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A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. (English) Zbl 1127.47053

Summary: We introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem. These results extended and improved the corresponding results of [G. Marino and H. K. Xu, J. Math. Anal. Appl. 318, No. 1, 43–52 (2006; Zbl 1095.47038); S. Takahashi and W. Takahashi, ibid. 331, No. 1, 506–515 (2007; Zbl 1122.47056)], and many others.

MSC:

47J25 Iterative procedures involving nonlinear operators
49J40 Variational inequalities
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators
90C47 Minimax problems in mathematical programming
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References:

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[6] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. math. anal. appl., 331, 1, 506-515, (2007) · Zbl 1122.47056
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