The authors consider the existence of almost automorphic mild solutions to the fractional differential equation
where is a linear sectorial operator densely defined in a Banach space of type , is the fractional derivative in the Riemann-Liouville sense with , and is an almost automorphic function in satisfying some Lipschitz type conditions.
These results go back to the reviewer’s paper [“Existence and uniqueness of almost automorphic mild solutions of some semilinear abstract differential equations”, Semigroup Forum 69, No. 1, 80–86 (2004; Zbl 1077.47058)]. The technique used is the Banach fixed point theorem. An example is given to illustrate the abstract results.