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Transactions of the American Mathematical Society
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
Original Article: Tran. Amer. Math. Soc. 350 (1998), no. 7, 2953-2972.
MathSciNet review: 1443892
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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.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02045-5
PII: S 0002-9947(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