Improved results for testing the oscillation of functional differential equations with multiple delays

Amira Essam; Osama Moaaz; Moutaz Ramadan; Ghada AlNemer; Ibrahim M. Hanafy; Amira Essam; Osama Moaaz; Moutaz Ramadan; Ghada AlNemer; Ibrahim M. Hanafy

doi:10.3934/math.20231435

AIMS Mathematics

2023, Volume 8, Issue 11: 28051-28070. doi: 10.3934/math.20231435

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Research article

Improved results for testing the oscillation of functional differential equations with multiple delays

1.
Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, Egypt
2.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
4.
Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia

Received: 09 May 2023 Revised: 23 August 2023 Accepted: 24 September 2023 Published: 12 October 2023
MSC : 34C10, 34K11

In this article, we test whether solutions of second-order delay functional differential equations oscillate. The considered equation is a general case of several important equations, such as the linear, half-linear, and Emden-Fowler equations. We can construct strict criteria by inferring new qualities from the positive solutions to the problem under study. Furthermore, we can incrementally enhance these characteristics. We can use the criteria more than once if they are unsuccessful the first time thanks to their iterative nature. Sharp criteria were obtained with only one condition that guarantees the oscillation of the equation in the canonical and noncanonical forms. Our oscillation results effectively extend, complete, and simplify several related ones in the literature. An example was given to show the significance of the main results.
- functional differential equations,
- delay,
- second-order,
- oscillation,
- differential equations with multiple delays
Citation: Amira Essam, Osama Moaaz, Moutaz Ramadan, Ghada AlNemer, Ibrahim M. Hanafy. Improved results for testing the oscillation of functional differential equations with multiple delays[J]. AIMS Mathematics, 2023, 8(11): 28051-28070. doi: 10.3934/math.20231435

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Abstract

In this article, we test whether solutions of second-order delay functional differential equations oscillate. The considered equation is a general case of several important equations, such as the linear, half-linear, and Emden-Fowler equations. We can construct strict criteria by inferring new qualities from the positive solutions to the problem under study. Furthermore, we can incrementally enhance these characteristics. We can use the criteria more than once if they are unsuccessful the first time thanks to their iterative nature. Sharp criteria were obtained with only one condition that guarantees the oscillation of the equation in the canonical and noncanonical forms. Our oscillation results effectively extend, complete, and simplify several related ones in the literature. An example was given to show the significance of the main results.

References

[1]	G. A. Bocharov, F. A. Rihan, Numerical modelling in bio sciences using delay differential equations, J. Comput. Appl. Math., 125 (2000), 183–199. https://doi.org/10.1016/s0377-0427(00)00468-4 doi: 10.1016/s0377-0427(00)00468-4
[2]	S. Lakshmanan, F. A. Rihan, R. Rakkiyappan, J. H. Park, Stability analysis of the diferential genetic regulatory networks model with time-varying delays and Markovian jumping parameters, Nonlinear Anal. Hybrid Syst., 14 (2014), 1–15. https://doi.org/10.1016/j.nahs.2014.04.003 doi: 10.1016/j.nahs.2014.04.003
[3]	F. A. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim, M. A. Abdeen, Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control, Comput. Math. Methods Med., 14 (2014), 1–15. https://doi.org/10.1155/2014/982978 doi: 10.1155/2014/982978
[4]	F. A. Rihan, D. H. Abdel Rahman, S. Lakshmanan, A. S. Alkhajeh, Time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606–623. https://doi.org/10.1016/j.amc.2014.01.111 doi: 10.1016/j.amc.2014.01.111
[5]	J. S. W. Wong, A second order nonlinear oscillation theorems, Proc. Amer. Math. Soc., 40 (1973), 487–491. https://doi.org/10.1090/s0002-9939-1973-0318585-6 doi: 10.1090/s0002-9939-1973-0318585-6
[6]	I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford: The Clarenden Press, 1991.
[7]	T. Li, N. Pintus, G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Zeitschrift für angewandte Mathematik und Physik, 70(3) (2019), 1–18. https://doi.org/10.1007/s00033-019-1130-2 doi: 10.1007/s00033-019-1130-2
[8]	T. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differenrial Integral Equations, 4 (2021), 315–336. https://doi.org/10.57262/die034-0506-315 doi: 10.57262/die034-0506-315
[9]	J. C. F. Sturm, Memoire sur les equations differentielles lineaires du second ordre, J. Math. Pures Appl., 1 (1836), 106–186. https://doi.org/10.1007/978-3-7643-7990-2 doi: 10.1007/978-3-7643-7990-2
[10]	A. Kneser, Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen, Math. Ann., 42 (1893), 409–435. https://doi.org/10.1007/bf01444165 doi: 10.1007/bf01444165
[11]	W. B. Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc., 19 (1918), 341–352. https://doi.org/10.1090/s0002-9947-1918-1501107-2 doi: 10.1090/s0002-9947-1918-1501107-2
[12]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Dordrecht: Kluwer Academic Publishers, 2002. https://doi.org/10.1007/978-94-017-2515-6
[13]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for second order dynamic equa tions, Series in Mathematical Analysis and Applications, 5 Eds., London: Taylor & Francis, Ltd., 2003. https://doi.org/10.4324/9780203222898
[14]	R. P. Agarwal, M. Bohner, W. T. Li, Nonoscillation and oscillation: theory for functional differential equations, Monographs and Textbooks in Pure and Applied Mathematics, 267 Eds., New York: Marcel Dekker, Inc., 2004. https://doi.org/10.1201/9780203025741
[15]	O. Dosly, P. Rehak, Half-linear differential equations, Handbook of Differential Equations, 1 Eds., North-Holland: Elsevier, 2004. https://doi.org/10.1016/s1874-5725(00)80005-x
[16]	I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford: Clarendon Press, 1991.
[17]	J. Dzurina, I. Jadlovska, A note on oscillation of second-order delay differential equations, Appl. Math. Lett., 69 (2017), 126–132. https://doi.org/10.1016/j.aml.2017.02.003 doi: 10.1016/j.aml.2017.02.003
[18]	J. Dzurina, I. Jadlovska, I. P. Stavroulakis, Oscillatory results for second-order noncanonical delay differential equations, Opuscula Math., 39 (2019), 483–495. https://doi.org/10.7494/opmath.2019.39.4.483 doi: 10.7494/opmath.2019.39.4.483
[19]	J. Dzurina, I. Jadlovska, A sharp oscillation result for second-order half-linear noncanonical delay differential equations, Electron. J. Qual. Theory Differ. Equ., 46 (2020), 1–14. https://doi.org/10.14232/ejqtde.2020.1.46 doi: 10.14232/ejqtde.2020.1.46
[20]	M. Bohner, S. R. Grace, I. Jadlovska, Oscillation criteria for second-order neutral delay differential equations, Electron. J. Qual. Theory Differ. Equ., 60 (2017), 1–12. https://doi.org/10.14232/ejqtde.2017.1.60 doi: 10.14232/ejqtde.2017.1.60
[21]	M. Bohner, S. R. Grace, I. Jadlovska, Sharp oscillation criteria for second-order neutral delay differential equations, Math. Meth. Appl. Sci., 43 (2020), 1–13. https://doi.org/10.1002/mma.6677 doi: 10.1002/mma.6677
[22]	T. S. Hassan, O. Moaaz, A. Nabih, M. B. Mesmouli, A. M. El-Sayed, New Sufficient Conditions for Oscillation of Second-Order Neutral Delay Differential Equations, Axioms, 10 (2021), 281. https://doi.org/10.3390/axioms10040281 doi: 10.3390/axioms10040281
[23]	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
[24]	J. Džurina, S. R. Grace, I. Jadlovska, T. Li, Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term, Math. Nachr., 293 (2020), 910–922. https://doi.org/10.1002/mana.201800196 doi: 10.1002/mana.201800196
[25]	I. Jadlovska, New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations, Mathematics, 9 (2021), 2089. https://doi.org/10.3390/math9172089 doi: 10.3390/math9172089
[26]	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
[27]	T. Li, Y. V. Rogovchenko, Oscillation of second-order neutral differential equations, Math. Nachr., 288 (2015), 1150–1162. https://doi.org/10.1002/mana.201300029 doi: 10.1002/mana.201300029
[28]	T. Li, Y. V. Rogovchenko, Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations, Monatsh. Math., 184 (2017), 489–500. https://doi.org/10.1007/s00605-017-1039-9 doi: 10.1007/s00605-017-1039-9
[29]	R. P. Agarwal, C. Zhang, T. Li, New Kamenev-type oscillation criteria for second-order nonlinear advanced dynamic equations, Appl. Math. Comput., 225 (2013), 822–828. https://doi.org/10.1016/j.amc.2013.09.072 doi: 10.1016/j.amc.2013.09.072
[30]	G. E. Chatzarakis, J. Dzurina, I. Jadlovska, New oscillation criteria for second-order half-linear advanced differential equations, Appl. Math. Comput., 347 (2019), 404–416. https://doi.org/10.1016/j.amc.2018.10.091 doi: 10.1016/j.amc.2018.10.091
[31]	G. E. Chatzarakis, O. Moaaz, T. Li, B.Qaraad, Some oscillation theorems for nonlinear second-order differential equations with an advanced argument, Adv. Difference Equ., 1 (2020). https://doi.org/10.1186/s13662-020-02626-9 doi: 10.1186/s13662-020-02626-9
[32]	T. S. Hassan, Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales, Appl. Math. Comput., 217 (2011), 5285–5297. https://doi.org/10.1016/j.amc.2010.11.052 doi: 10.1016/j.amc.2010.11.052
[33]	T. Li, Y. V. Rogovchenko, On asymptotic behavior of solutions to higher-order sublinear Emden–Fowler delay differential equations, Appl. Math. Lett., 67 (2017), 53–59. https://doi.org/10.1016/j.aml.2016.11.007 doi: 10.1016/j.aml.2016.11.007
[34]	T. Li, Y. V. Rogovchenko, scillation criteria for even-order neutral differential equations, Appl. Math. Lett., 61 (2016), 35–41. https://doi.org/10.1016/j.aml.2016.04.012 doi: 10.1016/j.aml.2016.04.012
[35]	O. Moaaz, S. Furuichi, A. Muhib, New comparison theorems for the nth order neutral differential equations with delay inequalities, Mathematics, 8 (2020), 454. https://doi.org/10.3390/math8030454 doi: 10.3390/math8030454
[36]	O. Moaaz, A. Muhib, New oscillation criteria for nonlinear delay differential equations of fourth-order, Appl. Math. Comput., 377 (2020), 125192. https://doi.org/10.1016/j.amc.2020.125192 doi: 10.1016/j.amc.2020.125192
[37]	O. Moaaz, P. Kumam, O. Bazighifan, On the oscillatory behavior of a class of fourth-order nonlinear differential equation, Symmetry, 12 (2020), 524. https://doi.org/10.3390/sym12040524 doi: 10.3390/sym12040524
[38]	R. Koplatadze, G. Kvinikadze, I. P. Stavroulakis, Oscillation of second order linear delay differential equations, Funct. Differ. Equ., 7 (2000), 121–145. https://doi.org/10.1515/gmj.1999.553 doi: 10.1515/gmj.1999.553
[39]	R. Koplatadze, Oscillation criteria of solutions of second order linear delay differential in equalities with a delayed argument, Trudy Inst. Prikl. Mat. I.N. Vekua., 17 (1986), 104–120. https://doi.org/10.21136/mb.2011.141582 doi: 10.21136/mb.2011.141582
[40]	J. J. Wei, Oscillation of second order delay differential equation, Ann. Differential Equations, 4 (1988), 473–478.
[41]	G. E. Chatzarakis, I. Jadlovska, Improved oscillation results for second-order half-linear delay differential equations, Hacet. J. Math. Stat., 48 (2019), 170–179. https://doi.org/10.15672/hjms.2017.522 doi: 10.15672/hjms.2017.522
[42]	R. Marik, Remarks on the paper by Sun and Meng, Appl. Math. Comput., 248 (2014), 309–313. https://doi.org/10.1016/j.amc.2014.09.100 doi: 10.1016/j.amc.2014.09.100
[43]	G. E. Chatzarakis, S. R. Grace, I. Jadlovska, On the sharp oscillation criteria for half-linear second-order differential equations with several delay arguments, Appl. Math. Comput., 397 (2021), 125915. https://doi.org/10.1016/j.amc.2020.125915 doi: 10.1016/j.amc.2020.125915

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