New oscillation criteria for third-order nonlinear functional differential equations with distributed delay

Zuhur Alqahtani; Belgees Qaraad; Areej Almuneef; Cheon Seoung Ryoo; Zuhur Alqahtani; Belgees Qaraad; Areej Almuneef; Cheon Seoung Ryoo

doi:10.3934/math.20261318

AIMS Mathematics

2026, Volume 11, Issue 10: 33727-33744. doi: 10.3934/math.20261318

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Research article

New oscillation criteria for third-order nonlinear functional differential equations with distributed delay

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
3.
Department of Mathematics, Hannam University, Daejeon 34430, South Korea

Received: 06 July 2026 Revised: 08 September 2026 Accepted: 16 September 2026 Published: 15 October 2026
MSC : 34C10, 34K11

This paper addresses the oscillatory behavior of solutions to a general class of third-order functional differential equations. The analysis is carried out within a nonstandard framework, where new oscillation criteria are established via multiple analytical techniques, including the Riccati transformation method, the integral method, and comparison principles. Furthermore, refined oscillation conditions are obtained by improving several key inequalities that play a central role in the qualitative analysis of the considered equations. The applicability and sharpness of the proposed results are illustrated through a representative special case, demonstrating enhanced oscillation tests compared to existing criteria. Finally, the obtained results extend and unify several previously reported findings in the literature.
- oscillation,
- nonoscillation,
- delay differential equation,
- comparison method
Citation: Zuhur Alqahtani, Belgees Qaraad, Areej Almuneef, Cheon Seoung Ryoo. New oscillation criteria for third-order nonlinear functional differential equations with distributed delay[J]. AIMS Mathematics, 2026, 11(10): 33727-33744. doi: 10.3934/math.20261318

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Abstract

This paper addresses the oscillatory behavior of solutions to a general class of third-order functional differential equations. The analysis is carried out within a nonstandard framework, where new oscillation criteria are established via multiple analytical techniques, including the Riccati transformation method, the integral method, and comparison principles. Furthermore, refined oscillation conditions are obtained by improving several key inequalities that play a central role in the qualitative analysis of the considered equations. The applicability and sharpness of the proposed results are illustrated through a representative special case, demonstrating enhanced oscillation tests compared to existing criteria. Finally, the obtained results extend and unify several previously reported findings in the literature.

References

[1]	J. K. Hale, Functional differential equations, New York: Springer, 1971. https://doi.org/10.1007/978-1-4615-9968-5
[2]	F. A. Rihan, Delay differential equations and applications to biology, Singapore: Springer, 2021. https://doi.org/10.1007/978-981-16-0626-7
[3]	R. Agarwal, M. Bohner, T. Li, C. Zhang, Oscillation of the third order nonlinear delay differential equations, Taiwanese J. Math., 17 (2013), 545–558. https://doi.org/10.11650/tjm.17.2013.2095 doi: 10.11650/tjm.17.2013.2095
[4]	M. F. Aktaş, A. Tiryaki, A. Zafer, Oscillation criteria for third-order nonlinear functional differential equations, Appl. Math. Lett., 23 (2010), 756–762. https://doi.org/10.1016/j.aml.2010.03.003 doi: 10.1016/j.aml.2010.03.003
[5]	J. Alzabut, S. R. Grace, S. S. Santra, G. N. Chhatria, Asymptotic and oscillatory behavior of third order non-linear differential equations with canonical operator and mixed neutral terms, Qual. Theory Dyn. Syst., 22 (2023), 15. https://doi.org/10.1007/s12346-022-00715-6 doi: 10.1007/s12346-022-00715-6
[6]	R. T. Q. Chen, Y. Rubanova, J. Bettencourt, D. Duvenaud, Neural ordinary differential equations, In: 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), 2018.
[7]	M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. https://doi.org/10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
[8]	H. L. Smith, An introduction to delay differential equations with applications to the life sciences, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-7646-8
[9]	M. Bohner, J. R. Graef, I. Jadlovská, Asymptotic properties of Kneser solutions to third-order delay differential equations, J. Appl. Anal. Comput., 12 (2022), 2024–2032. https://doi.org/10.11948/20210439 doi: 10.11948/20210439
[10]	B. Baculíková, J. Džurina, Oscillation of third order nonlinear differential equations, Appl. Math. Lett., 24 (2011), 466–470. https://doi.org/10.1016/j.aml.2010.10.043 doi: 10.1016/j.aml.2010.10.043
[11]	B. Baculíková, E. M. Elabbasy, S. H. Saker, J. Džurina, Oscillation criteria for third order nonlinear differential equations, Math. Slovaca, 58 (2008), 201–220. https://doi.org/10.2478/s12175-008-0068-1 doi: 10.2478/s12175-008-0068-1
[12]	G. S. Ladde, V. Lakshmikantham, B. G. Zhang, Oscillation theory of differential equations with deviating arguments, New York: Marcel Dekker, 1987.
[13]	S. Saker, Oscillation theory of delay differential and difference equations: second and third orders, LAP Lambert, 2021.
[14]	J. Džurina, I. Jadlovská, B. Baculíková, On asymptotic properties of solutions to third-order delay differential equations, Electron. J. Qual. Theory Differ. Equ., 7 (2019), 1–11. https://doi.org/10.14232/ejqtde.2019.1.7 doi: 10.14232/ejqtde.2019.1.7
[15]	J. Džurina, R. Kotorová, Properties of the third order trinomial differential equations with delay argument, Nonlinear Anal.-Theor., 71 (2009), 1995–2002. https://doi.org/10.1016/j.na.2009.01.070 doi: 10.1016/j.na.2009.01.070
[16]	F. Masood, B. Batiha, O. Bazighifan, Asymptotic and oscillatory characteristics of solutions of neutral differential equations, J. Math. Comput. Sci., 39 (2025), 418–436. https://doi.org/10.22436/jmcs.039.04.02 doi: 10.22436/jmcs.039.04.02
[17]	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 Mathematics, 10 (2025), 29607–29626. https://doi.org/10.3934/math.20251301 doi: 10.3934/math.20251301
[18]	G. E. Chatzarakis, S. R. Grace, I. Jadlovská, Oscillation criteria for third-order delay differential equations, Adv. Differ. Equ., 2017 (2017), 330. https://doi.org/10.1186/s13662-017-1384-y doi: 10.1186/s13662-017-1384-y
[19]	I. Jadlovská, T. X. Li, A note on the oscillation of third-order delay differential equations, Appl. Math. Lett., 167 (2025), 109555. https://doi.org/10.1016/j.aml.2025.109555 doi: 10.1016/j.aml.2025.109555
[20]	S. H. Saker, Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca, 56 (2006), 433–450.
[21]	I. Jadlovská, G. E. Chatzarakis, J. Džurina, S. R. Grace, On sharp oscillation criteria for general third-order delay differential equations, Mathematics, 9 (2021), 1675. https://doi.org/10.3390/math9141675 doi: 10.3390/math9141675
[22]	S. R. Grace, I. Jadlovská, A. Zafer, On oscillation of third-order noncanonical delay differential equations, Appl. Math. Comput., 362 (2019), 124530. https://doi.org/10.1016/j.amc.2019.06.044 doi: 10.1016/j.amc.2019.06.044
[23]	F. Masood, C. Cesarano, O. Moaaz, S. S. Askar, A. M. Alshamrani, H. El-Metwally, Kneser-type oscillation criteria for half-linear delay differential equations of third order, Symmetry, 15 (2023), 1994. https://doi.org/10.3390/sym15111994 doi: 10.3390/sym15111994
[24]	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 doi: 10.1016/j.aml.2018.08.016
[25]	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), 2847. https://doi.org/10.3390/math12182847 doi: 10.3390/math12182847
[26]	G. E. Chatzarakis, J. Džurina, I. Jadlovská, Oscillatory and asymptotic properties of third order quasilinear delay differential equations, J. Inequal. Appl., 2019 (2019), 23. https://doi.org/10.1186/s13660-019-1967-0 doi: 10.1186/s13660-019-1967-0
[27]	E. M. Elabbasy, T. S. Hassan, B. M. Elmatary, Oscillation criteria for third order delay nonlinear differential equations, Electron. J. Qual. Theory Differ. Equ., 5 (2012), 1–9. https://doi.org/10.14232/ejqtde.2012.1.5 doi: 10.14232/ejqtde.2012.1.5
[28]	J. R. Graef, I. Jadlovská, E. Tunç, Sharp asymptotic results for third-order linear delay differential equations, J. Appl. Anal. Comput., 11 (2021), 2459–2472. https://doi.org/10.11948/20200417 doi: 10.11948/20200417
[29]	J. R. Graef, S. H. Saker, Oscillation theory of third-order nonlinear functional differential equations, Hiroshima Math. J., 43 (2013), 49–72. https://doi.org/10.32917/hmj/1368217950 doi: 10.32917/hmj/1368217950
[30]	Y. Kitamura, T. Kusano, Oscillation of first-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc., 78 (1980), 64–68. https://doi.org/10.2307/2043041 doi: 10.2307/2043041
[31]	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 doi: 10.1112/S0024610701002678

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