On the variational representation of monotone operators
Augusto Visintin · Discrete and Continuous Dynamical Systems - S · 2017
Let $V$ be a Banach space, $z'\in V'$, and $\alpha: V\to {\mathcal P}(V')$ be a maximal monotone operator. A large number of phenomena can be modelled by inclusions of the form $\alpha(u) i z'$, or by the associated flow $D_tu + \alpha(u) i z'$. Fitzpatrick proved that there exists a lower semicontinuous, convex representative function$f_\alpha: V \!\times\! V'\to \mathbb{R}\cup \{+\infty\}$ such that\begin{document}$f_\alpha(v,v') \ge \langle v',v\rangle\quad\;\forall (v,v'), \qquad\quadf_\alpha(v,v') = \langle v',v\rangle\;\;\Leftrightarrow\;\;\; v'\in \alpha(v).$\end{document}This provides a variational formulation for the above inclusions. Here we use this approach to prove two results of existence of a solution, without using the classical theory of maximal monotone operators. This is based on a minimax theorem, and on the duality theory of convex optimization.