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Nehari-type Oscillation Theorems for Second Order Functional Dynamic Equations

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Abstract

This paper is devoted to studying the half-linear functional dynamic equations of second-order on an unbounded above time scale \(\mathbb {T}\). We present some Nehari-type oscillation criteria for a class of second-order dynamic equations. The obtained results show that there is a substantial improvement in the literature on second-order dynamic equations. We include some examples illustrating the significance of our results.

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References

  1. Agarwal, R.P., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: A survey. J. Comput. Appl. Math. 141, 1–26 (2002)

    Article  MATH  Google Scholar 

  2. Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation criteria for second-order dynamic equations on time scales. Appl. Math. Lett. 31, 34–40 (2014)

    Article  MATH  Google Scholar 

  3. Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Hille and Nehari type criteria for third order delay dynamic equations. J. Differ. Equ. Appl. 19, 1563–1579 (2013)

    Article  MATH  Google Scholar 

  4. Agarwal, R.P., Bohner, M., Li, T.: Oscillatory behavior of second-order half-linear damped dynamic equations. Appl. Math. Comput. 254, 408–418 (2015)

    MATH  Google Scholar 

  5. Baculikova, B.: Oscillation of second-order nonlinear noncanonical differential equations with deviating argument. Appl. Math. Lett. 91, 68–75 (2019)

    Article  MATH  Google Scholar 

  6. Bazighifan, O., El-Nabulsi, E.M.: Different techniques for studying oscillatory behavior of solution of differential equations. Rocky Mountain J. Math. 51(1), 77–86 (2021)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Bohner, M., Li, T.: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math. 58(7), 1445–1452 (2015)

    Article  MATH  Google Scholar 

  9. Bohner, M., Peterson, A.: Dynamic equations on time scales: an introduction with applications. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  10. Bohner, M., Peterson, A.: Advances in dynamic equations on time scales. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  11. Bohner, M., Hassan, T.S.: Oscillation and boundedness of solutions to first and second order forced functional dynamic equations with mixed nonlinearities. Appl. Anal. Discrete Math. 3, 242–252 (2009)

    Article  MATH  Google Scholar 

  12. 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. 29, 548–560 (2018)

    Article  MATH  Google Scholar 

  13. Chatzarakis, G.E., Moaaz, O., Li, T., Qaraad, B.: Oscillation theorems for nonlinear second-order differential equations with an advanced argument. Adv. Differ. Equ. 2020, 160 (2020)

    Article  MATH  Google Scholar 

  14. Džurina, J., Jadlovská, I.: A sharp oscillation result for second-order half-linear noncanonical delay differential equations. Electron. J. Qual. Theo. 46, 1–14 (2020)

    MATH  Google Scholar 

  15. Džurina, J., Jadlovská, I.: A note on oscillation of second-order delay differential equations. Appl. Math. Lett. 69, 126–132 (2017)

    Article  MATH  Google Scholar 

  16. Elabbasy, E.M., El-Nabulsi, R.A., Moaaz, O., Bazighifan, O.: Oscillatory properties of solutions of even order differential equations. Symmetry 12(2), 212 (2020)

    Article  Google Scholar 

  17. Erbe, L.: Oscillation criteria for second order nonlinear delay equations. Canad. Math. Bull. 16, 49–56 (1973)

    Article  MATH  Google Scholar 

  18. Erbe, L., Hassan, T.S., Peterson, A.: Oscillation criteria for second order sublinear dynamic equations with damping term. J. Differ. Equ. Appl. 17, 505–523 (2011)

    Article  MATH  Google Scholar 

  19. Erbe, L., Hassan, T.S.: New oscillation criteria for second order sublinear dynamic equations. Dynam. Syst. Appl. 22, 49–63 (2013)

    MATH  Google Scholar 

  20. Erbe, L., Hassan, T.S., Peterson, A., Saker, S.H.: Oscillation criteria for half-linear delay dynamic equations on time scales. Nonlinear Dynam. Sys. Th. 9, 51–68 (2009)

    MATH  Google Scholar 

  21. Erbe, L., Hassan, T.S., Peterson, A., Saker, S.H.: Oscillation criteria for sublinear half-linear delay dynamic equations on time scales. Int. J. Differ. Equ. 3, 227–245 (2008)

    Google Scholar 

  22. Erbe, L., Peterson, A., Saker, S.H.: Hille and Nehari type criteria for third order dynamic equations. J. Math. Anal. Appl. 329, 112–131 (2007)

    Article  MATH  Google Scholar 

  23. Frassu, S., Viglialoro, G.: Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent. Nonlinear Anal. 213, 112505 (2021)

    Article  MATH  Google Scholar 

  24. 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)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Hille, E.: Non-oscillation theorems. Trans. Amer. Math. Soc. 64, 234–252 (1948)

    Article  MATH  Google Scholar 

  27. Kac, V.; Chueng, P. Quantum Calculus; Universitext, 2002

  28. Leighton, W.: The detection of the oscillation of solutions of asecond order linear differential equation. Duke J. Math. 17, 57–62 (1950)

    Article  MATH  Google Scholar 

  29. Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(3), 1–18 (2019)

    Article  MATH  Google Scholar 

  30. Li, T., Viglialoro, G.: Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differ. Integral Equ. 34(5–6), 315–336 (2021)

    MATH  Google Scholar 

  31. Karpuz, B.: Hille-Nehari theorems for dynamic equations with a time scale independent critical constant. Appl. Math. Comput. 346, 336–351 (2019)

    MATH  Google Scholar 

  32. Nehari, Z.: Oscillation criteria for second-order linear differential equations. Trans. Amer. Math. Soc. 85, 428–445 (1957)

    Article  MATH  Google Scholar 

  33. Řehak, P.: New results on critical oscillation constants depending on a graininess. Dynam. Syst. Appl. 19, 271–288 (2010)

    MATH  Google Scholar 

  34. Sun, S., Han, Z., Zhao, P., Zhang, C.: Oscillation for a class of second-order Emden-Fowler delay dynamic equations on time scales. Adv. Differ. Equ. 2010, 642356 (2010)

    Article  MATH  Google Scholar 

  35. Sun, Y., Hassan, T.S.: Oscillation criteria for functional dynamic equations with nonlinearities given by Riemann-Stieltjes integral. Abstr. Appl. Anal. 2014, 9 (2014)

    Article  MATH  Google Scholar 

  36. Wong, J.S.: Second order oscillation with retarded arguments, In: Ordinary differential equations, 581–596; Washington, 1971. Academic press, New York and London (1972)

  37. 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(2), 761–778 (2015)

    Article  MATH  Google Scholar 

  38. Zhang, Q., Gao, L., Wang, L.: Oscillation of second-order nonlinear delay dynamic equations on time scales. Comput. Math. Appl. 61, 2342–2348 (2011)

    Article  MATH  Google Scholar 

  39. Erbe, L., Higgins, R.: Some Oscillation Results for Second Order Functional Dynamic Equations1. Adv. Dyn. Syst. Appl. 3, 73–88 (2008)

    Google Scholar 

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Acknowledgements

This research has been funded by Scientific Research Deanship at University of Ha’il – Saudi Arabia through project number RG-21 011. R.A. El-Nabulsi would like to thank Jaume Giné for inviting him to submit a work to QTDS.

Author information

Author notes

  1. E. M. Elabbasy, Rabie A. Ramadan, H. Saber, A. E. Matouk and Ismoil Odinaev have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, College of Science, University of Ha’il, Ha’il, 2440, Saudi Arabia

    Taher S. Hassan & H. Saber

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Taher S. Hassan & E. M. Elabbasy

  3. Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, Roma, 00186, Italy

    Taher S. Hassan

  4. Center of Excellence in Quantum Technology, Faculty of Engineering, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi

  5. Mathematics and Physics Divisions, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi

  6. Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Rami Ahmad El-Nabulsi

  7. College of Computer Science and Engineering, University of Ha’il, Ha’il, 81481, Saudi Arabia

    Rabie A. Ramadan

  8. Quantum-Atom Optics Laboratory and Research Center for Quantum Technology, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    A. E. Matouk

  9. Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, Yekaterinburg, 620002, Russian Federation

    Ismoil Odinaev

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  1. Taher S. Hassan
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Contributions

Hassan directed the study and help inspection. All the authors carried out the the main results of this article and drafted the manuscript and read and approved the final manuscript.

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Correspondence to Taher S. Hassan or Rami Ahmad El-Nabulsi.

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Hassan, T.S., Elabbasy, E.M., El-Nabulsi, R.A. et al. Nehari-type Oscillation Theorems for Second Order Functional Dynamic Equations. Qual. Theory Dyn. Syst. 22, 13 (2023). https://doi.org/10.1007/s12346-022-00711-w

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