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On the variational representation of monotone operators

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  • 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 $\alpha(u) \ni z'$, or by the associated flow $D_tu + \alpha(u) \ni z'$. Fitzpatrick proved that there exists a lower semicontinuous, convex representative function$f_\alpha: V \!\times\! V'\to \mathbb{R}\cup \{+\infty\}$ such that

    $f_\alpha(v,v') \ge \langle v',v\rangle\quad\;\forall (v,v'), \qquad\quadf_\alpha(v,v') = \langle v',v\rangle\;\;\Leftrightarrow\;\;\; v'\in \alpha(v).$

    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.

    Mathematics Subject Classification: 47H05, 49J40, 58E.


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