The objective of this paper is to study the oscillation criteria for odd-order neutral differential equations with several delays. We establish new oscillation criteria by using Riccati transformation. Our new criteria are interested in complementing and extending some results in the literature. An example is considered to illustrate our results.
Citation: Clemente Cesarano, Osama Moaaz, Belgees Qaraad, Ali Muhib. Oscillatory and asymptotic properties of higher-order quasilinear neutral differential equations[J]. AIMS Mathematics, 2021, 6(10): 11124-11138. doi: 10.3934/math.2021646
The objective of this paper is to study the oscillation criteria for odd-order neutral differential equations with several delays. We establish new oscillation criteria by using Riccati transformation. Our new criteria are interested in complementing and extending some results in the literature. An example is considered to illustrate our results.
[1] | J. K. Hale, Functional differential equations, In: Analytic theory of differential equations, Lecture Notes in Mathematics, New York: Springer-Verlag, 183 (1971), 9-22. |
[2] | K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, London: Kluwer Academic Publishers, 1992. |
[3] | O. Moaaz, D. Chalishajar, O. Bazighifan, Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7 (2019), 585. doi: 10.3390/math7070585 |
[4] | J. K. Hale, Partial neutral functional differential equations, Rev. Roum. Math. Pures Appl., 39 (1994), 339-344. |
[5] | N. MacDonald, Biological delay systems: Linear stability theory, Cambridge: Cambridge University Press, 1989. |
[6] | I. Dassios, D. Baleanu, Optimal solutions for singular linear systems of Caputo fractional differential equations, Math. Methods Appl. Sci., 44 (2021), 7884-7896. doi: 10.1002/mma.5410 |
[7] | I. K. Dassios, D. I. Baleanu, Caputo and related fractional derivatives in singular systems, Appl. Math. Comput., 337 (2018), 591-606. |
[8] | 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 |
[9] | G. E. Chatzarakis, S. R. Grace, I. Jadlovská, T. X. Li, E. Tunç, Oscillation criteria for third-order Emden-Fowler differential equations with unbounded neutral coefficients, Complexity, 2019 (2019), 5691758. |
[10] | T. X. 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. doi: 10.1007/s00033-019-1130-2 |
[11] | T. X. Li, G. Viglialoro, Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime, Differ. Integr. Equations, 34 (2021), 315-336. |
[12] | O. Moaaz, H. Ramos, J. Awrejcewicz, Second-order Emden-Fowler neutral differential equations: A new precise criterion for oscillation, Appl. Math. Lett., 118 (2021), 107172. doi: 10.1016/j.aml.2021.107172 |
[13] | J. Džurina, S. R. Grace, I. Jadlovská, T. X. Li, Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term, Math. Nachr., 293 (2020), 910-922. doi: 10.1002/mana.201800196 |
[14] | G. E. Chatzarakis, O. Moaaz, T. X. Li, B. Qaraad, Some oscillation theorems for nonlinear second-order differential equations with an advanced argument, Adv. Differ. Equations, 2020 (2020), 160. doi: 10.1186/s13662-020-02626-9 |
[15] | T. X. Li, Y. V. Rogovchenko, Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations, Monatsh. Math., 184 (2017), 489-500. doi: 10.1007/s00605-017-1039-9 |
[16] | O. Moaaz, M. Anis, D. Baleanu, A. Muhib, More effective criteria for oscillation of second-order differential equations with neutral arguments, Mathematics, 8 (2020), 986. doi: 10.3390/math8060986 |
[17] | O. Moaaz, E. M. Elabbasy, B. Qaraad, An improved approach for studying oscillation of generalized Emden-Fowler neutral differential equation, J. Inequal. Appl., 2020 (2020), 69. doi: 10.1186/s13660-020-2304-3 |
[18] | S. Y. Zhang, Q. R. Wang, Oscillation of second-order nonlinear neutral dynamic equations on time scales, Appl. Math. Comput., 216 (2010), 2837-2848. |
[19] | R. P. Agarwal, S. R. Grace, D. O'Regan, The oscillation of certain higher-order functional differential equations, Math. Comput. Modell., 37 (2003), 705-728. doi: 10.1016/S0895-7177(03)00079-7 |
[20] | S. R. Grace, Oscillation theorems for nth-order differential equations with deviating arguments, J. Math. Appl. Anal., 101 (1984), 268-296. doi: 10.1016/0022-247X(84)90066-0 |
[21] | B. Karpuz, Ö. Öcalan, S. Öztürk, Comparison theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations, Glasgow Math J., 52 (2010), 107-114. doi: 10.1017/S0017089509990188 |
[22] | G. S. Ladde, V. Lakshmikantham, B. G. Zhang, Oscillation theory of differential equations with deviating arguments, New York: Marcel Dekker, 1987. |
[23] | T. X. Li, Y. V. Rogovchenko, On asymptotic behavior of solutions to higher-order sublinear Emden-Fowler delay differential equations, Appl. Math. Lett., 67 (2017), 53-59. doi: 10.1016/j.aml.2016.11.007 |
[24] | O. Moaaz, R. A. El-Nabulsi, O. Bazighifan, Oscillatory behavior of fourth-order differential equations with neutral delay, Symmetry, 12 (2020), 371. doi: 10.3390/sym12030371 |
[25] | O. Moaaz, I. Dassios, O. Bazighifan, Oscillation criteria of higher-order neutral differential equations with several deviating arguments, Mathematics, 8 (2020), 412. doi: 10.3390/math8030412 |
[26] | G. J. Xing, T. X. Li, C. H. Zhang, Oscillation of higher-order quasi-linear neutral differential equations, Adv. Differ. Equations, 2011 (2011), 45. doi: 10.1186/1687-1847-2011-45 |
[27] | M. K. Yıldız, Ö. Öcalan, Oscillation results of higher-order nonlinear neutral delay differential equations, Selcuk J. Appl. Math., 11 (2010), 55-62. |
[28] | B. G. Zhang, G. S. Ladde, Oscillation of even order delay differential equations, J. Math. Appl. Anal., 127 (1987), 140-150. doi: 10.1016/0022-247X(87)90146-6 |
[29] | B. Baculíková, J. Džurina, Oscillation of third-order nonlinear differential equations, Appl. Math. Lett., 24 (2011), 466-470. doi: 10.1016/j.aml.2010.10.043 |
[30] | T. X. 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. doi: 10.1016/j.aml.2020.106293 |
[31] | T. X. Li, E. Thandapani, Oscillation of solutions to odd-order nonlinear neutral functional differential equations, Electron. J. Differ. Equations, 23 (2011), 1-12. doi: 10.1007/s10884-010-9200-3 |
[32] | O. Moaaz, D. Baleanu, A. Muhib, New aspects for non-existence of kneser solutions of neutral differential equations with odd-order, Mathematics, 8 (2020), 494. doi: 10.3390/math8040494 |
[33] | O. Moaaz, E. M. Elabbasy, E. Shaaban, Oscillation criteria for a class of third order damped differential equations, Arab J. Math. Sci., 24 (2018), 16-30. |
[34] | E. Thandapani, T. X. Li, On the oscillation of third-order quasi-linear neutral functional differential equations, Arch. Math., 47 (2011), 181-199. |
[35] | R. P. Agarwal, S. R. Grace, D. Regan, Oscillation theory for difference and functional differential equations, Kluwer Academic Publishers, 2000. |