An extension of the Fitzpatrick theory
Augusto Visintin · Communications on Pure & Applied Analysis · 2014
In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotoneoperator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space)there exists a lower semicontinuous, convex representative function$f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that\begin{eqnarray}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{eqnarray}Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$,and extend the Fitzpatrick theory to provide a new variational formulation foreither stationary or evolutionary (nonmonotone) inclusions of the form$\alpha_v(v) i v'$.For any $v'\in V'$, we prove existence of a solution via the classical minimax theoremof Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators,and variational inequalities.