Bulletin of the Australian Mathematical Society

Table of Contents - 27 January 2011 - Volume 83, Issue 01  

Research Article

NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

FILOMENA CIANCIARUSOa1, GIUSEPPE MARINOa2, LUIGI MUGLIAa3 and HONG-KUN XUa4a5 c1

a1 Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: cianciaruso@unical.it)

a2 Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: gmarino@unical.it)

a3 Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: muglia@mat.unical.it)

a4 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

a5 Department of Mathematics, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia (email: xuhk@math.nsysu.edu.tw)

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}.

(Received November 12 2009)

(Online publication November 26 2010)

2010 Mathematics subject classification

  • primary 47H05; secondary 47H09;
  • 47H10

Keywords and phrases

  • maximal monotone operator;
  • graph-convergence;
  • Mann iteration

Correspondence:

c1 For correspondence; e-mail: xuhk@math.nsysu.edu.tw

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.