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 delaysy˙(t)=β(t)[y(t−δ)−y(t−τ)] with β:[t0−τ,∞)→R+, τ>δ>0τ>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.