367 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Some Oscillatory Results of Nonlinear Neutral Differential Equation Zahrah Abdul abbas Hussein 1 and Aqeel Falih Jaddoa 2* 1,2 Mathematics, College of Education for Pure Sciences/ Ibn-Al-Haitham, University of Baghdad, Baghdad, Iraq. *Corresponding Author. Received: 7 January 2025 Accepted: 9 April 2025 Published: 20 July 2025 doi.org/10.30526/38.3.4063 Abstract In the last decades, functional differential equations have attracted the attention of many researchers; they were interested in the theory and its applications. The most common differential equations of functional type are advanced, neutral, and delay DEs. The theory of oscillatory DEs with retarded arguments has a paramount effect on the qualitative properties of DEs. It is essential to deduce conditions for oscillatory and non-oscillatory solutions. The objective of this paper is to obtain oscillatory conditions for differential equations with retarded arguments. So, the oscillatory behavior has been considered in the nonlinear differential equations (DEs) of neutral type with three delays. Some important conditions of all functions have been submitted. The sufficient conditions to secure the oscillatory property have been deduced. We dealt with special cases for delays to obtain some desired conditions for oscillation property. In fact, all new results and conditions innovate, and improved some oscillation properties that appeared in the literature. Some application examples with specific functions for important results have been illustrated and applied to all conditions. Two results with some different conditions have been obtained to get oscillatory behavior for DE. A new relationship between delays and other functions to get desired property has been formulated. Some application examples explained to ensure the importance of our results compared with other previous studies. Keywords: property of oscillation, multiple delays, NDE, nonlinear case. 1. Introduction The continuous development in the fields of science has led to the formulation of many physical laws, which often appear in the form of DEs when reformulated in mathematical form. We can also describe applied problems mathematically using DEs; thus, DEs play a critical role in solving practical problems (1, 2). The theory of DEs is a great tool in modeling many scientific problems in population dynamics, optimal control, and nonlinear problems (3). There are several methods for solving this kind of equation, such as the variational iteration method and the homotopy transforms analysis method (2, 4). Furthermore, some researchers considered some properties that describe all solutions, such as asymptotic behavior, oscillatory property, and stability to different kinds of DEs (5-8). In publications, https://orcid.org/0009-0009-8387-9316 mailto:Zahra.Abd2203m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5590-7991 mailto:aqeel.f.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0009-8387-9316 mailto:Zahra.Abd2203m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5590-7991 mailto:aqeel.f.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0009-8387-9316 mailto:Zahra.Abd2203m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5590-7991 mailto:aqeel.f.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0009-8387-9316 mailto:Zahra.Abd2203m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5590-7991 mailto:aqeel.f.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0009-8387-9316 mailto:Zahra.Abd2203m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5590-7991 mailto:aqeel.f.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0009-8387-9316 mailto:Zahra.Abd2203m@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-5590-7991 mailto:aqeel.f.j@ihcoedu.uobaghdad.edu.iq IHJPAS. 2025, 38(3) 368 different authors established some conditions for oscillation behavior and asymptotic property for nonlinear NDEs (9-17). 2. Materials and Methods Consider nonlinear NDE with several delays: [ ( ( ))] ( ) ( ) Supposing the below conditions hold: 1: , , [ [ with ∫ ∫ 3: [ with and The delays are continuous strictly increasing and invertible. 4: , for and are positive constants. 5: Let ( ) twice continuously differentiable in (1) Definition 1.1 (18) A nontrivial solution to eq. (1) is called non-oscillatory if it is either eventually a positive solution or it is eventually a negative, if it doesn’t, so satisfies oscillatory property. 3. Results Theorem1: Assume that H1- H5 are true with condition: ∫ ∫ (2) min, ( ( )) ( ( ))- mi , ( ( )) ( ( ))- Then the oscillation property holds. Proof: Let a solution has non-oscillatory behaviour to eq. (1). So, without loss of generality, we suppose that the solution is eventually positive It is obvious that . From equation (1), we get: [ ] , so is nonincreasing eventually. [ ( )] ( ( )) ( ) ( ) is nonincreasing, we assume that ( ) Otherwise, there exists ( ) for Then * ( ) ( )+ . ∫ (3) When in (3), then contradiction. So, we have ( ) IHJPAS. 2025, 38(3) 369 We replace ( ) instead of in (1) once and ( ) instead of in (1) a second time. So, we get: * ( ) ( ( ( ) ) ( ( ) ) ( ( ( ) )))+ ( ( ) ) ( ( ( ) )) ( ( ) ) ( ( ( ) )) (4) * ( ) ( ( ( ) ) ( ( ) ) ( ( ) ))+ ( ( ) ) ( ( ( ) )) ( ( ) ) ( ( ( ) )) (5) By multiplying equations and by and summing equations and we get: * ( ( ))+ ( ) ( ) * ( ) ( ( ( ) ) ( ( ) ) ( ( ( ) )))+ ( ( ) ) ( ( ( ))) ( ( ) ) ( ( ( ) )) * ( ) ( ( ( ) ) ( ( ) ) ( ( ) ))+ ( ( ) ) ( ( ( ) )) ( ( ) ) ( ( ( ))) (6) Substituting by and in (6) with deleting some terms: * ( )+ ( ) ( ) * ( ) ( ( ( )))+ ( ( ( ))) * ( ) ( ( ( )))+ ( ( ( ))) (7) From condition H4, we have: [ ( )] [ ( ) ( ( ( )))] ( ( )) [ ( ) ( ( ( )))] ( ( )) Or * ( )+ * ( ) ( ( ( )))+ * ( ) ( ( ( )))+ (8) We integrate the inequality (8) from to : IHJPAS. 2025, 38(3) 370 ∫ ∫ ( ) ( ) ( ( )) ( ( ( ))) ( ( )) ( ( ( ))) ( ) ( ( )) ( ( )) ( ) is nonincreasing function, so as goes to infinity, we conclude that: ∫ ∫ a contradiction ! Theorem2: Assume that H1- H5 hold, with condition (2) in addition to: ∫ ( ) ∫ . ( ( ( )))/ ( ( ( ))) (9) Then the oscillation property holds. Proof: Let a solution has non-oscillatory behaviour to eq. (1). So, without loss of generality, we suppose that the solution is eventually positive It is obvious that . In the same steps as theorem (1), we get inequality (8). Let ( ) ( ( )) ( ( ( ))) ( ( )) ( ( ( ))) (10) Substituting about in (8), we get: ( ) Or ( ) (11) But ( ) is nonincreasing, so: ( ) ( ) Now, we divide last inequality by and integrates it from to , we have: ( ) ∫ IHJPAS. 2025, 38(3) 371 ( ) Or ( ) (12) But ( ) , so we have: ( ( )) ( ( ( ))) ( ( )) ( ( ( ))) ( ( )) ( ( ( ))) And So, ( ( ( ))) ( ( ( ))) (13) From (11) and (12), we get: ( ( )) ( ( )) Or ( ( )) ( ( )) (14) ( ) ( ) ( ( ( ))) From (10), we get: ( ) ( ( ( ( ))) ) (15) We combine (13) and (14), reducing: ( ) ( ( ( ( ))) ) (16) From (13) and (15), we get: ( ) . ( ( ( )))/ ( ( ( ))) (17) By theorem 2.2.9 in (30) the inequality (17) does not have eventually positive solution, a contradiction ! 4.1. Illustrative Examples In this section, we present some examples to support the obtained results IHJPAS. 2025, 38(3) 372 4.1.1. Example 1 Let us consider nonlinear neutral differential equation as follows: 0 . /1 (18) To satisfy all conditions of theorem (1): ∫ ∫ min, ( ( )) ( ( ))- mi , ( ( )) ( ( ))- { } ∫ ∫ ] Also, we get: ∫ ∫ We satisfied all conditions for theorem (1) are satisfied, so all solutions of eq. (1) oscillate. 4.1.2. Example 2 Let us consider nonlinear neutral differential equation as follows: 0 . /1 (19) To satisfy all conditions of theorem (2): ∫ ∫ ∫ ∫ . ( ( ( )))/ mi , ( ( )) ( ( ))- { } To satisfy the condition (9) with ∫ IHJPAS. 2025, 38(3) 373 ∫ All the conditions of theorem (2) are satisfied, so all solutions of eq. (1) oscillate. 5. Discussion Through the paper, qualitative properties are studied as oscillatory properties. It can be noted that the influence of changing delays in main results takes a huge role to control in sufficient conditions to obtain the oscillation property; it is explained in application examples. 6. Conclusion In this search, oscillatory behavior of second-order NDE is considered in the nonlinear case. Specific conditions have been established for known functions in the delay differential eq. (1). A new relationship between the deviating arguments and known functions to get oscillatory property has been built. Application examples are presented to show the importance of our results compared with other previous studies. In theorem 2, a new condition is submitted to get oscillatory behavior of equation (1) with an application example. We improved some previous theorems in the literature by taking the delays as functions. As future works, it would be interesting to extend the results of this article to higher-order nonlinear Des. Acknowledgment Our acknowledgments are to all authors whose results we have improved and to reviewers for their remarks. Conflict of Interest The authors declare that they have no conflicts of interest. Funding No funding. References 1. Esuabana I.M., Ugboh J.A. Survey of impulsive differential equations with continuous delay. Int J Math Trends Technol 2018;60(1):22–28. https://doi.org/10.14445/22315373/IJMTT-V60P504 2. Nemah E.M. Homotopy transforms analysis method for solving fractional Navier-Stokes equations with applications. Iraqi J Sci 2020;61(8):2048–2054. https://doi.org/10.24996/ijs.2020.61.8.20 3. Nemah E.M. Efficiency algorithm for solving some models of nonlinear problems. J Phys Conf Ser 2021;1897(1):012051. https://doi.org/10.1088/1742-6596/1897/1/012051 4. Nemah E.M. Variational approximate solutions of fractional delay differential equations with integral transform. Iraqi J Sci 2021;62(10):3679–3689. https://doi.org/10.24996/ijs.2021.62.10.26 5. Jaddoa A.F. On oscillatory to nonlinear impulsive differential equation of second-order with damping term. J Phys Conf Ser 2021;1897(1):012047. https://doi.org/10.1088/1742- 6596/1897/1/012047 6. Mohamad H.A., Jaddoa A.F. Oscillation criteria for solutions of neutral differential equations of impulses effects with positive and negative coefficients. Baghdad Sci J 2020;17(2):537–544. https://doi.org/10.21123/bsj.2020.17.2.0537 https://doi.org/10.14445/22315373/IJMTT-V60P504 https://doi.org/10.24996/ijs.2020.61.8.20 https://doi.org/10.1088/1742-6596/1897/1/012051 https://doi.org/10.24996/ijs.2021.62.10.26 https://doi.org/10.1088/1742-6596/1897/1/012047 https://doi.org/10.1088/1742-6596/1897/1/012047 https://doi.org/10.21123/bsj.2020.17.2.0537 IHJPAS. 2025, 38(3) 374 7. Santra S.S. Existence of positive solution and new oscillation criteria for nonlinear first-order neutral delay differential equations. Differ Equ Appl 2016;8(1):33–51. http://dx.doi.org/10.7153/dea-08-03 8. Helal M.M. Qualitative analysis of some types of neutral delay differential equations. Iraqi J Sci 2021;62(10):3634–3641. https://doi.org/10.24996/ijs.2021.62.10.21 9. Xiao H., Zheng B. The existence of multiple periodic solutions of nonautonomous delay differential equations. J Appl Math 2011;2011:829107. https://doi.org/10.1155/2011/829107 10. Sharba B.A., Jaddoa A.F. On the existence and oscillatory solutions of multiple delay differential equation. Iraqi J Sci 2023;64(2):878–892. https://doi.org/10.24996/ijs.2023.64.2.33 11. Mushtt I.Z., Hameed D.M., Mohamad H.A. Nonoscillatory properties of fourth order nonlinear neutral differential equation. Iraqi J Sci 2023;64(2):798–803. https://doi.org/10.24996/ijs.2023.64.2.25 12. Taher S., Osama M., Amany N., Mouataz B., Ahmed M.A. New sufficient conditions for oscillation of second-order neutral delay differential equations. Axioms 2021;10(4):281. https://doi.org/10.3390/axioms10040281 13. Elmetwally M., Ethiraju T., Osama M., Omar B. Oscillation of solutions to fourth-order delay differential equations with middle term. Open J Math Sci 2019;3:191–197. https://doi.org/10.30538/oms2019.0062 14. Yingzhu W., Yuanhong Y., Jinsen X. Oscillation of second order nonlinear neutral differential equations. Mathematics 2022;10(15):2739. https://doi.org/10.3390/math10152739 15. Marianna R., Shyam S.S., Andrea S. Oscillatory behavior of second-order neutral differential equations. Bull Braz Math Soc 2022;53:665–675. https://doi.org/10.1007/s00574-021-00276-3 16. Osama M., Ali M., Saud O., Emad E.M., Aml A. Second-order neutral differential equations: Improved criteria for testing the oscillation. J Math 2021;2021:6665103. https://doi.org/10.1155/2021/6665103 17. Mohamad H.A., Ketab S.N. Asymptotic behavior criteria for solutions of nonlinear n-th order neutral differential equations. IOP Conf Ser Mater Sci Eng 2019;571(1):012034. https://doi.org/10.1088/1757-899X/571/1/012034 18. Mohamad H.A., Mushtt I.Z. Oscillation of second order nonlinear neutral differential equations. Pure Appl Math J 2015;4(2):62–65. https://doi.org/10.11648/j.pamj.20150402.16 19. Santra S.S., Bazighifan O., Ahmad H. Second-order differential equation with multiple delays: Oscillation theorems and applications. Complexity 2020;2020:8853745. https://doi.org/10.1155/2020/8853745 20. Agarwal R.P., Grace S.R., Regan D.O. Oscillation theory for difference and functional differential equations. Dordrecht: Kluwer Academic Publishers; 2000. http://dx.doi.org/10.7153/dea-08-03 https://doi.org/10.24996/ijs.2021.62.10.21 https://doi.org/10.1155/2011/829107 https://doi.org/10.24996/ijs.2023.64.2.33 https://doi.org/10.24996/ijs.2023.64.2.25 https://doi.org/10.3390/axioms10040281 https://doi.org/10.30538/oms2019.0062 https://doi.org/10.3390/math10152739 https://doi.org/10.1007/s00574-021-00276-3 https://doi.org/10.1155/2021/6665103 https://doi.org/10.1088/1757-899X/571/1/012034 https://doi.org/10.11648/j.pamj.20150402.16 https://doi.org/10.1155/2020/8853745