On antiperiodic boundary value problem for semilinear fractional differential inclusion with deviating argument in Banach space

Abstract

We consider a boundary value problem for a semi-linear differential inclusion of Caputo fractional derivative and a deviating coefficient in a Banach space. We assume that the linear part of the inclusion generates a bounded $C_0$-semigroup. A nonlinear part of the inclusion is a multi-valued mapping depending on the time and the prehistory of the function before a current time. The boundary condition is functional and anti-periodic in the sense that one function is equals to another with an opposite sign. To solve the considered problem, we employ the theory of fractional mathematical analysis, the properties of Mittag-Leffler as well as the theory of topological power for multi-valued condensing maps. The idea is as follws: the original problem is reduced to the existence of fixed points of a corresponding resolving multi-valued integral operator in the space of continuous functions. To prove the existence of the fixed points of the resolving multi-operator we employ a generalized theorem of Sadovskii type on a fixed point. This is why we show that the resolving integral multi-operator is condensing with respect to a vector measure of non-compactness in the space of continuous functions and maps a closed ball in this space into itself.