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

Author

Listed:
  • 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
    Department of Mathematics, College of Science, Jouf University, Sakaka 42421, Saudi Arabia)

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.

Suggested Citation

  • Mahmoud Abdel-Aty & Musa E. Kavgaci & Ioannis P. Stavroulakis & Nour Zidan, 2020. "A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays," Mathematics, MDPI, vol. 8(9), pages 1-12, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1492-:d:408367
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    References listed on IDEAS

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    1. G. M. Moremedi & I. P. Stavroulakis, 2018. "Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument," Discrete Dynamics in Nature and Society, Hindawi, vol. 2018, pages 1-13, June.
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