Third-order neutral differential equations of the mixed type: Oscillatory and asymptotic behavior

B. Qaraad; O. Moaaz; D. Baleanu; S. S. Santra; R. Ali; E. M. Elabbasy; B. Qaraad; O. Moaaz; D. Baleanu; S. S. Santra; R. Ali; E. M. Elabbasy

doi:10.3934/mbe.2022077

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 2: 1649-1658. doi: 10.3934/mbe.2022077

Previous Article Next Article

Research article Special Issues

Third-order neutral differential equations of the mixed type: Oscillatory and asymptotic behavior

1.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2.
Department of Mathematics, Faculty of Science, Amran University, Amran, Yemen
3.
Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, Roma 39, 00186, Italy
4.
Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Çankaya University Ankara, Etimesgut 06790, Turkey
5.
Instiute of Space Sciences, Magurele-Bucharest, Magurele 077125, Romania; Department of Medical Research, China
6.
Medical University Hospital, China Medical University, Taiwan, China
7.
Department of Mathematics, JIS College of Engineering, Kalyani, West Bengal – 741235, India
8.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

Received: 11 July 2021 Accepted: 18 October 2021 Published: 14 December 2021

In this work, by using both the comparison technique with first-order differential inequalities and the Riccati transformation, we extend this development to a class of third-order neutral differential equations of the mixed type. We present new criteria for oscillation of all solutions, which improve and extend some existing ones in the literature. In addition, we provide an example to illustrate our results.
- thrid-order differential equations,
- delay,
- mixed-neutral term,
- oscillation criteria
Citation: B. Qaraad, O. Moaaz, D. Baleanu, S. S. Santra, R. Ali, E. M. Elabbasy. Third-order neutral differential equations of the mixed type: Oscillatory and asymptotic behavior[J]. Mathematical Biosciences and Engineering, 2022, 19(2): 1649-1658. doi: 10.3934/mbe.2022077

Related Papers:

Abstract

In this work, by using both the comparison technique with first-order differential inequalities and the Riccati transformation, we extend this development to a class of third-order neutral differential equations of the mixed type. We present new criteria for oscillation of all solutions, which improve and extend some existing ones in the literature. In addition, we provide an example to illustrate our results.

References

[1]	J. Hale, Theory of Functional Differential Equations, Springer, 1977.
[2]	O. Bazighifan, T. Abdeljawad, Q. M. Al-Mdallal, Differential equations of even-order with p-Laplacian like operators: qualitative properties of the solutions, Adv. Differ. Equations, 2021 (2021), 1–10. doi: 10.1186/s13662-021-03254-7. doi: 10.1186/s13662-021-03254-7
[3]	O. Bazighifan, A. F. Aljohani, Explicit criteria for the qualitative properties of differential equations with p-Laplacian-like operator, Adv. Differ. Equ., 2020 (2020), 454. doi: 10.1186/s13662-020-02907-3. doi: 10.1186/s13662-020-02907-3
[4]	O. Bazighifan, P. Kumam, Oscillation theorems for advanced differential equations with p-Laplacian like operators, Mathematics, 8 (2020), 821. doi: 10.3390/math8050821. doi: 10.3390/math8050821
[5]	M. Bohner, T. Li, Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient, Appl. Math. Lett., 37 (2014), 72–76. doi: 10.1016/j.aml.2014.05.012. doi: 10.1016/j.aml.2014.05.012
[6]	J. Dzurina, Oscillation of second order differential equations with advanced argument, Math. Slovaca, 45 (1995), 263–268.
[7]	S. R. Grace, J. R. Graef, E. Tunc, Oscillatory behavior of second order damped neutral differential equations with distributed deviating arguments, Miskolc Math. Notes, 18 (2017), 759–769. doi: 10.18514/MMN.2017.2326. doi: 10.18514/MMN.2017.2326
[8]	T. Li, E. Thandapani, J. R. Graef, E. Tunc, Oscillation of second-order Emden–Fowler neutral differential equations, Nonlinear Stud., 20 (2013), 1–8.
[9]	O. Moaaz, E. M. Elabbasy, A. Muhib, Oscillation criteria for even-order neutral differential equations with distributed deviating arguments, Adv. Differ. Equations, 297 (2019). doi: 10.1186/s13662-019-2240-z. doi: 10.1186/s13662-019-2240-z
[10]	T. Candan, Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations, Adv. Differ. Equations, 2014 (2014), 35. doi: 10.1186/1687-1847-2014-35. doi: 10.1186/1687-1847-2014-35
[11]	O. Moaaz, New criteria for oscillation of nonlinear neutral differential equations, Adv. Differ. Eqs., 2019 (2019), 484. doi: 10.1186/s13662-019-2418-4. doi: 10.1186/s13662-019-2418-4
[12]	O. Moaaz, E. M. Elabbasy, B. Qaraad, An improved approach for studying oscillation of generalized Emden–Fowler neutral differential equation, J. Inequal. Appl., 69 (2020), doi: 10.1186/s13660-020-02332-w. doi: 10.1186/s13660-020-02332-w
[13]	O. Moaaz, B. Qaraad, R. El-Nabulsi, O. Bazighifan, New results for kneser solutions of third-order nonlinear neutral differential equations, Mathematics, 8 (2020), 686. doi: 10.3390/math8050686. doi: 10.3390/math8050686
[14]	B. Baculikova, J. Dzurina, Oscillation of third-order neutral differential equations, Math. Comput. Model., 52 (2010), 215–226. doi: 10.1016/j.mcm.2010.02.011. doi: 10.1016/j.mcm.2010.02.011
[15]	J. Dzurina, E. Thandapani, S. Tamilvanan, Oscillation of solutions to third-order half-linear neutral differential equations, Electron. J. Differ. Equations, 2012 (2012), 1–9. doi: 10.21136/MB.2013.143232. doi: 10.21136/MB.2013.143232
[16]	J. Graef, E. Tunc, S. Grace, Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation, Opusc. Math., 37 (2017), 839–852. doi: 10.7494/OpMath.2017.37.6.839. doi: 10.7494/OpMath.2017.37.6.839
[17]	T. Li, C. Zhang, G. Xing, Oscillation of third-order neutral delay differential equations, Abstr. Appl. Anal., 2012 (2012). doi: 10.1155/2012/569201. doi: 10.1155/2012/569201
[18]	T. Li, Yu. V. Rogovchenko, On asymptotic behavior of solutions to higher-order sublinear Emden-Fowler delay differential equations, Appl. Math. Lett., 67 (2017), 53–59, doi.org/10.1016/j.aml.2016.11.007 doi: 10.1016/j.aml.2016.11.007
[19]	E. Thandapani, T. Li, On the oscillation of third-order quasi-linear neutral functional differential equations, Arch. Math., 47 (2011), 181–199.
[20]	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, doi: 10.1016/j.aml.2011.04.015. doi: 10.1016/j.aml.2011.04.015
[21]	R. P. Agarwal, M. Bohner, T. Li, C. Zhang, Oscillation of third-order nonlinear delay differential equations, Taiwanese J. Math., 17 (2013), 545–558. doi: 10.11650/tjm.17.2013.2095. doi: 10.11650/tjm.17.2013.2095
[22]	G. E. Chatzarakis, S. R. Grace, I. Jadlovska, T. Li, E. Tunc, Oscillation criteria for third-order Emden-Fowler differential equations with unbounded neutral coefficients, Complexity, 2019 (2019), 1–7. doi: 10.1155/2019/5691758. doi: 10.1155/2019/5691758
[23]	J. Dzurina, S. R. Grace, I. Jadlovska, On nonexistence of Kneser solutions of third-order neutral delay differential equations, Appl. Math. Letter., 88 (2019), 193–200. doi: 10.1016/j.aml.2018.08.016. doi: 10.1016/j.aml.2018.08.016
[24]	B. Baculikova, J. Dzurina, Some properties of third-order differential equations with mixed arguments, J. Math., 2013 (2013). doi: 10.1155/2013/528279. doi: 10.1155/2013/528279
[25]	S. R. Grace, On the oscillations of mixed neutral equations, J. Math. Anal. Appl., 194 (1995), 377–388. doi: 10.1006/jmaa.1995.1306. doi: 10.1006/jmaa.1995.1306
[26]	E. Thandapani, R. Rama, Oscillatory behavior of solutions of certain third order mixed neutral differential equations, Tamkang J. Math., 44 (2013), 99–112. doi: 10.5556/J.TKJM.44.2013.1150. doi: 10.5556/J.TKJM.44.2013.1150
[27]	Z. Han, T. Li, C. Zhang, S. Sun, Oscillatory behavior of solutions of certain third-order mixed neutral functional differential equations, Bull. Malays. Math., 35 (2012), 611–620.
[28]	O. Moaaz, D. Chalishajar, O. Bazighifan, Asymptotic behavior of solutions of the third order nonlinear mixed type neutral differential equations, Mathematics., 8 (2020), 485. doi: 10.3390/math8040485. doi: 10.3390/math8040485
[29]	B. Baculikova, J. Dzurina, Oscillation of third-order neutral differential equations, Math. Comput. Model., 52 (2010), 215–226. doi: 10.1016/j.mcm.2010.02.011. doi: 10.1016/j.mcm.2010.02.011
[30]	S. Y. Zhang, Q. Wang, Oscillation of second-order nonlinear neutral dynamic equations on time scales, Appl. Math. Comput., 216 (2010), 2837–2848, doi: 10.1016/j.amc.2010.03.134. doi: 10.1016/j.amc.2010.03.134
[31]	E. M. Elabbasy, T. S. Hassan, O. Moaaz, Oscillation behavior of second-order nonlinear neutral differential equations with deviating arguments, Opusc. Math., 32 (2012), 719–730. doi: 10.7494/OpMath.2012.32.4.719. doi: 10.7494/OpMath.2012.32.4.719
[32]	C. Philos, On the existence of nonoscillatory solutions tending to zero at $\infty $for differential equations with positive delay, Arch. Math., 36 (1981), 168–178. doi: 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., 78 (1980), 64–68. doi: 10.1090/S0002-9939-1980-0548086-5. doi: 10.1090/S0002-9939-1980-0548086-5

Reader Comments

Your name:*

Email:*
© 2022 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)