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)


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