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NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  26 November 2010

FILOMENA CIANCIARUSO ,
GIUSEPPE MARINO ,
LUIGI MUGLIA  and
HONG-KUN XU
FILOMENA CIANCIARUSO
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: [email protected])
GIUSEPPE MARINO
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: [email protected])
LUIGI MUGLIA
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: [email protected])
HONG-KUN XU*
Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan Department of Mathematics, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We construct a sequence {An} of maximal monotone operators with a common domain and converging, uniformly on bounded subsets, to another maximal monotone operator A; however, the sequence {t−1nAn} fails to graph-converge for some null sequence {tn}.

Keywords

maximal monotone operatorgraph-convergenceMann iteration

MSC classification

Secondary: 47H05: Monotone operators and generalizations 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 1 , February 2011 , pp. 22 - 29
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The second author was supported in part by Ministero dell’Universitá e della Ricerca of Italy. The fourth author was supported in part by NSC 97-2628-M-110-003-MY3.

References

[1]Attouch, H., Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman (Advanced Publishing Program), Boston, MA, 1984).Google Scholar
[2]Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, 5. Notas de Matemática (50) (North Holland, Amsterdam, 1973).Google Scholar
[3]Lions, P. L., ‘Two remarks on the convergence of convex functions and monotone operators’, Nonlinear Anal. 2(5) (1978), 553–562.CrossRefGoogle Scholar
[4]Maingé, P. E. and Moudafi, A., ‘Strong convergence of an iterative method for hierarchical fixed point problems’, Pac. J. Optim. 3(3) (2007), 529–538.Google Scholar
[5]Marino, G. and Xu, H. K., ‘A general iterative method for nonexpansive mappings in Hilbert spaces’, J. Math. Anal. Appl. 318(1) (2006), 43–52.CrossRefGoogle Scholar
[6]Moudafi, A. and Maingé, P. E., ‘Towards viscosity approximations of hierarchical fixed-point problems’, Fixed Point Theory Appl. 2006 (2006), 10; Article ID 95453.CrossRefGoogle Scholar
[7]Xu, H. K., ‘An iterative approach to quadratic optimization’, J. Optim. Theory Appl. 116(3) (2003), 659–678.CrossRefGoogle Scholar