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

We study in this paper the properties of C(n)C(n)-almost automorphic and asymptoticallyC(n)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 Dαx(t)=Ax(t)+F(t,x(t),Bx(t))Dαx(t)=Ax(t)+F(t,x(t),Bx(t)) with t∈R,0<α<1t∈R,0<α<1, where AA generates a family of αα-resolvent family Sα(t)Sα(t) and ff satisfies some Lipschitz conditions.