Functional differential inclusions of Hale type with fractional order of derivative in a Banach space

Abstract

Over the past decades, the theory of functional differential inclusions, primarily, the delayed
functional differential inclusion, has received significant development. Scientists from different
countries conduct research in the theory of initial-boundary value problems for various classes of
differential, integro-differential and functional differential inclusions in partial derivatives with
integer and fractional orders of derivatives.
The present work is devoted to fractional functional-differential and integro-differential
inclusions of Hale type, which occupy an intermediate place between functional-differential
inclusions with delay and inclusions of a neutral type. Sufficient conditions for the existence of
weak solutions of inclusions of Hale type with fractional order of the derivative are established.
The methods of fractional integro-differential calculus and the theory of fixed points of
multivalued mappings are the basis of this study. It is known that the dynamics of economic,
social, and ecological macrosystems is a multi-valued dynamic process, and fractional differential
and integro-differential inclusions are natural models of macrosystem dynamics. Such inclusions
are also used to describe some physical and mechanical systems with hysteresis. At the end of
the paper, an example illustrates abstract results.