In this paper we study the existence of periodic solutions to the partial functional differential equation
{
d
y
(
t
)
d
t
=
B
y
(
t
)
+
L
^
(
y
t
)
+
f
(
t
,
y
t
)
,
∀
t
≥
0
,
y
0
=
φ
∈
C
B
.
\begin{equation*} \left \{ \begin {array}{l} \frac {dy(t)}{dt}=By(t)+\hat {L}(y_{t})+f(t,y_{t}), \;\forall t\geq 0,\\ y_{0}=\varphi \in C_{B}. \end{array} \right . \end{equation*}
where
B
:
Y
→
Y
B: Y \rightarrow Y
is a Hille-Yosida operator on a Banach space
Y
Y
. For
C
B
≔
{
φ
∈
C
(
[
−
r
,
0
]
;
Y
)
:
φ
(
0
)
∈
D
(
B
)
¯
}
C_{B}≔\{\varphi \in C([-r,0];Y): \varphi (0)\in \overline {D(B)}\}
,
y
t
∈
C
B
y_{t}\in C_{B}
is defined by
y
t
(
θ
)
=
y
(
t
+
θ
)
y_{t}(\theta )=y(t+\theta )
,
θ
∈
[
−
r
,
0
]
\theta \in [-r,0]
,
L
^
:
C
B
→
Y
\hat {L}: C_{B}\rightarrow Y
is a bounded linear operator, and
f
:
R
×
C
B
→
Y
f:\mathbb {R}\times C_{B}\rightarrow Y
is a continuous map and is
T
T
-periodic in the time variable
t
t
. Sufficient conditions on
B
B
,
L
^
\hat {L}
and
f
(
t
,
y
t
)
f(t,y_{t})
are given to ensure the existence of
T
T
-periodic solutions. The results then are applied to establish the existence of periodic solutions in a reaction-diffusion equation with time delay and the diffusive Nicholson’s blowflies equation.