Nonlinear Analysis 74 (2011) 5286–5302 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Some iterative methods for finding fixed points and for solving constrained convex minimization problems L.-C. Ceng a,b , Q.H. Ansari c , J.-C. Yao d,∗ a Department of Mathematics, Shanghai Normal University, Shanghai 200234, China b Scientific Computing Key Laboratory of Shanghai Universities, China c Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India d Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan article info Article history: Received 12 February 2011 Accepted 3 May 2011 Communicated by Ravi Agarwal Keywords: Iterative schemes Variational inequality Fixed point Constrained convex minimization Nonexpansive mapping abstract The present paper is divided into two parts. In the first part, we introduce implicit and ex- plicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real Hilbert space. We establish results on the strong conver- gence of the sequences generated by the proposed schemes to a fixed point of a nonexpan- sive mapping. Such a fixed point is also a solution of a variational inequality defined on the set of fixed points. In the second part, we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping. The results of this paper extend and improve several results presented in the literature in the recent past. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction In 2000, Moudafi [1] proposed a viscosity approximation method for finding the fixed points of nonexpansive mappings and proved the convergence of the sequence generated by the proposed method. Xu [2] proved the strong convergence of the sequence generated by the viscosity approximation method to a unique solution of a certain variational inequality problem defined on the set of fixed points of a nonexpansive map. It is well known that the iterative methods for finding the fixed points of nonexpansive mappings can also be used to solve a convex minimization problem; see, for example, [3–5] and the references therein. In 2003, Xu [4] introduced an iterative method for computing the approximate solutions of a quadratic minimization problem over the set of fixed points of a nonexpansive mapping defined on a real Hilbert space. He proved that the sequence generated by the proposed method converges strongly to a unique solution of the quadratic minimization problem. By combining the iterative schemes proposed by Maudafi [1] and Xu [4], Marino and Xu [6] considered a general iterative method and proved that the sequence generated by the method converges strongly to a unique solution of a certain variational inequality problem which is the optimality condition for a particular minimization problem. Liu [7] and Qin et al. [8] also studied some applications of the iterative method considered in [6]. Yamada [5] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem and also studied the convergence of the sequence generated by the proposed method. Very recently, Tian [9] combined the iterative methods of [6,5] in order to propose implicit and explicit schemes for constructing a fixed point of a nonexpansive mapping T defined on a real Hilbert space. He ∗ Corresponding author. Tel.: +866 7 3229746x29; fax: +866 7 3111739. E-mail addresses: zenglc@hotmail.com (L.-C. Ceng), qhansari@gmail.com, qhansari@sancharnet.in (Q.H. Ansari), yaojc@kmu.edu.tw (J.-C. Yao). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.05.005