On the oscillation of first-order differential equations with deviating arguments and oscillatory coefficients

Emad R. Attia; George E. Chatzarakis; Emad R. Attia; George E. Chatzarakis

doi:10.3934/math.2023341

AIMS Mathematics

2023, Volume 8, Issue 3: 6725-6736. doi: 10.3934/math.2023341

Previous Article Next Article

Research article

On the oscillation of first-order differential equations with deviating arguments and oscillatory coefficients

Emad R. Attia ^{1,2
,
,},
George E. Chatzarakis ³

1.
Department of Mathematics, College of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
3.
Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), Marousi 15122, Athens, Greece

Received: 26 October 2022 Revised: 08 December 2022 Accepted: 29 December 2022 Published: 09 January 2023
MSC : 34K11, 34K06

In this work, we study the oscillation problem of first-order differential equations with deviating arguments and oscillatory coefficients. We generalize and improve the work of Kwong ^[30] such that the delay (advanced) and the coefficient functions do not need to be monotone and nonnegative, respectively. This method essentially improves many known oscillation conditions. The significance and the substantial improvement of our results are shown by two illustrative examples.
- oscillation,
- differential equations,
- nonmonotone delays,
- oscillatory coefficients
Citation: Emad R. Attia, George E. Chatzarakis. On the oscillation of first-order differential equations with deviating arguments and oscillatory coefficients[J]. AIMS Mathematics, 2023, 8(3): 6725-6736. doi: 10.3934/math.2023341

Related Papers:

Abstract

In this work, we study the oscillation problem of first-order differential equations with deviating arguments and oscillatory coefficients. We generalize and improve the work of Kwong ^[30] such that the delay (advanced) and the coefficient functions do not need to be monotone and nonnegative, respectively. This method essentially improves many known oscillation conditions. The significance and the substantial improvement of our results are shown by two illustrative examples.

References

[1]	R. P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Non-oscillation theory of functional differential equations with applications, Springer, New York, Dordrecht Heidelberg London, 2012. https://doi.org/10.1007/978-1-4614-3455-9
[2]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for difference and functional differential equations, Springer Science and Business Media, 2013.
[3]	H. Akca, G. E. Chatzarakis, I. P. Stavroulakis, An oscillation criterion for delay differential equations with several non-monotone arguments, Appl. Math. Lett., 59 (2016), 101–108. https://doi.org/10.1016/j.aml.2016.03.013 doi: 10.1016/j.aml.2016.03.013
[4]	E. R. Attia, B. M. El-Matary, New explicit oscillation criteria for first-order differential equations with several non-monotone delays, Mathematics, 11 (2023), 64. https://doi.org/10.3390/math11010064 doi: 10.3390/math11010064
[5]	E. R. Attia, H. A. El-Morshedy, Improved oscillation criteria for first order differential equations with several non-monotone delays, Mediterr. J. Math., 156 (2021), 1–16. https://doi.org/10.1007/s00009-021-01807-4 doi: 10.1007/s00009-021-01807-4
[6]	E. R. Attia, H. A. El-Morshedy, I. P. Stavroulakis, Oscillation criteria for first order differential equations with non-monotone delays, Symmetry, 12 (2020), 718. https://doi.org/10.3390/sym12050718 doi: 10.3390/sym12050718
[7]	H. Bereketoglu, F. Karakoc, G. S. Oztepe, I. P. Stavroulakis, Oscillation of first order differential equations with several non-monotone retarded arguments, Georgian Math. J., 27 (2019), 341–350. https://doi.org/10.1515/gmj-2019-2055 doi: 10.1515/gmj-2019-2055
[8]	E. Braverman, B. Karpuz, On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput., 218 (2011), 3880–3887. https://doi.org/10.1016/j.amc.2011.09.035 doi: 10.1016/j.amc.2011.09.035
[9]	G. E. Chatzarakis, I. Jadlovská, Explicit criteria for the oscillation of differential equations with several arguments, Dyn. Syst. Appl., 28 (2019), 217–242. http:/doi.org/10.12732/dsa.v28i2.1 doi: 10.12732/dsa.v28i2.1
[10]	G. E. Chatzarakis, H. P$\acute{e}$ics, Differential equations with several non-monotone arguments: an oscillation result, Appl. Math. Lett., 68 (2017), 20–26. https://doi.org/10.1016/j.aml.2016.12.005 doi: 10.1016/j.aml.2016.12.005
[11]	G. E. Chatzarakis, I. K. Purnaras, I. P. Stavroulakis, Oscillations of deviating difference equations with non- monotone arguments, J. Differ. Equ. Appl., 23 (2017), 1354–1377. https://doi.org/10.1080/10236198.2017.1332053 doi: 10.1080/10236198.2017.1332053
[12]	K.-S. Chiu, T. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr., 292 (2019), 2153–2164. https://doi.org/10.1002/mana.201800053 doi: 10.1002/mana.201800053
[13]	J. Dzurina, 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
[14]	Á. Elbert, I. P. Stavroulakis, Oscillations of first order differential equations with deviating arguments, Univ of Ioannina T.R. No 172, 1990, Recent trends in differential equations, (1992), 163–178. https://doi.org/10.1142/9789812798893_0013
[15]	H. A. El-Morshedy, E. R. Attia, New oscillation criterion for delay differential equations with non-monotone arguments, Appl. Math. Lett., 54 (2016), 54–59. https://doi.org/10.1016/j.aml.2015.10.014 doi: 10.1016/j.aml.2015.10.014
[16]	L. H. Erbe, B. G. Zhang, Oscillation theory for functional differential equations, Dekker: New York, NY, USA, 1995. https://doi.org/10.1201/9780203744727
[17]	Á. Garab, I. P Stavroulakis, Oscillation criteria for first order linear delay differential equations with several variable delays, Appl. Math. Let., 106 (2020), 106366. https://doi.org/10.1016/j.aml.2020.106366 doi: 10.1016/j.aml.2020.106366
[18]	K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic Publishers, 1992.
[19]	I. Gyori, G. Ladas, Oscillation theory of delay differential equations with applications, Clarendon Press, Oxford, 1991.
[20]	B. R. Hunt, J. A. Yorke, When all solutions of $x'(t) = -\sum q_l(t) x(t-T_l(t))$ oscillate, J. Differ. Equ., 53 (1984), 139–145. https://doi.org/10.1016/0022-0396(84)90036-6 doi: 10.1016/0022-0396(84)90036-6
[21]	G. Infante, R. Koplatadze, I. P. Stavroulakis, Oscillation criteria for differential equations with several retarded arguments, Funkcial. Ekvac., 58 (2015), 347–364. https://doi.org/10.1619/fesi.58.347 doi: 10.1619/fesi.58.347
[22]	J. Jaroš, I. P. Stavroulakis, Oscillation tests for delay equations, Rocky Mt. J. Math., 29 (1999), 197–207. https://www.jstor.org/stable/44238259
[23]	V. Kolmanovskii, A. Myshkis, Applied theory of functional differential equations, Kluwer, Boston, 1992.
[24]	M. Kon, Y. G. Sficas, I. P. Stavroulakis, Oscillation criteria for delay equations, Proc. Amer. Math. Soc., 128 (2000), 2989-–2997. https://doi.org/10.1090/S0002-9939-00-05530-1 doi: 10.1090/S0002-9939-00-05530-1
[25]	R. G. Koplatadze, Specific properties of solutions of first order differential equations with several delay arguments, J. Contemp. Math. Anal., 50 (2015), 229–235. https://doi.org/10.3103/s1068362315050039 doi: 10.3103/s1068362315050039
[26]	R. G. Koplatadze, T. A. Chanturija, On oscillatory and monotonic solutions of first order differential equations with deviating arguments, Differential'nye Uravnenija., 18 (1982), 1463–1465, (in Russian).
[27]	R. G. Koplatadze, G. Kvinikadze, On the oscillation of solutions of first order delay differential inequalities and equations, Georgian Math. J., 1 (1994), 675–685. https://doi.org/10.1515/GMJ.1994.675 doi: 10.1515/GMJ.1994.675
[28]	Y. Kuang, Delay differential equations with applications in population dynamics, in: mathematics in science and engineering, Academic Press, Boston, MA, 1993.
[29]	M. R. Kulenovic, M. Grammatikopoulos, First order functional differential inequalities with oscillating coefficients, Nonlinear Anal., 8 (1984), 1043–1054. https://doi.org/10.1016/0362-546X(84)90098-1
[30]	M. K. Kwong, Oscillation of first order delay equations, J. Math. Anal. Appl., 156 (1991), 274–286. https://doi.org/10.1016/0022-247X(91)90396-H doi: 10.1016/0022-247X(91)90396-H
[31]	G. Ladas, Sharp conditions for oscillations caused by delays, Appl. Anal., 9 (1979), 93–98. https://doi.org/10.1080/00036817908839256 doi: 10.1080/00036817908839256
[32]	G. Ladas, V. Lakshmikantham, L. S. Papadakis, Oscillations of higher-order retarded differential equations generated by the retarded arguments, in Delay and functional differential equations and their applications, Academic Press, New York, 1972.
[33]	G. Ladas, Y. G. Sficas, I. P. Stavroulakis, Functional differential inequalities and equations with oscillating coefficients, In: V. Lakshmikantham, ed., Trends in Theory and Practice of Nonlinear Differential Equations, Marcel Dekker, New York, Basel, 1984.
[34]	G. S. Ladde, Oscillations caused by retarded perturbations of first order linear ordinary differential equations, Atti Acad. Naz. Lincei, Rend. Lincei, 6 (1978), 351–359.
[35]	T. Li, N. Pintus, G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70 (2019), 1–18. https://doi.org/10.1007/s00033-019-1130-2 doi: 10.1007/s00033-019-1130-2
[36]	T. Li, Y. V. Rogovchenko, Oscillation 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
[37]	T. Li, Y. V. Rogovchenko, On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations, Appl. Math. Lett., 105 (2020), 106293. https://doi.org/10.1016/j.aml.2020.106293 doi: 10.1016/j.aml.2020.106293
[38]	T. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differential Integral Equations, 34 (2021), 315–336. 10.57262/die034-0506-315 doi: 10.57262/die034-0506-315
[39]	G. M. Moremedi, H. Jafari, I. P. Stavroulakis, Oscillation criteria for differential equations with several non-monotone deviating arguments, J. Comput. Anal. Appl., 28 (2020), 136–151.
[40]	J. D. Murray, Mathematical biology I: an introduction, interdisciplinary applied mathematics, Springer, New York, NY, USA, 2002.
[41]	A. D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspekhi Mat. Nauk, 5 (1950), 160–162 (Russian).
[42]	Y. G. Sficas, I. P. Stavroulakis, Oscillation criteria for first-order delay equations, Bull. Lond. Math. Soc., 35 (2003), 239–246. https://doi.org/10.1112/S0024609302001662 doi: 10.1112/S0024609302001662
[43]	I. P. Stavroulakis, Oscillation criteria for delay and difference equations with non-monotone arguments, Appl. Math. Comput., 226 (2014), 661–672. https://doi.org/10.1016/j.amc.2013.10.041 doi: 10.1016/j.amc.2013.10.041
[44]	T. Xianhua, Oscillation of first order delay differential equations with oscillating coefficients, Appl. Math. J. Chinese Univ. Ser. B, 15 (2000), 252–258. https://doi.org/10.1007/s11766-000-0048-x doi: 10.1007/s11766-000-0048-x
[45]	J. S. Yu, Z. C. Wang, B. G. Zhang, X. Z. Qian, Oscillations of differential equations with deviating arguments, Panamer. Math. J., 2 (1992), 59–78.

Reader Comments

Your name:*

Email:*
© 2023 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)