Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems

We consider a mild solution u of a well-posed, inhomogeneous, Cauchy problem, u(t)=A(t) u(t)+f(t), on a Banach space X, where A(•) is periodic. For a problem on R+, we show that u is asymptotically almost periodic if f is asymptotically almost periodic, u is bounded, uniformly continuous and totally ergodic, and the spectrum of the monodromy operator V contains only countably many points of the unit circle. For a problem on R, we show that a bounded, uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied. We also show that there is a unique bounded solution subject to certain spectral assumptions on V, f and u.