In this paper we investigate the following fractional order in time integrodifferential problem $$ \mathbb{D}_{t}^{\alpha}u(t)+Au(t)=f \bigl(t,u(t) \bigr)+ \int _{-\infty}^{t} k(t-s)g \bigl(s,u(s) \bigr)\,ds, \quad t \in \mathbb{R}. $$
D
t
α
u
(
t
)
+
A
u
(
t
)
=
f
(
t
,
u
(
t
)
)
+
∫
−
∞
t
k
(
t
−
s
)
g
(
s
,
u
(
s
)
)
d
s
,
t
∈
R
.
Here, $\mathbb{D}_{t}^{\alpha}$
D
t
α
is the Caputo derivative. We obtain results on the existence and uniqueness of $(\omega ,c)$
(
ω
,
c
)
-periodic mild solutions assuming that −A generates an analytic semigroup on a Banach space X and f, g, and k satisfy suitable conditions. Finally, an interesting example that fits our framework is given.