Topological Structure Of The Solution Sets For A Class Of Fractional Evolution Inclusions

The paper is concerned with the factional evolution inclusion in Banach space X, where, c Dtq,0<q<1, is the regularized Caputo fractional derivative of orderq, A is the infinitesimal generator of a Co semigroup{T(t)}t≥0 on X, and F is a multi-valued function with convex, closed values. We study the qualitative properties of solutions of such inclusion as follows.Firstly, A generates a compact semigroup. Constructing a suitable directionally Lp-integrable selection from F, we study the compactness and Rδ-structure of the set of trajectories on a closed domain. Moreover, we discuss the Rδ-structure of the set of trajectories to the control problem corresponding to the inclusion above. We apply our abstract theory to a boundary value problem of fractional diffusion inclusion.Secondly, it is shown that the solution set is a compact Rδ-set in the space of continuous functions and the corresponding solution operator is an Rδ-map if, in particular, X is reflexive, F is weakly upper semicontinuous with respect to the second variable and the semigroup generated by A is noncompact. An example is also given to illustrate our abstract results.