Oscillation theorems for second-order nonlinear delay differential equations of neutral type

Year 2019, Volume: 48 Issue: 3, 633 - 643, 15.06.2019

Başak Karpuz Shyam S. Santra

Abstract

In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form
\[\frac{d}{dt}\bigg[r(t)\frac{d}{dt}[x(t)+p(t)x(\tau(t))]\bigg]+q(t)G\bigl(x(\sigma(t))\bigr)=0\: for\: t\geq t_{0},\]
under the assumptions $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ and $\int^{\infty}\frac{1}{r(\eta)}d\eta<\infty$ for various ranges of the bounded neutral coefficient $p$. Unlike most of the previous results, $\tau^{\prime}$ is allowed to be oscillatory. Further, some illustrative examples showing applicability of the new results are included.

Keywords

Oscillation, nonoscillation, nonlinear, delay argument, second-order neutral differential equation

References

• B. Baculíková and J. Džurina,Oscillation theorems for second order neutral differential equations, Comput. Math. Appl. 61 (1), 94-99, 2011.
• B. Baculíková and J. Džurina, Oscillation theorems for second-order nonlinear neutral differential equations, Comput. Math. Appl. 62 (12), 4472-4478, 2011.
• B. Baculíková, T.X. Li and J. Džurina, Oscillation theorems for second order neutral differential equations, Electron. J. Qual. Theory Differ. Equ. 2011 (74), 1-13, 2011.
• B. Baculíková, T.X. Li and J. Džurina, Oscillation theorems for second-order super- linear neutral differential equations, Math. Slovaca, 63 (1), 123-134, 2013.
• J. Džurina, Oscillation theorems for second order advanced neutral differential equa- tions, Tatra Mt. Math. Publ. 48, 61-71, 2011.
• I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Ap- plications, Oxford, Clarendon Press, 1991.
• J. Hale, Theory of Functional Differential Equations, New York, Springer-Verlag, 1977.
• M. Hasanbulli and Y.V. Rogovchenko, Oscillation criteria for second order nonlinear neutral differential equations, Appl. Math. Comput. 215 (12), 4392-4399, 2010.
• T.H. Hildebrandt, Introduction to the Theory of Integration, New York, Academic Press, 1963.
• B. Karpuz, Ö. Öcalan and S. Öztürk, Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations, Glasg. Math. J. 52 (1), 107-114, 2010.
• B. Karpuz, L.N. Padhy and R.N. Rath, Oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients, Elec- tron. J. Differ. Equ. 113, 15 pp., 2008.
• T.X. Li and Y.V. Rogovchenko, Oscillation theorems for second-order nonlinear neu- tral delay differential equations Abstr. Appl. Anal. Article ID: 594190, 1-5, 2014.
• T.X. Li, Y.V. Rogovchenko and C.H. Zhang, Oscillation results for second-order non- linear neutral differential equations, Adv. Difference Equ. 336, 13 pp., 2013.
• Y.J. Liu, J.W. Zhang and J. Yan, Existence of oscillatory solutions of second order delay differential equations, J. Comput. Appl. Math. 277, 17-22, 2015.
• S.S. Santra, Existence of positive solution and new oscillation criteria for nonlinear first-order neutral delay differential equations, Differ. Equ. Appl. 8 (1), 33-51, 2016.
• S. Sun, T.X. Li, Z.L. Han and C. Zhang, On oscillation of second-order nonlinear neutral functional differential equations, Bull. Malays. Math. Sci. Soc. (2), 36 (3), 541-554, 2013.
• J. Yan, Existence of oscillatory solutions of forced second order delay differential equations, Appl. Math. Lett. 24 (8), 1455-1460, 2011.