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Some problems about the representation of monotone operators by convex functions

Published online by Cambridge University Press:  17 February 2009

Jean-Paul Penot  and
Constantin Zᾰlinescu
Jean-Paul Penot
Affiliation:
Laboratoire de Mathématiques appliquées, ERS CNRS 2570, Faculté des sciences, av. de l'Université, 64000 PAU, France; e-mail: [email protected].
Constantin Zᾰlinescu
Affiliation:
University “Al. I. Cuza” laşi, Faculty of Mathematics, Bd. Carol 1, Nr. 11, 700506 laşi, Romania; e-mail: [email protected].
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Abstract

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We answer a few questions raised by S. Fitzpatrick concerning the representation of maximal monotone operators by convex functions. We also examine some other questions concerning this representation and other ones which have recently emerged.

Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 1 , July 2005 , pp. 1 - 20
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Barbu, V., Nonlinear semigroups and differential equations in Banach spaces (Noordhoff, Leyden, 1976).CrossRefGoogle Scholar
[2]Brezis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies 5 (North-Holland, Amsterdam, 1973).Google Scholar
[3]Browder, F. E., Nonlinear operators and nonlinear equations of evolution in Banach spaces Proc. Symp. Pure Math. Vol. 18, Part 2 (American Mathematical Society, Providence, RI, 1976).CrossRefGoogle Scholar
[4]Burachik, R. S. and Svaiter, B. F., “Maximal monotone operators, convex functions and a special family of enlargements”, Set-Valued Anal. 10 (2002) 297–316.CrossRefGoogle Scholar
[5]Burachik, R. S. and Svaiter, B. F., “Maximal monotonicity, conjugation and the duality product”, Proc. Amer. Math. Soc. 131 (2005) 2379–2383.CrossRefGoogle Scholar
[6]Fitzpatrick, S., “Representing monotone operators by convex functions”, in Workshop and Miniconference on Functional Analysis and Optimization (Canberra, 1988), (Austral. Nat. Univ., Canberra, 1988) 59–65.Google Scholar
[7]Kinderlehrer, D. and Stampacchia, G., An introduction to variational inequalities and their applications, Classics in Appl. Math. 31 (SIAM, Philadelphia, 2000).CrossRefGoogle Scholar
[8]Krauss, E., “A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions”, Nonlinear Anal. Theory Methods Appl. 9 (1985) 1381–1399.CrossRefGoogle Scholar
[9]Krauss, E., “A representation of maximal monotone operators by saddle functions”, Rev. Roum. Math. Pures Appl. 30 (1985) 823–836.Google Scholar
[10]Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires (Dunod-Gauthier-Villars, Paris, 1969).Google Scholar
[11]Martínez-Legaz, J. E. and Svaiter, B., “Monotone operators representable by l.s.c. functions”, Set-Valued Anal. 13 (2005) 21–46.CrossRefGoogle Scholar
[12]Martínez-Legaz, J. E. and Théra, M., “A convex representation of maximal monotone operators”, J. Nonlinear Convex Anal. 2 (2001) 243–247.Google Scholar
[13]Pennanen, T., “Dualization of generalized equations of maximal monotone type”, SIAM J. Optim. 10 (2000) 809–835.CrossRefGoogle Scholar
[14]Penot, J.-P., “Autoconjugate functions and representations of monotone operators”, Bull. Austral. Math. Soc. 67 (2005) 277–284.CrossRefGoogle Scholar
[15]Penot, J.-P., “Is convexity useful for the study of monotonicity?”, in Nonlinear Analysis and Applications (eds. Agarwal, R. P. and O'Regan, D.), (Kluwer, Dordrecht, 2005) 807–822.Google Scholar
[16]Penot, J.-P., “The relevance of convex analysis for the study of monotonicity”, Nonlinear Anal. Theory Methods Appl. 58 (2004) 855–871.CrossRefGoogle Scholar
[17]Penot, J.-P. and Zᾰlinescu, C., “On the convergence of maximal monotone operators”, Proc. Amer. Math. Soc. (to appear).Google Scholar
[18]Simons, S., Minimax and monotonicity, Lecture Notes in Mathematics 1693 (Springer, New York, 1998).CrossRefGoogle Scholar
[19]Simons, S. and Zᾰlinescu, C., “A new proof for Rockafellar's characterization of maximal monotone operators”, Proc. Amer Math. Soc. 132 (2004) 2969–2972.CrossRefGoogle Scholar
[20]Simons, S. and Zᾰlinescu, C., “Fenchel duality, Fitzpatrick functions and maximal monotonicity”, J. Nonlinear Convex Anal. 6 (2005) 1–22.Google Scholar
[21]Svaiter, B. F., “Fixed points in the family of convex representations of a maximal, monotone operator”, Proc. Amer. Math. Soc. 131 (2005) 3851–3859.CrossRefGoogle Scholar
[22]Zᾰlinescu, C., Convex analysis in general vector spaces (World Scientific, Singapore, 2002).CrossRefGoogle Scholar
[23]Zᾰlinescu, C., “A new proof of the maximal monotonicity of the sum using the Fitzpatrick function”, in Variational Analysis and Applications (eds. Giannessi, F. and Maugeri, A.), (Springer, Berlin, 2005) (to appear).Google Scholar
[24]Zeidler, E., Nonlinear functional analysis and its applications. II/B: Nonlinear monotone operators (Springer, New York, 1990).Google Scholar