Some new unified iteration schemes with errors for nonexpansive mappings and variational inequalities

In this paper, we suggest and analyze a class of unified iteration schemes with errors for the variational inequalities coupled with nonexpansive mappings in Hilbert spaces. The main results of this paper extend the main results of Iiduka and Takahashi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341-350], and Takahasi and Toyoda [W. Takahasi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2) (2003) 417-428] to the case of strong convergence for relaxed (γ, r)-cocoercive mappings involved in variational inequalities. Results obtained in this paper may be viewed as an refinement of the previously known results in this field.