Abstract

The paper has a double aim. On the one hand, it studies some monotone extensions to the bidual of monotone operators which are not of type (D), paying particular attention to the issue of representability of these extensions by means of convex functions. On the other hand, it derives a density property for maximal monotone operators that, not only are of type (D), but have their unique maximal monotone extensions to the bidual which are of type (D) as well, as it is the case for subdifferentials of lower semicontinuous convex functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Marco Rocco

Dept. of Mathematics, University of Bergamo, Via dei Caniana 2, 24127 Bergamo, Italy

marco.rocco@unibg.it

Keywords

  • Monotone operator
  • type (D)
  • type (BR)
  • convex representation
  • bidual

Mathematics Subject Classification

47H05, 46T99, 47N10

References

  1. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10(4) (2002) 297–316.https://doi.org/10.1023/A:1020639314056Google Scholar
  2. S. Fitzpatrick: Representing monotone operators by convex functions, in: Functional Analysis and Optimization, Workshop / Miniconference (Canberra, 1988), Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, Australian National University, Canberra (1988) 59–65.Google Scholar
  3. S. Fitzpatrick, R. R. Phelps: Bounded approximants to monotone operators on Banach spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9(5) (1992) 573–595.https://doi.org/10.1016/s0294-1449(16)30230-xGoogle Scholar
  4. J.-P. Gossez: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971) 371–395.https://doi.org/10.1016/0022-247X(71)90119-3Google Scholar
  5. J.-P. Gossez: On the range of a coercive maximal monotone operator in a nonreflexive Banach space, Proc. Amer. Math. Soc. 35 (1972) 88–92.https://doi.org/10.1090/S0002-9939-1972-0298492-7Google Scholar
  6. J.-P. Gossez: On a convexity property of the range of a maximal monotone operator, Proc. Amer. Math. Soc. 55 (1976) 359–360.https://doi.org/10.1090/S0002-9939-1976-0397485-2Google Scholar
  7. J.-P. Gossez: On the extensions to the bidual of a maximal monotone operator, Proc. Amer. Math. Soc. 62(1) (1977) 67–71.https://doi.org/10.1090/S0002-9939-1977-0428121-5Google Scholar
  8. M. Marques Alves, B. F. Svaiter: Brønsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces, J. Convex Analysis 15(4) (2008) 693–706.https://doi.org/10.68381/jca15047Google Scholar
  9. M. Marques Alves, B. F. Svaiter: A new old class of maximal monotone operators, J. Convex Analysis 16(3&4) (2009) 881–890.https://doi.org/10.68381/jca16053Google Scholar
  10. M. Marques Alves, B. F. Svaiter: Maximal monotone operators with a unique extension to the bidual, J. Convex Analysis 16(2) (2009) 409–421.https://doi.org/10.68381/jca16022Google Scholar
  11. M. Marques Alves, B. F. Svaiter: On Gossez type (D) maximal monotone operators, J. Convex Analysis 17(3&4) (2010) 1077–1088.https://doi.org/10.68381/jca17066Google Scholar
  12. M. Marques Alves, B. F. Svaiter: On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces, J. Convex Analysis 18(1) (2011) 209–226.https://doi.org/10.68381/jca18012Google Scholar
  13. J. E. Martínez-Legaz: Some generalizations of Rockafellar’s surjectivity theorem, Pac. J. Optim. 4(3) (2008) 527–535.Google Scholar
  14. J. E. Martínez-Legaz: On maximally q-positive sets, J. Convex Analysis 16(3&4) (2009) 891–898.https://doi.org/10.68381/jca16054Google Scholar
  15. J. E. Martínez-Legaz, B. F. Svaiter: Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13(1) (2005) 21–46.https://doi.org/10.1007/s11228-004-4170-4Google Scholar
  16. J. E. Martínez-Legaz, M. Théra: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2(2) (2001) 243–247.Google Scholar
  17. R. R. Phelps: Lectures on maximal monotone operators, Extr. Math. 12(3) (1997) 193–230.Google Scholar
  18. M. Rocco, J. E. Martínez-Legaz: On surjectivity results for maximal monotone operators of type (D), J. Convex Analysis 18(2) (2011) 545–576.https://doi.org/10.68381/jca18035Google Scholar
  19. S. Simons: The range of a monotone operator, J. Math. Anal. Appl. 199(1) (1996) 176–201.https://doi.org/10.1006/jmaa.1996.0135Google Scholar
  20. S. Simons: Minimax and Monotonicity, Springer, Berlin (1998).https://doi.org/10.1007/BFb0093633Google Scholar
  21. S. Simons: The Fitzpatrick function and nonreflexive spaces, J. Convex Analysis 13(3&4) (2006) 861–881.https://doi.org/10.68381/jca13064Google Scholar
  22. S. Simons: From Hahn-Banach to Monotonicity, Springer, Berlin (2008).Google Scholar
  23. A. Verona, M. E. Verona: Remarks on subgradients and ε-subgradients, Set-Valued Anal. 1(3) (1993) 261–272.https://doi.org/10.1007/BF01027637Google Scholar

Suggested citation

M. Rocco. “A Note on Some Extensions of Monotone Operators to the Bidual.” Journal of Convex Analysis 19 (2012), No. 1, 125–140. https://doi.org/10.68381/jca19007

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.