DOI: 10.68381/jca07006 ISSN: 0944-6532
Regular Maximal Monotone Operators and the Sum Theorem
Andrei Verona, Maria Elena Verona
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
\mathcal{C}_0
C
0
(resp.
\mathcal{C}_1
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
\mathcal{C}_1
C
1
-regular and all linear maximal monotone operators are
\mathcal{C}_0
C
0
-regular. We prove that the sum of a
\mathcal{C}_0
C
0
(or
\mathcal{C}_1
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