Continuity in functional differential equations with infinite delay

In this paper, we investigate the continuous dependence of solutions of the functional differential equation with infinite delayx(t)=f(t,x_t) on initial functions. Endowing the phase space ag-norm as well as a supremum norm, we show that if the equation satisfies a mild fading memory dondition, then the continuity off in respect to the topology induced by the supremum norm can yield the continuity of solutions of the equation in respect to the topology induced by theg-norm which is stronger than the ahead one.This research was supported in part by an NSF grant with number NSF-DMS-8521408.On leave from South China Normal University, Guangzhou, PRC. This research was supported in part by the National Science Foundation of PRC.