On the oscillation of fourth-order neutral differential equations with multiple delays

Salma Aljawi; Fahd Masood; Omar Bazighifan; Salma Aljawi; Fahd Masood; Omar Bazighifan

doi:10.3934/math.2025536

AIMS Mathematics

2025, Volume 10, Issue 5: 11880-11898. doi: 10.3934/math.2025536

Previous Article Next Article

Research article

On the oscillation of fourth-order neutral differential equations with multiple delays

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, Mansoura 35516, Egypt
3.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 39, 00186 Rome, Italy
4.
Jadara Research Center, Jadara University, Irbid 21110, Jordan

Received: 11 March 2025 Revised: 02 May 2025 Accepted: 12 May 2025 Published: 22 May 2025
MSC : 34C10, 34K11

This work focuses on the canonical scenario and examines the oscillatory and asymptotic features of fourth-order differential equations with numerous delays and mixed neutral terms. The Riccati methodology is employed as a useful mathematical tool to simplify the theoretical analysis and derive stringent conditions that rule out the existence of positive solutions satisfying the examined equation. By systematically combining these conditions, precise criteria ensuring the oscillation of all solutions are obtained. These findings contribute qualitatively to the scientific literature by advancing the theoretical understanding of the oscillatory behavior of such equations. Furthermore, to highlight the practical importance of the established results, two applied examples are provided to demonstrate the effectiveness of the derived criteria in handling relevant mathematical models.
- oscillation properties,
- neutral differential equations,
- Riccati method,
- multiple delays,
- canonical case
Citation: Salma Aljawi, Fahd Masood, Omar Bazighifan. On the oscillation of fourth-order neutral differential equations with multiple delays[J]. AIMS Mathematics, 2025, 10(5): 11880-11898. doi: 10.3934/math.2025536

Related Papers:

Abstract

This work focuses on the canonical scenario and examines the oscillatory and asymptotic features of fourth-order differential equations with numerous delays and mixed neutral terms. The Riccati methodology is employed as a useful mathematical tool to simplify the theoretical analysis and derive stringent conditions that rule out the existence of positive solutions satisfying the examined equation. By systematically combining these conditions, precise criteria ensuring the oscillation of all solutions are obtained. These findings contribute qualitatively to the scientific literature by advancing the theoretical understanding of the oscillatory behavior of such equations. Furthermore, to highlight the practical importance of the established results, two applied examples are provided to demonstrate the effectiveness of the derived criteria in handling relevant mathematical models.

References

[1]	J. K. Hale, Theory of functional differential equations, New York: Springer, 1977. http://doi.org/10.1007/978-1-4612-9892-2
[2]	M. Braun, Qualitative theory of differential Equations, In: Differential equations and their applications, New York, NY: Springer, 1978,348–445. https://doi.org/10.1007/978-1-4684-9360-3_4
[3]	W. Snow, Existence, uniqueness and stability for nonlinear differential-difference equations in the neutral case, New York: Courant Institute of Mathematical Sciences, New York University, 1965.
[4]	R. K. Brayton, R. A. Willoughby, On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. Math. Anal. Appl., 18 (1967), 182–189. https://doi.org/10.1016/0022-247X(67)90191-6 doi: 10.1016/0022-247X(67)90191-6
[5]	I. Gyori, G. Ladas, Oscillation theory of delay differential equations with applications, Oxford: Clarendon Press, 1991. https://doi.org/10.1093/oso/9780198535829.001.0001
[6]	B. Almarri, F. Masood, O. Moaaz, A. Muhib, Amended criteria for testing the asymptotic and oscillatory behavior of solutions of higher-order functional differential equations, Axioms, 11 (2022), 718. https://doi.org/10.3390/axioms11120718 doi: 10.3390/axioms11120718
[7]	F. Masood, S. Aljawi, O. Bazighifan, Novel iterative criteria for oscillatory behavior in nonlinear neutral differential equations, AIMS Mathematics, 10 (2025), 6981–7000. https://doi.org/10.3934/math.2025319 doi: 10.3934/math.2025319
[8]	B. Baculíková, J. Džurina, Oscillation theorems for second order neutral differential equations, Comput. Math. Appl., 62 (2011), 4472–4478. https://doi.org/10.1016/j.camwa.2011.10.024 doi: 10.1016/j.camwa.2011.10.024
[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 doi: 10.3390/sym15091630
[10]	G. E. Chatzarakis, S. R. Grace, I. Jadlovská, T. Li, E. Tunç, Oscillation criteria for third-order Emden-Fowler differential equations with unbounded neutral coefficients, Complexity, 2019 (2019), 5691758. https://doi.org/10.1155/2019/5691758 doi: 10.1155/2019/5691758
[11]	S. R. Grace, J. Dzurina, I. Jadlovska, T. Li, On the oscillation of fourth-order delay differential equations, Adv. Differ. Equ., 2019 (2019), 118. https://doi.org/10.1186/s13662-019-2060-1 doi: 10.1186/s13662-019-2060-1
[12]	J. R. Graef, S. R. Grace, E. Tunç, Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term, Opuscula math., 39 (2019), 39–47. https://doi.org/10.7494/OpMath.2019.39.1.39 doi: 10.7494/OpMath.2019.39.1.39
[13]	G. Xing, T. Li, C. Zhang, Oscillation of higher-order quasi-linear neutral differential equations, Adv. Differ. Equ., 2011 (2011), 45. https://doi.org/10.1186/1687-1847-2011-45 doi: 10.1186/1687-1847-2011-45
[14]	Y. Alnafisah, F. Masood, A. Muhib, O. Moaaz, Improved oscillation theorems for even-order quasi-linear neutral differential equations, Symmetry, 15 (2023), 1128. https://doi.org/10.3390/sym15051128 doi: 10.3390/sym15051128
[15]	Z. Alqahtani, B. Qaraad, A. Almuneef, H. Ramos, Asymptotic and oscillatory analysis of second-order differential equations with distributed deviating arguments, Mathematics, 12 (2024), 3542. https://doi.org/10.3390/math12223542 doi: 10.3390/math12223542
[16]	I. Jadlovská, J. Džurina, J. R. Graef, S. R. Grace, Sharp oscillation theorem for fourth-order linear delay differential equations, J. Inequal. Appl., 2022 (2022), 122. https://doi.org/10.1186/s13660-022-02859-0 doi: 10.1186/s13660-022-02859-0
[17]	C. Zhang, T. Li, S. H. Saker, Oscillation of fourth-order delay differential equations, J. Math. Sci., 201 (2014), 296–309.
[18]	T. Kusano, J. Manojlovic, T. Tanigawa, Sharp oscillation criteria for a class of fourth order nonlinear differential equations, Rocky Mt. J. Math., 41 (2011), 249–274.
[19]	K. Kamo, H. Usami, Nonlinear oscillations of fourth order quasilinear ordinary differential equations, Acta Math. Hung., 132 (2011), 207–222. https://doi.org/10.1007/s10474-011-0127-x doi: 10.1007/s10474-011-0127-x
[20]	O. Bazighifan, C. Cesarano, A Philos-type oscillation criteria for fourth-order neutral differential equations, Symmetry, 12 (2020), 379. https://doi.org/10.3390/sym12030379 doi: 10.3390/sym12030379
[21]	M. Alatwi, O. Moaaz, W. Albalawi, F. Masood, H. El-Metwally, Asymptotic and oscillatory analysis of fourth-order nonlinear differential equations with $p$-Laplacian-like operators and neutral delay arguments, Mathematics, 12 (2024), 470. https://doi.org/10.3390/math12030470 doi: 10.3390/math12030470
[22]	F. Masood, W. Albalawi, O. Moaaz, H. El-Metwally, Oscillatory features of fourth-order Emden–Fowler differential equations with sublinear neutral terms, Symmetry, 16 (2024), 933. https://doi.org/10.3390/sym16070933 doi: 10.3390/sym16070933
[23]	L. Erbe, Oscillation theory for functional differential equations, New York: Routledge, 2017. https://doi.org/10.1201/9780203744727
[24]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for difference and functional differential equations, Springer Dordrecht, 2000. https://doi.org/10.1007/978-94-015-9401-1
[25]	G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities (Cambridge mathematical library), Cambridge university press, 1934.
[26]	C. Zhang, R. P. Agarwal, M. Bohner, T. Li, New results for oscillatory behavior of even-order half-linear delay differential equations, Appl. Math. Lett., 16 (2013), 179–183. https://doi.org/10.1016/j.aml.2012.08.004 doi: 10.1016/j.aml.2012.08.004
[27]	I. T. Kiguradze, T. A. Chanturiya, Asymptotic properties of solutions of nonautonomous ordinary differential equations, Springer Dordrecht, 1993. https://doi.org/10.1007/978-94-011-1808-8

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)