  In this paper we study the asymptotic behavior of solutions of fractional
differential equations of the form $ D^{\alpha}_Cu(t)=Au(t)+f(t), u(0)=x,
0<\alpha\le1, ( *) $ where $D^{\alpha}_Cu(t)$ is the derivative of the function
$u$ in the Caputo's sense, $A$ is a linear operator in a Banach space $\X$ that
may be unbounded and $f$ satisfies the property that $\lim_{t\to \infty}
(f(t+1)-f(t))=0$ which we will call asymptotic $1$-periodicity. By using the
spectral theory of functions on the half line we derive analogs of
Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions
in terms of spectral properties of the operator $A$ for all asymptotic mild
solutions of Eq. (*) to be asymptotic $1$-periodic, or there exists an
asymptotic mild solution that is asymptotic $1$-periodic.
