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 an upper Carathéodory 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.