On the averaging principle for semilinear fractional differential inclusions in a Banach space with a deviating argument and a small parameter

The this paper, we considers the Cauchy problem for a class of semilinear differential inclusions in a separable Banach space involving a fractional Caputo derivative of order $q\in(0,1)$, a small parameter, and a deviant argument. We assume that the linear part of the inclusion generates a $C_0$-semigroup. In the space of continuous functions, we construct a multivalued integral operator whose fixed points are solutions. An analysis of the dependence of this operator on a parameter allows one to establish an analog of the averaging principle. We apply methods of the theory of fractional analysis and the theory of topological degree for condensing set-valued mappings.