Functional differential equations of the neutral type: Oscillatory features of solutions

Osama Moaaz; Asma Al-Jaser; Osama Moaaz; Asma Al-Jaser

doi:10.3934/math.2024802

AIMS Mathematics

2024, Volume 9, Issue 6: 16544-16563. doi: 10.3934/math.2024802

Previous Article Next Article

Research article

Functional differential equations of the neutral type: Oscillatory features of solutions

Osama Moaaz ^1,2,
Asma Al-Jaser ^{3
,
,}

1.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 27 February 2024 Revised: 24 April 2024 Accepted: 30 April 2024 Published: 13 May 2024
MSC : 34C10, 34K11

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature.
- differential equations,
- oscillation theory,
- neutral delay argument,
- Philos-type criteria
Citation: Osama Moaaz, Asma Al-Jaser. Functional differential equations of the neutral type: Oscillatory features of solutions[J]. AIMS Mathematics, 2024, 9(6): 16544-16563. doi: 10.3934/math.2024802

Related Papers:

Abstract

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature.

References

[1]	V. Volterra, Sur la theorie mathematique des phenomenes hereditaires, J. Math. Pure. Appl., 7 (1928), 249–298.
[2]	A. D. Mishkis, Lineare differentialgleichungen mit nacheilenden argumentom, Berlin: Verlag, 1955.
[3]	R. Bellman, J. M. Danskin, A survey of the mathematical theory of time lag, retarded control, and hereditary processes, United States: RAND Corporation, 1954.
[4]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Dordrecht: Springer, 2002. https://doi.org/10.1007/978-94-017-2515-6
[5]	I. Gyori, G. Ladas, Oscillation theory of delay differential equations with applications, Oxford University Press, 1991. https://doi.org/10.1093/oso/9780198535829.001.0001
[6]	L. H. Erbe, Q. Kong, B. G. Zhong, Oscillation theory for functional differential equations, Routledge, 1995. https://doi.org/10.1201/9780203744727
[7]	O. Moaaz, W. Albalawi, Differential equations of the neutral delay type: More efficient conditions for oscillation, AIMS Math., 8 (2023), 12729–12750. https://doi.org/10.3934/math.2023641 doi: 10.3934/math.2023641
[8]	A. Palanisamy, J. Alzabut, V. Muthulakshmi, S. S. Santra, K. Nonlaopon, Oscillation results for a fractional partial differential system with damping and forcing terms, AIMS Math., 8 (2023), 4261–4279. https://doi.org/10.3934/math.2023212 doi: 10.3934/math.2023212
[9]	S. S. Santra, A. K. Sethi, O. Moaaz, K. M. Khedher, S. W. Yao, New oscillation theorems for second-order differential equations with canonical and non-canonical operator via riccati transformation, Mathematics, 9 (2021), 1111. https://doi.org/10.3390/math9101111 doi: 10.3390/math9101111
[10]	J. K. Hale, Functional differential equations, In: Analytic theory of differential equations, Berlin: Springer, 1971. https://doi.org/10.1007/BFb0060406
[11]	C. Huang, B. Liu, C. Qian, J. Cao, Stability on positive pseudo almost periodic solutions of HPDCNNs incorporating D operator, Math. Comput. Simulat., 190 (2021), 1150–1163. https://doi.org/10.1016/j.matcom.2021.06.027 doi: 10.1016/j.matcom.2021.06.027
[12]	B. Liu, Finite-time stability of CNNs with neutral proportional delays and time-varying leakage delays, Math. Method. Appl. Sci., 40 (2017), 167–174. https://doi.org/10.1002/mma.3976 doi: 10.1002/mma.3976
[13]	C. Huang, B. Liu, H. Yang, J. Cao, Positive almost periodicity on SICNNs incorporating mixed delays and D operator, Nonlinear Anal. Model., 27 (2022) 719–739. https://doi.org/10.15388/namc.2022.27.27417
[14]	C. G. Philos, Oscillation theorems for linear differential equation of second order, Arch. Math., 53 (1989), 483–492. https://doi.org/10.1007/BF01324723 doi: 10.1007/BF01324723
[15]	I. V. Kamenev, Oscillation criteria, connected with averaging, for the solutions of second order ordinary differential equations, Differentsial'nye Uravneniya, 10 (1974), 246–252.
[16]	J. Sugie, K. Ishihara, Philos-type oscillation criteria for linear differential equations with impulsive effects, J. Math. Anal. Appl., 470 (2019), 911–930. https://doi.org/10.1016/j.jmaa.2018.10.041 doi: 10.1016/j.jmaa.2018.10.041
[17]	Z. Xu, X. Liu, Philos-type oscillation criteria for Emden-Fowler neutral delay differential equations, J. Comput. Appl. Math., 206 (2007), 1116-1126. https://doi.org/10.1016/j.cam.2006.09.012 doi: 10.1016/j.cam.2006.09.012
[18]	O. Moaaz, E. M. Elabbasy, A. Muhib, Oscillation criteria for even-order neutral differential equations with distributed deviating arguments, Adv. Differ. Equ., 2019 (2019), 297. https://doi.org/10.1186/s13662-019-2240-z doi: 10.1186/s13662-019-2240-z
[19]	J. Manojlović, Oscillation theorems for nonlinear differential equations, Electron. J. Qual. Theory Differ. Equ., 2000 (2000), 1–21. https://doi.org/10.14232/ejqtde.2000.1.1 doi: 10.14232/ejqtde.2000.1.1
[20]	J. Džurina, D. Lacková, Oscillation results for second order nonlinear differential equations, Centr. Eur. J. Math., 2 (2004), 57–66. https://doi.org/10.2478/BF02475950 doi: 10.2478/BF02475950
[21]	Y. Şahiner, On oscillation of second order neutral type delay differential equations, Appl. Math. Comput., 150 (2004), 697–706. https://doi.org/10.1016/S0096-3003(03)00300-X doi: 10.1016/S0096-3003(03)00300-X
[22]	Z. Xu, P. Weng, Oscillation of second order neutral equations with distributed deviating argument, J. Comput. Appl. Math., 202 (2007), 460–477. https://doi.org/10.1016/j.cam.2006.03.001 doi: 10.1016/j.cam.2006.03.001
[23]	L. Ye, Z. Xu, Oscillation criteria for second order quasilinear neutral delay differential equations, Appl. Math. Comput., 207 (2009), 388–396. https://doi.org/10.1016/j.amc.2008.10.051 doi: 10.1016/j.amc.2008.10.051
[24]	Q. Li, R. Wang, F. Chen, T. Li, Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients, Adv. Differ. Equ., 2015 (2015), 35. https://doi.org/10.1186/s13662-015-0377-y doi: 10.1186/s13662-015-0377-y
[25]	R. Arul, V. S. Shobha, Improvement results for oscillatory behavior of second order neutral differential equations with nonpositive neutral term, J. Adv. Math. Comput. Sci., 12 (2015), 1–7. https://doi.org/10.9734/BJMCS/2016/20641 doi: 10.9734/BJMCS/2016/20641
[26]	S. R. Grace, Oscillatory behavior of second-order nonlinear differential equations with a nonpositive neutral term, Mediterr. J. Math., 14 (2017), 229. https://doi.org/10.1007/s00009-017-1026-3 doi: 10.1007/s00009-017-1026-3
[27]	S. R. Grace, J. R. Graef, I. Jadlovská, Oscillatory behavior of second order nonlinear delay differential equations with positive and negative neutral terms. Differ. Equ. Appl., 12 (2020), 201–211. https://doi.org/10.7153/dea-2020-12-13
[28]	B. Baculíková, B. Sudha, K. Thangavelu, E. Thandapani, Oscillation of second order delay differential equations with nonlinear nonpositive neutral term, Math. Slovaca, 72 (2022), 103–110. https://doi.org/10.1515/ms-2022-0007 doi: 10.1515/ms-2022-0007
[29]	S. Grace, J. Dzurina, I. Jadlovska, T. Li, An improved approach for studying oscillation of second-order neutral delay differential equations, J. Inequal. Appl., 2018 (2018), 193. https://doi.org/10.1186/s13660-018-1767-y doi: 10.1186/s13660-018-1767-y
[30]	O. Moaaz, M. Anis, D. Baleanu, A. Muhib, More effective criteria for oscillation of second-order differential equations with neutral arguments, Mathematics, 8 (2020), 986. https://doi.org/10.3390/math8060986 doi: 10.3390/math8060986
[31]	O. Moaaz, New criteria for oscillation of nonlinear neutral differential equations, Adv. Differ. Equ., 2019 (2019), 484. https://doi.org/10.1186/s13662-019-2418-4 doi: 10.1186/s13662-019-2418-4
[32]	O. Moaaz, H. Ramos, J. Awrejcewicz, Second-order Emden-Fowler neutral differential equations: A new precise criterion for oscillation, Appl. Math. Lett., 118 (2021), 107172. https://doi.org/10.1016/j.aml.2021.107172 doi: 10.1016/j.aml.2021.107172
[33]	T. S. Hassan, O. Moaaz, A. Nabih, M. B. Mesmouli, A. M. A. El-Sayed, New sufficient condi tions for oscillation of second-order neutral delay differential equations, Axioms, 10 (2021), 281. https://doi.org/10.3390/axioms10040281 doi: 10.3390/axioms10040281
[34]	O. Moaaz, A. Muhib, S. Owyed, E. E. Mahmoud, A. Abdelnaser, Second-order neutral differential equations: Improved criteria for testing the oscillation, J. Math., 2021 (2021), 6665103. https://doi.org/10.1155/2021/6665103 doi: 10.1155/2021/6665103
[35]	O. Moaaz, C. Cesarano, B. Almarri, An improved relationship between the solution and its corresponding function in fourth-order neutral differential equations and its applications, Mathematics, 11 (2023), 1708. https://doi.org/10.3390/math11071708 doi: 10.3390/math11071708
[36]	Q. Feng, B. Zheng, Oscillation criteria for nonlinear third-order delay dynamic equations on time scales involving a super-linear neutral term, Fractal Fract., 8 (2024), 115. https://doi.org/10.3390/fractalfract8020115 doi: 10.3390/fractalfract8020115
[37]	K. S. Vidhyaa, R. Deepalakshmi, J. R. Graef, E. Thandapani, Oscillatory behavior of semi-canonical third-order delay differental equations with a superlinear neutral term, Appl. Anal. Discr. Math., 00 (2024), 6. https://doi.org/10.2298/AADM210812006V doi: 10.2298/AADM210812006V
[38]	N. Prabaharan, E. Thandapani, E. Tunç, Asymptotic behavior of semi-canonical third-order delay differential equations with a superlinear neutral term, Palestine J. Math., 12 (2023).
[39]	C. Huang, H. Yang, J. Cao, Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator, Discrete Cont. Dyn. S, 14 (2020), 1259–1272. https://doi.org/10.3934/dcdss.2020372 doi: 10.3934/dcdss.2020372
[40]	S. Y. Zhang, Q. R. Wang, Oscillation of second-order nonlinear neutral dynamic equations on time scales, Appl. Math. Comput., 216 (2010), 2837–2848. https://doi.org/10.1016/j.amc.2010.03.134 doi: 10.1016/j.amc.2010.03.134
[41]	J. Alzabut, S. R. Grace, S. S. Santra, G. N. Chhatria, Asymptotic and oscillatory behaviour of third order non-linear differential equations with canonical operator and mixed neutral terms, Qual. Theory Dyn. Syst., 22 (2023), 15. https://doi.org/10.1007/s12346-022-00715-6 doi: 10.1007/s12346-022-00715-6
[42]	J. Alzabut, S. R. Grace, G. N. Chhatria, New oscillation results for higher order nonlinear differential equations with a nonlinear neutral terms, J. Math. Comput. Sci., 28 (2023), 294–305. https://doi.org/10.22436/jmcs.028.03.07 doi: 10.22436/jmcs.028.03.07
[43]	I. Jadlovská, J. Džurina, J. R. Graef, S. R. Grace, Sharp oscillation theorem for fourth-order linear delay differential equations, J. Inequal. Appl., 2022 (2022), 122. https://doi.org/10.1186/s13660-022-02859-0 doi: 10.1186/s13660-022-02859-0
[44]	S. Janaki, V. Ganesan, On the oscillatory behavior of a class of even order nonlinear damped delay differential equations with distributed deviating arguments, Int. J. Appl. Math., 36 (2023), 391. https://doi.org/10.12732/ijam.v36i3.6 doi: 10.12732/ijam.v36i3.6
[45]	H. Ramos, O. Moaaz, A. Muhib, J. Awrejcewicz, More effective results for testing oscillation of non-canonical neutral delay differential equations, Mathematics, 9 (2021), 1114. https://doi.org/10.3390/math9101114 doi: 10.3390/math9101114

Reader Comments

Your name:*

Email:*
© 2024 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)