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 ( τ j ) = B j x ( τ j − ) + α j , we study the type of stability which can be deduced if a solution is bounded for any bounded sequence { α j } . 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.