Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sum theorems for monotone
operators and convex functions


Author: S. Simons
Journal: Trans. Amer. Math. Soc. 350 (1998), 2953-2972
MSC (1991): Primary 47H05, 46B10; Secondary 49J35, 46A30
DOI: https://doi.org/10.1090/S0002-9947-98-02045-5
Original Article: Tran. Amer. Math. Soc. 350 (1998), no. 7, 2953-2972.
MathSciNet review: 1443892
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.


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

  • 1. H. Attouch and H. Brézis, Duality for the sum of convex funtions in general Banach spaces, Aspects of Mathematics and its Applications, J. A. Barroso, ed., Elsevier Science Publishers, 1986, pp. 125-133. MR 87m:90095
  • 2. H. Attouch, H. Riahi and M. Théra, Somme ponctuelle d'opérateurs maximaux monotones, Serdica 22 (1996), 165-190. CMP 97:14
  • 3. J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. MR 87a:58002
  • 4. J. M. Borwein, A Lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand. 48 (1981), 198-204. MR 83d:49027
  • 5. J. M. Borwein, Adjoint process duality, Math. Oper. Res. 8 (1983), 403-434. MR 85h:90092
  • 6. J. M. Borwein and S. Fitzpatrick, Local boundedness of monotone operators under minimal hypotheses, Bull. Australian Math. Soc. 39 (1988), 439-441. MR 90c:47093
  • 7. H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. Pure. Appl. Math. 23 (1970), 123-144. MR 41:2454
  • 8. F. E. Browder, Nonlinear maximal monotone operators in Banach spaces, Math. Annalen 175 (1968), 89-113. MR 36:6989
  • 9. M. Coodey and S. Simons, The convex function determined by a multifunction, Bull. Austral. Math. Soc. 54 (1996), 87-97. CMP 96:16
  • 10. L.-J. Chu, On the sum of monotone operators, Michigan Math. J. 43 (1996), 273-289. CMP 96:15
  • 11. K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47. MR 14:1109f
  • 12. R. B. Holmes, Geometric functional analysis and its applications, Springer-Verlag, Graduate Texts in Mathematics, 24, New York-Heidelberg, 1975. MR 53:14085
  • 13. H. König, Über das Von Neumannsche Minimax-Theorem, Arch. Math. 19 (1968), 482-487. MR 39:1947
  • 14. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics 1364 (Second Edition), Springer-Verlag, Berlin, 1993. MR 94f:46055
  • 15. R. R. Phelps, Lectures on Maximal Monotone Operators, 2nd Summer School on Banach Spaces, Related Areas and Applications, Prague and Paseky, August 15-28, 1993. (Preprint, 30 pages.), TeX file: $<$math.okstate.edu/pub/banach/phelpsmaxmonop.tex$>$ Banach space bulletin board archive, Posted Nov. 1993.
  • 16. S. M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), 130-143; 2 (1977), 382. MR 55:3188; MR 57:12801
  • 17. R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969), 397-407. MR 40:6229
  • 18. R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88. MR 43:7984
  • 19. S. Simons, Critères de faible compacité en termes du théorème de minimax, Seminaire Choquet, 1970/1971, Fasc. 2, Exposé 24, Secrétariat Math., Paris, 1971. MR 57:17218
  • 20. S. Simons, The range of a monotone operator, J. Math. Anal. Appl. 199 (1996), 176-201. MR 97e:47088
  • 21. C. Ursescu, Multifunctions with convex closed graph, Czechoslovak Math. J. 25 (1975), 438-441. MR 52:8869
  • 22. C. Zalinescu, Letter to the editor: on J. M. Borwein's paper: ``Adjoint process duality'', Math. Oper. Res. 11 (1986), 692-698. MR 88h:90172

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47H05, 46B10, 49J35, 46A30

Retrieve articles in all journals with MSC (1991): 47H05, 46B10, 49J35, 46A30


Additional Information

S. Simons
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106-3080
Email: simons@math.ucsb.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02045-5
Keywords: Banach space, reflexivity, maximal monotone operator, sum theorem, constraint qualification, proper convex lower semicontinuous function, uniform boundedness theorem, Fenchel Duality Theorem, minimax theorem
Received by editor(s): July 16, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society