Some new unified iteration schemes with errors for nonexpansive mappings and variational inequalities Zhenyu Huang a, * ,1 , Muhammad Aslam Noor b,2 a Department of Mathematics, Nanjing University, Nanjing 210093, PR China b Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract In this paper, we suggest and analyze a class of unified iteration schemes with errors for the variational inequalities cou- pled 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 (c, 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. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Unified iteration scheme with errors; Strong convergence; Common elements; Nonexpansive mappings; a-inverse strongly monotonic mappings; Relaxed (c, r)-cocoercive mappings; Nonlinear variational inequalities; Projection mappings 1. Preliminaries and basic results Variational inequality problems have been found with an explosive growth in theoretical advances, algo- rithmic development and applications across all the discipline of pure and applied sciences, see [1–8] and the references therein. Analysis of these problems requires a blend of techniques from convex analysis, func- tional analysis and numerical analysis. As a result of the interaction between different branches of mathemat- ical and engineering sciences, we now have a variety of techniques to suggest and analyze various algorithms for solving variational inequalities and related optimization problems. Using the projection technique, one can establish the equivalence between the variational inequalities and fixed point problems. This alternative equiv- alent formulation has played an important role in developing some efficient numerical techniques for solving variational inequalities and related optimization problems. 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.056 * Corresponding author. E-mail addresses: zhenyunju1@jsmail.com.cn, junhome@jsmail.com.cn, zhenyu@nju.edu.cn (Z. Huang), noormaslam@hotmail.com, noormaslam@gmail.com (M.A. Noor). 1 The Project is sponsored by the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry. 2 This research is supported by the Higher Education Commission, Pakistan, through Grant No: 1-28/HEC/HRD/2005/90. Available online at www.sciencedirect.com Applied Mathematics and Computation 194 (2007) 135–142 www.elsevier.com/locate/amc