Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 174318, 20 pages doi:10.1155/2012/174318 Research Article General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities Nopparat Wairojjana and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 24 March 2012; Accepted 16 May 2012 Academic Editor: Zhenyu Huang Copyright q 2012 N. Wairojjana and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with new methods for approximating a solution to the fixed point problem; find x ∈ FT , where H is a Hilbert space, C is a closed convex subset of H, f is a ρ-contraction from C into H,0 <ρ< 1, A is a strongly positive linear-bounded operator with coefficient γ> 0, 0 <γ< γ/ρ, T is a nonexpansive mapping on C, and P FT denotes the metric projection on the set of fixed point of T . Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality 〈A-γf xτ I -S x, x - x〉≥ 0 for x ∈ FT , where τ ∈ 0, ∞. Our results generalize and improve the corresponding results of Yao et al. 2010 and some authors. Furthermore, we give an example which supports our main theorem in the last part. 1. Introduction Throughout this paper, we assume that H is a real Hilbert space where inner product and norm are denoted by 〈·, ·〉 and ‖·‖, respectively, and let C be a nonempty closed convex subset of H. A mapping T : C → C is called nonexpansive if Tx - Ty ≤ x - y , ∀x, y ∈ C. 1.1 We use FT to denote the set of fixed points of T , that is, FT {x ∈ C : Tx x}. It is assumed throughout the paper that T is a nonexpansive mapping such that FT / ∅. Recall that a mapping f : C → H is a contraction on C if there exists a constant ρ ∈ 0, 1 such that f x - f ( y ) ≤ ρ x - y , ∀x, y ∈ C. 1.2