Some new results for multi-valued fractional evolution equations

Given a closed linear operator A in a Banach space X and a multi-valued function with convex, closed values F:[0,b]×C([-τ,0];X)→2X, we consider the Cauchy problemcDtαu(t)∈Au(t)+F(t,ut),t∈[0,b],u(t)=φ(t),t∈[-τ,0],where τ⩾0,cDtα,0<α<1, represents the regularized Caputo fractional derivative of order α. We concentrate on the case when the semigroup generated by A is noncompact and obtain nonemptyness of the solution set if, in particular, X is reflexive and F is weakly upper semicontinuous with respect to the second variable. Furthermore, in this situation topological properties of the set of all solutions are considered.