Stability of noninstantaneous impulsive evolution equations

In this paper, we introduce a notation of noninstantaneous impulsive evolution operator, an extension of the classical impulsive evolution operator for linear evolution equations with fixed impulses, which help us to give the concepts of mild solutions to noninstantaneous impulsive Cauchy problems. By characterizing the estimation of noninstantaneous impulsive evolution operator, we establish sufficient conditions to guarantee asymptotic stability of linear and semilinear problems.