Let u be a bounded, uniformly continuous, mild solution
of an inhomogeneous Cauchy problem on R+:
u′(t)=Au(t)+ϕ(t) (t[ges ]0).
Suppose that u has uniformly convergent means,
σ(A)∩iR is countable, and
ϕ is asymptotically almost periodic. Then u is asymptotically
almost periodic. Related results have been
obtained by Ruess and Vũ, and by Basit, using different methods.
A direct proof is given of a Tauberian
theorem of Batty, van Neerven and Räbiger, and applications to
Volterra equations are discussed.