Multi-valued nonlinear perturbations of time fractional evolution equations in Banach spaces

Abstract

The paper is concerned with the fractional evolution inclusion \(^\mathrm{c}D_t^q u(t)\in Au(t)+F(t,u(t))\) in Banach spaces, where \(^\mathrm{c}D_t^q\), \(0<q<1\), is the regularized Caputo fractional derivative of order q, A generates a compact semigroup, and \(F\) is a multi-valued function with convex, closed values. Constructing a suitable directionally \(L^p\)-integrable selection from \(F\), we study the compactness and \(R_\delta \)-structure of the set of trajectories on a closed domain. Moreover, we discuss the \(R_\delta \)-structure of the set of trajectories to the control problem corresponding to the inclusion above. Finally, we apply our abstract theory to boundary value problems of fractional diffusion inclusions.