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

Meijiao Zhang; Kai Zhou; Zhi Zhang; Duoduo Zhao; Meijiao Zhang; Kai Zhou; Zhi Zhang; Duoduo Zhao

doi:10.3934/era.2026201

Electronic Research Archive

2026, Volume 34, Issue 7: 4560-4576. doi: 10.3934/era.2026201

Previous Article Next Article

Research article

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

1.
School of Big Data and Artificial Intelligence, Center of Applied Mathematics, Chizhou University, Chizhou 247000, China
2.
School of Mathematics and Statistics, Huaibei Normal University, Huaibei 235000, China

Received: 30 January 2026 Revised: 18 April 2026 Accepted: 15 May 2026 Published: 28 May 2026

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 $ \alpha $ and $ \beta $. 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 $ \alpha $ with the rate of change of the delay, quantified by $ \beta $. These results are shown to be both sharp and easily applicable, as illustrated by examples that also highlight their improvement over existing results.
- damped term,
- positive and negative,
- coefficient variable time delay,
- Riccati transformation,
- oscillation
Citation: Meijiao Zhang, Kai Zhou, Zhi Zhang, Duoduo Zhao. Oscillation criteria for damped second-order equations with time delays based on the parameter-Riccati method[J]. Electronic Research Archive, 2026, 34(7): 4560-4576. doi: 10.3934/era.2026201

Related Papers:

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 $ \alpha $ and $ \beta $. 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 $ \alpha $ with the rate of change of the delay, quantified by $ \beta $. These results are shown to be both sharp and easily applicable, as illustrated by examples that also highlight their improvement over existing results.

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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.3390/math10152739

Reader Comments

Your name:*

Email:*
© 2026 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)