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;
2009-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
