Stepanov-like almost automorphic functions and monotone evolution equations

In this paper we are concerned with a (new) class of (Stepanov-like) almost automorphic (SpSp-a.a.) functions with values in a Banach space XX. This class contains the space AA(X) of all (Bochner) almost automorphic functions. We use the results obtained to prove the existence and uniqueness of a weak SpSp-a.a. solution to the parabolic equation u′(t)+A(t)u=f(t)u′(t)+A(t)u=f(t) in a reflexive Banach space, assuming some appropriate conditions of monotonicity, coercitivity of the operators A(t)A(t) and Sp′Sp′-almost automorphy of the forced term f(t)f(t). This result extends a known result in the case of almost periodicity. An application is also given.