On a result of Brezis and Mawhin

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.