Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.

*(English)*Zbl 1130.90055Summary: 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

##### Keywords:

Extragradient method; fixed points; monotone mappings; nonexpansive mappings; variational inequalities
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\textit{N. Nadezhkina} and \textit{W. Takahashi}, J. Optim. Theory Appl. 128, No. 1, 191--201 (2006; Zbl 1130.90055)

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##### References:

[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. · Zbl 1013.49005 |

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