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
Irena Jadlovská
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.
- 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.
- 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.
- 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.
- R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Springer Science & Business Media, 2013.
- R.P. Agarwal, C. Zhang, T. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Comput. 274 (2016), 178-181.
- O. Došlý, P. Rehák, Half-linear Differential Equations, vol. 202, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2005.
- J. Džurina, I. Jadlovská, A note on oscillation of second-order delay differential equations, Appl. Math. Lett. 69 (2017), 126-132.
- J. Džurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 2-3, 445-453.
- 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.
- 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.
- I. Győri, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991.
- T.S. Hassan, Oscillation criteria for second-order nonlinear dynamic equations, Adv. Difference Equ. 171 (2012) 13, 2012.
- R. Mařík, Remarks on the paper by Sun and Meng, Appl. Math. Comput. 174 (2006), Appl. Math. Comput. 248 (2014), 309-313.
- S.H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 2, 375-387.
- 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.
- 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
- Irena Jadlovská
- 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.
- Received: 2019-01-24.
- Revised: 2019-03-01.
- Accepted: 2019-03-01.
- Published online: 2019-05-23.
