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NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

  • FILOMENA CIANCIARUSO (a1), GIUSEPPE MARINO (a2), LUIGI MUGLIA (a3) and HONG-KUN XU (a4) (a5)

Abstract

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}.

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Copyright

Corresponding author

For correspondence; e-mail: xuhk@math.nsysu.edu.tw

Footnotes

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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.

Footnotes

References

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[1]Attouch, H., Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman (Advanced Publishing Program), Boston, MA, 1984).
[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).
[3]Lions, P. L., ‘Two remarks on the convergence of convex functions and monotone operators’, Nonlinear Anal. 2(5) (1978), 553562.
[4]Maingé, P. E. and Moudafi, A., ‘Strong convergence of an iterative method for hierarchical fixed point problems’, Pac. J. Optim. 3(3) (2007), 529538.
[5]Marino, G. and Xu, H. K., ‘A general iterative method for nonexpansive mappings in Hilbert spaces’, J. Math. Anal. Appl. 318(1) (2006), 4352.
[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.
[7]Xu, H. K., ‘An iterative approach to quadratic optimization’, J. Optim. Theory Appl. 116(3) (2003), 659678.
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NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

  • FILOMENA CIANCIARUSO (a1), GIUSEPPE MARINO (a2), LUIGI MUGLIA (a3) and HONG-KUN XU (a4) (a5)

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