Skip to main content
SpringerLink
Log in

Oscillation Criteria for Even-Order Nonlinear Dynamic Equations with Sublinear and Superlinear Neutral Terms on Time Scales

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

Abstract

The symmetrical properties of dynamic and/or differential equations are kind of oscillation properties that allow us to conclude the character of solutions for dynamic equations. In this paper, we obtain some symmetrical properties of solutions to an even-order nonlinear dynamic equations with superlinear and sublinear neutral terms on time scales. Our approach is based on linearizing the considered equation in the sense that we would deduce the properties of the considered equation from that of the linear form and provide new oscillation results via comparing with first order as well as \(n\text {th}\) order non-neutral delay dynamic inequalities. The new obtained results outfit a general podium that enables to analyse the oscillatory behaviour for many types of even-order nonlinear dynamic equations. An example is provided to demonstrate the validity of the theoretical outcomes.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation of certain fourth-order functional differential equations. Ukr. Math. J. 59, 315–342 (2007)

    Article  MathSciNet  Google Scholar 

  2. Hajiseyedazizi, S.N., Samei, M.E., Alzabut, J., Chu, Y.: On multi-step methods for singular fractional \(q\)-integro-differential equations. Open Math. 19, 1378–1405 (2021). https://doi.org/10.1515/math-2021-0093

    Article  MathSciNet  Google Scholar 

  3. Erbe, L.H., Karpuz, B., Peterson, A.C.: Kamenev-type oscillation criteria for higher-order neutral delay dynamic equations. Int. J. Differ. Equ. 6, 1–16 (2011)

    MathSciNet  Google Scholar 

  4. Grace, S.R., Graef, J.R., El-Beltagy, M.A.: On the oscillation of third order neutral delay dynamic equations on time scales. Comput. Math. Appl. 63, 775–782 (2012)

    Article  MathSciNet  Google Scholar 

  5. Grace, S.R., Jadlovská, I.: Oscillatory behavior of odd-order nonlinear differential equations with a nonpositive neutral term. Dyn. Syst. Appl. 27, 125–136 (2018)

    Google Scholar 

  6. Graef, J.R., Grace, S.R., Tunç, E.: Oscillation criteria for even-order differential equations with unbounded neutral coefficients and distributed deviating arguments. Funct. Differ. Equ. 25, 143–153 (2018)

    MathSciNet  Google Scholar 

  7. Yue, X.G., Samei, M.E., Fathipour, A., Kaabar, M.K.A., Kashuri, A.: Using Krasnoselskii’s theorem to investigate the Cauchy and neutral fractional \(q\)-integro-differential equation via numerical technique. Nonlinear Eng. Model. Appl. 11, 207–227 (2022). https://doi.org/10.1515/nleng-2022-0023

    Article  Google Scholar 

  8. Li, T., Rogovchenko, Y.V.: Oscillation criteria for even-order neutral differential equations. Appl. Math. Lett. 61, 35–41 (2016)

    Article  MathSciNet  Google Scholar 

  9. Zafer, A.: Oscillation criteria for even order neutral differential equations. Appl. Math. Lett. 11(3), 21–25 (1998). https://doi.org/10.1016/S0893-9659(98)00028-7

    Article  MathSciNet  Google Scholar 

  10. Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (2010)

    Google Scholar 

  11. Grace, S.R., Agarwal, R.P., Wang, C.: New oscillation results for non-canonical higher order nonlinear neutral dynamic equations. Boletín de la Sociedad Matemática Mexicana 27, 1–33 (2021)

    Article  MathSciNet  Google Scholar 

  12. Grace, S.R., Alzabut, J., Abodayeh, K.: Oscillation theorems for higher order dynamic equations with superlinear neutral term. AIMS Math. 6, 5493–5501 (2021)

    Article  MathSciNet  Google Scholar 

  13. Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of second-order differential equations with a sublinear neutral term. Carpathian J. Math. 30(1), 6 (2014)

    Article  MathSciNet  Google Scholar 

  14. Grace, S.R., Graef, J.R., Jadlovská, I.: Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms. Math. Slovaca 69, 1117–1126 (2019)

    Article  MathSciNet  Google Scholar 

  15. Graef, J.R., Grace, S.R., Jadlovská, I., Tunç, E.: Some new oscillation results for higher-order nonlinear differential equations with a nonlinear neutral term. Mathematics 10, 2997 (2022)

    Article  Google Scholar 

  16. Graef, J.R., Grace, S.R., Tunç, E.: Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term. Opusc. Math. 39, 39–47 (2019)

    Article  MathSciNet  Google Scholar 

  17. Liu, Q., Grace, S.R., Jadlovskà, I., Tunc, E., Li, T.: On the asymptotic behavior of noncanonical third-order Emden-fowler delay differential equations with a superlinear neutral term. Mathematics 10(16), 2902 (2022). https://doi.org/10.3390/math10162902

    Article  Google Scholar 

  18. Alzabut, J., Grace, S.R., Chhatria, G.N.: New oscillation results for higher order nonlinear differential equations with a nonlinear neutral terms. J. Math. Comput. Sci. 28, 294–305 (2023). https://doi.org/10.22436/jmcs.028.03.0

    Article  Google Scholar 

  19. Džurina, J., Grace, S.R., Jadlovskà, I., Li, T.: Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293(5), 910–922 (2020). https://doi.org/10.1002/mana.201800196

    Article  MathSciNet  Google Scholar 

  20. Tunç, C., Tunç, O.: On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order. J. Adv. Res. 7, 165–168 (2016). https://doi.org/10.1016/j.jare.2015.04.005

    Article  Google Scholar 

  21. TunÇ, C.: Convergence of solutions of nonlinear neutral differential equations with multiple delays. Boletín de la Sociedad Matemática Mexicana (2015). https://doi.org/10.1007/s40590-014-0050-6

    Article  Google Scholar 

  22. Saker, S.H., Tunç, C., Mahmoud, R.R.: New Carlson-Bellman and Hardy-Littlewood dynamic inequalities. Math. Inequal. Appl. 21(4), 967–983 (2018). https://doi.org/10.7153/mia-2018-21-66

    Article  MathSciNet  Google Scholar 

  23. Moaaz, O., Chatzarakis, G.E., Abdeljawad, T., Cesarano, C., Nabih, A.: Amended oscillation criteria for second-order neutral differential equations with damping term. Adv. Differ. Equ. 2020, 553 (2020). https://doi.org/10.1186/s13662-020-03013-0

    Article  MathSciNet  Google Scholar 

  24. Hassan, T.S., Cesarano, C., El-Nabulsi, R.A., Anukool, W.: Improved Hille-type oscillation criteria for second-order quasilinear dynamic equations. Mathematics 10, 3675 (2022). https://doi.org/10.3390/math10193675

    Article  Google Scholar 

  25. Nabih, A., Cesarano, C., Moaaz, O., Anis, M., Elabbasy, E.M.: Non-canonical functional differential equation of fourth-order: new monotonic properties and their applications in oscillation theory. Axioms 11, 636 (2022). https://doi.org/10.3390/axioms11110636

    Article  Google Scholar 

  26. El-Deeb, A.A., Rashid, S.: On some new double dynamic inequalities associated with Leibniz integral rule on time scales. Adv. Differ. Equ. 2021, 125 (2021). https://doi.org/10.1186/s13662-021-03282-3

    Article  MathSciNet  Google Scholar 

  27. El-Deeb, A.A., Elsennary, H.A., Baleanu, D.: Some new Hardy-type inequalities on time scales. Adv. Differ. Equ. 2020, 441 (2020). https://doi.org/10.1186/s13662-020-02883-8

    Article  MathSciNet  Google Scholar 

  28. El-Deeb, A.A., Xu, E., Abdeldaim, A., Wang, G.: Some dynamic inequalities on time scales and their applications. Adv. Differ. Equ. 2019, 130 (2019). https://doi.org/10.1186/s13662-019-2023-6

    Article  MathSciNet  Google Scholar 

  29. Zeeshan, Ahammad, N.A., Rasheed, H.U., El-Deeb, A.A., Almarri, B., Shah, N.A.: A numerical intuition of activation energy in transient micropolar nanofluid flow configured by an exponentially extended plat surface with thermal radiation effects. Mathematics 10(21), 4046 (2022). https://doi.org/10.3390/math10214046

  30. El-Deeb, A.A., El-Sennary, H.A., Khan, Z.A.: Some reverse inequalities of Hardy type on time scales. Adv. Differ. Equ. 2020, 402 (2020). https://doi.org/10.1186/s13662-020-02857-w

    Article  MathSciNet  Google Scholar 

  31. El-Deeb, A.A., Makharesh, S.D., Baleanu, D.: Dynamic Hilbert-type inequalities with Fenchel–Legendre transform. Adv. Differ. Equ. 12(4), 582 (2020). https://doi.org/10.3390/sym12040582

    Article  Google Scholar 

  32. Mohamed Hassan, A., Ramos, H., Moaaz, O.: Second-order dynamic equations with noncanonical operator: oscillatory behavior. Fractal Fract. 7, 134 (2023). https://doi.org/10.3390/fractalfract7020134

    Article  Google Scholar 

  33. Alzabut, J., Grace, S.R., Jonnalagadda, J.M., Santra, S.S., Abdalla, B.: Higher-order Nabla difference equations of arbitrary order with forcing, positive and negative terms: Non-oscillatory solutions. Axioms 12(4), 325 (2023). https://doi.org/10.3390/axioms12040325

    Article  Google Scholar 

  34. Santra, S.S., Priyadharshini, S., Sadhasivam, V., Kavitha, J., Fernandez-Gamiz, U., Noeiaghdam, S., Khedher, K.M.: On the oscillation of certain class of conformable Emden-Fowler type elliptic partial differential equations. AIMS Math. 8(6), 12622–12636 (2023). https://doi.org/10.3934/math.2023634

    Article  MathSciNet  Google Scholar 

  35. Asad, J., Mallick, P., Samei, M.E., Rath, B., Mohapatra, P., Shanak, H., Jarrar, R.: Asymmetric variation of a finite mass harmonic like oscillator. Results Phys. 19, 103335 (2020). https://doi.org/10.1016/j.rinp.2020.1033

    Article  Google Scholar 

  36. Sangeetha, S., Thamilvanan, S.K., Santra, S.S., Noeiaghdam, S., Abdollahzadeh, M.: Property a of third-order noncanonical functional differential equations with positive and negative terms. AIMS Math. 8(6), 14167–14179 (2023). https://doi.org/10.3934/math.2023724

    Article  MathSciNet  Google Scholar 

  37. Masood, F., Moaaz, O., Santra, S.S., Fernandez Gamiz, U., Elabbasy, E.M.: Oscillation theorems for fourth-order quasi-linear delay differential equations. AIMS Math. 8(7), 16291–16307 (2023). https://doi.org/10.3934/math.2023834

    Article  MathSciNet  Google Scholar 

  38. Thabet, S.T.M., Vivas-Cortez, M., Kedim, I., Samei, M.E., Iadh Ayari, M.: Solvability of \(\varrho -\)Hilfer fractional snap dynamic system on unbounded domains. Fractals Fract. 7(8), 607 (2023). https://doi.org/10.3390/fractalfract7080607

    Article  Google Scholar 

  39. Santra, S.S., Mondal, P., Samei, M.E., Alotaibi, H., Altanji, M., Botmart, T.: Study on the oscillation of solution to second-order impulsive systems. AIMS Math. 8(9), 22237–22255 (2023). https://doi.org/10.3934/math.20231134

    Article  MathSciNet  Google Scholar 

  40. Moaaz, O., Muhib, A., Abdeljawad, T., Santra, S.S., Anis, M.: Asymptotic behavior of even-order noncanonical neutral differential equations. Demonstratio Math. 55(1), 28–39 (2023). https://doi.org/10.1515/dema-2022-0001

    Article  MathSciNet  Google Scholar 

  41. Altanji, M., Chhatria, G.N., Santra, S.S., Scapellato, A.: Oscillation criteria for sublinear and superlinear first-order difference equations of neutral type with several delays. AIMS Math. 7(10), 17670–17684 (2022). https://doi.org/10.3934/math.2022973

    Article  MathSciNet  Google Scholar 

  42. Palanisamy, A., Alzabut, J., Muthulakshmi, V., Santra, S.S., Nonlaopon, K.: Oscillation results for a fractional partial differential system with damping and forcing. AIMS Math. 8(2), 4261–4279 (2023). https://doi.org/10.3934/math.2023212

    Article  MathSciNet  Google Scholar 

  43. Hammad, H.A., Rashwan, R.A., Nafea, A., Samei, M.E., De la Sen, M.: Stability and existence of solutions for a tripled problem of fractional hybrid delay differential equations. Symmetry 14, 2579 (2022). https://doi.org/10.3390/sym14122579

    Article  Google Scholar 

  44. Alzabut, J., Grace, S.R., Santra, S.S., Chhatria, G.N.: Asymptotic and oscillatory behaviour of third order non-linear differential equations with canonical operator and mixed neutral terms. Qual Theory Dyn Syst 22, 15 (2023). https://doi.org/10.1007/s12346-022-00715-6

    Article  MathSciNet  Google Scholar 

  45. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2010)

    Google Scholar 

  46. Chhatria, G.N., Grace, S.R., Graef, J.R.: Oscillation of nonlinear neutral dynamic equations on time scales. J Egypt. Math. Soc. 29, 308 (2021)

    Article  MathSciNet  Google Scholar 

  47. Agarwal, R.P., Bohner, M.: Basic calculus on time scales and some of its applications. Results Math. 35, 3–22 (1999)

    Article  MathSciNet  Google Scholar 

  48. Hardy, G.H., Littlewood, I.E., Polya, G.: Inequalities, Reprint of the 52 Cambridge University Press, Cambridge (1988)

    Google Scholar 

Download references

Acknowledgements

J. Alzabut is thankful to Prince Sultan University and OSTİM Technical University for their endless support for writing this paper.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics and Sciences, Prince Sultan University, 11586, Riyadh, Saudi Arabia

    Jehad Alzabut

  2. Department of Industrial Engineering, OSTİM Technical University, 06374, Ankara, Turkey

    Jehad Alzabut

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

    Said R. Grace

  4. Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal, 741235, India

    Shyam Sundar Santra

  5. Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, 65178-38695, Iran

    Mohammad Esmael Samei

Authors
  1. Jehad Alzabut
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Said R. Grace
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shyam Sundar Santra
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mohammad Esmael Samei
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors have equal contributions. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Mohammad Esmael Samei.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alzabut, J., Grace, S.R., Santra, S.S. et al. Oscillation Criteria for Even-Order Nonlinear Dynamic Equations with Sublinear and Superlinear Neutral Terms on Time Scales. Qual. Theory Dyn. Syst. 23, 103 (2024). https://doi.org/10.1007/s12346-024-00961-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-024-00961-w

Keywords

Mathematics Subject Classification

Search

Navigation