Multivalued Linear Operators and Differential Inclusions in Banach Spaces

In this paper, we study multivalued linear operators (MLO’s) and their resolvents in non reflexive Banach spaces, introducing a new condition of a minimal growth at infinity, more general than the HilleYosida condition. Then we describe the generalized semigroups induced by MLO’s. We present a criterion for an MLO to be a generator The work of A. Baskakov and V. Obukhovskii is partially supported by US CRDF-RF Ministry of Education Award VZ-010-0. The work of the second author is supported also by GNAMPA and RFBR Grants 02-01-00189 and 01-01-00425. The work of P. Zecca was partially supported by GNAMPA and MIUR. 54 A. Baskakov, V. Obukhovskii and P. Zecca of a generalized semigroup in an arbitrary Banach space. Finally, we obtain some existence results for differential inclusions with MLO’s and various types of multivalued nonlinearities. As a consequence, we give theorems on the existence of local, global and bounded solutions of the Cauchy problem for degenerate differential inclusions.