On some classes of almost automorphic functions and applications to fractional differential equations

We study in this paper the properties of C.n/-almost automorphic and asymptotically C.n/-almost automorphic functions (a new concept) with values in a Banach space. We then give a new result related to the existence and uniqueness of an asymptotically almost automorphic solution of a semilinear fractional differential equation of the form D x.t/ D Ax.t/CF .t; x.t/; Bx.t// with t 2 R; 0 < < 1, where A generates a family of -resolvent family S .t/ and f satisfies some Lipschitz conditions.