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August  2017, 10(4): 909-918. doi: 10.3934/dcdss.2017046

## On the variational representation of monotone operators

 Dipartimento di Matematica, dell'Università degli Studi di Trento, via Sommarive 14,38050 Povo di Trento, Italy

Received  May 2016 Revised  May 2016 Published  April 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
 $\alpha(u) \ni z'$
, 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.
Citation: Augusto VisintiN. On the variational representation of monotone operators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046
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