Existence of solutions on compact and non-compact intervals for semilinear impulsive differential inclusions with delay

Abstract

In this paper we deal with
impulsive Cauchy problems in Banach spaces governed by a delay
semilinear differential inclusion $y'\in A(t)y$ $ + F(t,y_t)$. The
family $\{A(t)\}_{t\in [0,b]}$ of linear operators is supposed to
generate an evolution operator and $F$ is a upper Carath\`eodory
type multifunction. We first provide the existence of mild
solutions on a compact interval and the compactness of the
solution set. Then we apply this result to obtain the existence of
mild solutions for the impulsive Cauchy problem on non-compact
intervals.