Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains

Abstract

This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x ′ ( t ) ∈ A ( t ) x ( t ) + F ( t , x ( t ) ) , where { A ( t ) } t ∈ [ 0 , b ] is a family of linear operators (not necessarily bounded) in a Banach space E generating an evolution operator and F is a Carathéodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains.