Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case

A scalar linear differential equation with time-dependent delay ẋ(t)=−a(t)x(t−τ(t)) is considered, where t∈I≔[t0,∞)t∈I≔[t0,∞), t0∈Rt0∈R, a:I→R+≔(0,∞)a:I→R+≔(0,∞) is a continuous function and τ:I→R+τ:I→R+ is a continuous function such that t−τ(t)>t0−τ(t0)t−τ(t)>t0−τ(t0) if t>t0t>t0. The goal of our investigation is to give sufficient conditions for the existence of positive solutions as t→∞t→∞ in the critical case in terms of inequalities on aa and ττ. A generalization of one known final (in a certain sense) result is given for the case of ττ being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay since it does not give the best possible result. This is demonstrated on a suitable example.