On Systems of Semilinear Fractional Differential Inclusions of Fractional Order with Nondensely Defined Operators in Banach Spaces

In this paper, systems of semilinear differential inclusions of fractional orders are studied. It is assumed that the linear parts of inclusions are represented by Hille–Yosida operators in Banach spaces. The nonlinear parts of inclusions are multivalued Carathéodory type maps, depending on time and a finite set of functions. To study the problem of the existence of solutions to such a system, the theory of fractional mathematical analysis, the theory of generalized metric spaces, and also the theory of topological degree for multivalued condensing maps are used. We present a multivalued resolving operator for this system and describe its properties. It is shown, in particular, that this multioperator is condensing with respect to a special vector measure of noncompactness. This makes it possible, using some fixed point theorems for the specified multioperators, to prove local and global existence theorems for integral solutions to a given system. In the latter case, the compactness of the set of such solutions and the upper semicontinuous dependence of the set of solutions on the initial data are also justified.