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ON AN IMPLICIT HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES

Part of: Existence theories Nonlinear operators and their properties

Published online by Cambridge University Press:  19 June 2009

FILOMENA CIANCIARUSO ,
VITTORIO COLAO ,
LUIGI MUGLIA  and
HONG-KUN XU
FILOMENA CIANCIARUSO
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: cianciaruso@unical.it)
VITTORIO COLAO
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: colao@mat.unical.it)
LUIGI MUGLIA
Affiliation:
Dipartimento di Matematica, Universitá della Calabria, 87036 Arcavacata di Rende (CS), Italy (email: muglia@mat.unical.it)
HONG-KUN XU*
Affiliation:
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan (email: xuhk@math.nsysu.edu.tw)
*
✉For correspondence; e-mail: xuhk@math.nsysu.edu.tw
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Abstract

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Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math.13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

Keywords

implicit methodvariational inequalityhierarchical fixed pointnonexpansive mappingprojection

MSC classification

Secondary: 49J40: Variational methods including variational inequalities 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 117 - 124
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The first three authors were 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 (Taiwan).

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