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Hille and Nehari-Type Oscillation Criteria for Third-Order Emden–Fowler Neutral Delay Dynamic Equations

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Abstract

We establish some oscillation criteria for the third-order Emden–Fowler neutral delay dynamic equations of the form:

$$\begin{aligned} (a(t)(x(t)+r(t)x(\tau (t)))^{\Delta \Delta })^\Delta +p(t)x^\gamma (\delta (t))=0 \end{aligned}$$

on a time scale \(\mathbb {T}\), where \(\gamma >0\) is a quotient of odd positive integers, and a and p are real-valued positive rd-continuous functions defined on \(\mathbb {T}\). Due to the different values of \(\gamma \), we give not only the oscillation criteria for superlinear neutral delay dynamic equations, but also the oscillation criteria for sublinear neutral delay dynamic equations based on the Hille and Nehari-type oscillation criteria. Our results extend and improve some known results in the literature and are new even for the corresponding third-order differential equations and difference equations as our special cases.

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Acknowledgments

This research is supported by the Natural Science Foundation of China (61374074, 11571202), and the Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003). 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.

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Authors and Affiliations

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

    Yizhuo Wang, Zhenlai Han & Shurong Sun

  2. School of Control Science and Engineering, University of Jinan, Jinan, Shandong, 250022, People’s Republic of China

    Ping Zhao

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  1. Yizhuo Wang
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  2. Zhenlai Han
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  3. Shurong Sun
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  4. Ping Zhao
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Correspondence to Zhenlai Han.

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Communicated by Shangjiang Guo.

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Wang, Y., Han, Z., Sun, S. et al. Hille and Nehari-Type Oscillation Criteria for Third-Order Emden–Fowler Neutral Delay Dynamic Equations. Bull. Malays. Math. Sci. Soc. 40, 1187–1217 (2017). https://doi.org/10.1007/s40840-016-0354-y

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  • DOI: https://doi.org/10.1007/s40840-016-0354-y

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