Brezis and Mawhin proved the existence of at least one
T
T
periodic solution for differential equations of the form
(0.1)
(
ϕ
(
u
′
)
)
′
−
g
(
t
,
u
)
=
h
(
t
)
\begin{equation}\notag (\phi (u^{\prime }))^{\prime }-g(t,u)=h(t)\tag *{(0.1)} \end{equation}
when
ϕ
:
(
−
a
,
a
)
→
R
,
\phi :(-a,a)\rightarrow \mathbb {R},
0
>
a
>
∞
0>a>\infty
, is an increasing homeomorphism with
ϕ
(
0
)
=
0
\phi (0)=0
,
g
g
is a Carathéodory function
T
T
periodic with respect to
t
t
,
2
π
2\pi
periodic with respect to
u
u
, of mean value zero with respect to
u
u
, and
h
∈
L
l
o
c
1
(
R
)
h\in L_{loc}^{1}(\mathbb {R})
is
T
T
periodic and has mean value zero. Their proof was partly variational. First it was shown that the corresponding action integral had a minimum at some point
u
0
u_{0}
in a closed convex subset
K
\mathcal {K}
of the space of
T
T
periodic Lipschitz functions. However,
u
0
u_{0}
may not be an interior point of
K
\mathcal {K}
, so it may not be a critical point of the action integral. The authors used an ingenious argument based on variational inequalities and uniqueness of a
T
T
periodic solution to (0.1) when
g
(
t
,
u
)
=
u
g(t,u)=u
to show that
u
0
u_{0}
is indeed a
T
T
periodic solution of (0.1). Here we make full use of the variational structure of the problem to obtain Brezis and Mawhin’s result.