In this nicely written paper, the authors give sufficient conditions for the existence of an almost automorphic mild solution of the semilinear differential equation
in a Banach space , where is the infinitesimal generator of an exponentially stable -semigroup and is jointly continuous. The main result improves a recent theorem proved by G. M. N’Guérékata [Semigroup Forum 69, 80–86 (2004; Zbl 1077.47058)].
Throughout the paper, denotes the Banach space of all almost automorphic functions , endowed with the sup-norm and is a Banach space algebraically contained in with compact injection. The authors suppose that , for all and , where is continuous with for every and and is a continuous mapping such that there is with with the property that for every . Under these hypotheses, the main result states that equation (1) has a mild solution in .
The presentation is very interesting, the central idea of the proof is based on Schauder’s fixed-point theorem. By an illustrative example, the authors show that generally, the almost automorphic mild solution of the above equation is not uniquely determined.