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

Abstract

A scalar linear differential equation with time-dependent delay x ̇ ( t ) = − a ( t ) x ( t − τ ( t ) ) is considered, where t ∈ I ≔ [ t 0 , ∞ ) , t 0 ∈ R , a : I → R + ≔ ( 0 , ∞ ) is a continuous function and τ : I → R + is a continuous function such that t − τ ( t ) > t 0 − τ ( t 0 ) if t > t 0 . The goal of our investigation is to give sufficient conditions for the existence of positive solutions as t → ∞ in the critical case in terms of inequalities on a and τ . A generalization of one known final (in a certain sense) result is given for the case of τ 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.