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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 ${\left\{{T}_{n}\right\}}_{n=1}^{\infty }$ of nonexpansive mappings ${T}_{n}:E\to E,\phantom{\rule{0.166667em}{0ex}}n=1,2,\cdots$, and denote by Fix $\left({T}_{n}\right)$ the set of all fixed points of ${T}_{n}$, i.e.,

$\text{Fix}\phantom{\rule{0.166667em}{0ex}}\left({T}_{n}\right)=\left\{x\in E:{T}_{n}\left(x\right)=x\right\}·$

An equilibrium function $F:E×E\to ℝ$, which defines the following equilibrium problem: find $x\in E$ such that $F\left(x,y\right)\ge 0,\phantom{\rule{0.166667em}{0ex}}\forall y\in E$. Denote by $EP\left(F\right)$ the set of equilibrium points of this problem.

A nonlinear mapping $B:E\to H$ which defines the following variational inequality: find $x\in E$ such that $〈Bx,y-x〉\ge 0,\phantom{\rule{0.166667em}{0ex}}\forall y\in E$, and denote by $VI\left(E,B\right)$ the set of all solutions of this variational problem.

In order to approximate an element

$z\in \bigcap _{n=1}^{\infty }\text{Fix}\phantom{\rule{0.166667em}{0ex}}\left({T}_{n}\right)\cap EP\left(F\right)\cap VI\left(E,B\right),$

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.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J20 Inequalities involving nonlinear operators
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