Differential equations of the neutral delay type: More efficient conditions for oscillation

Osama Moaaz; Wedad Albalawi; Osama Moaaz; Wedad Albalawi

doi:10.3934/math.2023641

AIMS Mathematics

2023, Volume 8, Issue 6: 12729-12750. doi: 10.3934/math.2023641

Previous Article Next Article

Research article Special Issues

Differential equations of the neutral delay type: More efficient conditions for oscillation

Osama Moaaz ^{1,2
,
,},
Wedad Albalawi 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: 07 January 2023 Revised: 27 February 2023 Accepted: 09 March 2023 Published: 29 March 2023
MSC : 34C10, 34K11

In this article, we derive an optimized relationship between the solution and its corresponding function for second- and fourth-order neutral differential equations (NDE) in the canonical case. Using this relationship, we obtain new monotonic properties of the second-order equation. The significance of this paper stems from the fact that the asymptotic behavior and oscillation of solutions to NDEs are substantially affected by monotonic features. Based on the new relationships and properties, we obtain oscillation criteria for the studied equations. Finally, we present examples and review some previous theorems in the literature to compare our results with them.
- delay differential equation,
- neutral delay,
- monotonic properties,
- oscillation
Citation: Osama Moaaz, Wedad Albalawi. Differential equations of the neutral delay type: More efficient conditions for oscillation[J]. AIMS Mathematics, 2023, 8(6): 12729-12750. doi: 10.3934/math.2023641

Related Papers:

Abstract

In this article, we derive an optimized relationship between the solution and its corresponding function for second- and fourth-order neutral differential equations (NDE) in the canonical case. Using this relationship, we obtain new monotonic properties of the second-order equation. The significance of this paper stems from the fact that the asymptotic behavior and oscillation of solutions to NDEs are substantially affected by monotonic features. Based on the new relationships and properties, we obtain oscillation criteria for the studied equations. Finally, we present examples and review some previous theorems in the literature to compare our results with them.

References

[1]	C. Sturm, Mémoire sur les Équations différentielles linéaires du second ordre, J. Math. Pure. Appl., 1 (2009), 392–472. http://dx.doi.org/10.1007/978-3-7643-7990-2_30 doi: 10.1007/978-3-7643-7990-2_30
[2]	A. Kneser, Untersuchungen über die reellen nullstellen der integrale linearer differentialgleichungen, Math. Ann., 42 (1893), 409–435. http://dx.doi.org/10.1007/BF01444165 doi: 10.1007/BF01444165
[3]	W. B. Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc., 19 (1918), 341–352. http://dx.doi.org/10.1090/S0002-9947-1918-1501107-2 doi: 10.1090/S0002-9947-1918-1501107-2
[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. http://dx.doi.org/10.1007/978-94-017-2515-6
[5]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for second order dynamic equations, In: Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Dordrecht: Springer, 2002. http://dx.doi.org/10.1007/978-94-017-2515-6_5
[6]	R. P. Agarwal, M. Bohner, W. T. Li, Nonoscillation and oscillation: Theory for functional differential equations, Boca Raton: CRC Press, 2004.
[7]	O. Došlý, P. Rehák, Half-linear differential equations, Amsterdam: Elsevier, 2005.
[8]	I. Gyori, G. Ladas, Oscillation theory of delay differential equations: with applications, New York: Oxford University Press, 1991.
[9]	J. K. Hale, Functional differential equations, In: Analytic theory of differential equations, 183 (1971), 9–22. http://dx.doi.org/10.1007/BFb0060406
[10]	A. Goldbeter, Dissipative structures in biological systems: bistability, oscillations, spatial patterns and waves, Phil. Trans. R. Soc., 376 (2018), 20170376. http://dx.doi.org/10.1098/rsta.2017.0376 doi: 10.1098/rsta.2017.0376
[11]	K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Dordrecht: Springer, 1992. http://dx.doi.org/10.1007/978-94-015-7920-9
[12]	M. C. Mackey, L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287–289. http://dx.doi.org/10.1126/science.267326 doi: 10.1126/science.267326
[13]	F. A. Rihan, Delay differential equations and applications to biology, Singapore: Springer, 2021. http://dx.doi.org/10.1007/978-981-16-0626-7
[14]	B. Baculikova, J. Dzurina, Oscillation theorems for second-order nonlinear neutral differential equations, Comput. Math. Appl., 62 (2011), 4472–4478. http://dx.doi.org/10.1016/j.camwa.2011.10.024 doi: 10.1016/j.camwa.2011.10.024
[15]	H. Liu, F. Meng, P. Liu, Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation, Appl. Math. Comput., 219 (2012), 2739–2748. http://dx.doi.org/10.1016/j.amc.2012.08.106 doi: 10.1016/j.amc.2012.08.106
[16]	Y. Wu, Y. Yu, J. Zhang, J. Xiao, Oscillation criteria for second order Emden-Fowler functional differential equations of neutral type, J. Inequal. Appl., 2016 (2016), 328. http://dx.doi.org/10.1186/s13660-016-1268-9 doi: 10.1186/s13660-016-1268-9
[17]	Y. H. Zeng, L. P. Lou, Y. H. Yu, Oscillation for Emden-Fowler delay differential equations of neutral type, Acta Math. Sci., 35 (2015), 803-814.
[18]	S. R. 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. http://dx.doi.org/10.1186/s13660-018-1767-y doi: 10.1186/s13660-018-1767-y
[19]	Z. Pátíková, S. Fišnarová, Use of the modified Riccati technique for neutral half-linear differential equations, Mathematics, 9 (2021), 235. http://dx.doi.org/10.3390/math9030235 doi: 10.3390/math9030235
[20]	I. Jadlovská, New criteria for sharp oscillation of second-order neutral delay differential equations, Mathematics, 9 (2021), 2089. http://dx.doi.org/10.3390/math9172089 doi: 10.3390/math9172089
[21]	A. Zafer, Oscillation criteria for even order neutral differential equations, Appl. Math. Lett., 11 (1998), 21–25. http://dx.doi.org/10.1016/S0893-9659(98)00028-7 doi: 10.1016/S0893-9659(98)00028-7
[22]	B. Karpuz, Ö. Öcalan, S. Öztürk, Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations, Glasgow Math. J., 52 (2010), 107–114. http://dx.doi.org/10.1017/S0017089509990188 doi: 10.1017/S0017089509990188
[23]	Q. Zhang, J. Yan, L. Gao, Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients, Comput. Math. Appl., 59 (2010), 426–430. http://dx.doi.org/10.1016/j.camwa.2009.06.027 doi: 10.1016/j.camwa.2009.06.027
[24]	Q. Zhang, J. Yan, Oscillation behavior of even order neutral differential equations with variable coefficients, Appl. Math. Lett., 19 (2006), 1202–1206. http://dx.doi.org/10.1016/j.aml.2006.01.003 doi: 10.1016/j.aml.2006.01.003
[25]	R. P. Agarwal, M. Bohner, T. Li, C. Zhang, A new approach in the study of oscillatory behavior of even-order neutral delay differential equations, Appl. Math. Comput., 225 (2013), 787–794. http://dx.doi.org/10.1016/j.amc.2013.09.037 doi: 10.1016/j.amc.2013.09.037
[26]	C. Zhang, T. Li, B. Sun, E. Thandapani, On the oscillation of higher-order half-linear delay differential equations, Appl. Math. Lett., 24 (2011), 1618–1621. http://dx.doi.org/10.1016/j.aml.2011.04.015 doi: 10.1016/j.aml.2011.04.015
[27]	C. Zhang, R. P. Agarwal, M. Bohner, T. Li, New results for oscillatory behavior of even-order half-linear delay differential equations, Appl. Math. Lett., 26 (2013), 179–183. http://dx.doi.org/10.1016/j.aml.2012.08.004 doi: 10.1016/j.aml.2012.08.004
[28]	B. Baculikova, J. Dzurina, J. R. Graef, On the oscillation of higher-order delay differential equations, J. Math. Sci., 187 (2012), 387–400. http://dx.doi.org/10.1007/s10958-012-1071-1 doi: 10.1007/s10958-012-1071-1
[29]	J. Graef, S. Grace, E. Tunç, Oscillation criteria for even-order differential equations with unbounded neutral coefficients and distributed deviating arguments, Funct. Differ. Equ., 25 (2018), 143–153.
[30]	O. Moaaz, C. Park, A. Muhib, O. Bazighifan, Oscillation criteria for a class of even-order neutral delay differential equations, J. Appl. Math. Comput., 63 (2020), 607–617. http://dx.doi.org/10.1007/s12190-020-01331-w doi: 10.1007/s12190-020-01331-w
[31]	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. http://dx.doi.org/10.1186/s13662-019-2240-z doi: 10.1186/s13662-019-2240-z
[32]	C. G. Philos, On the existence of nonoscillatory solutions tending to zero at $\infty $ for differential equations with positive delays, Arch. Math., 36 (1981), 168–178. http://dx.doi.org/10.1007/BF01223686 doi: 10.1007/BF01223686
[33]	Y. Kitamura, T. Kusano, Oscillation of first-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc., 78 (1980), 64–68.
[34]	C. G. Philos, A new criterion for the oscillatory and asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci., 29 (1981), 367–370.
[35]	R. P. Agarwal, S. R. Grace, D. ÓRgean, Oscillation theory for difference and functional differential equations, Dordrecht: Springer, 2000. http://dx.doi.org/10.1007/978-94-015-9401-1
[36]	R. P. Agarwal, S. L. Shieh, C. C. Yeh, Oscillation criteria for second order retarded differential equations, Math. Comput. Model., 26 (1997), 1–11. http://dx.doi.org/10.1016/S0895-7177(97)00141-6 doi: 10.1016/S0895-7177(97)00141-6
[37]	P. J. Y. Wong, R. P. Agarwal, Oscillation theorems and existence criteria of asymptotically monotone solutions for second order differential equations, Dynamic Syst. Appl., 4 (1995), 477–496.
[38]	I. Kiguradze, T. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations, Dordrecht: Springer, 1993. http://dx.doi.org/10.1007/978-94-011-1808-8

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)