Asymptotical Convergence of the Solutions of a Linear Differential Equation with Delays

The asymptotic behavior of the solutions of the first-order differential equation y (t)=∑i=1nβi(t)[y(t-δi)-y(t-τi)] containing delays is studied with βi:[t0-τ,∞)→[0,∞), τ=max {τ1,…,τn}, ∑i=1nβi(t)>0, τi>δi>0. The attention is focused on an analysis of the asymptotical convergence of solutions. A criterion for the asymptotical convergence of all solutions, characterized by the existence of a strictly increasing bounded solution, is proved. Relationships with the previous results are discussed, too.