On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces

The main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear Volterra integral equation in Banach spaces. We obtain a collection of existence results of such solutions to these equations. We investigate also a topological structure of such solution sets. Moreover, we prove Aronszajn-type theorems for solutions of the initial value problem as well as the nonlinear Volterra integral equation, defined on the whole real line. (C) 2004 Elsevier Ltd. All rights reserved.