On the existence of mild solutions of semilinear evolution differential inclusions

In this paper we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion: x′(t)∈A(t)x(t)+F(t,x(t)) and with initial data x(0)=x0∈E, where {A(t)}t∈[0,d] is a family of linear operators in the Banach space E generating an evolution operator and F is a Carathèodory type multifunction. We prove the existence of local and global mild solutions of the problem. Moreover, we obtain the compactness of the set of all global mild solutions. In order to obtain these results, we define a generalized Cauchy operator. Our existence theorems respectively contain the analogous results provided by Kamenskii, Obukhovskii and Zecca [Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 7, de Gruyter, Berlin, 2001] for inclusions with constant operator.