On a result of Brezis and Mawhin

Abstract:

Brezis and Mawhin proved the existence of at least one $T$ periodic solution for differential equations of the form \begin{equation}\notag (\phi (u^{\prime }))^{\prime }-g(t,u)=h(t)\tag *{(0.1)} \end{equation} when $\phi :(-a,a)\rightarrow \mathbb {R},$ $0<a<\infty$, is an increasing homeomorphism with $\phi (0)=0$, $g$ is a Carathéodory function $T$ periodic with respect to $t$, $2\pi$ periodic with respect to $u$, of mean value zero with respect to $u$, and $h\in L_{loc}^{1}(\mathbb {R})$ is $T$ 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 $u_{0}$ in a closed convex subset $\mathcal {K}$ of the space of $T$ periodic Lipschitz functions. However, $u_{0}$ may not be an interior point of $\mathcal {K}$, 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 $T$ periodic solution to (0.1) when $g(t,u)=u$ to show that $u_{0}$ is indeed a $T$ periodic solution of (0.1). Here we make full use of the variational structure of the problem to obtain Brezis and Mawhin’s result.