Oscillation criterion of Kneser type for half-linear second-order dynamic equations with deviating arguments

Taher S. Hassan; Amir Abdel Menaem; Yousef Jawarneh; Naveed Iqbal; Akbar Ali; Taher S. Hassan; Amir Abdel Menaem; Yousef Jawarneh; Naveed Iqbal; Akbar Ali

doi:10.3934/math.2024947

AIMS Mathematics

2024, Volume 9, Issue 7: 19446-19458. doi: 10.3934/math.2024947

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Oscillation criterion of Kneser type for half-linear second-order dynamic equations with deviating arguments

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, Roma 00186, Italy
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
4.
Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, Russia

Received: 18 April 2024 Revised: 28 May 2024 Accepted: 05 June 2024 Published: 13 June 2024
MSC : 39A10, 39A99, 34N05, 34K11

This paper employed the well-known Riccati transformation method to deduce a Kneser-type oscillation criterion for second-order dynamic equations. These results are considered an extension and improvement of the known Kneser results for second-order differential equations and are new for other time scales. We have included examples to highlight the significance of the results we achieved.
- Kneser-type,
- oscillation,
- second-order,
- half-linear,
- dynamic equation,
- time scales
Citation: Taher S. Hassan, Amir Abdel Menaem, Yousef Jawarneh, Naveed Iqbal, Akbar Ali. Oscillation criterion of Kneser type for half-linear second-order dynamic equations with deviating arguments[J]. AIMS Mathematics, 2024, 9(7): 19446-19458. doi: 10.3934/math.2024947

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Abstract

This paper employed the well-known Riccati transformation method to deduce a Kneser-type oscillation criterion for second-order dynamic equations. These results are considered an extension and improvement of the known Kneser results for second-order differential equations and are new for other time scales. We have included examples to highlight the significance of the results we achieved.

References

[1]	S. Hilger, Analysis on measure chains–A unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56. https://doi.org/10.1007/BF03323153
[2]	V. Kac, P. Chueng, Quantum calculus, New York: Springer, 2002. https://doi.org/10.1007/978-1-4613-0071-7
[3]	M. Bohner, A. Peterson, Dynamic equations on time scales: an introduction with applications, Boston: Birkhäuser, 2001. https://doi.org/10.1007/978-1-4612-0201-1
[4]	M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Boston: Birkhäuser, 2003. https://doi.org/10.1007/978-0-8176-8230-9
[5]	R. P. Agarwal, S. L. Shieh, C. C. Yeh, Oscillation criteria for second-order retarded differential equations, Math. Comput. Modell., 26 (1997), 1–11. https://doi.org/10.1016/S0895-7177(97)00141-6
[6]	L. Erbe, T. S. Hassan, A. Peterson, S. H. Saker, Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Differ. Equ., 3 (2008), 227–245.
[7]	B. Baculikova, Oscillation of second-order nonlinear noncanonical differential equations with deviating argument, Appl. Math. Lett., 91 (2019), 68–75. https://doi.org/10.1016/j.aml.2018.11.021 doi: 10.1016/j.aml.2018.11.021
[8]	O. Bazighifan, E. M. El-Nabulsi, Different techniques for studying oscillatory behavior of solution of differential equations, Rocky Mountain J. Math., 51 (2021), 77–86. https://doi.org/10.1216/rmj.2021.51.77 doi: 10.1216/rmj.2021.51.77
[9]	T. S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345 (2008), 176–185. https://doi.org/10.1016/j.jmaa.2008.04.019 doi: 10.1016/j.jmaa.2008.04.019
[10]	S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math., 177 (2005), 375–387. https://doi.org/10.1016/j.cam.2004.09.028 doi: 10.1016/j.cam.2004.09.028
[11]	B. Almarri, A. H. Ali, A. M. Lopes, O. Bazighifan, Nonlinear differential equations with distributed delay: some new oscillatory solutions, Mathematics, 10 (2022), 995. https://doi.org/10.3390/math10060995 doi: 10.3390/math10060995
[12]	I. Jadlovská, Iterative oscillation results for second-order differential equations with advanced argument, Electron. J. Differ. Eq., 2017 (2017), 1–11.
[13]	M. Bohner, K. S. Vidhyaa, E. Thandapani, Oscillation of noncanonical second-order advanced differential equations via canonical transform, Constructive Math. Anal., 5 (2022), 7–13. https://doi.org/10.33205/cma.1055356 doi: 10.33205/cma.1055356
[14]	G. E. Chatzarakis, J. Džurina, I. Jadlovská, 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
[15]	T. S. Hassan, C. Cesarano, R. A. El-Nabulsi, W. Anukool, Improved Hille-type oscillation criteria for second-order quasilinear dynamic equations, Mathematics, 10 (2022), 3675. https://doi.org/10.3390/math10193675 doi: 10.3390/math10193675
[16]	G. E. Chatzarakis, O. Moaaz, T. Li, B. Qaraad, Some oscillation theorems for nonlinear second-order differential equations with an advanced argument, Adv. Differ. Equ., 2020 (2020), 160. https://doi.org/10.1186/s13662-020-02626-9 doi: 10.1186/s13662-020-02626-9
[17]	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
[18]	S. Frassu, G. Viglialoro, Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent, Nonlinear Anal., 213 (2021), 112505. https://doi.org/10.1016/j.na.2021.112505 doi: 10.1016/j.na.2021.112505
[19]	T. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differ. Integral Equ., 34 (2021), 315–336. https://doi.org/10.57262/die034-0506-315 doi: 10.57262/die034-0506-315
[20]	R. P. Agarwal, M. Bohner, T. Li, Oscillatory behavior of second-order half-linear damped dynamic equations, Appl. Math. Comput., 254 (2015), 408–418. https://doi.org/10.1016/j.amc.2014.12.091 doi: 10.1016/j.amc.2014.12.091
[21]	M. Bohner, T. S. Hassan, T. Li, Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments, Indag. Math., 29 (2018), 548–560. https://doi.org/10.1016/j.indag.2017.10.006
[22]	M. Bohner, T. Li, Oscillation of second-order $p$-Laplace dynamic equations with a nonpositive neutral coefficient, Appl. Math. Lett., 37 (2014), 72–76. https://doi.org/10.1016/j.aml.2014.05.012 doi: 10.1016/j.aml.2014.05.012
[23]	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), 86. https://doi.org/10.1007/s00033-019-1130-2 doi: 10.1007/s00033-019-1130-2
[24]	T. S. Hassan, M. Bohner, I. L. Florentina, A. A. Menaem, M. B. Mesmouli, New criteria of oscillation for linear Sturm-Liouville delay noncanonical dynamic equations, Mathematics, 11 (2023), 4850. https://doi.org/10.3390/math11234850 doi: 10.3390/math11234850
[25]	G. V. Demidenko, I. I. Matveeva, Asymptotic stability of solutions to a class of second-order delay differential equations, Mathematics, 9 (2021), 1847. https://doi.org/10.3390/math9161847 doi: 10.3390/math9161847
[26]	A. Kneser, Untersuchungen über die reellen nullstellen der integrale linearer differentialgleichungen, Math. Ann., 42 (1893), 409–435.
[27]	I. Jadlovská, J. Džurina, Kneser-type oscillation criteria for second-order half-linear delay differential equations, Appl. Math. Comput., 380 (2020), 125289. https://doi.org/10.1016/j.amc.2020.125289 doi: 10.1016/j.amc.2020.125289
[28]	O. Došlý, P. Řehák, Half-linear differential equations, Elsevier, 2005.
[29]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for second-order linear, half-linear, superlinear and sublinear dynamic equations, Springer Dordrecht, 2002. https://doi.org/10.1007/978-94-017-2515-6
[30]	T. S. Hassan, Y. Sun, A. A. Menaem, Improved oscillation results for functional nonlinear dynamic equations of second-order, Mathematics, 8 (2020), 1897. https://doi.org/10.3390/math8111897

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