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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
Published electronically: July 16, 2003
MathSciNet review: 1998179
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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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  • 1. H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1984. MR 86f:49002
  • 2. H. Attouch, J.-L. Ndoutoune and M. Théra, On the equivalence between the convergence of functions and the graph-convergence of their derivatives in Banach spaces, Séminaire d'Analyse Convexe de Montpellier Exposé $ N^0$ 9 (1990), pp. 9.1-9.45.
  • 3. H. Attouch, A. Moudafi and H. Riahi, Quantitative stability analysis for maximal monotone operators and semi-groups of contractions, Nonlinear Anal., 21 (1993), pp. 697-723. MR 94i:47084
  • 4. H. Attouch and R.J.-B. Wets, Quantitative stability of variational systems, I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), pp. 695-729. MR 92c:90111
  • 5. G. Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, 268, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 95k:49001
  • 6. G. Beer and R. Lucchetti, The epi-distance topology: Continuity and stability results with applications to convex optimization problems, Math. Oper. Res 17 (1992), pp. 715-726. MR 93k:49011
  • 7. H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Studies, Amsterdam, London, 1973. MR 50:1060
  • 8. R. R. Phelps, Lectures on maximal monotone operators, Extracta Math., 12(3) (1997), 193-230. MR 99c:47083
  • 9. T. Pennanen, Dualization of generalized equations of maximal monotone type, SIAM J. Optim., 10 (2000), pp. 809-835. MR 2002e:90121
  • 10. T. Pennanen, R. T. Rockafellar, and M. Théra, Graphical convergence of sums of monotone mappings, Proc. Amer. Math. Soc., 130 (2002), 2261-2269.
  • 11. J.-P. Penot, Topologies and convergences on the space of convex functions, Nonlinear Anal., TMA, 18(1992), 905-916. MR 93f:49016
  • 12. R. T. Rockafellar, Local boundedness of nonlinear monotone operators, Michigan Math. J., 16 (1969), pp. 397-407. MR 40:6229
  • 13. R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), pp. 209-216. MR 41:7432
  • 14. R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), pp. 75-88. MR 43:7984
  • 15. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, 1998. MR 98m:49001
  • 16. S. Simons, Minimax and Monotonicity, Lect. Notes in Math., Vol. 1693, Springer-Verlag, Berlin, Heidelberg, 1998. MR 2001h:49002

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Additional Information

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

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

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

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

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