##
**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].

(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 |