Almost automorphic mild solutions to fractional differential equations

We introduce the concept of alpha-resolvent families to prove the existence of almost automorphic mild solutions to the differential equation D(1)(alpha)u(t) = Au(t) + t(n)f(t), 1 <= alpha <= 2, n is an element of Z(+) considered in a Banach space X, where f : R -> X is almost automorphic. We also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation D(1)(alpha)u(t) = Au(t) + f(t, u(t)), 1 <= alpha <= 2 assuming f(t, x) is almost automorphic in t for each x is an element of X, satisfies a global Lipschitz condition and takes values on X. Finally, we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation D(1)(alpha)u(t) = Au(t) + f(t, u(t), u'(t)), 1 <= alpha <= 2, under analogous conditions as in the previous case. (C) 2007 Elsevier Ltd. All rights reserved.