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Zbl 1175.47058
Colao, Vittorio; Acedo, Genaro López; Marino, Giuseppe
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)
[J] Nonlinear Anal., Theory Methods Appl. 71, No. 7-8, A, 2708-2715 (2009). ISSN 0362-546X

Let $H$ be a Hilbert space and $C$ a closed convex subset of $H$. An equilibrium function is a mapping $G:H\times H\to \Bbb R$ such that (A$_1$) $G(x,x)=0$ for all $x\in H$. A strongly positive operator is a bounded linear operator $A:H\to H$ such that for all $x\in H$, $\langle Ax,x\rangle\geq \bar{\gamma}\|x\|^2$ for some $\bar{\gamma}>0$. Supposing that the equilibrium function $G$ satisfies further the conditions (A$_2$) for all $x,y\in C$, $G(x,y)+G(y,x)\leq 0$ (i.e., $G$ is monotone); (A$_3$) for all $x,y,z \in C$ $\limsup_{t\to 0}G(tz+(1-t)x,y)\leq G(x,y)$, and (A$_4$) for all $x\in C$, the mapping $G(x,\cdot)$ is convex and lsc. {\it S.\,Plubtieng} and {\it R.\,Punpaeng} [J.~Math.\ Anal.\ Appl.\ 336, No.\,1, 455--469 (2007; Zbl 1127.47053)] proposed an iteration procedure to find the unique solution $z\in$ $\text{Fix}(T)\cap\text{SEP}(G)$ of the variational inequality: (1) $\langle(A-\gamma f)z,z-x\rangle\leq 0, $ for all $x\in\text{Fix}(T)\cap\text{SEP}(G)$. 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 $G_i,\, i=1,\dots,K,$ of equilibrium functions satisfying (A$_2$)--(A$_4$) and a family $(T_n)_{n\in \Bbb N}$ of nonexpansive mappings. Supposing that $D:=\cap_{i=1}^K$SEP$(G_i)\cap\cap_{n\in \Bbb N}$Fix$(T_n)\neq \varnothing$. 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.
[Stefan Cobzaş (Cluj-Napoca)]

MSC 2000:: *47J20 Inequalities involving nonlinear operators
47J25 Methods for solving nonlinear operator equations (general)
41A65 Abstract approximation theory
47H09 Mappings defined by "shrinking" properties

Keywords: equilibrium points; nonexpansive mapping; variational inequality; implicit iterative method; fixed point; minimization problem

Citations: Zbl 1127.47053

Cited in: Zbl 1225.47098 Zbl 1204.41028

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