Opuscula Math. 45, no. 4 (2025), 543-558
https://doi.org/10.7494/OpMath.2025.45.4.543

 
Opuscula Mathematica

Novel oscillation criteria for third-order semi-canonical differential equations with an advanced neutral term

Kumar S. Vidhyaa
Ethiraju Thandapani
Ercan Tunç

Abstract. The main purpose of this paper is to present new oscillation results for nonlinear semi-canonical third-order differential equations with an advanced neutral term. The main idea is first by reducing the studied semi-canonical equation into standard canonical type equation without assuming any extra conditions. Then, by using the comparison method and integral averaging technique, sufficient conditions are established to ensure the oscillation of the reduced canonical equation, which in turn leads to the oscillation of the original equation. Therefore, the technique used here is very useful since the results already known for the canonical equations can be applied to obtain the oscillation of the semi-canonical equations. Two examples are provided to illustrate the importance of the main results.

Keywords: oscillation, third-order, neutral differential equations, semi-canonical.

Mathematics Subject Classification: 34C10, 34K11, 34K40.

Full text (pdf)

  1. R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2010.
  2. B. Baculíková, J. Džurina, Property (A) of third-order advanced differential equations, Math. Slovaca 64 (2014), 339-346. https://doi.org/10.2478/s12175-014-0208-8
  3. B. Baculíková, J. Džurina, Remarks on properties of Kneser solutions for third-order neutral differential equations, Appl. Math. Lett. 63 (2017), 1-5. https://doi.org/10.1016/j.aml.2016.07.005
  4. B. Baculíková, B. Rani, S. Selvarangam, E. Thandapani, Properties of Kneser's solution for half-linear third order neutral differential equations, Acta Math. Hungar. 152 (2017), 525-533. https://doi.org/10.1007/s10474-017-0721-7
  5. E. Chandrasekaran, G.E. Chatzarakis, R. Sakthivel, E. Thandapani, Third-order nonlinear semi-canonical functional differential equations: Oscillation via new canonical tranform, Mathematics 12 (2024), Art. 3113, 1-11. https://doi.org/10.3390/math12193113
  6. G.E. Chatzarakis, J. Džurina, I. Jadlovská, Oscillation properties of third-order neutral delay differential equations with noncanonical operators, Mathematics 7 (2019), Art. 1177, 1-12. https://doi.org/10.3390/math7121177
  7. G.E. Chatzarakis, S.R. Grace, I. Jadlovská, T. Li, E. Tunç, Oscillation criteria for third-order Emdon-Fowler differential equations with unbounded neutral coefficients, Complexity 2019 (2019), Article ID 5691758, 1-7. https://doi.org/10.1155/2019/5691758
  8. Z. Došlá, P. Liška, Oscillation of third-order nonlinear neutral differential equations, Appl. Math. Lett. 56 (2016), 42-48. https://doi.org/10.1016/j.aml.2015.12.010
  9. J. Džurina, S.R. Grace, I. Jadlovská, On nonexistence of Kneser solutions of third-order neutral delay differential equations, Appl. Math. Lett. 88 (2019), 193-200. https://doi.org/10.1016/j.aml.2018.08.016
  10. L. Feng, Z. Han, Oscillation of a class of third-order neutral differential equations with noncanonical operators, Bull. Malays. Math. Sci. Soc. 44 (2021), 2519-2530. https://doi.org/10.1007/s40840-021-01079-x
  11. V. Ganesan, M. Sathishkumar, On the oscillation of a third-order nonlinear differential equations with neutral type, Ural Math. J. 3 (2017), 122-129. https://doi.org/10.15826/umj.2017.2.013
  12. J.R. Graef, Canonical, noncanonical, and semicanonical third order dynamic equations on time scales, Results Nonlinear Anal. 5 (2022), 273-278. https://doi.org/10.53006/rna.1075859
  13. J.R. Graef, I. Jadlovská, Canonical representation of third-order delay dynamic equations on time scales, Differ. Equ. Appl. 16 (2024), 1-18. https://doi.org/10.7153/dea-2024-16-01
  14. J. Jiang, C. Jiang, T. Li, Oscillatory behavior of third-order nonlinear neutral delay differential equations, Adv. Differ. Equ. 2016 (2016), Art. 171, 1-12. https://doi.org/10.1186/s13662-016-0902-7
  15. I.T. Kiguradze, On the oscillatory character of solutions of the equation \(d^{m}u/dt^{m} +a(t)|u|^{n}\operatorname{sign} u = 0\), Mat. Sb. (N.S.) 65 (1964), 172-187.
  16. I.T. Kiguradze, T.A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations, [in:] Mathematics and its Applications (Soviet Series), vol. 89, Kluwer Academic Publishers Group, Dordrecht, 1993. https://doi.org/10.1007/978-94-011-1808-8
  17. R.G. Koplatadze, T.A. Chanturiya, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differ. Uravn. 18 (1982), 1463-1465 [in Russian].
  18. T. Li, Y.V. Rogovchenko, On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations, Appl. Math. Lett. 105 (2020), Art. 106293, 1-7. https://doi.org/10.1016/j.aml.2020.106293
  19. T. Li, C. Zhang, G. Xing, Oscillation of third-order neutral delay differential equations, Abstr. Appl. Anal. 2012 (2012), Article ID 569201, 1-11. https://doi.org/10.1155/2012/569201
  20. G. Nithyakala, G.E. Chatzarakis, G. Ayyappan, E. Thandapani, Third-order noncanonical neutral delay differential equations: Nonexistence of Kneser solutions via Myshkis type criteria, Mathematics 12 (2024), Art. 2847, 1-10. https://doi.org/10.3390/math12182847
  21. O. Ozdemir, S. Kaya, Comparison theorems on the oscillation of third-order functional differential equations with mixed deviating arguments in neutral term, Differ. Equ. Appl. 14 (2022), 17-30. https://doi.org/10.7153/dea-2022-14-02
  22. S. Padhi, S. Pati, Theory of Third-Order Differential Equations, Springer, New Delhi, 2014.
  23. C.G. Philos, On the existence of nonoscillatory solutions tending to zero at \(\infty\) for differential equations with positive delays, Arch. Math. (Basel) 36 (1981), 168-178. https://doi.org/10.1007/bf01223686
  24. N. Prabaharan, M. Madhan, E. Thandapani, E. Tunç, Remarks on the oscillation of nonlinear third-order noncanonical delay differential equations, Appl. Math. Comput. 481 (2024), Art. 128950, 1-8. https://doi.org/10.1016/j.amc.2024.128950
  25. G. Purushothaman, K. Suresh, E. Thandapani, E. Tunç, Existence and bounds for Kneser-type solutions to noncanonical third-order neutral differential equations, Electron. J. Differ. Equ. 2024 (2024), Art. 55, 1-13. https://doi.org/10.58997/ejde.2024.55
  26. S. Salem, M.M.A. El-Sheikh, A.M. Hassan, On the oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with mixed nonlinear neutral terms, Turk. J. Math. 48 (2024), 221-247. https://doi.org/10.55730/1300-0098.3503
  27. R.A. Sallam, S. Salem, M.M.A. El-Sheikh, Oscillation of solutions of third-order nonlinear neutral differential equations, Adv. Differ. Equ. 2020 (2020), Art. 314, 1-25. https://doi.org/10.1186/s13662-020-02777-9
  28. K. Saranya, V. Piramanantham, E. Thandapani, Oscillation results for third-order semi-canonical quasi-linear delay differential equations, Nonauton. Dyn. Systs. 8 (2021), 228-238. https://doi.org/10.1515/msds-2020-0135
  29. K. Saranya, V. Piramanantham, E. Thandapani, E. Tunç, Asymptotic behavior of Kneser's solution for semi-canonical third-order half-linear advanced neutral differential equations, Func. Differ. Equ. 29 (2022), 115-125. https://doi.org/10.26351/fde/29/1-2/6
  30. K. Saranya, V. Piramanantham, E. Thandapani, E. Tunç, Asymptotic behavior of semi-canonical third-order nonlinear functional differential equations, Palestine J. Math. 11 (2022), 433-442.
  31. K. Saranya, V. Piramanantham, E. Thandapani, E. Tunç, Oscillation criteria for third-order semi-canonical differential equations with unbounded neutral coefficients, Stud. Univ. Babes-Bolyai Math. 69 (2024), 115-125 https://doi.org/10.24193/subbmath.2024.1.08
  32. M. Su, Z. Xu, Oscillation criteria of certain third order neutral differential equations, Differ. Equ. Appl. 4 (2012), 221-232. https://doi.org/10.7153/dea-04-13
  33. K. Suresh, G. Purushothaman, E. Thandapani, E. Tunç, New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform, Math. Slovaca 75 (2025), 329-339. https://doi.org/10.1515/ms-2025-0025
  34. X.H. Tang, Oscillation for first order superlinear delay differential equations, J. London Math. Soc. 65 (2002), 115-122. https://doi.org/10.1112/s0024610701002678
  35. E. Thandapani, T. Li, On the oscillation of third-order quasi-linear neutral functional differential equations, Arch. Math. (Brno) 47 (2011), 181-199.
  36. K.S. Vidhyaa, R. Deepalakshmi, J.R. Graef, E. Thandapani, Oscillatory behavior of semi-canonical third-order delay differential equations with a superlinear neutral term, Appl. Anal. Discrete. Math. 18 (2024), 477-490. https://doi.org/10.2298/aadm210812006v
  • Ercan Tunç (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-8860-608X
  • Department of Mathematics, Faculty of Arts and Sciences, Tokat Gaziosmanpaşa University, 60240, Tokat, Türkiye
  • Communicated by Josef Diblík.
  • Received: 2025-02-24.
  • Revised: 2025-05-27.
  • Accepted: 2025-05-27.
  • Published online: 2025-07-16.
Opuscula Mathematica - cover

Cite this article as:
Kumar S. Vidhyaa, Ethiraju Thandapani, Ercan Tunç, Novel oscillation criteria for third-order semi-canonical differential equations with an advanced neutral term, Opuscula Math. 45, no. 4 (2025), 543-558, https://doi.org/10.7494/OpMath.2025.45.4.543

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.