Global stability of nonautonomous logistic equations with a piecewise constant delay

Consider the following nonautonomous differential equation with a piecewise constant delay: {dN(t)dt=N(t){r(t)−a(t)N(t)−b(t)N([t])},n≤t0, where r(t)r(t) is a nonnegative continuous function on [0,+∞),r(t)≢0, a(t)≥0a(t)≥0 and b(t)>0b(t)>0.In this paper, using a recent result of a full affirmative answer to the Gopalsamy and Liu’s conjecture for the autonomous case of the above equation with a(t)≡a≥0a(t)≡a≥0 and b(t)≡b>0b(t)≡b>0, we derive three types of sufficient conditions for the global asymptotic stability of solution for the above nonautonomous logistic equation with a piecewise constant delay.