Antiperiodic boundary value problem for a semilinear differential equation of fractional order

The present paper is concerned with an antiperiodic boundary value problem for a semilinear differential equation with Caputo fractional derivative of order $ q \in (1,2) $ considered in a separable Banach space. To prove the existence of a solution to our problem, we construct the Green's function corresponding to the problem employing the theory of fractional analysis and properties of the Mittag-Leffler function . Then, we reduce the original problem to the problem on existence of fixed points of a resolving integral operator. To prove the existence of fixed points of this operator we investigate its properties based on topological degree theory for condensing mappings and use a generalized B.N. Sadovskii-type fixed point theorem.