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An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. (English) Zbl 1175.47058
Let $H$ be a Hilbert space and $C$ a closed convex subset of $H$. An equilibrium function is a mapping $G:H×H\to ℝ$ such that (A${}_{1}$) $G\left(x,x\right)=0$ for all $x\in H$. A strongly positive operator is a bounded linear operator $A:H\to H$ such that for all $x\in H$, $〈Ax,x〉\ge \overline{\gamma }{\parallel x\parallel }^{2}$ for some $\overline{\gamma }>0$. Supposing that the equilibrium function $G$ satisfies further the conditions (A${}_{2}$) for all $x,y\in C$, $G\left(x,y\right)+G\left(y,x\right)\le 0$ (i.e., $G$ is monotone); (A${}_{3}$) for all $x,y,z\in C$ ${lim sup}_{t\to 0}G\left(tz+\left(1-t\right)x,y\right)\le G\left(x,y\right)$, and (A${}_{4}$) for all $x\in C$, the mapping $G\left(x,·\right)$ is convex and lsc. S. Plubtieng and R. Punpaeng [J. Math. Anal. Appl. 336, No. 1, 455–469 (2007; Zbl 1127.47053)] proposed an iteration procedure to find the unique solution $z\in$ $\text{Fix}\left(T\right)\cap \text{SEP}\left(G\right)$ of the variational inequality: (1) $〈\left(A-\gamma f\right)z,z-x〉\le 0,$ for all $x\in \text{Fix}\left(T\right)\cap \text{SEP}\left(G\right)$. Here, $T$ is a nonexpansive mapping on $H$, $A$ is a strongly positive operator on $H$, $f$ an $\alpha$-contraction on $H$ and $\gamma >0$ an appropriate constant. In this paper, the authors extend the above result by considering a family ${G}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,\cdots ,K,$ of equilibrium functions satisfying (A${}_{2}$)–(A${}_{4}$) and a family ${\left({T}_{n}\right)}_{n\in ℕ}$ of nonexpansive mappings. Supposing that $D:={\cap }_{i=1}^{K}$SEP$\left({G}_{i}\right)\cap {\cap }_{n\in ℕ}$Fix$\left({T}_{n}\right)\ne ⌀$. They propose an implicit iteration procedure for finding the unique solution of the variational inequality (1) on the set $D$ and prove the strong convergence of this procedure.

##### MSC:
 47J20 Inequalities involving nonlinear operators 47J25 Iterative procedures (nonlinear operator equations) 41A65 Abstract approximation theory 47H09 Mappings defined by “shrinking” properties