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Oscillation criteria for quasi-linear neutral delay dynamic equations on time scale

  • Qiaoshun Yang EMAIL logo , Zhiting Xu and Ping Long
Published/Copyright: April 28, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 1

Abstract

In this paper, we consider the oscillation for the second-order quasi-linear neutral dynamic equation

(r(t)|ZΔ(t)|α−1ZΔ(t))Δ+q(t)|x(δ(t))|β−1x(δ(t))=0.

on time scale 𝕋, where Z(t) = x(t) + p(t)x(τ(t)), α,β > 0 are constants. We establish some new oscillation criteria and give sufficient conditions to insure that all solutions of quasi-linear neutral dynamic equation are oscillatory on time scale 𝕋. The new oscillation criteria are presented that improve some known results in the literature.

Key words: oscillation; neutral dynamic equation; time scale
MSC: Primary 34K11; 34C10; 39A11; 34C15

This project was supported by Foundation of Hunan Educational Committee (Grant No. 13C753).

Acknowledgement

We are grateful to the anonymous referees for their careful reading and helpful comments which led to an improvement of our original manuscript.

  1. This paper has been communicated by Michal Fečkan

References

[1] Hilger, S.: Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56.10.1007/BF03323153Search in Google Scholar

[2] Bohner, M. —Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.10.1007/978-1-4612-0201-1Search in Google Scholar

[3] Bohner, M. —Peterson, A.: Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.10.1007/978-0-8176-8230-9Search in Google Scholar

[4] Agwo, H. A.: Oscillation of nonlinear second order neutral delay dynamic equations on time scales, Bull. Korean Math. Soc. 45 (2008), 299–312.10.4134/BKMS.2008.45.2.299Search in Google Scholar

[5] Agarwal, R. P. —O’Regan, D. —Saker, S. H.: Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl. 300 (2004), 203–217.10.1016/j.jmaa.2004.06.041Search in Google Scholar

[6] Agarwal, R. P. —O’Regan, D. —Saker, S. H.: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales, Appl. Anal. 86 (2007), 1–17.10.1081/00036810601091630Search in Google Scholar

[7] Chen, D: Oscillation of second-order Emden-Fowler neutral delay dynamic equations on time scales, Math. Comput. Modelling 51 (2010), 1221–1229.10.1016/j.mcm.2010.01.004Search in Google Scholar

[8] Han, Z. —Sun, S. —Li, T, —Zhang, C: Oscillatory behavior of quasilinear neutral delay dynamic equations on time scales, Adv. Difference Equ. 2010 (2010), Article ID 450264, 24 pp.10.1186/1687-1847-2010-450264Search in Google Scholar

[9] Saker, S. H.: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, J. Comput. Appl. Math. 187 (2006), 123–141.10.1016/j.cam.2005.03.039Search in Google Scholar

[10] Saker, S. H.: Oscillation of second-order neutral delay dynamic equations of Emden-Fowler type, Dynam. Systems Appl. 15 (2006), 629–644.Search in Google Scholar

[11] Saker, S. H.: Oscillation of superlinear and sublinear neutral delay dynamic equations, Commun. Appl. Anal. 12 (2008), 173–188.Search in Google Scholar

[12] Zhang, S, —Wang, Q. R: Oscillation of second-order nonlinear neutral dynamic equations on time scales, Appl. Math. Comput. 216 (2010), 2837–2848.10.1016/j.amc.2010.03.134Search in Google Scholar

[13] Saker, S. H.: Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005), 375–387.10.1016/j.cam.2004.09.028Search in Google Scholar

[14] Hassan, T. S.: Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl. 345 (2008), 176–185.10.1016/j.jmaa.2008.04.019Search in Google Scholar

[15] Grace, S. R. —Agarwal, R. P. —Kaymakçalan, B. —Sae-Jie, W.: Oscillation criteria for second-order nonlinear dynamic equations, Can. Appl. Math. Q. 16 (2008), 59–76.Search in Google Scholar

[16] Saker, S. H. —O’Regan, D.: New oscillation criteria for second-order neutral dynamic equations on time scales via Riccati substitution, Hiroshima Math. J. 42 (2012), 77–98.10.32917/hmj/1333113007Search in Google Scholar

[17] Saker, S. H. —Grace, S. R.: Oscillation criteria for quasi-linear functional dynamic equations on time scales, Math. Slovaca 62 (2012), 501–524.10.2478/s12175-012-0026-9Search in Google Scholar

Received: 2013-7-24
Accepted: 2013-10-5
Published Online: 2016-4-28
Published in Print: 2016-2-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0128
Keywords for this article
oscillation; neutral dynamic equation; time scale

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  5. Research Article
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  9. Research Article
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  11. Research Article
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  13. Research Article
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