Sum theorems for monotone operators and convex functions

Simons, S.

doi:10.1090/S0002-9947-98-02045-5

Sum theorems for monotone operators and convex functions
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by S. Simons

Trans. Amer. Math. Soc. 350 (1998), 2953-2972

DOI: https://doi.org/10.1090/S0002-9947-98-02045-5

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Original Article: Tran. Amer. Math. Soc. 350 (1998), no. 7, 2953-2972.

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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