Abstract

We introduce the concept of α-resolvent families to prove the existence of almost automorphic mild solutions to the differential equation Dt α u (t) = A u (t) + tn f (t), 1 ≤ α ≤ 2, n ∈ 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 Dt α u (t) = A u (t) + f (t, u (t)), 1 ≤ α ≤ 2 assuming f (t, x) is almost automorphic in t for each x ∈ 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 Dt α u (t) = A u (t) + f (t, u (t), u′ (t)), 1 ≤ α ≤ 2, under analogous conditions as in the previous case. © 2007 Elsevier Ltd. All rights reserved.