Mild solutions for impulsive semilinear evolution differential inclusions

In this paper we prove a sufficient condition for the existence of mild solutions for an impulsive Cauchy problem monitored by the semilinear evolution differential inclusion $x^\prime(t)\in A(t)x(t)+F(t,x(t))$ where $\{A(t)\}_{t\in [0,b]}$ is a family of linear operators in a Banach space $E$ generating an evolution operator and $F$ is a Carathèodory type multifunction. Since we do not assume any hypothesis on the impulse functions, our existence theorem extends in a broad sense a proposition obtained by Benchohra, Henderson and Ntouyas.