Graph-distance convergence and uniform local boundedness of monotone mappings

Authors:
Teemu Pennanen, Julian P. Revalski and Michel Théra

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3721-3729

MSC (2000):
Primary 47H05; Secondary 54B20

DOI:
https://doi.org/10.1090/S0002-9939-03-07179-X

Published electronically:
July 16, 2003

MathSciNet review:
1998179

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Abstract | References | Similar Articles | Additional Information

Abstract: In this article we study graph-distance convergence of monotone operators. First, we prove a property that has been an open problem up to now: the limit of a sequence of graph-distance convergent maximal monotone operators in a Hilbert space is a maximal monotone operator. Next, we show that a sequence of maximal monotone operators converging in the same sense in a reflexive Banach space is uniformly locally bounded around any point from the interior of the domain of the limit mapping. The result is an extension of a similar one from finite dimensions. As an application we give a simplified condition for the stability (under graph-distance convergence) of the sum of maximal monotone mappings in Hilbert spaces.

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Additional Information

**Teemu Pennanen**

Affiliation:
Department of Economics and Management Science, Helsinki School of Economics, PL 1210, 00101 Helsinki, Finland

Email:
pennanen@hkkk.fi

**Julian P. Revalski**

Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Email:
revalski@math.bas.bg

**Michel Théra**

Affiliation:
Département de Mathématiques, LACO UMR-CNRS 6090, Université de Limoges, 123, Av. A. Thomas, 87060 Limoges Cedex, France

Email:
michel.thera@unilim.fr

DOI:
https://doi.org/10.1090/S0002-9939-03-07179-X

Received by editor(s):
May 14, 2002

Published electronically:
July 16, 2003

Additional Notes:
The first author’s research was partially supported by LACO (Laboratoire d’Arithmétique, Calcul Formel et Optimisation), UMR-CNRS 6090 of the University of Limoges, as well as by the Région Limousin under a Research Grant

The second author’s research was partially supported by the LACO (Laboratoire d’Arithmétique, Calcul Formel et Optimisation), UMR-CNRS 6090 of the University of Limoges, by Bulgarian NFSR under grant No. MM-1105/01 and by NATO-CLG 978488

The third author’s research was partially supported by the French Chilean Scientific Cooperation Programme ECOS under grant C00E05 and by NATO-CLG 978488

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2003
American Mathematical Society