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)+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.