Local mild solutions and impulsive mild solutions for semilinear Cauchy problems involving lower Scorza-Dragoni multifunctions

In this note we investigate in Banach spaces the existence of mild solutions for initial problems, also in presence of impulses, governed by semilinear differential inclusions where the non-linear part is a Scorza-Dragoni multifunction. All the results are obtained via a generalization of Artstein-Prikry selection theorem that we obtain in the first part of the paper.