Research (OTKA K109782).

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  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    L. Hatvani

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  1. L. Hatvani
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Hatvani, L. On the Asymptotic Stability for Nonlinear Oscillators with Time-Dependent Damping. Qual. Theory Dyn. Syst. 18, 441–459 (2019). https://doi.org/10.1007/s12346-018-0294-z

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