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On an implicit hierarchical fixed point approach to variational inequalities. (English) Zbl 1168.49005
Summary: {\it A. Moudafi} and {\it P.-E. Maingé} [Fixed Point Theory Appl. 2006, Article ID 95453, 10 p. (2006; Zbl 1143.47305)] and {\it 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 $s,t\in(0,1)$, and under certain conditions, the iterated $\lim_{t\to 0} \lim_{s\to 0} x_{s,t}$ exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of $(x_{s,t})$ as $(s,t)\to (0,0)$ 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.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47H09Mappings defined by “shrinking” properties
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