Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line

Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u'(t)= Au(t)+phi(t) (t greater than or equal to O). Suppose that u has uniformly convergent means, sigma(A) boolean AND iR is c ountable, and phi is asymptotically almost periodic. Then u is asymptotical ly almost periodic. Related results have been obtained by Ruess and Vu, and by Basit, using different methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Rabiger, and applications to Volterra equ ations are discussed.