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 C1 functions considered by Spingarn and Rockafellar.
|Journal||Transactions of the American Mathematical Society|
|Publication status||Published - 1 Jan 2003|
- Submonotone operator
- Subsmooth function
Daniilidis, A., Georgiev, P., & Penot, J. P. (2003). Integration of multivalued operators and cyclic submonotonicity. Transactions of the American Mathematical Society, 355, 177-195. https://doi.org/10.1090/S0002-9947-02-03118-5