The problem of a lazy tester, or exponential dichotomy for impulsive differential equations revisited

For the impulsive equation in a Banach space x′(t)+A(t)x(t)=0,t≥0,x(τj)=Bjx(τj−)+αj, we study the type of stability which can be deduced if a solution is bounded for any bounded sequence {αj}. Under certain restrictions on the distance between impulses we can obtain either exponential or asymptotic stability, with a guaranteed polynomial degree (as t−r) of solution decay. A similar scheme is applied to equations with piecewise constant arguments and to integrodifferential equations.