Fourth-order differential equations with neutral delay: Investigation of monotonic and oscillatory features

H. Salah; M. Anis; C. Cesarano; S. S. Askar; A. M. Alshamrani; E. M. Elabbasy; H. Salah; M. Anis; C. Cesarano; S. S. Askar; A. M. Alshamrani; E. M. Elabbasy

doi:10.3934/math.20241630

AIMS Mathematics

2024, Volume 9, Issue 12: 34224-34247. doi: 10.3934/math.20241630

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Fourth-order differential equations with neutral delay: Investigation of monotonic and oscillatory features

1.
Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt
2.
Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele Ⅱ, 39, 00186 Roma, Italy
3.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 03 August 2024 Revised: 27 October 2024 Accepted: 11 November 2024 Published: 04 December 2024
MSC : 34C10, 34K11

For fourth-order neutral differential equations (NDE) in the canonical case, we present new relationships between the solution and its corresponding function in two casses: $ p < 1 $ and $ p > 1 $. Through these relationships, we discover new monotonic properties for this equation of fourth order. Using the new relationships and properties, we derive some oscillation conditions for the equation under study. By using the Comparison and Ricatti technique, the positive solutions are excluded by providing some conditions. Lastly, we provide examples and review previous theorems from the literature to compare our findings.
- functional differential equation,
- neutral,
- oscillation,
- fourth-order,
- canonical
Citation: H. Salah, M. Anis, C. Cesarano, S. S. Askar, A. M. Alshamrani, E. M. Elabbasy. Fourth-order differential equations with neutral delay: Investigation of monotonic and oscillatory features[J]. AIMS Mathematics, 2024, 9(12): 34224-34247. doi: 10.3934/math.20241630

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Abstract

For fourth-order neutral differential equations (NDE) in the canonical case, we present new relationships between the solution and its corresponding function in two casses: $ p < 1 $ and $ p > 1 $. Through these relationships, we discover new monotonic properties for this equation of fourth order. Using the new relationships and properties, we derive some oscillation conditions for the equation under study. By using the Comparison and Ricatti technique, the positive solutions are excluded by providing some conditions. Lastly, we provide examples and review previous theorems from the literature to compare our findings.

References

[1]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation of functional differential equations, Springer eBooks, 2000, 166–317. https://doi.org/10.1007/978-94-015-9401-1_2
[2]	J. K. Hale, Theory of functional differential equations, New York: Springer, 1977. https://doi.org/10.1007/978-1-4612-9892-2
[3]	M. R. Roussel, The use of delay differential equations in chemical kinetics, J. Phys. Chem., 100 (1996), 8323–8330. https://doi.org/10.1021/jp9600672 doi: 10.1021/jp9600672
[4]	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. https://doi.org/10.1016/j.amc.2013.09.037 doi: 10.1016/j.amc.2013.09.037
[5]	M. Botros, E. A. Ziada, I. L. El-Kalla, Semi-analytic solutions of nonlinear multidimensional fractional differential equations, Math. Biosci. Eng., 19 (2022), 13306–13320. https://doi.org/10.3934/mbe.2022623 doi: 10.3934/mbe.2022623
[6]	S. R. Grace, J. Džurina, I. Jadlovská, T. Li, An improved approach for studying oscillation of second-order neutral delay differential equations, J. Inequal. Appl., 2018 (2018). https://doi.org/10.1186/s13660-018-1767-y doi: 10.1186/s13660-018-1767-y
[7]	J. K. Hale, Asymptotic behavior of dissipative systems, American Mathematical Soc., 25 (2010).
[8]	R. P. Agarwal, M. Bohner, W. T. Li, Nonoscillation and oscillation: Theory for functional differential equations, New York: Marcel Dekker, Inc., 2004. https://doi.org/10.1201/9780203025741
[9]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Springer Science & Business Media, 2013. http://dx.doi.org/10.1007/978-94-017-2515-6 5
[10]	B. Almarri, B. Batiha, O. Bazighifan, F. Masood, Third-order neutral differential equations with non-canonical forms: Novel oscillation theorems, Axioms, 13 (2024), 755. https://doi.org/10.3390/axioms13110755 doi: 10.3390/axioms13110755
[11]	I. Jadlovská, 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
[12]	M. Wei, C. Jiang, T. Li, Oscillation of third-order neutral differential equations with damping and distributed delay, Adv. Differential Equ., 2019 (2019). http://dx.doi.org/10.1186/s13662-019-2363-2
[13]	Y. C. Qiu, I. Jadlovská, K. S. Chiu, T. Li, Existence of nonoscillatory solutions tending to zero of third-order neutral dynamic equations on time scales, Adv. Differential Equ., 2020 (2020). https://doi.org/10.1186/s13662-020-02678-x
[14]	I. Jadlovská, 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
[15]	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. https://doi.org/10.1016/j.amc.2013.09.037.0722 doi: 10.1016/j.amc.2013.09.037.0722
[16]	L. Ye, Z. Xu, Oscillation criteria for second order quasilinear neutral delay differential equations, Appl. Math. Comput., 207 (2008), 388–396. https://doi.org/10.1016/j.amc.2008.10.051 doi: 10.1016/j.amc.2008.10.051
[17]	J. Džurina, I. Jadlovská, 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. Alzabut, S. R. Grace, S. S. Santra, G. N. Chhatria, Asymptotic and oscillatory behaviour of third order non-linear differential equations with canonical operator and mixed neutral terms, Qual. Theor. Dyn. Syst., 22 (2022). https://doi.org/10.1007/s12346-022-00715-6
[19]	S. R. Grace, Oscillatory and asymptotic behavior of damped functional differential equations, Math. Nachr., 142 (1989), 297–305. https://doi.org/10.1002/mana.19891420121 doi: 10.1002/mana.19891420121
[20]	M. C. Mackey, L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (4300), 287–289. http://dx.doi.org/10.1126/science.267326 doi: 10.1126/science.267326
[21]	K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Springer Dordrecht, 1992. https://doi.org/10.1007/978-94-015-7920-9
[22]	L. Erbe, Q. Kong, B. G. Zhang, Oscillation theory for functional differential eqautions, New York: Marcel Dekker, Inc., 1995.
[23]	S. R. Grace, B. S. Lalli, Oscillation of nonlinear second order neutral differential equations, Rad. Mat., 3 (1987), 34277–34284.
[24]	G. Xing, T. Li, C. Zhang, Oscillation of higher-order quasi-linear neutral differential equations, Adv. Differential Equ., 2011 (2011). https://doi.org/10.1186/1687-1847-2011-45
[25]	L. Erbe, T. S. Hassan, A. Peterson, Oscillation of second order neutral delay differential equations, Adv. Dynam. Syst. Appl., 1 (2008), 53–71.
[26]	F. Masood, O. Moaaz, G. AlNemer, H. El-Metwally, New monotonic properties for solutions of a class of functional differential equations and their applications, Symmetry, 15 (2023), 1956. https://doi.org/10.3390/sym15101956 doi: 10.3390/sym15101956
[27]	G. E. Chatzarakis, S. R. Grace, I. Jadlovská, Oscillation criteria for third-order delay differential equations, Adv. Differential Equ., 2017 (2017). https://doi.org/10.1186/s13662-017-1384-y
[28]	R. P. Agarwal, S. R. Grace, D. O'Regan, The oscillation of certain higher-order functional differential equations, Math. Comput. Model., 37 (2003), 705–728. https://doi.org/10.1016/s0895-7177(03)00079-7 doi: 10.1016/s0895-7177(03)00079-7
[29]	Q. Zhang, J. Yan, L. Gao, Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients, Comput. Math. Appl., 59 (2009), 426–430. https://doi.org/10.1016/j.camwa.2009.06.027 doi: 10.1016/j.camwa.2009.06.027
[30]	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
[31]	O. Moaaz, R. A. El-Nabulsi, A. Muhib, S. K. Elagan, M. Zakarya, New improved results for oscillation of fourth-order neutral differential equations, Mathematics, 9 (2021), 2388. https://doi.org/10.3390/math9192388 doi: 10.3390/math9192388
[32]	O. Bazighifan, M. Ruggieri, A. Scapellato, An improved criterion for the oscillation of fourth-order differential equations, Mathematics, 8 (2020), 610. https://doi.org/10.3390/math8040610 doi: 10.3390/math8040610
[33]	R. P. Agarwal, C. Zhang, T. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Comput., 274 (2015), 178–181. https://doi.org/10.1016/j.amc.2015.10.089 doi: 10.1016/j.amc.2015.10.089
[34]	M. Bohner, S. R. Grace, I. Jadlovska, Oscillation criteria for second-order neutral delay differential equations, Electron. J. Qual. Theo., 60 (2017), 1–12. https://doi.org/10.14232/ejqtde.2017.1.60 doi: 10.14232/ejqtde.2017.1.60
[35]	O. Moaaz, A. Muhib, S. Owyed, E. E. Mahmoud, A. Abdelnaser, Second-order neutral differential equations: Improved criteria for testing the oscillation, J. Math., 2021 (2021), 1–7. https://doi.org/10.1155/2021/6665103 doi: 10.1155/2021/6665103
[36]	T. Hassan, O. Moaaz, A. Nabih, M. Mesmouli, A. 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
[37]	O. Moaaz, E. E. Mahmoud, W. R. Alharbi, Third-order neutral delay differential equations: New iterative criteria for oscillation, J. Funct. Space., 2020 (2020), 1–8. https://doi.org/10.1155/2020/6666061 doi: 10.1155/2020/6666061
[38]	O. Moaaz, Y. Alnafisah, An improved approach to investigate the oscillatory properties of third-order neutral differential equations, Mathematics, 11 (2023), 2290. https://doi.org/10.3390/math11102290 doi: 10.3390/math11102290
[39]	M. Bohner, S. R. Grace, I. Jadlovská, Sharp results for oscillation of second-order neutral delay differential equations, Electron. J. Qual. Theo., 2023, 1–23. https://doi.org/10.14232/ejqtde.2023.1.4
[40]	Y. Alnafisah, F. Masood, A. Muhib, O. Moaaz, Improved oscillation theorems for even-order Quasi-linear neutral differential equations, Symmetry, 15 (2023), 1128. https://doi.org/10.3390/sym15051128 doi: 10.3390/sym15051128
[41]	I. T. Kiguradze, T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations, 1993. https://doi.org/10.1007/978-94-011-1808-8
[42]	R. P. Agarwal, S. R. Grace, D. O'Regan, Oscillation theory for difference and functional differential equations, Springer Dordrecht, 2000. https://doi.org/10.1007/978-94-015-9401-1
[43]	O. Moaaz, C. Cesarano, B. Almarri, An improved relationship between the solution and its corresponding function in fourth-order neutral differential equations and its applications, Mathematics, 11 (2023), 1708. https://doi.org/10.3390/math11071708 doi: 10.3390/math11071708
[44]	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. https://doi.org/10.1007/bf01223686 doi: 10.1007/bf01223686

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