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We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If T : C → H is a non-self and non-expansive mapping, we can define a map h : C → R by h(x) := inf{λ ≥ 0 : λx + (1 -λ)Tx ∈ C}. Then, for a fixed x 0 ∈ C and for α 0 := max{1/2, h(x 0 )}, we define the Krasnoselskii-Mann algorithm x n+1 = α n x n + (1 -α n )Tx n , where α n+1 = max{α n , h(x n+1 )}. We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.

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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 (x s,t ) of two parameters s, t ∈ (0, 1), and under certain conditions, the iterated lim t→0 lim s→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) → (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.2000 Mathematics subject classification: primary 49J40; secondary 47J20, 47H09.

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