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A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings. (English) Zbl 1207.47071

Let H be a Hilbert space and let E be a nonempty closed convex subset of H. Consider the following:

An infinite family {T n } n=1 of nonexpansive mappings T n :EE,n=1,2,, and denote by Fix (T n ) the set of all fixed points of T n , i.e.,

Fix(T n )=xE:T n (x)=x·

An equilibrium function F:E×E, which defines the following equilibrium problem: find xE such that F(x,y)0,yE. Denote by EP(F) the set of equilibrium points of this problem.

A nonlinear mapping B:EH which defines the following variational inequality: find xE such that Bx,y-x0,yE, and denote by VI(E,B) the set of all solutions of this variational problem.

In order to approximate an element

z n=1 Fix(T n )EP(F)VI(E,B),

a priori supposed to exist, the authors introduce a (very complicated) hybrid algorithm for which they state and prove a strong convergence theorem (Theorem 3.1).

Alas, no examples are given to illustrate that the hypotheses in Theorem 3.1 are feasible.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
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