Stepanov-like almost automorphic functions and monotone evolution equations

In this paper we are concerned with a (new) class of (Stepanov-like) almost automorphic (S-p-a.a.) functions with values in a Banach space X. 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 S-p-a.a. solution to the parabolic equation u'(t) + A(t)u = f (t) in a reflexive Banach space, assuming some appropriate conditions of monotonicity, coercitivity of the operators A(t) and S-p'-almost automorphy of the forced term f (t). This result extends a known result in the case of almost periodicity. An application is also given. (C) 2007 Elsevier Ltd. All rights reserved.