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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 xC such that


where C is a nonempty, closed and convex subset of a real Hilbert space H and f:C×C. 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 xC;

(A2) f is monotone;

(A3) x,y,zC, lim sup t0 f(tz+(1-t)x,y)f(x,y);

(A4) f(x,·) is convex and lower semicontinuous for all xC.

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:HH. Indeed, given such a map S whose fixed points are denoted by F(S) and under the assumption EP(f)F(S), they find a suitable sequence {x n }, generated starting from a point xH, such that {x n } converges strongly to the projection of x onto F(S)EP(f).

47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
49J40Variational methods including variational inequalities
47H09Mappings defined by “shrinking” properties
47N10Applications of operator theory in optimization, convex analysis, programming, economics