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.
- Delay differential equation
- S-asymptotically periodic function
- Slowly varying function
ASJC Scopus subject areas
- Applied Mathematics