Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. (English) Zbl 1142.47350

Takahashi, Satoru; Takahashi, Wataru

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 3, 1025-1033 (2008).

Summary: We introduce an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets. Using this result, we prove three new strong convergence theorems in fixed point problems, variational inequalities and equilibrium problems.

MSC:

47J05	Equations involving nonlinear operators (general)
47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

nonexpansive mapping; fixed point; inverse-strongly monotone mapping; variational inequality; equilibrium problem

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