Moudafi and Mainge [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 (x(s,t)) of two parameters s, t is an element of (0, 1), and under certain conditions, the iterated lim(t -> 0) lim(s -> 0) x(s,t) exists in the norm topology. Moudafi, Mainge and Xu stated the problem of convergence of (x(s,t)) as (s, t) -> (0, 0) jointly in the norm topology. In this paper we further study the behaviour of the net (x(s,t)); in particular, we give a negative answer to this problem.

On an Implicit Hierarchical Fixed Point Approach to Variational Inequalities

CIANCIARUSO, Filomena;COLAO, Vittorio;MUGLIA L;
2009

Abstract

Moudafi and Mainge [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 (x(s,t)) of two parameters s, t is an element of (0, 1), and under certain conditions, the iterated lim(t -> 0) lim(s -> 0) x(s,t) exists in the norm topology. Moudafi, Mainge and Xu stated the problem of convergence of (x(s,t)) as (s, t) -> (0, 0) jointly in the norm topology. In this paper we further study the behaviour of the net (x(s,t)); in particular, we give a negative answer to this problem.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11770/152439
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