Fixed Point Theory, 16(2015), No. 1, 67-90 http://www.math.ubbcluj.ro/ ∼ nodeacj/sfptcj.html APPROXIMATION METHODS FOR TRIPLE HIERARCHICAL VARIATIONAL INEQUALITIES (I) L.-C. CENG * , Q.H. ANSARI ** , A. PETRUS ¸EL *** AND J.-C. YAO **** * Department of Mathematics, Shanghai Normal University, Shanghai 200234 and Scientific Computing Key Laboratory of Shanghai Universities, China E-mail: zenglc@hotmail.com ** Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India E-mail: qhansari@gmail.com *** Department of Applied Mathematics, Babes-Bolyai University Cluj-Napoca Kog˘alniceanu Street No.1, 400084 Cluj-Napoca, Romania E-mail: petrusel@math.ubbcluj.ro **** Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan and Department of Mathematics, King Abdulaziz University, P.O. Box 80203 Jeddah 21589, Saudi Arabia E-mail: yaojc@cc.kmu.edu.tw Abstract. In this work, we consider two types of triple hierarchical variational inequalities (in short, THVI), one with a single nonexpansive mapping and another one with a finite family of nonexpansive mappings. In this paper, by combining the viscosity approximation method, hybrid steepest-descent method and Mann’s iteration method, we propose the hybrid steepest-descent viscosity approxima- tion method for solving the THVI. The strong convergence of this method to a unique solution of the THVI is studied under some appropriate assumptions. Another iterative algorithm for solving THVI is also presented. Under some mild conditions, we prove that the sequence generated by the proposed algorithm converges strongly to a unique solution of THVI. The case of a finite family of nonexpansive mappings will ve presented in the second part of this work. Key Words and Phrases: Triple hierarchical variational inequalities, hybrid steepest-descent viscosity approximation method, monotone operators, nonexpansive mappings, fixed points, strong convergence theorems. 2010 Mathematics Subject Classification: 49J40, 47H10, 65J20, 65J25, 49J30, 47H05, 47H09. 1. Introduction and formulations Let H be a real Hilbert space with its inner product and norm are denoted by 〈·, ·〉 and ‖·‖, respectively. The set of all fixed points of a mapping T : H → H is denoted by Fix(T ), that is, Fix(T )= {x ∈ H : Tx = x}. The mapping T : H → H is called L-Lipschitzian if there exists a constant L ≥ 0 such that ‖Tx − Ty‖≤ L‖x − y‖ for all x, y ∈ H. In particular, if L ∈ [0, 1), T is called a contraction mapping, while if L = 1, then T is called a nonexpansive mapping. 67