Abstract

In this article we show sufficient conditions ensuring the existence and uniqueness of a mild solution to the equation D alpha u(t) = Au(t) + D alpha-1f (t, u(t)), 1 < alpha <= 2; t is an element of R, (*) in the same space where f belongs. Here A is a sectorial operator defined in a Banach space X; Da is the fractional derivative in the Riemann-Liouville sense and f (x); is an element of Omega(X) X is an element of X for each x 2 X, where X is an element of X is a vector-valued subspace of the space of continuous and bounded functions. The subspaces XdX_ 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 d in the papers Agarwal et al. (2010), Cuevas et al. (2010) and Cuevas and Lizama (2008). (C) 2013 Elsevier Inc. All rights reserved.