Abstract

We consider the almost automorphy of bounded mild solutions to equations of the form $\displaystyle (*)\quad\qquad\qquad\qquad\qquad\qquad dx/dt = A(t)x + f(t) \quad\qquad\qquad\qquad\qquad\qquad\qquad $ with (generally unbounded) $ \tau$-periodic $ A(\cdot )$ and almost automorphic $ f(\cdot )$ in a Banach space $ \mathbb{X}$. Under the assumption that $ \mathbb{X}$ does not contain $ c_0$, the part of the spectrum of the monodromy operator associated with the evolutionary process generated by $ A(\cdot )$ on the unit circle is countable. We prove that every bounded mild solution of $ (*)$ on the real line is almost automorphic.