Opuscula Math. 39, no. 4 (2019), 483-495
https://doi.org/10.7494/OpMath.2019.39.4.483

Opuscula Mathematica

# Oscillatory results for second-order noncanonical delay differential equations

Jozef Džurina
Ioannis P. Stavroulakis

Abstract. The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation $\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,$ under the condition $\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.$ Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given.

Keywords: linear differential equation, delay, second-order, noncanonical, oscillation.

Mathematics Subject Classification: 34C10, 34K11.

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1. R.P. Agarwal, M. Bohner, W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, vol. 267, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2004.
2. R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic Publishers, Dordrecht, 2002.
3. R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, vol. 5, Series in Mathematical Analysis and Applications, Taylor & Francis, Ltd., London, 2003.
4. R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Springer Science & Business Media, 2013.
5. R.P. Agarwal, C. Zhang, T. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Comput. 274 (2016), 178-181.
6. O. Došlý, P. Rehák, Half-linear Differential Equations, vol. 202, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2005.
7. J. Džurina, I. Jadlovská, A note on oscillation of second-order delay differential equations, Appl. Math. Lett. 69 (2017), 126-132.
8. J. Džurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 2-3, 445-453.
9. L. Erbe, T.S. Hassan, A. Peterson, Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comput. 203 (2008) 1, 343-357.
10. L. Erbe, A. Peterson, S.H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl. 333 (2007) 1, 505-522.
11. I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991.
12. T.S. Hassan, Oscillation criteria for second-order nonlinear dynamic equations, Adv. Difference Equ. 171 (2012) 13, 2012.
13. R. Mařík, Remarks on the paper by Sun and Meng, Appl. Math. Comput. 174 (2006), Appl. Math. Comput. 248 (2014), 309-313.
14. S.H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 2, 375-387.
15. Y.G. Sun, F.W. Meng, Note on the paper of J. Džurina and I. P. Stavroulakis: "Oscillation criteria for second-order delay differential equations", Appl. Math. Comput. 174 (2006) 2, 1634-1641.
16. L. Ye, Z. Xu, Oscillation criteria for second order quasilinear neutral delay differential equations, Appl. Math. Comput. 207 (2009) 2, 388-396.
• Jozef Džurina
• https://orcid.org/0000-0002-6872-5695
• Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Mathematics and Theoretical Informatics, B. Němcovej 32, 042 00 Košice, Slovakia
• https://orcid.org/0000-0003-4649-5611
• Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Mathematics and Theoretical Informatics, B. Němcovej 32, 042 00 Košice, Slovakia
• Ioannis P. Stavroulakis
• https://orcid.org/0000-0002-4810-0540
• Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
• Al-Farabi Kazakh National University, Faculty of Mathematics and Mechanics, Almaty, 050040 Kazakhstan
• Communicated by Josef Diblík.
• Revised: 2019-03-01.
• Accepted: 2019-03-01.
• Published online: 2019-05-23.