Integration of multivalued operators and cyclic submonotonicity

Abstract

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.

In Asplund Spaces, Approximately Convex Functions and Regular Functions Are Generically Differentiable

“…The class of approximately convex functions has interesting stability properties; see for instance [12,Prop. 3.1], [6,Section 6], [1].…”