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 over dot (t)=beta(t)[y(t-delta)-y(t-tau)] with beta: [t(0) - tau, infinity) -> R+, tau > delta > 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 beta. 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. (c) 2006 Elsevier Inc. All rights reserved.