A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

Mahmoud Abdel-Aty, Musa E. Kavgaci, Ioannis P. Stavroulakis and Nour Zidan
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Mahmoud Abdel-Aty: Department of Mathematics, Faculty of Science, Sohag University, Sohag 82749, Egypt
Musa E. Kavgaci: Department of Mathematics, Faculty of Science, Ankara University, Tandogan Ankara 06100, Turkey
Ioannis P. Stavroulakis: Department of Mathematics, Faculty of Science, University of Ioannina, 451 10 Ioannina, Greece
Nour Zidan: Department of Mathematics, Faculty of Science, Sohag University, Sohag 82749, Egypt

Mathematics, 2020, vol. 8, issue 9, 1-12

Abstract: This paper deals with the oscillation of the first-order differential equation with several delay arguments x ′ t + ∑ i = 1 m p i t x τ i t = 0 , t ≥ t 0 , where the functions p i , τ i ∈ C t 0 , ∞ , R + , for every i = 1 , 2 , … , m , τ i t ≤ t for t ≥ t 0 and lim t → ∞ τ i t = ∞ . In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.

Keywords: oscillation; delay arguments; differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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