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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

Mathematical Citation Quotient (MCQ) 2017: 0.71

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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


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


  • [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.

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© 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

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