Integration of multivalued operators and cyclic submonotonicity

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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