Convergence of the solutions of the equation ˙ y(t) = β(t)[y(t −δ)−y(t −τ)] in the critical case

We study the asymptotic behavior of the solutions of the first order differential equation containing two delays ˙ y(t) = β(t) y(t −δ) − y(t − τ) with β : [t0 − τ,∞)→R +, τ >δ>0. The convergence of all solutions is characterized by the existence of a strictly increasing bounded solution. A critical case is found for the coefficient function β. For coefficients below the critical function a strictly increasing and bounded solution is constructed, and thus the convergence of all solutions is shown. Relations with known results are discussed, too. © 2006 Elsevier Inc. All rights reserved.