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Oscillation of third-order nonlinear delay dynamic equation with damping term on time scales

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Abstract

In this article, we consider the oscillation of a class of third-order nonlinear damped delay dynamic equation on time scales of the form

$$\begin{aligned} \left( r_2(r_1(y^\Delta )^\alpha )^\Delta \right) ^\Delta (t)+p(t)(y^\Delta )^\alpha (\sigma (t))+q(t)f(y(g(t)))=0. \end{aligned}$$

We offer a new description of oscillation of the third-order equation in terms of the nonoscillation of a related well studied second-order dynamic equation

$$\begin{aligned} \left( r_2z^\Delta \right) ^\Delta (t)+\frac{p(t)}{r_1(\sigma (t))}z(\sigma (t))=0. \end{aligned}$$

Using generalized Riccati transformation and integral averaging technique, some new sufficient conditions which insure that any solution of the equation oscillates are established.

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Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61703180), and supported by Shandong Provincial Natural Science Foundation ( ZR2016AM17, ZR2017MA04317).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Jinan, Jinan, 250022, Shandong, People’s Republic of China

    Ying Sui & Zhenlai Han

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  1. Ying Sui
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  2. Zhenlai Han
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Correspondence to Zhenlai Han.

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Sui, Y., Han, Z. Oscillation of third-order nonlinear delay dynamic equation with damping term on time scales. J. Appl. Math. Comput. 58, 577–599 (2018). https://doi.org/10.1007/s12190-017-1158-4

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  • DOI: https://doi.org/10.1007/s12190-017-1158-4

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