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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

f(x,y)0yC,

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).


MSC:
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