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Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. (English) Zbl 1122.47056

The authors provide a result strictly related to Theorem 1 in [A. Tada and W. Takahashi, Proc. NACA (Okinawa, 2005), 609–617 (2007; Zbl 1122.47055), reviewed above]. Indeed, they consider the 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).

Under the same assumptions of Theorem 1, and given in addition a contraction f:HH, they find a way to generate two sequences of points, namely {x n } and {u n }, approximating in the viscosity sense the equilibria that are also the fixed points F(S) of a nonexpansive map S, i.e., both of them converge strongly to a point zEP(F)F(S), where z is the projection of f(z) onto EP(F)F(S). As corollaries, they get results previously obtained by R. Wittman [Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)] and P. L. Combettes and S. A. Hirstoaga [J. Nonlinear Convex Anal. 6, No. 1, 117–136 (2005; Zbl 1109.90079)].


MSC:
47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
49J40Variational methods including variational inequalities
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
90C47Minimax problems