Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the convergence of maximal monotone operators


Authors: Jean-Paul Penot and Constantin Zalinescu
Journal: Proc. Amer. Math. Soc. 134 (2006), 1937-1946
MSC (2000): Primary 47H05; Secondary 26B25
DOI: https://doi.org/10.1090/S0002-9939-05-08275-4
Published electronically: December 16, 2005
MathSciNet review: 2215762
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.


References [Enhancements On Off] (What's this?)

  • 1. H. Attouch, A. Moudafi, and H. Riahi, Quantitative stability analysis for maximal monotone operators and semi-groups of contractions, Nonlinear Anal. 21 (1993), 697-723. MR 1246288 (94i:47084)
  • 2. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Bucharest, Noordhoff, Leyden (1976). MR 0390843 (52:11666)
  • 3. G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht (1993). MR 1269778 (95k:49001)
  • 4. H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies. 5. Notas de matematica (50). North-Holland, Amsterdam (1973).
  • 5. F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proceedings of the symposium in pure mathematics of the American Mathematical Society, Vol. XVIII, Part 2. Am. Math. Soc., Providence (1976). MR 0405188 (53:8982)
  • 6. R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002), 297-316. MR 1934748 (2003k:90085)
  • 7. R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Am. Math. Soc. 131 (2003), 2379-2383. MR 1974634 (2004a:49037)
  • 8. S. Fitzpatrick, Representing monotone operators by convex functions, Workshop and Miniconference on Functional Analysis and Optimization (Canberra, 1988), Austral. Nat. Univ., Canberra, 1988, pp. 59-65. MR 1009594 (90i:47054)
  • 9. J. E. Martínez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001), 243-247. MR 1848704 (2002e:49035)
  • 10. T. Pennanen, Dualization of generalized equations of maximal monotone type. SIAM J. Optim. 10 (2000), 809-835. MR 1774774 (2002e:90121)
  • 11. T. Pennanen, J. Revalski and M. Théra, Graph-distance convergence and uniform local boundedness of monotone mappings, Proc. Am. Math. Soc. 131 (2003), 3721-3729. MR 1998179 (2004d:49024)
  • 12. T. Pennanen, R. T. Rockafellar and M. Théra, Graphical convergence of sums of monotone mappings, Proc. Am. Math. Soc. 130 (2002), 2261-2269. MR 1896407 (2003k:49042)
  • 13. J.-P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules, C. Rendus Acad. Sci. Paris, Ser. I 338 (2004), 853-858. MR 2059661 (2005f:49054)
  • 14. J.-P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal. 58 (2004), 855-871. MR 2086060 (2005g:49026)
  • 15. J.-P. Penot and C. Zalinescu, Continuity of usual operations and variational convergences, Set-Valued Anal. 11 (2003), 225-256. MR 1992063 (2004c:49029)
  • 16. J.-P. Penot and C. Zalinescu, Bounded (Hausdorff) convergence: basic facts and applications, preprint, October 2003; in Variational Analysis and Applications, F. Giannessi and A. Maugeri, Eds., Kluwer Acad. Publ., Dordrecht, 2005, 827-854.
  • 17. J.-P. Penot and C. Zalinescu, Continuity of the Legendre-Fenchel transform for some variational convergences, Optimization 53 (5-6) (2004), 549-562. MR 2115270
  • 18. J.-P. Penot and C. Zalinescu, Bounded convergence for perturbed optimization problems, Optimization 53 (5-6) (2004), 625-640. MR 2115995
  • 19. J.-P. Penot and C. Zalinescu, Some problems about the representation of monotone operators by convex functions, ANZIAM J. 47 (2005), 1-20. MR 2159848
  • 20. S. Simons, Minimax and Monotonicity, Springer, New York, 1998. MR 1723737 (2001h:49002)
  • 21. S. Simons and C. Zalinescu, Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005), 1-22.
  • 22. C. Zalinescu, A new proof of the maximal monotonicity of the sum using the Fitzpatrick function, in Variational Analysis and Applications, F. Giannessi and A. Maugeri, Eds., Kluwer Acad. Publ., Dordrecht, 2005, 1159-1172.
  • 23. E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators. Springer-Verlag, New York (1990). MR 1033498 (91b:47002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H05, 26B25

Retrieve articles in all journals with MSC (2000): 47H05, 26B25


Additional Information

Jean-Paul Penot
Affiliation: Laboratoire de Mathématiques Appliquées, ERS CNRS 2055, Faculté des Sciences, av. de l’Université, 64000 Pau, France
Email: jean-paul.penot@univ-pau.fr

Constantin Zalinescu
Affiliation: Faculty of Mathematics, University “Al. I. Cuza” Iaşi, Bd. Carol I, Nr. 11, 700506 Iaşi, Romania
Email: zalinesc@uaic.ro

DOI: https://doi.org/10.1090/S0002-9939-05-08275-4
Keywords: Bounded convergence, convergence, convex function, maximal monotone operator, monotone operator, qualification condition
Received by editor(s): January 28, 2005
Received by editor(s) in revised form: February 1, 2005
Published electronically: December 16, 2005
Additional Notes: The second author was partially supported by Grant CEEX-05-D11-36.
Dedicated: Dedicated to F.E. Browder for the impact of his work on nonlinear analysis
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society