## 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 $f(x,y)\geq 0\quad \forall y\in C,$ where $$C$$ is a nonempty, closed and convex subset of a real Hilbert space $$H$$ and $$f:C\times C\to {\mathbb R}$$. The set of solutions is denoted by $$EP(f)$$. Within the present paper, the function $$f$$ is supposed to satisfy the following assumptions:
(A1) $$f(x,x)=0$$ for all $$x\in C;$$
(A2) $$f$$ is monotone;
(A3) $$\forall x,y,z\in C$$, $$\limsup_{t\downarrow 0}f(tz+(1-t)x,y)\leq f(x,y)$$;
(A4) $$f(x,\cdot)$$ 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 $$EP(f)$$ 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 $$F(S)$$ and under the assumption $$EP(f)\cap F(S)\neq \varnothing$$, they find a suitable sequence $$\{x_n\}$$, generated starting from a point $$x\in H$$, such that $$\{x_n\}$$ converges strongly to the projection of $$x$$ onto $$F(S)\cap EP(f)$$.
For the entire collection see [Zbl 1104.47002].
Reviewer: Rita Pini (Milano)

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47J20 Variational and other types of inequalities involving nonlinear operators (general) 49J40 Variational inequalities 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics