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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Integration of multivalued operators and cyclic submonotonicity


Authors: Aris Daniilidis, Pando Georgiev and Jean-Paul Penot
Journal: Trans. Amer. Math. Soc. 355 (2003), 177-195
MSC (2000): Primary 49J52, 47H05; Secondary 58C20
Published electronically: September 6, 2002
MathSciNet review: 1928084
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Abstract: We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space $X$ to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset $U$ of $X$, then it is also maximal cyclically submonotone on $U$, and, conversely, that every maximal cyclically submonotone operator on $U$ is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if $U$ is connected. In finite dimensions these functions are exactly the lower C$^{1}$ functions considered by Spingarn and Rockafellar.


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Additional Information

Aris Daniilidis
Affiliation: Laboratoire de Mathématiques Appliquées, CNRS ERS 2055, Université de Pau et des Pays de l’Adour, avenue de l’Université, 64000 Pau, France
Address at time of publication: CODE - Edifici B, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain
Email: aris.daniilidis@univ-pau.fr

Pando Georgiev
Affiliation: Sofia University “St. Kl. Ohridski”, Faculty of Mathematics and Informatics, 5 J. Bourchier Blvd., 1126 Sofia, Bulgaria
Address at time of publication: Laboratory for Advanced Brain Signal Processing, Brain Science Institute, The Institute of Physical and Chemical Research (RIKEN), 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan
Email: georgiev@bsp.brain.riken.go.jp

Jean-Paul Penot
Affiliation: Laboratoire de Mathématiques Appliquées, CNRS ERS 2055, Université de Pau et des Pays de l’Adour, avenue de l’Université, 64000 Pau, France
Email: jean-paul.penot@univ-pau.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03118-5
Keywords: Integration, subdifferential, submonotone operator, subsmooth function
Received by editor(s): May 4, 2000
Published electronically: September 6, 2002
Additional Notes: The research of the first author was supported by the TMR grant ERBFMBI CT 983381
A major part of this work was accomplished while the second author was visiting the University of Pau under the NATO grant CB/JB SC105 N$^{0}$ 44/96165
Article copyright: © Copyright 2002 American Mathematical Society