Oscillation conditions of nonlinear neutral differential equations with several delays

Maryam AlKandari; Maryam AlKandari

doi:10.3934/math.2025622

AIMS Mathematics

2025, Volume 10, Issue 6: 13825-13835. doi: 10.3934/math.2025622

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Oscillation conditions of nonlinear neutral differential equations with several delays

Maryam AlKandari ^,

Department of Mathematics, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Received: 17 February 2025 Revised: 16 May 2025 Accepted: 20 May 2025 Published: 16 June 2025
MSC : 34C10, 34K11

In this paper, we examine whether oscillatory solutions to an even-order differential equation with multiple delays exist. We create new oscillation criteria using the comparison method. When comparing the results obtained in this paper with some of the results in the literature, we find that the results we obtained give better values for the oscillation of the studied equation, and thus we find that the results we obtained expand some of the results in the literature. To further emphasize the relevance of our proposed criteria, we offer some instances to support and exemplify our results.
- oscillatory solutions,
- even order neutral differential equations,
- several delays
Citation: Maryam AlKandari. Oscillation conditions of nonlinear neutral differential equations with several delays[J]. AIMS Mathematics, 2025, 10(6): 13825-13835. doi: 10.3934/math.2025622

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Abstract

In this paper, we examine whether oscillatory solutions to an even-order differential equation with multiple delays exist. We create new oscillation criteria using the comparison method. When comparing the results obtained in this paper with some of the results in the literature, we find that the results we obtained give better values for the oscillation of the studied equation, and thus we find that the results we obtained expand some of the results in the literature. To further emphasize the relevance of our proposed criteria, we offer some instances to support and exemplify our results.

References

[1]	J. K. Hale, Theory of Functional Differential Equations, Berlin/Heidelberg: Springer, 1977. http://dx.doi.org/10.1007/978-1-4612-9892-2
[2]	J. Dzurina, S. R. Grace, I. Jadlovska, T. Li, Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term, Math. Nachr., 293 (2020), 910–922. https://doi.org/10.1002/mana.201800196 doi: 10.1002/mana.201800196
[3]	S. R. Grace, Oscillation of certain neutral difference equations of mixed type, J. Math. Anal. Appl., 224 (1998), 241–254. https://doi.org/10.1006/jmaa.1998.6001 doi: 10.1006/jmaa.1998.6001
[4]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Appl. Math. Lett., 18 (2005), 1201–1207. https://doi:10.1007/978-94-017-2515-6 doi: 10.1007/978-94-017-2515-6
[5]	S. H. Saker, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, J. Comput. Appl. Math., 187 (2006), 123–141. https://doi.org/10.1016/j.cam.2005.03.039 doi: 10.1016/j.cam.2005.03.039
[6]	B. Batiha, N. Alshammari, F. Aldosari, F. Masood, O. Bazighifan, Asymptotic and oscillatory properties for even-order nonlinear neutral differential equations with damping term, Symmetry, 17 (2025), 87. https://doi.org/10.3390/sym17010087
[7]	C. G. Philos, Oscillation theorems for linear differential equation of second order, Arch. Math., 53 (1989), 483–492. http://dx.doi.org/10.1007/BF01324723 doi: 10.1007/BF01324723
[8]	Z. Han, T. Li, S. Sun, Y. Sun, Remarks on the paper, Appl. Math. Comput., 215 (2010), 3998–4007. https://doi.org/10.1016/j.amc.2009.12.006
[9]	B. Baculikova, 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 doi: 10.1016/j.aml.2018.11.021
[10]	B. Almarri, A. H. Ali, A. M. Lopes, O. Bazighifan, Nonlinear differential equations with distributed delay: Some new oscillatory solutions, Mathematics, 10 (2022), 995. https://doi.org/10.3390/math10060995 doi: 10.3390/math10060995
[11]	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
[12]	Y. G. Sun, F. W. Meng, Note on the paper of Dzurina and Stavroulakis, Appl. Math. Comput., 164 (2006), 1634–1641.
[13]	T. Kusano, Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hung., 76 (1997), 81–99. https://doi.org/10.1007/bf02907054 doi: 10.1007/bf02907054
[14]	M. Bohner, S. R. Grace, I. Jadlovska, Oscillation criteria for second-order neutral delay differential equations, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 60. http://doi:10.14232/ejqtde.2017.1.60 doi: 10.14232/ejqtde.2017.1.60
[15]	B. Batiha, N. Alshammari, F. Aldosari, F. Masood, O. Bazighifan, Nonlinear neutral delay differential equations: Novel criteria for oscillation and asymptotic behavior, Mathematics, 13 (2025), 147. https://doi.org/10.3390/math13010147 doi: 10.3390/math13010147
[16]	S. Sun, T. Li, Z. Han, H. Li, Oscillation theorems for Second-Order quasilinear neutral functional differential equations, Abstr. Appl. Anal., 2012 (2012), 819342. https://doi.org/10.1155/2012/819342 doi: 10.1155/2012/819342
[17]	A. Zafer, Oscillation criteria for even order neutral differential equations, Appl. Math. Lett., 11 (1998), 21–25. https://doi.org/10.1016/S0893-9659(98)00028-7 doi: 10.1016/S0893-9659(98)00028-7
[18]	Q. Zhang, J. Yan, Oscillation behavior of even order neutral differential equations with variable coefficients, Appl. Math. Lett., 19 (2006), 1202–1206. https://doi.org/10.1016/j.aml.2006.01.003 doi: 10.1016/j.aml.2006.01.003
[19]	G. Xing, T. Li, C. Zhang, Oscillation of higher-order quasi linear neutral differential equations, Adv. Difference Equ., 2011 (2011), 1–10. https://doi.org/10.1186/1687-1847-2011-45 doi: 10.1186/1687-1847-2011-45
[20]	O. Bazighifan, T. Abdeljawad, Improved approach for studying oscillatory properties of fourth-order advanced differential equations with $ p$-Laplacian like operator, Mathematics, 8 (2020), 656. https://doi.org/10.3390/math8050656 doi: 10.3390/math8050656
[21]	T. Li, B. Baculikova, J. Dzurina, C. Zhang, Oscillation of fourth order neutral differential equations with $p$-Laplacian like operators, Bound. Value Probl., 56 (2014), 41–58. https://doi.org/10.1186/1687-2770-2014-56 doi: 10.1186/1687-2770-2014-56
[22]	S. Liu, Q. Zhang, Y. Yu, Oscillation of even-order half-linear functional differential equations with damping, Comput. Math. Appl., 61 (2011), 2191–2196. https://doi.org/10.1016/j.camwa.2010.09.011 doi: 10.1016/j.camwa.2010.09.011
[23]	B. Baculikova, J. Dzurina, Oscillation theorems for second-order nonlinear 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
[24]	R. P. Agarwal, M. Bohner, T. Li, C. Zhang, A new approach in the study of oscillatory behavior of even-order neutral delay diferential equations, Appl. Math. Comput., 225 (2013), 787–794. https://doi.org/10.1016/j.amc.2013.09.037 doi: 10.1016/j.amc.2013.09.037
[25]	Ch. G. Philos, A new criterion for the oscillatory and asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math., 39 (1981), 61–64.
[26]	Ch.G. Philos, On the existence of non-oscillatory solutions tending to zero at $\infty $ for differential equations with positive delays, Arch. Math. 36 (1981), 168–178. https://doi.org/10.1007/BF01223686
[27]	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/10.1090/S0002-9939-1980-0548086-5 doi: 10.1090/S0002-9939-1980-0548086-5

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