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On an implicit hierarchical fixed point approach to variational inequalities. (English) Zbl 1168.49005
Summary: A. Moudafi and P.-E. Maingé [Fixed Point Theory Appl. 2006, Article ID 95453, 10 p. (2006; Zbl 1143.47305)] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math. 13, No. 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(0,1), and under certain conditions, the iterated lim t0 lim s0 x s,t exists in the norm topology. Moudafi, Maingé 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.
MSC:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47H09Mappings defined by “shrinking” properties