Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose


Mathematical Citation Quotient (MCQ) 2018: 0.62

Open Access
Online
ISSN
2353-0626
See all formats and pricing
More options …

Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

Shekhar Singh Negi / Syed Abbas / Muslim Malik
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/msds-2018-0008

Abstract

By using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

Keywords: Time scale; Oscillation; dynamic equation; Dynamic inequality

MSC 2010: 34N05; 34K11; 39A10; 26D10

References

  • [1] Abbas, S. : Qualitative analysis of dynamic equations on time scales, Electron. J. Differential Equations, 2018(51), (2018), 1 − 13.Google Scholar

  • [2] Agarwal, R. P., O’Regan, D., Saker, S. H. : Oscillation criteria for nonlinear perturbed dynamic equation of second-order on time scales, J. Appl. Math. & Computing, 20(1 − 2), (2006), 133 − 147.Google Scholar

  • [3] Agarwal, R. P., O’Regan, D., Saker, S. H. : Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. App, 300(1), (2004), 203 − 217.Google Scholar

  • [4] Agwa, H. A., Khodier, A. M. M., Hassan, H. A. : Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equationswith Damping and Oscillatory Potential on Time Scales, Int. J. Differ. Equ., 2016, (2016), pp-12, Article ID 3298289.Google Scholar

  • [5] Annaby, M.H., Mansour, Z. S.: q−Fractional Calculus and Equations, Springer, Berlin (2012).CrossrefGoogle Scholar

  • [6] Bohner, M., Peterson, A. : Dynamic equations on time scales: An introduction with applications, Birkhauser, Boston, MA, (2001).Google Scholar

  • [7] Bohner. M., Saker, S. H. : Oscillation Criteria for Perturbed Nonlinear Dynamic Equations, Mathl. Comput. Modelling, 40, (2004), 249 − 280.Google Scholar

  • [8] Erbe, L., Peterson, A., Saker, S. H. : Oscillation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl., 333(1), (2007), 505 − 522.Web of ScienceGoogle Scholar

  • [9] Grace, S. R., Graef, J. R., Zafer, A. : Oscillation of integro−dynamic equations on time scales, Appl. Math. Lett., 26, (2013), 383 − 386.CrossrefGoogle Scholar

  • [10] Hardy, G. H, Littlewood, J. E., Polya, G. : Inequalities, 2nd edn. Cambridge Univ. Press, Cambridge, (1988).Google Scholar

  • [11] Hilger, S. : Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math., 18(1), (1990), 18 − 56.CrossrefGoogle Scholar

  • [12] Negi, S. S., Abbas, S.,Malik, M. : Oscillation criteria of second-order non-linear dynamic equations with integro forcing term on time scales, Vestnik YuUrGU. Ser. Mat. Model. Progr., 10(1), (2017), 35 − 47.Google Scholar

  • [13] Negi, S. S., Abbas, S., Malik, M. et al. : Oscillation Criteria of Special Type Second-Order Delayed Dynamic Equations on Time Scales, Math. Sci., 12(1), (2018), 25 − 39.CrossrefGoogle Scholar

  • [14] Saker, S. H. : Oscillation of second-order perturbed nonlinear difference equations, Appl.Math. Comput., 144, (2003), 305−324.Google Scholar

  • [15] Saker, S. H. : Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math., 177(2), (2005), 375 − 387.Google Scholar

  • [16] Saker, S. H. : Oscillation Theory of Dynamic Equations on Time Scales, Lambert Academic Publisher, (2010).Google Scholar

  • [17] Saker, S. H. : New inequalities of Opial’s type on time scales and some of their applications, Discrete Dyn. Nat. Soc., 2012, (2012), pp − 23, Art.ID − 362526.Google Scholar

  • [18] Tunç, E. : Oscillation results for even order functional dynamic equations on time scales, Electron. J. Qual. Theory Differ. Equ., 2014(27), (2014), 1 − 14.Google Scholar

  • [19] Özütrk, Ö., Akın, E. : Nonoscillation Criteria for Two-Dimensional Time-Scale Systems, Nonauton. Dyn. Syst., 3(1), (2016) pp. 1 − 13.Google Scholar

About the article

Received: 2017-12-20

Accepted: 2018-05-28

Published Online: 2018-07-20


Citation Information: Nonautonomous Dynamical Systems, Volume 5, Issue 1, Pages 102–112, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2018-0008.

Export Citation

© 2018 Shekhar Singh Negi, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Yong Zhou, Ahmed Alsaedi, and Bashir Ahmad
Advances in Difference Equations, 2019, Volume 2019, Number 1
[2]
Yong Zhou, Bashir Ahmad, and Ahmed Alsaedi
Mathematics, 2019, Volume 7, Number 8, Page 680
[3]
Yong Zhou, Bashir Ahmad, and Ahmed Alsaedi
Mathematical Methods in the Applied Sciences, 2019, Volume 42, Number 13, Page 4488
[4]
Shekhar Singh Negi, Syed Abbas, and Muslim Malik
Mathematical Methods in the Applied Sciences, 2019, Volume 42, Number 12, Page 4146

Comments (0)

Please log in or register to comment.
Log in