Periodicity of mild solutions to higher order differential equations in Banach spaces

We give necessary and sufficient conditions for the periodicity of mild solutions to the the higher order differential equation $u^{(n)}(t)=Au(t)+f(t)$, $0le t le T$, in a Banach space $E$. Applications are made to the cases, when $A$ generates a $C_0$-semigroup or a cosine family, and when $E$ is a Hilbert space.