Local mild solutions and impulsive mild solutions for semilinear Cauchy problems involving lower Scorza-Dragoni multifunctions
The authors consider the semilinear differential inclusion \[ x'\in A(t)x+F(t,x),\qquad x(0)=x_0\tag{1} \] in a Banach space, where {A(t)} is a family of linear operators generating an evolution system, and F is a set-valued function satisfying the lower Scorza-Dragoni property. The authors first prove a Carathéodory selection theorem, which is an extension of the one in [\textit{Z. Artstein} and \textit{K. Prikry}, J. Math. Anal. Appl. 127, 540--547 (1987; Zbl 0649.28011)]. Applying this result and known existence theorems, the authors prove several theorems on the existence of local and global mild solutions to problem (1), and also existence of mild solutions to an impulsive version of (1). A closely related paper is [\textit{G. Bezzini} and \textit{T. Cardinali}, ``Local mild solutions for semilinear evolution differential inclusions with lower semicontinuous type multifunctions'', P. U. M. A. 18, 1--17 (2007)].
scientific article; zbMATH DE number 5567741