Abstract

Following some older ideas of ours and some more recent ones due to Yao, we give a self contained (except some well known results) and relatively short proof of the Rockafellar's sum theorem for the largest possible class of maximally monotone operators on a non reflexive Banach space.

Author information

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

Andrei Verona

Dept. of Mathematics, California State University, Los Angeles, U.S.A.

averona10@yahoo.com

Maria E. Verona

Dept. of Mathematics, University of Southern California, Los Angeles, U.S.A.

verona@usc.edu

Keywords

  • C0-maximally monotone operator
  • convex function
  • convex set
  • Fitzpatrick function
  • maximally monotone operator
  • monotone operator
  • operator of type (FPV)
  • sum theorem

Mathematics Subject Classification

47H05; 49N15, 52A41, 90C25

References

  1. J. M. Borwein, L. Yao: Structure theory for maximally monotone operators with points of continuity, J. Optimization Theory Appl. 156 (2013) 1–24.https://doi.org/10.1007/s10957-012-0162-yGoogle Scholar
  2. J. M. Borwein, L. Yao: Sum theorems for maximally monotone operators of type (FPV), J. Aust. Math. Soc. 97 (2014) 1–26.https://doi.org/10.1017/S1446788714000056Google Scholar
  3. H. Debrunner, P. Flor: Ein Erweiterungssatz für monotone Mengen, Arch. Math. 15 (1964) 445–447.https://doi.org/10.1007/BF01589229Google Scholar
  4. S. P. Fitzpatrick: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization (Canberra, Australia 1988), Proc. Centre Math. Anal., Australian Nat. Univ. 20 (1988) 59–65.Google Scholar
  5. S. P. Fitzpatrick, R. R. Phelps: Some properties of maximal monotone operators on nonreflexive Banach spaces, Set-Valued Analysis 3 (1995) 51–69.https://doi.org/10.1007/BF01033641Google Scholar
  6. R. T. Rockafellar: Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969) 397–407.https://doi.org/10.1307/mmj/1029000324Google Scholar
  7. R. T. Rockafellar: On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75–88.https://doi.org/10.1090/S0002-9947-1970-0282272-5Google Scholar
  8. S. Simons: From Hahn-Banach to Monotonicity, 2nd edition, Lecture Notes in Mathematics 1693, Spinger, Berlin (2008).Google Scholar
  9. A. Verona, M. E. Verona: Remarks on subgradients and ε-subgradients, Set-Valued Analysis 1 (1993) 61–272.https://doi.org/10.1007/BF01027637Google Scholar
  10. A. Verona, M. E. Verona: Regular maximally monotone operators, Set-Valued Analysis 6 (1998) 302–312.https://doi.org/10.1023/A:1008679427671Google Scholar
  11. A. Verona, M. E. Verona: Regular maximally monotone operators and the sum theorem, J. Convex Analysis 7 (2000) 115–128.https://doi.org/10.68381/jca07006Google Scholar
  12. A. Verona, M. E. Verona: Regular maximally monotone multifunctions and enlargements, J. Convex Analysis 16 (2009) 1003–1009.https://doi.org/10.68381/jca16062Google Scholar
  13. M. D. Voisei: The sum and chain rules for maximally monotone operators, Set-Valued Analysis 16 (2008) 461–476.https://doi.org/10.1007/s11228-007-0042-zGoogle Scholar
  14. L. Yao: Maximality of the sum of the subdifferential operator and a maximally monotone operator, Linear Nonlinear Analysis 1 (2015) 95–108.Google Scholar

Suggested citation

A. Verona, M. E. Verona. “On Rockafellar's Sum Theorem in General Banach Spaces.” Journal of Convex Analysis 29 (2022), No. 2, 381–390. https://doi.org/10.68381/jca29024

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.