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Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. (English) Zbl 1130.90055
Summary: In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process

MSC:
90C52Methods of reduced gradient type
49J40Variational methods including variational inequalities
References:
[1] · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[2] · Zbl 0924.49009 · doi:10.1023/A:1008643727926
[3]
[4] · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[5]
[6] · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[7] · doi:10.1090/S0002-9947-1970-0282272-5
[8] · Zbl 0709.47051 · doi:10.1017/S0004972700028884
[9]Yamada I. The Hybrid Steepest-Descent Method for the Variational Inequality Problem over the Intersection of Fixed-Point Sets of Nonexpansive Mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Edited by D. Butnariu, Y. Censor, and S. Reich, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 473–504, 2001.