## 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 $$G:H\times H\to \mathbb R$$ 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. 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 $$z\in$$ $$\text{Fix}(T)\cap\text{SEP}(G)$$ 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 $$(T_n)_{n\in \mathbb N}$$ of nonexpansive mappings. Supposing that $$D:=\cap_{i=1}^K$$SEP$$(G_i)\cap\cap_{n\in \mathbb N}$$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.

### MSC:

 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.

Zbl 1127.47053
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