On the existence of a solution of a periodic boundary value problem for semilinear differential inclusions of fractional order from the interval (3,4) in Banach spaces

In this paper we study a periodic boundary value problem for a class of semilinear differential inclusions of fractional order from the interval (3,4) in a Banach space for which the multivalued nonlinearity satisfies the regularity condition expressed in terms of measures of noncompactness. We prove the existence of a solution to the problem, we first construct the corresponding Green's function. Then we introduce into consideration a multivalued resolving operator in the space of continuous functions and reduce the problem posed to the problem of the existence of fixed points of a resolving multioperator. We prove the existence of fixed points, by using a generalized B.N. Sadovskii type theorem.