In this paper, we study the structure of projection methods for
variational inequality problems and then prove weak convergence theorems
which generalize Takahashi and Toyoda [W. Takahashi and M. Toyoda, Weak
convergence theorems for nonepxansive mappings and monotone mappings,
J. Optim. Theory Appl. 118 (2003), 417–428] and Nadezhkina and Takahashi
[N. Nadezhkina and W. Takahashi, Weak convergence theorem by an
extragradient method for nonexpansive mappings and monotone mappings, J.
Optim. Theory Appl. 128 (2006), 191-201]. Our proofs are different from
them. Furthermore, using these weak convergence theorems, we obtain some
new results.
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