A spectral countability condition for almost automorphy of solutions of differential equations

Abstract:

We consider the almost automorphy of bounded mild solutions to equations of the form \begin{equation*} (*)\quad \qquad \qquad \qquad \qquad \qquad dx/dt = A(t)x + f(t) \quad \qquad \qquad \qquad \qquad \qquad \qquad \end{equation*} 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.