Oscillatory behavior of $n$-th order nonlinear delay differential equations with a nonpositive neutral term

Yıl 2020, Cilt: 49 Sayı: 2, 766 - 776, 02.04.2020

Said R. Grace , İrena Jadlovska , Zafer Ağacık

https://doi.org/10.15672/hujms.471023

Cited By: 1

Öz

We study the oscillation problem for solutions of a class of $n$-th order nonlinear delay differential equations with nonpositive neutral terms. The obtained results improve and correlate many of the known oscillation criteria in the literature for neutral and non-neutral equations.

Anahtar Kelimeler

Oscillation, neutral term, second order differential equation

Kaynakça

• [1] R.P. Agarwal, S.R. Grace, and D. O’Regan, Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Springer Science & Business Media, 2002.
• [2] R.P. Agarwal, S.R. Grace, and P.J.Y. Wong, Oscillation theorems for certain higher order nonlinear functional differential equations, Appl. Anal. Discr. Math. 2, 1–30, 2008.
• [3] R.P. Agarwal, M. Bohner, T. Li, and C. Zhang, A new approach in the study of oscillatory behavior of even-order neutral delay differential equations, Appl. Math. Comput. 225, 787–794, 2013.
• [4] R.P. Agarwal, M. Bohner, T. Li, and C. Zhang, Oscillation of second order differential equations with a sublinear neutral term, Carpathian J. Math. 30 (1), 1–6, 2014.
• [5] J.G. Dong, Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments, Comput. Math. Appl. 59, 3710 – 3717, 2010.
• [6] S.R. Grace and B.S. Lalli, Oscillation of nonlinear second order neutral differential equations, Rat. Math. 3, 77 – 84, 1987.
• [7] S.R. Grace, J.R. Graef, and M.A. El-Beltagy, On the oscillation of third order neutral delay dynamic equations on time scales, Comput. Math. Appl. 63 (4), 775–782, 2012.
• [8] S.R. Grace and I. Jadlovská, Oscillation Criteria for second-order neutral damped differential equations with delay argument, in: Dynamical Systems - Analytical and Computational Techniques, InTech, 2017.
• [9] S.R. Grace,Oscillatory behavior of second-order nonlinear differential equations with a nonpositive neutral term, Mediterr. J. Math. 14 (6), Art. 229, 2017.
• [10] G.H. Hardy, I.E. Littlewood, and G. Polya, Inequalities, University Press, Cambridge, 1959.
• [11] B. Karpuz, O. Ocalan, and S. Ozturk, Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations, Glasgow Math. J. 52 (1), 107–114, 2010.
• [12] I.T. Kiguradze, On the oscillation of solutions of the Eq. $d^mu/dt^m+a(t)|u|^n {\rm sgn}u = 0$, Mat. Sb. 65, 172–187, 1964 (in Russian).
• [13] T. Li, Z. Han, C. Zhang, and H. Li, Oscillation criteria for second-order superlinear neutral differential equations, Abstr. Appl. Anal. 2011, 2011.
• [14] T. Li, Yu.V. Rogovchenko, and C. Zhang, Oscillation results for second-order nonlinear neutral differential equations, Adv. Differ. Equ. 2013, 1 – 13, 2013.
• [15] Q. Li, R. Wang, F. Chen, and T. Li , Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients, Adv. Differ. Equ. 2015, 1–15, 2015.
• [16] Ch. G. Philos, A new criterion for the oscillatory and asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Mat. 39 (1), 61–64, 1981.
• [17] H. Qin, N. Shang, and Y. Lu, A note on oscillation criteria of second order nonlinear neutral delay differential equations, Comput. Math. Appl. 56, 2987–299, 2008.
• [18] V. Staikos and I. Stavroulakis, Bounded oscillations under the effect of retardations for differential equations of arbitrary order, P. Roy. Soc. Edinb. 77 (1), 129–136, 1977.
• [19] H. Wu, L. Erbe, and A. Peterson, Oscillation of solution to second-order half-linear delay dynamic equations on time scales, Electron. J. Differ. Eq. 2016 (71), 1–15, 2016.
• [20] J.S.W. Wong, Necessary and sufficient conditions for oscillation of second order neutral differential equations, J. Math. Anal. Appl. 252, 342–352, 2000.
• [21] Q. Yang, l. Yang, and S. Zhu, Interval criteria for oscillation of second-order nonlinear neutral differential equations, Comput. Math. Appl. 46 (5), 903–918, 2003.