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On the existence of C (n) -almost automorphic mild solutions of certain differential equations in Banach spaces

In this paper we study the existence and uniqueness of almost automorphic mild solutions to the semilinear equation u (t) = A(t)u(t) + f (t, u(t)), t ∈ R where A(t) satisfies the Acquistapace-Terreni conditions and generates an exponentially stable family of operators (U (t, s) t≥s and f (t, u) is almost automorphic in t ∈ R for any u ∈ X. This extends a well-known result by G.M. N'Guerekata in the case where A(t) = A does not depend on t. We also investigate the existence of mild C n-almost automorphic solutions of the linear version of the above equation when A is an operator of simplest type.