In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an (Formula presented.) -set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented. © 2017 Informa UK Limited, trading as Taylor & Francis Group.