Oscillation criteria for damped second-order equations with time delays based on the parameter-Riccati method

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Abstract

This work presents a comprehensive oscillation theory for a class of second-order nonlinear neutral differential equations with a variable damping term and variable delay functions. The primary contribution consists of a set of new oscillation criteria whose validity depends on the interplay between the key parameters α and β. To address the limitations of existing approaches, a novel framework is introduced based on the construction of parameterized auxiliary equations and the use of a generalized Riccati transformation. This approach reduces the problem to the analysis of an associated integro- differential inequality, which is then analyzed via refined inequality techniques and iterative integration. The main results reveal that oscillatory behavior is dictated by a threshold condition comparing α with the rate of change of the delay, quantified by β. These results are shown to be both sharp and easily applicable, as illustrated by examples that also highlight their improvement over existing results.

Keywords

damped term / positive and negative / coefficient variable time delay / Riccati transformation / oscillation

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References

[1] I. Newton, Philosophiæ Naturalis Principia Mathematica , Royal Society of London, 1687. https://doi.org/10.5479/sil.52126.39088015628399

[2] J. L. R. D'Alembert, Recherches sur la courbe que forme une corde tenduë mise en vibration, Hist. Acad. R. Sci. Belles Lett. Berlin, 3 (1747), 214-219.

[3] C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl., 1 (1836), 106-186.

[4] C. Sturm, Sur une classe d'équations à différences partielles, J. Math. Pures Appl., 1 (1836), 373-444.

[5] N. Guglielmi, E. Iacomini, A. Viguerie, Delay differential equations for the spatially resolved simulation of epidemics with specific application to COVID-19, Math. Methods Appl. Sci., 45 (2022), 4752-4771. https://doi.org/10.1002/mma.8068

[6] I. Kovacic, L. Teofanov, Z. Kanovic, J. Zhao, R. Zhu, V. Rajs, On the influence of internal oscillators on the performance of metastructures: Modelling and tuning conditions, Mech. Syst. Signal Process., 205 (2023), 110861. https://doi.org/10.1016/j.ymssp.2023.110861

[7] B. Baculíková, Oscillatory behavior of the second order functional differential equations, Appl. Math. Lett., 72 (2017), 35-41. https://doi.org/10.1016/j.aml.2017.04.003

[8] J. Yang, J. Wang, X. Qin, T. Li, Oscillation of nonlinear second-order neutral delay differential equations, J. Nonlinear Sci. Appl., 10 (2017), 2727-2734. https://doi.org/10.22436/jnsa.010.05.39

[9] H. Tian, R. Guo, Some oscillatory criteria for second-order Emden-Fowler neutral delay differential equations, Mathematics, 12 (2024), 1559. https://doi.org/10.3390/math12101559

[10] S. R. Grace, G. N. Chhatria, Improved oscillation criteria for second order quasilinear dynamic equations of noncanonical type, Rend. Circ. Mat. Palermo, II., 73 (2024), 127-140. https://doi.org/10.1007/s12215-023-00905-4

[11] M. Alkilayh, Nonlinear neutral differential equations of second-order: Oscillatory properties, AIMS Math., 10 (2025), 1589-1601. https://doi.org/10.3934/math.2025073

[12] 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

[13] X. H. Deng, X. Huang, Q. R. Wang, Oscillation and asymptotic behavior of third-order nonlinear delay differential equations with positive and negative terms, Appl. Math. Lett., 129 (2022), 107927. https://doi.org/10.1016/j.aml.2022.107927

[14] E. Braverman, A. Domoshnitsky, J. I. Stavroulakis, Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations, J. Math. Anal. Appl., 531 (2024), 127875. https://doi.org/10.1016/j.jmaa.2023.127875

[15] G. Purushothaman, E. Chandrasekaran, G. E. Chatzarakis, E. Thandapani, Oscillatory analysis of third-order hybrid trinomial delay differential equations via binomial transform, Mathematics, 13 (2025), 2520. https://doi.org/10.3390/math13152520

[16] A. Al-Jaser, E. Alluqmani, B. Qaraad, H. Ramos, Third-order functional differential equations with damping term: Oscillatory behavior of solutions, Axioms, 14 (2025), 877. https://doi.org/10.3390/axioms14120877

[17] S. R. Grace, I. Jadlovská, A. Zafer, Oscillation criteria for odd-order nonlinear delay differential equations with a middle term, Math. Methods Appl. Sci., 40 (2017), 5147-5160. https://doi.org/10.1002/mma.4377

[18] O. Moaaz, R. A. El-Nabulsi, O. Bazighifan, Oscillatory behavior of fourth-order differential equations with neutral delay, Symmetry, 12 (2020), 371. https://doi.org/10.3390/sym12030371

[19] C. Cesarano, O. Moaaz, B. Qaraad, A. Muhib, Oscillatory and asymptotic properties of higher-order quasilinear neutral differential equations, AIMS Math., 6 (2021), 11124-11138. https://doi.org/10.3934/math.2021646

[20] D. Zhao, K. Zhou, F. Ye, X. Xu, A class of time-varying differential equations for vibration research and application, AIMS Math., 9 (2024), 28778-28791. https://doi.org/10.3934/math.20241396

[21] S. Abbas, J. R. Graef, S. S. Negi, Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term, Differ. Equations Dyn. Syst., 32 (2024), 819-829. https://doi.org/10.1007/s12591-022-00592-0

[22] S. R. Grace, G. N. Chhatria, Some novel conditions for the oscillation of second-order noncanonical dynamic equations with non-linear neutral terms, Mediterr. J. Math., 22 (2025), 25. https://doi.org/10.1007/s00009-024-02795-x

[23] H. Liu, F. Meng, P. Liu, Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation, Appl. Math. Comput., 219 (2012), 2739-2748. https://doi.org/10.1016/j.amc.2012.08.106

[24] R. P. Agarwal, C. Zhang, T. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Comput., 274 (2016), 178-181. https://doi.org/10.1016/j.amc.2015.10.089

[25] Y. Wu, Y. Yu, J. Xiao, Oscillation of second order nonlinear neutral differential equations, Mathematics, 10 (2022), 2739. https://doi.org/10.3390/math10152739