Oscillatory Behavior and Qualitative Properties of Second-Order Nonlinear Neutral Differential Equations of Noncanonical Type
DOI:
https://doi.org/10.65135/atsf.2026.10Keywords:
Oscillation, Neutral differential equations, Noncanonical case, Asymptotic behaviorAbstract
This study examines a class of neutral differential equations involving both delayed and advanced arguments, in which the present state of the system depends simultaneously on its past and future values. Such equations arise naturally in the mathematical modeling of various physical, biological, and engineering phenomena. By employing a refined analytical framework, new sufficient conditions are derived to ensure that all solutions exhibit oscillatory behavior. The proposed criteria generalize and improve several existing results by relaxing restrictive assumptions commonly imposed in the literature, thereby providing a more comprehensive approach to the qualitative analysis of these systems. A set of illustrative examples is presented to demonstrate the sharpness and effectiveness of the obtained conditions, highlighting their relevance for a deeper understanding of oscillatory dynamics in nonlinear neutral differential equations.
References
R. Bellman and K. L. Cooke, Differential-Difference Equations. Academic Press, New York, 1963. DOI: https://doi.org/10.1063/1.3050672
I. Györi and G. Ladas, Oscillations of Neutral Differential Equations. In: Oscillation Theory of Delay Differential Equations: With Applications, Oxford Academic, 1991, pp. 111--161. DOI: https://doi.org/10.1093/oso/9780198535829.003.0006 DOI: https://doi.org/10.1093/oso/9780198535829.003.0006
J. K. Hale, Theory of Functional Differential Equations. Springer, Berlin/Heidelberg, 1977. DOI: https://doi.org/10.1007/978-1-4612-9892-2 DOI: https://doi.org/10.1007/978-1-4612-9892-2_3
R. K. Brayton and 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. DOI: https://doi.org/10.1016/0022-247X(67)90191-6 DOI: https://doi.org/10.1016/0022-247X(67)90191-6
M. Slemrod and E. F. Infante, Asymptotic stability criteria for linear system of difference-differential equations of neutral type and their discrete analogues. J. Math. Anal. Appl. 38 (1972), 399--415. DOI: https://doi.org/10.1016/0022-247X(72)90098-4 DOI: https://doi.org/10.1016/0022-247X(72)90098-4
S. R. Grace, Oscillation of certain neutral difference equations of mixed type. J. Math. Anal. Appl. 224 (1998), no. 2, 241--254. DOI: https://doi.org/10.1006/jmaa.1998.6001 DOI: https://doi.org/10.1006/jmaa.1998.6001
R. P. Agarwal, S. R. Grace and D. O'Regan, Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic Publishers, Dordrecht, 2002. DOI: https://doi.org/10.1007/978-94-017-2515-6 DOI: https://doi.org/10.1007/978-94-017-2515-6
S. Aljawi, F. Masood and O. Bazighifan, Oscillation analysis in nonlinear differential equations with mixed neutral terms. Appl. Math. Sci. Eng. 33 (2025), no. 1, Paper No. 2508175, 17 pp. DOI: https://doi.org/10.1080/27690911.2025.2508175 DOI: https://doi.org/10.1080/27690911.2025.2508175
M. Alqhtani, F. Masood, K. M. Saad, O. Bazighifan and W. M. Hamanah, On the oscillation criteria for neutral differential equations with several delays. Scientific Rep. 15 (2025), Paper No. 34014, 16 pp. DOI: https://doi.org/10.1038/s41598-025-13746-1 DOI: https://doi.org/10.1038/s41598-025-13746-1
Z. Han, T. Li, S. Sun and Y. Sun, Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396]. Appl. Math. Comput. 215 (2010), no. 11, 3998--4007. DOI: https://doi.org/10.1016/j.amc.2009.12.006 DOI: https://doi.org/10.1016/j.amc.2009.12.006
B. Baculikova, Oscillation of second-order nonlinear noncanonical differential equations, Appl. Math. Lett. 91 (2019), 68--75. DOI: https://doi.org/10.1016/j.aml.2018.11.021 DOI: https://doi.org/10.1016/j.aml.2018.11.021
O. Bazighifan, H. Alotaibi and A. A. A. Mousa, Neutral delay differential equations: oscillation conditions for the solutions, Symmetry 13 (2021), no. 1, Paper No. 101, 11 pp. DOI: https://doi.org/10.3390/sym13010101 DOI: https://doi.org/10.3390/sym13010101
O. Moaaz, A. Muhib, S. Owyed, E. E. Mahmoud and A. Abdelnaser, Second-order neutral differential equations: improved criteria for testing the oscillation. J. Math. 2021 (2021), Paper No. 6665103, 7 pp. DOI: https://doi.org/10.1155/2021/6665103 DOI: https://doi.org/10.1155/2021/6665103
A. Muhib, On oscillation of second-order noncanonical neutral differential equations. J. Inequal. Appl. 2021, Paper No. 79, 11 pp. DOI: https://doi.org/10.1186/s13660-021-02595-x DOI: https://doi.org/10.1186/s13660-021-02595-x
M. Aldiaiji, B. Qaraad, L. F. Iambor and E. M. Elabbasy, New oscillation theorems for second-order superlinear neutral differential equations with variable damping terms. Symmetry 15 (2023), Paper No. 1630, 10 pp. DOI: https://doi.org/10.3390/sym15091630 DOI: https://doi.org/10.3390/sym15091630
M. Aldiaiji, B. Qaraad, L. F. Iambor and E. M. Elabbasy, On the asymptotic behavior of class of third-order neutral differential equations with symmetrical and advanced argument. Symmetry 15 (2023), Paper No. 1165, 13 pp. DOI: https://doi.org/10.3390/sym15061165 DOI: https://doi.org/10.3390/sym15061165
M. K. Grammatikopoulos, G. Ladas and A. Meimaridou, Oscillation of second order neutral delay differential equation. Rad. Mat. 1 (1985), no. 2, 267--274.
L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations. Monogr. Textbooks Pure Appl. Math., Vol. 190, Marcel Dekker, Inc., New York, 1995. DOI: https://doi.org/10.1201/9780203744727 DOI: https://doi.org/10.1201/9780203744727
S. R. Grace and B. S. Lalli, Oscillation of nonlinear second order neutral delay differential equations. Rad. Mat. 3 (1987), no. 1, 77--84. DOI: https://doi.org/10.1016/0022-247X(87)90035-7
B. Baculikova and J. Dzurina, Oscillation theorems for second order neutral differential equations. Comput. Math. Appl. 61 (2011), no. 1, 94--99. DOI: https://doi.org/10.1016/j.camwa.2010.10.035 DOI: https://doi.org/10.14232/ejqtde.2011.1.74
S. R. Grace, J. R. Graef, T. X. Li and E. Tunç, Oscillatory behavior of second-order nonlinear noncanonical neutral differential equations. Acta Univ. Sapientiae Math. 15 (2023), no. 2, 259--271. DOI: https://doi.org/10.2478/ausm-2023-0014 DOI: https://doi.org/10.2478/ausm-2023-0014
R. P. Agarwal, C. Zhang and T. Li, Some remarks on oscillation of second order neutral differential equations. Appl. Math. Comput. 274 (2016), 178--181. DOI: https://doi.org/10.1016/j.amc.2015.10.089 DOI: https://doi.org/10.1016/j.amc.2015.10.089
M. Bohner, S. R. Grace and I. Jadlovska, Oscillation criteria for second-order neutral delay differential equations. Electron. J. Qual. Theory Differ. Equ. 2017, Paper No. 60, 12 pp. DOI: https://doi.org/10.14232/ejqtde.2017.1.60 DOI: https://doi.org/10.14232/ejqtde.2017.1.60
M. Bohner, S. R. Grace and I. Jadlovska, Sharp results for oscillation of second-order neutral delay differential equations. Electron. J. Qual. Theory Differ. Equ. 2023, Paper No. 4, 23 pp. DOI: https://doi.org/10.14232/ejqtde.2023.1.4 DOI: https://doi.org/10.14232/ejqtde.2023.1.4
E. Tunç, K. Baş, Özdemir and E. Thandapani, Oscillation of second-order noncanonical neutral differential equations with distributed deviating arguments. Comput. Appl. Math. 44 (2025), no. 1, Paper No. 97, 14 pp. DOI: https://doi.org/10.1007/s40314-024-03065-y DOI: https://doi.org/10.1007/s40314-024-03065-y
I. Jadlovska, G. E. Chatzarakis and E. Tunç, Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients. Math. Slovaca 74 (2024), no. 3, 637--664. DOI: https://doi.org/10.1515/ms-2024-0049 DOI: https://doi.org/10.1515/ms-2024-0049
S. R. Grace and J. R. Graef, Oscillatory behavior of second-order non-linear differential equations with a sublinear neutral term. Math. Model. Anal 23 (2018), no. 2, 217--226. DOI: https://doi.org/10.3846/mma.2018.014 DOI: https://doi.org/10.3846/mma.2018.014
T. Li, Y. V. Rogovchenko and C. Zhang, Oscillation of second-order neutral differential equation, Funkcial. Ekvac. 56 (2013), no. 1, 111--120. DOI: https://doi.org/10.1619/fesi.56.111 DOI: https://doi.org/10.1619/fesi.56.111
H. S. Alrashdi, F. Masood, A. M. Alshamrani, S. S. Askar and M. Botros, New oscillation results for noncanonical quasilinear differential equations of neutral type. AIMS Math. 10 (2025), no. 6, 14372--14391. DOI: https://doi.org/10.3934/math.2021737 DOI: https://doi.org/10.3934/math.2025647
F. Masood, S. Aljawi and O. Bazighifan, Novel iterative criteria for oscillatory behavior in nonlinear neutral differential equations. AIMS Math. 10 (2025), no. 3, 6981--7000. DOI: https://doi.org/10.3934/math.2025319 DOI: https://doi.org/10.3934/math.2025319
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