### Abstract

It is well-known that for the oscillation of all solutions of the linear delay differential equation x
^{′}
(t)+p(t)x(t−τ)=0,t≥t
_{0}
, with p∈C([t
_{0}
,∞),R
^{+}
) and τ>0 it is necessary that B≔lim supt→∞A(t)≥[Formula presented],whereA(t)≔∫
_{t−τ}
^{t}
p(s)ds. Our main result shows that if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions B>[Formula presented] implies the oscillation of all solutions of the above linear delay differential equation. The applicability of the obtained results and the importance of the slowly varying assumption on A are illustrated by examples.

Original language | English |
---|---|

Pages (from-to) | 58-65 |

Number of pages | 8 |

Journal | Applied Mathematics Letters |

Volume | 93 |

DOIs | |

Publication status | Published - Jul 1 2019 |

### Fingerprint

### Keywords

- Delay differential equation
- Oscillation
- S-asymptotically periodic function
- Slowly varying function

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied Mathematics Letters*,

*93*, 58-65. https://doi.org/10.1016/j.aml.2019.01.042

**A sharp oscillation criterion for a linear delay differential equation.** / Garab, Ábel; Pituk, M.; Stavroulakis, Ioannis P.

Research output: Contribution to journal › Article

*Applied Mathematics Letters*, vol. 93, pp. 58-65. https://doi.org/10.1016/j.aml.2019.01.042

}

TY - JOUR

T1 - A sharp oscillation criterion for a linear delay differential equation

AU - Garab, Ábel

AU - Pituk, M.

AU - Stavroulakis, Ioannis P.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - It is well-known that for the oscillation of all solutions of the linear delay differential equation x ′ (t)+p(t)x(t−τ)=0,t≥t 0 , with p∈C([t 0 ,∞),R + ) and τ>0 it is necessary that B≔lim supt→∞A(t)≥[Formula presented],whereA(t)≔∫ t−τ t p(s)ds. Our main result shows that if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions B>[Formula presented] implies the oscillation of all solutions of the above linear delay differential equation. The applicability of the obtained results and the importance of the slowly varying assumption on A are illustrated by examples.

AB - It is well-known that for the oscillation of all solutions of the linear delay differential equation x ′ (t)+p(t)x(t−τ)=0,t≥t 0 , with p∈C([t 0 ,∞),R + ) and τ>0 it is necessary that B≔lim supt→∞A(t)≥[Formula presented],whereA(t)≔∫ t−τ t p(s)ds. Our main result shows that if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions B>[Formula presented] implies the oscillation of all solutions of the above linear delay differential equation. The applicability of the obtained results and the importance of the slowly varying assumption on A are illustrated by examples.

KW - Delay differential equation

KW - Oscillation

KW - S-asymptotically periodic function

KW - Slowly varying function

UR - http://www.scopus.com/inward/record.url?scp=85061574579&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061574579&partnerID=8YFLogxK

U2 - 10.1016/j.aml.2019.01.042

DO - 10.1016/j.aml.2019.01.042

M3 - Article

VL - 93

SP - 58

EP - 65

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

ER -