On the Stepanov-like Almost Automorphic Solutions of Abstract Differential Equations

In this paper we prove that any S-p-almost automorphic solution (1 <= p < infinity) of the first-order differential equation u'(t) = Au(t) + f(t, u(t)), t is an element of R, is almost automorphic in a Banach space X, where A : X -> X is a bounded linear operator, and f : R x X -> X is S-p-almost automorphic continuous function. We also prove the same result for the second-order differential equation u ''(t) = Au(t) + f(t, u(t)), t is an element of R.