Regularity of mild solutions for a class of fractional order differential equations

In this article we show sufficient conditions ensuring the existence and uniqueness of a mild solution to the equation equation(∗)Dαu(t)=Au(t)+Dα-1f(t,u(t)),1<α⩽2,t∈R,in the same space where f belongs. Here A   is a sectorial operator defined in a Banach space X,Dα is the fractional derivative in the Riemann–Liouville sense and f(·,x)∈Ω(X)f(·,x)∈Ω(X) for each x∈Xx∈X, where Ω(X)Ω(X) is a vector-valued subspace of the space of continuous and bounded functions. The subspaces Ω(X)Ω(X) that we will consider in this article are the space of periodic, almost periodic, almost automorphic and compact almost automorphic vector-valued functions, among others. In particular, we extend and unify recent results established for the equation (∗)(∗) in the papers Agarwal et al. (2010), Cuevas et al. (2010) and Cuevas and Lizama (2008).