Online ISSN 1552-4485; Print ISSN 0033-569X

A potential theory for monotone multivalued operators

Authors: G. Romano, L. Rosati, F. Marotti de Sciarra and P. Bisegna
Journal: Quart. Appl. Math. 51 (1993), 613-631
MSC: Primary 47H07; Secondary 31C45, 47H04, 49J52
DOI: https://doi.org/10.1090/qam/1247431
MathSciNet review: MR1247431
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Abstract: The concept of cyclic monotonicity of a multivalued map has been introduced by R. T. Rockafellar with reference to the subdifferential operator of a convex functional. He observed that cyclic monotonicity could be viewed heuristically as a discrete substitute for the classical condition of conservativity, i.e., the vanishing of all the circuital integrals of a vector field. In the present paper a potential theory for monotone multivalued operators is developed, and, in this context, an answer to Rockafellar's conjecture is provided. It is first proved that the integral of a monotone multivalued map along lines and polylines can be properly defined. This result allows us to introduce the concept of conservativity of a monotone multivalued map and to state its relation with cyclic monotonicity. Further, as a generalization of a classical result of integral calculus, it is proved that the potential of the subdifferential of a convex functional coincides, to within an additive constant, with the restriction of the functional on the domain of its subdifferential map. It is then shown that any conservative monotone graph admits a pair of proper convex potentials which meet a complementarity relation. Finally, sufficient conditions are given under which the complementary and the Fenchel's conjugate of the potential associated with a conservative maximal monotone graph do coincide.

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