Almost automorphic solutions for abstract functional differential equations. (Q1414231)

Let \(X\) be a Banach space and let \(B= B((-\infty,0], X)\) be the so-called fading memory space consisting of functions \(\varphi: (-\infty, 0]\to X\) which satisfy some conditions. The authors consider the abstract functional-differential equation (1) \(du(t)/dt= Au(t)+ F(t)u_t+ f(t)\), where \(A\) is the infinitesimal generator of a strongly continuous semigroup of linear operators on \(X\), \(F(t)\) is a bounded linear operator from \(B\) into \(X\) which is periodic in \(t\). Moreover, \(f\in BC(\mathbb{R}, X)\). A result on the existence of almost automorphic solutions of equation (1) is established. It is assumed that the function \(f\) is almost automorphic.