Optimal Oscillation Conditions for a Delay Differential Equation

Authors

I. P. Stavroulakis, Department of Mathematics, University of Ioannina, 45110 Ioannina, GreeceFollow
Zh. Kh. Zhunussova, Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, KazakhstanFollow
S. Sh. Ixanov, Faculty of Information Technology, Al-Farabi Kazakh National University, Almaty, 050040, KazakhstanFollow
Belal S. H. Rababh, Department of Business Administration, College of Administrative Sciences, Applied Science University, Kingdom of BahrainFollow

Author Country (or Countries)

Greece

Abstract

Consider the differential equation with a retarded argument of the form x′(t) + p(t)x(τ(t)) = 0, t ≥ t0, (1) where the functions p, τ ∈ C([t0,∞), R+), (here R+ = [0, ∞)), τ(t) ≤ t for t ≥ t0 and limt→∞ τ(t) = ∞ and the equation with a constant positive delay τ of the form x′(t) + p(t)x(t − τ) = 0, t ≥ t0, (2) Optimal conditions for the oscillation of all solutions to these equations are presented when the well-known oscillation conditions limsup t→∞ Zτt(t) p(s)ds > 1 and liminft→∞ Zτt(t) p(s)ds > 1e are not satisfied and also in the critical case where liminft→∞ p(t) = e1τ in Eq. (2). In the case that the function Rtt−τ p(s)ds is slowly varying at infinity, then under mild additional assumptions limsup t→∞ Zt−t τ p(s)ds > 1e is a sharp condition for the oscillation of all solutions to Eq. (2). Examples illustrating the results are given.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/130314

Recommended Citation

P. Stavroulakis, I.; Kh. Zhunussova, Zh.; Sh. Ixanov, S.; and S. H. Rababh, Belal (2019) "Optimal Oscillation Conditions for a Delay Differential Equation," Applied Mathematics & Information Sciences: Vol. 13: Iss. 3, Article 14.
DOI: http://dx.doi.org/10.18576/amis/130314
Available at: https://digitalcommons.aaru.edu.jo/amis/vol13/iss3/14