To the Theory of Differential Inclusions with Caputo Fractional Derivatives

Abstract

We consider the Cauchy problem for a differential inclusion with a Caputo fractional derivative of order $$ \alpha \in (0, 1)$$ . It is assumed that the set-valued mapping defining the right-hand side of the inclusion has nonempty, convex, and compact values, is upper semicontinuous, and satisfies the sublinear growth condition. We prove that the solution set of the Cauchy problem is nonempty and compact and establish the nature of the dependence of the solution set on the initial data, the right-hand side of the inclusion, and the order of differentiation. The semigroup property of the solution set is established.