Skip to main content
SpringerLink
Log in

Oscillation Properties of Solutions of Second Order Neutral Dynamic Equations of Non-canonical Type on Time Scales

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we study the oscillation for a class of second order sublinear and superlinear neutral dynamic equations on time scales. The tools used to prove results are the Krasnoselskii’s fixed point theorem and several inequalities. For results, the restriction that the solution be unbounded to make it oscillatory is required. But it is not required for the equation to be almost oscillatory. At the end, we give examples for illustrations. We point out that the results are new even for the cases \({\mathbb {T}}={\mathbb {R}}\) and \({\mathbb {T}}={\mathbb {Z}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  2. Agarwal, R.P., Bohner, M., Grace, S.R., O’Regan, D.: Discrete Oscillation Theory. Hindawi Publishing Corporation, New York (2005)

    MATH  Google Scholar 

  3. Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Comparison theorems for oscillation of second-order neutral dynamic equations. Mediterr. J. Math. 11, 1115–1127 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of second-order Emden–Fowler neutral delay differential equations. Ann. Mat. Pura Appl. 193, 1861–1875 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Agarwal, R.P., Zhang, C., Li, T.: Some remarks on oscillation of second order neutral differential equations. Appl. Math. Comput. 274, 178–181 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Arul, R., Shobha, V.S., Thandapani, E.: New oscillation criteria for second order delay differential equations with nonpositive neutral term. Nonlinear Stud. 25, 107–116 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston (2001)

    MATH  Google Scholar 

  8. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)

    MATH  Google Scholar 

  9. Bohner, M., Li, T.: Oscillation of second-order \(p\)-Laplace dynamic equations with a nonpositive neutral coefficient. Appl. Math. Lett. 37, 72–76 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Bohner, M., Grace, S.R., Jadlovska, I.: Oscillation criteria for second order neutral delay differential equation. Electron. J. Qual. Theory Differ. Equ. 60, 1–12 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Bohner, M., Hassan, T.S., Li, T.: Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag. Math. (N.S.) 29, 548–560 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Bohner, M., El-Morshedy, H.A., Grace, S.R., Sager, I.: Oscillation of second-order nonlinear difference equations with sublinear neutral term. Math. Moravica 23, 1–10 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Bohner, M., Grace, S.R., Jadlovska, I.: Sharp oscillation criteria for second-order neutral delay differential equations. Math. Methods Appl. Sci. 43, 10041–10053 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Dosly, O., Hilger, S.: A necessary and sufficient condition for oscillation of the Sturm–Liouville dynamic equation on time scales. J. Comput. Appl. Math. 141, 147–158 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Dzurina, J., Grace, S.R., Jadlovska, I., Li, T.: Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293, 910–922 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Gao, J., Wang, Q.R.: Existence of nonoscillatory solutions to second-order nonlinear neutral dynamic equations on time scales. Rocky Mt. J. Math. 43, 1521–1535 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Grace, S.R., Agarwal, R.P., Bohner, M., O’Regan, D.: Oscillation of second order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3463–3471 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Grace, S.R., Bohner, M., Agarwal, R.P.: On the oscillation of second-order half-linear dynamic equations. J. Differ. Equ. Appl. 15, 451–460 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Grace, S.R., Graef, J.R.: Oscillatory behaviour of second order nonlinear differential equations with a sublinear neutral term. Math. Model. Anal. 23, 217–226 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Grace, S.R., Jadlovsk, I., Zafer, A.: On oscillation of second order delay differential equations with a sublinear neutral term. Mediterr. J. Math. 17, 116 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Grace, S.R., Chhatria, G.N., Abbas, S.: Second order oscillation of non-canonical functional dynamical equations on time scales. Math. Methods Appl. Sci. 44, 9292–9301 (2020)

    MathSciNet  MATH  Google Scholar 

  22. Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Oxford University Press, London (1991)

    MATH  Google Scholar 

  23. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    MATH  Google Scholar 

  24. Hilger, S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18(1), 18–56 (1990)

    MathSciNet  MATH  Google Scholar 

  25. Ji, T., Tang, S., Thandapani, E.: On oscillation of second order neutral dynamic equations. Adv. Differ. Equ. 2013, 340 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Kusano, T., Akio, O., Hiroyuki, U.: On the oscillation of solutions of second order quasilinear ordinary differential equations. Hiroshima Math. J. 23, 645–667 (1993)

    MathSciNet  MATH  Google Scholar 

  27. Kusano, T., Marusiak, P.: Asymptotic properties of solutions of second order quasilinear functional differential equations of neutral type. Math. Bohem. 125, 199–213 (2000)

    MathSciNet  MATH  Google Scholar 

  28. Li, T., Rogovchenko, Yu.V.: Oscillation of second-order neutral differential equations. Math. Nachr. 288, 1150–1162 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Li, H., Zhao, Y., Han, Z.: New oscillation criterion for Emden–Fowler type nonlinear neutral delay differential equations. J. Appl. Math. Comput. 60, 191–200 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Meng, F., Zheng, Z.: Recent development in oscillatory properties of certain differential equations. J. Appl. Anal. Comput. 8, 1282–1306 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Moaaz, O., Bazighifan, O.: Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discret. Contin. Dyn. Syst-S. 13, 2465–2473 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Moaaz, O., Elabbasy, E.M., Qaraad, B.: An improved approach for studying oscillation of generalized Emden–Fowler neutral differential equation. Adv. Differ. Equ. 2020, 69 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Ponnuraj, D., Srinivasan, S., Thandapani, E.: Oscillation of second order neutral type Emden–Fowler delay difference equations. Int. J. Appl. Comput. Math. 5, 5–164 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Prabaharan, N., Dharuman, C., Thandapani, E., Pinelas, S.: New oscillation criteria for second order quasilinear neutral delay differential equations. Differ. Equ. Dyn. Syst. (2020). https://doi.org/10.1007/s12591-020-00550-8

    Article  Google Scholar 

  35. Saker, S.H.: Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales. Nelin. Kolyv. 13(3), 379–399 (2010). English translation: Nonlinear Oscillator 13(3), 407–428 (2011)

  36. Saker, S.H., Grace, S.R.: Oscillation criteria for quasi-linear functional dynamic equations on time scales. Math. Slovaca 69, 501–524 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Sudha, B., Tamilvanan, S., Rama, R., Thandapani, E.: Oscillation criteria of second order delay differential equations with nonpositive sublinear neutral term. AIP Conf. Proc. 2112, 020167 (2019)

    Google Scholar 

  38. Sui, Y., Han, Z.: Oscillation of second order neutral dynamic equations with deviating arguments on time scales. Adv. Differ. Equ. 2018, 337 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Sui, Y., Han, Z.: Oscillation of second order nonlinear dynamic equations with a nonlinear neutral term on time scales. J. Appl. Anal. Comput. 8, 1811–1820 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Sui, Y., Sun, S.: Oscillation of Emden–Fowler type nonlinear neutral delay dynamic equation on time scales. J. Appl. Math. Comput. 60, 291–301 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Tao, C., Sun, T., He, Q.: Nonoscillation for higher order nonlinear delay dynamic equations on time scales. Adv. Differ. Equ. 2016, 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Thandapani, E., Selvarangam, S.: Oscillation of second order Emden–Fowler type neutral difference equations. Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal. 19, 453–469 (2012)

    MathSciNet  MATH  Google Scholar 

  43. Thandapani, E., Selvarangam, S.: Oscillation theorems for second order quasilinear neutral difference equations. J. Math. Comput. Sci. 2, 866–879 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Thandapani, E., Balasubramanian, V.: Some oscillation results for second order neutral type difference equations. Differ. Equ. Appl. 3, 319–330 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Trench, W.F.: Canonical forms and principal systems for general disconjugate equations. Trans. Am. Math. Soc. 189, 319–327 (1973)

    MathSciNet  MATH  Google Scholar 

  46. Tripathy, A.K., Bhaskar, T.G.: Oscillation results for second order nonlinear neutral delay dynamic equations. Funct. Differ. Equ. 17, 329–344 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Wang, D., Xu, Z.: Oscillation of second order quasilinear neutral delay difference equations. Acta Math. Appl. Sin. Engl. Ser. 27, 93–104 (2011)

    MathSciNet  MATH  Google Scholar 

  48. Wu, H., Erbe, L., Peterson, A.: Oscillation of solution to second order half-linear delay dynamic equations on time scales. Electron. J. Differ. Equ. 2016, 1–15 (2016)

    MathSciNet  MATH  Google Scholar 

  49. Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of second order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. 38, 761–778 (2015)

    MathSciNet  MATH  Google Scholar 

  50. Zhang, M., Chena, W., El-Sheikhc, M.M.A., Sallamc, R.A., Hassand, A.M., Li, T.: New oscillation criteria for second-order nonlinear delay dynamic equations with nonpositive neutral coefficients on time scales. J. Comput. Anal. Appl. 27, 629–635 (2019)

    Google Scholar 

  51. Zhang, C., Li, T.: A note on the oscillation of second-order quasilinear neutral dynamic equations on time scales. J. Math. Sci. 246, 333–336 (2020)

    MathSciNet  MATH  Google Scholar 

  52. Zhu, Z.Q., Wang, Q.R.: Existence of nonoscillatory solutions to neutral dynamic equations on time scales. J. Math. Anal. Appl. 335, 751–762 (2007)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous reviewers for their constructive comments and suggestions which helped us to improve the manuscript considerably. G. N. Chhatria is supported by University Grants Commission (UGC), New Delhi, India under Grant/Award Number: F1-17.1/2017-18/RGNF-2017-18-SC-ORI-35849.

Author information

Authors and Affiliations

  1. Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza, 12221, Egypt

    Said R. Grace

  2. Department of Mathematics, Sambalpur University, Sambalpur, 768019, India

    G. N. Chhatria

  3. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H.P., 175005, India

    Syed Abbas

Authors
  1. Said R. Grace
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. N. Chhatria
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Syed Abbas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Syed Abbas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grace, S.R., Chhatria, G.N. & Abbas, S. Oscillation Properties of Solutions of Second Order Neutral Dynamic Equations of Non-canonical Type on Time Scales. Qual. Theory Dyn. Syst. 21, 17 (2022). https://doi.org/10.1007/s12346-021-00552-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-021-00552-z

Keywords

Mathematics Subject Classification

Search

Navigation