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Parameter diagram for global asymptotic stability of damped half-linear oscillators

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Abstract

We consider the half-linear differential equation with an unbounded damped term,

$$\begin{aligned} (\phi _p(x'))' + h(t)\!\,\phi _p(x') + \omega ^p\phi _p(x) = 0, \end{aligned}$$

where \(\omega > 0\) and \(\phi _p(z) = |z|^{p-2}z\) with \(p > 1\). The divergence speed of the damping coefficient \(h(t)\) is assumed to be determined by some parameters. By using the relations between the index number \(p\) and the parameters, we describe some criteria judging whether the equilibrium of this equation is globally asymptotically stable or not. We also present parameter diagrams to clarify the relations between them.

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Acknowledgments

This research was supported in part by a Grant-in-Aid for Scientific Research, No. 25400165, from the Japan Society for the Promotion of Science (J. S.).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Wei Zheng

  2. Department of Mathematics and Computer Science, Shimane University, Matsue, 690-8504, Japan

    Jitsuro Sugie

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  1. Wei Zheng
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  2. Jitsuro Sugie
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Correspondence to Jitsuro Sugie.

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Communicated by A. Constantin.

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Zheng, W., Sugie, J. Parameter diagram for global asymptotic stability of damped half-linear oscillators. Monatsh Math 179, 149–160 (2016). https://doi.org/10.1007/s00605-014-0695-2

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