Article
Ufa Mathematical Journal
Volume 12, Number 3, pp. 69-80
On antiperiodic boundary value problem for a semilinear differential inclusion of fractional order with a deviating argument in a Banach space
Petrosyan G.G.
DOI:10.13108/2020-12-3-69
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We consider a boundary value problem for a semi-linear differential inclusion of fractional Caputo 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 history of the function before the 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 resolve 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 issue on existence of fixed points of a corresponding resolving multi-valued integral operator in the space of continuous functions. To prove the existence of fixed points of the resolving multi-operator we employ a generalized theorem of Sadovsky 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.