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An extension of the Fitzpatrick theory

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  • In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ 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\quad f_\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 for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.
    Mathematics Subject Classification: 35K60, 47H05, 49J40, 58E.


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