Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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  • [1] Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
  • [2] N. Bourbaki, Éléments de mathématique, Hermann, Paris, 1976 (French). Fonctions d’une variable réelle; Théorie élémentaire; Nouvelle édition. MR 0580296
  • [3] Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
  • [4] W. Fenchel, On conjugate convex functions, Canadian J. Math. 1 (1949), 73–77. MR 0028365
  • [5] A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski. MR 528295
  • [6] A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 2: Measure. The Lebesgue integral. Hilbert space, Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm, Graylock Press, Albany, N.Y., 1961. MR 0118796
  • [7] George J. Minty, On the monotonicity of the gradient of a convex function, Pacific J. Math. 14 (1964), 243–247. MR 0167859
  • [8] J. J. Moreau, Fonctionelles convexes, lecture notes, séminaire: équationes aux dérivées partielles, Collègie de France, 1966
  • [9] R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497–510. MR 0193549
  • [10] R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969), 397–407. MR 0253014
  • [11] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 0262827
  • [12] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970
  • [13] M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR 0176364
  • [14] Kōsaku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. MR 1336382

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DOI: https://doi.org/10.1090/qam/1247431
Article copyright: © Copyright 1993 American Mathematical Society

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