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Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. (English) Zbl 1122.47055
Takahashi, Wataru (ed.) et al., Nonlinear analysis and convex analysis. Proceedings of the 4th international conference (NACA 2005), Okinawa, Japan, June 30–July 4, 2005. Yokohama: Yokohama Publishers (ISBN 978-4-946552-27-4/hbk). 609-617 (2007).

The authors consider the following equilibrium problem: find $x\in C$ such that

$f\left(x,y\right)\ge 0\phantom{\rule{1.em}{0ex}}\forall y\in C,$

where $C$ is a nonempty, closed and convex subset of a real Hilbert space $H$ and $f:C×C\to ℝ$. The set of solutions is denoted by $EP\left(f\right)$. Within the present paper, the function $f$ is supposed to satisfy the following assumptions:

(A1) $f\left(x,x\right)=0$ for all $x\in C;$

(A2) $f$ is monotone;

(A3) $\forall x,y,z\in C$, ${lim sup}_{t↓0}f\left(tz+\left(1-t\right)x,y\right)\le f\left(x,y\right)$;

(A4) $f\left(x,·\right)$ is convex and lower semicontinuous for all $x\in C$.

In their main result, the authors provide a strong convergence theorem which solves the problem of finding a common element of the set $EP\left(f\right)$ and the set of fixed points of a nonexpansive mapping $S:H\to H$. Indeed, given such a map $S$ whose fixed points are denoted by $F\left(S\right)$ and under the assumption $EP\left(f\right)\cap F\left(S\right)\ne ⌀$, they find a suitable sequence $\left\{{x}_{n}\right\}$, generated starting from a point $x\in H$, such that $\left\{{x}_{n}\right\}$ converges strongly to the projection of $x$ onto $F\left(S\right)\cap EP\left(f\right)$.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47J20 Inequalities involving nonlinear operators 49J40 Variational methods including variational inequalities 47H09 Mappings defined by “shrinking” properties 47N10 Applications of operator theory in optimization, convex analysis, programming, economics