Limits of sequences of maximal monotone operators.


We consider a sequence of maximal monotone operators on a reflexive
Banach space. In general, the (Kuratowski) lower limit of such a
sequence is not a maximal monotone operator. So, what can be said?
In the first part of the talk, we show that such a limit is a
representable monotone operator while its Mosco limit, when it exists,
is a maximal monotone operator. As an application of the former result,
we obtain that the variational sum of two maximal monotone operators is
a representable monotone operator.

In the second part of the talk, we consider a sequence  of
representative functions of  maximal monotone operators. We show that if
these functions epi-converges to a given function, then the lower limit
of the associated operators is representable by this limit function;
moreover, if such a sequence of functions  Mosco-converges, then the
lower limit of operators is maximal monotone. As an application, we
recover Attouch’s result:
if a sequence of convex lower semicontinuous functions  Mosco-converges,
then the lower limit of the Fenchel subdifferential coincides with the
subdifferential of the limit function.

Date: Nov 15, 2017 at 16:30 h
Venue: Sala de Seminarios John Von Neumann CMM, séptimo piso, torre norte, Beauchef 851.
Speaker: Prof. Marc Lassonde
Affiliation: Université des Antilles, Guadeloupe & LIMOS, Clermont-Ferrand, France
Coordinator: Prof. Abderrahim Hantoute

Posted on Nov 13, 2017 in Optimization and Equilibrium, Seminars