We prove an existence result for
T
T
-periodic mild solutions to nonlinear evolution equations of the form
\[
u
′
(
t
)
+
A
u
(
t
)
∍
F
(
t
,
u
(
t
)
)
,
t
∈
R
+
.
u’(t) + Au(t) \backepsilon F(t,u(t)),\quad t \in {R_ + }.
\]
Here
(
X
,
|
|
⋅
|
|
)
(X,|| \cdot ||)
is a real Banach space,
A
:
D
(
A
)
⊂
X
→
2
X
A:D(A) \subset X \to {2^X}
is an operator with
A
−
a
I
A - aI
m
m
-accretive for some
a
>
0
a > 0
and such that
−
A
- A
. generates a compact semigroup, while
F
:
R
+
×
D
(
A
)
¯
→
X
F:{R_ + } \times \overline {D(A)} \to X
is a Carathéodory mapping which is
T
T
-periodic with respect to its first argument and satisfies
\[
lim
r
→
+
∞
1
r
sup
{
|
|
F
(
t
,
v
)
|
|
;
t
∈
R
+
,
v
∈
D
(
A
)
¯
,
|
|
v
|
|
≤
r
}
>
a
.
\lim \limits _{r \to + \infty } \tfrac {1}{r}\sup \left \{ {||F(t,v)||;t \in {R_ + },v \in \overline {D(A)} ,||v|| \leq r} \right \} > a.
\]
. As a consequence, we obtain an existence theorem for
T
T
-periodic solutions to the porous medium equation.