Asymptotic Stability for one Dimensional Differential-Delay Equations*

G? = (4 E c* : II d II < PI. I f ~(a) is defined and continuous on [t q, t], we will write xt for the function for which x~(s) = x(t + S) for s E [-q, 01. Hence xt E C, . This paper shows that for a nonlinear one-dimensional differential delay equation a(t) = F;(t, x1(.)) (DDE) one can frequently determine (almost by inspection) if the 0 solution is asymptotically stable and give a region of attraction. Theorem 1.1 gives a simple, readily applicable criterion for asymptotic stability. No use is made of complicated criteria such as the existence of a Liapunov function. For 4 E Cq, define Y44 = sup{% $l& (b(s)>*