Oscillation in Advanced Neutral Differential Equations with Mixed Nonlinear Structures

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DOI: https://doi.org/10.28924/2291-8639-24-2026-322
Published: Oct 5, 2026
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Fahd Masood, Omar Bazighifan, Faten Aldosari, Alanoud Almutairi, Khalil S. Al-Ghafri, Oscillation in Advanced Neutral Differential Equations with Mixed Nonlinear Structures, Int. J. Anal. Appl., 24 (2026), 322.

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Fahd Masood, Omar Bazighifan, Faten Aldosari, Alanoud Almutairi, Khalil S. Al-Ghafri

Abstract

This paper investigates the oscillatory behavior of solutions to a class of second-order quasilinear advanced neutral differential equations with mixed nonlinearities. By applying the arithmetic–geometric mean inequality, we reduce the original equation to an associated half-linear advanced differential equation, providing an effective framework for analyzing its oscillatory properties. Through comparison principles and an iterative approach, several new sufficient conditions for oscillation are established, explicitly reflecting the effects of the advanced arguments and the neutral term. The obtained criteria describe the interplay between the neutral structure, advanced arguments, and mixed nonlinearities and extend several existing oscillation results in the literature. Illustrative examples demonstrate the applicability of the theoretical results and the effectiveness of the proposed criteria.

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References

  1. L.E. Elsgolts, S.B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Elsevier, 1973. https://doi.org/10.1016/s0076-5392(08)x6170-3.
  2. W.B. Fite, Properties of the Solutions of Certain Functional-Differential Equations, Trans. Am. Math. Soc. 22 (1921), 311–319. https://doi.org/10.1090/s0002-9947-1921-1501176-x.
  3. I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford University Press, Oxford, 1991. https://doi.org/10.1093/oso/9780198535829.001.0001.
  4. R.P. Agarwal, M. Bohner, W.-T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, CRC Press, 2004. https://doi.org/10.1201/9780203025741.
  5. O. Došlý, P. Řehák, Half-Linear Differential Equations, Elsevier, 2005. https://doi.org/10.1016/s0304-0208(05)x8001-x.
  6. O. Bazighifan, F. Masood, A. Almutairi, K.S. Al-Ghafri, New Results on Oscillation of Higher-Order Neutral Emden–Fowler Differential Equations with Sublinear Terms, J. Inequal. Appl. 2026 (2026), 66. https://doi.org/10.1186/s13660-026-03468-x.
  7. A. Muhib, On Oscillation of Second-Order Noncanonical Neutral Differential Equations, J. Inequal. Appl. 2021 (2021), 79. https://doi.org/10.1186/s13660-021-02595-x.
  8. B. Almarri, F. Masood, A. Muhib, O. Moaaz, New Comparison Results for Oscillation of Even-Order Delay Differential Equations, Symmetry 14 (2022), 946. https://doi.org/10.3390/sym14050946.
  9. M. Aldiaiji, B. Qaraad, L.F. Iambor, E.M. Elabbasy, New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms, Symmetry 15 (2023), 1630. https://doi.org/10.3390/sym15091630.
  10. F. Masood, A. Almutairi, L.F. Iambor, K.S. Al-Ghafri, O. Bazighifan, Novel Oscillation Criteria for General Third-Order Nonlinear Neutral Differential Equations, AIMS Math. 10 (2025), 29607–29626. https://doi.org/10.3934/math.20251301.
  11. F. Masood, M.N. Alshehri, L.F. Iambor, A.F. Aljohani, O. Bazighifan, Nonlinear Oscillation Analysis of Delay Differential Equations with Mixed Neutral Terms, AIMS Math. 10 (2025), 24580–24601. https://doi.org/10.3934/math.20251090.
  12. I. Jadlovská, Iterative Oscillation Results for Second-Order Differential Equations with Advanced Argument, Electron. J. Differ. Equ. 2017 (2017), 162.
  13. T. Kusano, M. Naito, Comparison Theorems for Functional Differential Equations with Deviating Arguments, J. Math. Soc. Japan 33 (1981), 509–532. https://doi.org/10.2969/jmsj/03330509.
  14. J. Džurina, Oscillation of Second Order Differential Equations with Advanced Argument, Math. Slovaca 45 (1995), 263–268.
  15. R.P. Agarwal, M. Bohner, T. Li, C. Zhang, Even-Order Half-Linear Advanced Differential Equations: Improved Criteria in Oscillatory and Asymptotic Properties, Appl. Math. Comput. 266 (2015), 481–490. https://doi.org/10.1016/j.amc.2015.05.008.
  16. M. Al-zailah, F. Masood, H. El-Metwally, Oscillation Criteria for Fourth-Order Neutral Quasilinear Differential Equations in the Noncanonical Case, Electron. J. Differ. Equ. 2026 (2026), 24. https://doi.org/10.58997/ejde.2026.24.
  17. F. Masood, A. Almutairi, K.S. Al-Ghafri, O. Bazighifan, Novel Oscillation Criteria for Second-Order Noncanonical Neutral Differential Equations with Sublinear Term, Bound. Value Probl. 2026 (2026), 127. https://doi.org/10.1186/s13661-026-02314-9.
  18. B. Baculíková, Oscillatory Behavior of the Second Order Functional Differential Equations, Appl. Math. Lett. 72 (2017), 35–41. https://doi.org/10.1016/j.aml.2017.04.003.
  19. J. Džurina, A Comparison Theorem for Linear Delay Differential Equations, Arch. Math. (Brno) 31 (1995), 113–120.
  20. B. Baculíková, Oscillatory Behavior of the Second Order Noncanonical Differential Equations, Electron. J. Qual. Theory Differ. Equ. 2019 (2019), 89. https://doi.org/10.14232/ejqtde.2019.1.89.
  21. Y.G. Sun, J.S.W. Wong, Oscillation Criteria for Second Order Forced Ordinary Differential Equations with Mixed Nonlinearities, J. Math. Anal. Appl. 334 (2007), 549–560. https://doi.org/10.1016/j.jmaa.2006.07.109.
  22. E. Thandapani, S. Murugadass, S. Pinelas, Oscillation Criteria for Second Order Nonlinear Differential Equations with Damping and Mixed Nonlinearities, Funct. Differ. Equ. 17 (2010), 319–328.
  23. R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, CRC Press, 2002. https://doi.org/10.4324/9780203222898.
  24. T.S. Hassan, Kamenev-Type Oscillation Criteria for Second Order Nonlinear Dynamic Equations on Time Scales, Appl. Math. Comput. 217 (2011), 5285–5297. https://doi.org/10.1016/j.amc.2010.11.052.
  25. B. Baculíková, Oscillation of Second-Order Nonlinear Noncanonical Differential Equations with Deviating Argument, Appl. Math. Lett. 91 (2019), 68–75. https://doi.org/10.1016/j.aml.2018.11.021.
  26. M. Bohner, K.S. Vidhyaa, E. Thandapani, Oscillation of Noncanonical Second-Order Advanced Differential Equations via Canonical Transform, Constr. Math. Anal. 5 (2022), 7–13. https://doi.org/10.33205/cma.1055356.
  27. Z. Alqahtani, B. Qaraad, A. Almuneef, F. Alharbi, Oscillatory Properties of Second-Order Differential Equations with Advanced Arguments in the Noncanonical Case, Symmetry 16 (2024), 1018. https://doi.org/10.3390/sym16081018.
  28. O. Bazighifan, A. Almutairi, K.S. Al-Ghafri, F. Masood, New Iterative Criteria for Oscillation and Asymptotic Properties of Second-Order Advanced Differential Equations, J. Nonlinear Math. Phys. 33 (2026), 78. https://doi.org/10.1007/s44198-026-00432-7.
  29. H. Shi, Y. Bai, Oscillatory Behavior of a Second Order Nonlinear Advanced Differential Equation with Mixed Neutral Terms, Adv. Differ. Equ. 2019 (2019), 468. https://doi.org/10.1186/s13662-019-2393-9.
  30. G. Purushothaman, E. Chandrasekaran, G.E. Chatzarakis, E. Thandapani, Oscillation Criteria for Second-Order Half-Linear Advanced Differential Equations Arising in Bacterial Population Dynamics, Math. Bohem. 151 (2026), 1–16. https://doi.org/10.21136/MB.2026.0066-25.
  31. I.T. Kiguradze, T.A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Springer Netherlands, 1993. https://doi.org/10.1007/978-94-011-1808-8.
  32. E.F. Beckenbach, R. Bellman, Inequalities, Springer Berlin Heidelberg, 1961. https://doi.org/10.1007/978-3-642-64971-4.
  33. R.P. Agarwal, S.R. Grace, D. O'Regan, The Oscillation of Certain Higher-Order Functional Differential Equations, Math. Comput. Model. 37 (2003), 705–728. https://doi.org/10.1016/s0895-7177(03)00079-7.
  34. T. Kusano, J. Wang, Oscillation Properties of Half-Linear Functional-Differential Equations of the Second Order, Hiroshima Math. J. 25 (1995), 371–385. https://doi.org/10.32917/hmj/1206127717.
  35. I. Jadlovská, J. Džurina, Kneser-Type Oscillation Criteria for Second-Order Half-Linear Delay Differential Equations, Appl. Math. Comput. 380 (2020), 125289. https://doi.org/10.1016/j.amc.2020.125289.
  36. G.E. Chatzarakis, S.R. Grace, I. Jadlovská, A Sharp Oscillation Criterion for Second-Order Half-Linear Advanced Differential Equations, Acta Math. Hung. 163 (2021), 552–562. https://doi.org/10.1007/s10474-020-01110-w.