Abstract

In this note, which is a continuation of a previous paper of the authors [Set-Valued Analysis 6 (1998) 302-312], we study two classes of maximal monotone operators on general Banach spaces which we call C0\mathcal{C}_0 (resp. C1\mathcal{C}_1)-regular. All maximal monotone operators on a reflexive Banach space, all subdifferential operators, and all maximal monotone operators with domain the whole space are C1\mathcal{C}_1-regular and all linear maximal monotone operators are C0\mathcal{C}_0-regular. We prove that the sum of a C0\mathcal{C}_0 (or C1\mathcal{C}_1)-regular maximal monotone operator with a maximal monotone operator which is locally inf bounded and whose domain is closed and convex is again maximal monotone provided that they satisfy a certain "dom–dom" condition. From this result one can obtain most of the known sum theorem type results in general Banach spaces. We also prove a local boundedness type result for pairs of monotone operators

Author information

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Andrei Verona

Dept. of Mathematics, California State University, Los Angeles, CA 90032, U.S.A.

Maria Elena Verona

Dept. of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, U.S.A.

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Suggested citation

A. Verona, M. E. Verona. “Regular Maximal Monotone Operators and the Sum Theorem.” Journal of Convex Analysis 7 (2000), No. 1, 115–128. https://doi.org/10.68381/jca07006

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Published by Heldermann Verlag, 2000. Rights now held by Banach Press LLC.