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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Integration of multivalued operators and cyclic submonotonicity
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by Aris Daniilidis, Pando Georgiev and Jean-Paul Penot PDF
Trans. Amer. Math. Soc. 355 (2003), 177-195 Request permission

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.
References
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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
  • MR Author ID: 613204
  • 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
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 177-195
  • MSC (2000): Primary 49J52, 47H05; Secondary 58C20
  • DOI: https://doi.org/10.1090/S0002-9947-02-03118-5
  • MathSciNet review: 1928084