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 such that
where is a nonempty, closed and convex subset of a real Hilbert space and . The set of solutions is denoted by . Within the present paper, the function is supposed to satisfy the following assumptions:
(A1) for all
(A2) is monotone;
(A3) , ;
(A4) is convex and lower semicontinuous for all .
In their main result, the authors provide a strong convergence theorem which solves the problem of finding a common element of the set and the set of fixed points of a nonexpansive mapping . Indeed, given such a map whose fixed points are denoted by and under the assumption , they find a suitable sequence , generated starting from a point , such that converges strongly to the projection of onto .