Integration of multivalued operators and cyclic submonotonicity

We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset of , then it is also maximal cyclically submonotone on , and, conversely, that every maximal cyclically submonotone operator on is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if is connected. In finite dimensions these functions are exactly the lower C functions considered by Spingarn and Rockafellar.