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Some problems about the representation of monotone operators by convex functions

  • Jean-Paul Penot (a1) and Constantin Zᾰlinescu (a2)
Abstract
Abstract

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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] V. Barbu , Nonlinear semigroups and differential equations in Banach spaces (Noordhoff, Leyden, 1976).

[3] F. E. Browder , Nonlinear operators and nonlinear equations of evolution in Banach spaces Proc. Symp. Pure Math. Vol. 18, Part 2 (American Mathematical Society, Providence, RI, 1976).

[4] R. S. Burachik and B. F. Svaiter , “Maximal monotone operators, convex functions and a special family of enlargements”, Set-Valued Anal. 10 (2002) 297316.

[5] R. S. Burachik and B. F. Svaiter , “Maximal monotonicity, conjugation and the duality product”, Proc. Amer. Math. Soc. 131 (2005) 23792383.

[7] D. Kinderlehrer and G. Stampacchia , An introduction to variational inequalities and their applications, Classics in Appl. Math. 31 (SIAM, Philadelphia, 2000).

[8] E. Krauss , “A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions”, Nonlinear Anal. Theory Methods Appl. 9 (1985) 13811399.

[11] J. E. Martínez-Legaz and B. Svaiter , “Monotone operators representable by l.s.c. functions”, Set-Valued Anal. 13 (2005) 2146.

[13] T. Pennanen , “Dualization of generalized equations of maximal monotone type”, SIAM J. Optim. 10 (2000) 809835.

[16] J.-P. Penot , “The relevance of convex analysis for the study of monotonicity”, Nonlinear Anal. Theory Methods Appl. 58 (2004) 855871.

[19] S. Simons and C. Zᾰlinescu , “A new proof for Rockafellar's characterization of maximal monotone operators”, Proc. Amer Math. Soc. 132 (2004) 29692972.

[21] B. F. Svaiter , “Fixed points in the family of convex representations of a maximal, monotone operator”, Proc. Amer. Math. Soc. 131 (2005) 38513859.

[22] C. Zᾰlinescu , Convex analysis in general vector spaces (World Scientific, Singapore, 2002).

[23] C. Zᾰlinescu , “A new proof of the maximal monotonicity of the sum using the Fitzpatrick function”, in Variational Analysis and Applications (eds. F. Giannessi and A. Maugeri ), (Springer, Berlin, 2005) (to appear).

[24] E. Zeidler , Nonlinear functional analysis and its applications. II/B: Nonlinear monotone operators (Springer, New York, 1990).

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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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