On semilinear differential inclusions in Banach spaces with nondensely defined operators

Abstract

We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille–Yosida operator whereas Caratheodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.