In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form
D
C
α
u
(
t
)
=
A
u
(
t
)
+
f
(
t
)
,
u
(
0
)
=
x
,
0
>
α
≤
1
,
(
∗
)
D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0>\alpha \le 1, ( *)
where
D
C
α
u
(
t
)
D^{\alpha }_Cu(t)
is the derivative of the function
u
u
in the Caputo’s sense,
A
A
is a linear operator in a Banach space
X
\mathbb {X}
that may be unbounded and
f
f
satisfies the property that
lim
t
→
∞
(
f
(
t
+
1
)
−
f
(
t
)
)
=
0
\lim _{t\to \infty } (f(t+1)-f(t))=0
which we will call asymptotic
1
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
A
for all asymptotic mild solutions of Eq. (*) to be asymptotic
1
1
-periodic, or there exists an asymptotic mild solution that is asymptotic
1
1
-periodic.