Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points

Abstract In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.

[1]  W. Oettli.  A remark on vector-valued equilibria and generalized monotonicity , 1997 .

[2]  Jian-Wen Peng.  Iterative Algorithms for Mixed Equilibrium Problems, Strict Pseudocontractions and Monotone Mappings , 2009 .

[3]  Giuseppe Marino,et al.  A general iterative method for nonexpansive mappings in Hilbert spaces , 2006 .

[4]  P. L. Combettes,et al.  Equilibrium programming in Hilbert spaces , 2005 .

[5]  H. Riahi,et al.  Variational Methods in Partially Ordered Spaces , 2003 .

[6]  Peter K. F. Kuhfittig.  Common fixed points of nonexpansive mappings by iteration. , 1981 .

[7]  Giuseppe Marino,et al.  An iterative method for finding common solutions of equilibrium and fixed point problems , 2008 .

[8]  Ronald E. Bruck.  Properties of fixed-point sets of nonexpansive mappings in Banach spaces , 1973 .

[9]  Giuseppe Marino,et al.  Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems , 2009 .

[10]  Vittorio Colao,et al.  ON AN IMPLICIT HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES , 2009 .

[11]  A. Moudafi.  Viscosity Approximation Methods for Fixed-Points Problems , 2000 .

[12]  Paul-Emile Maingé,et al.  Towards viscosity approximations of hierarchical fixed-point problems , 2006 .

[13]  Wataru Takahashi,et al.  Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings , 2003 .

[14]  Hong-Kun Xu.  VISCOSITY APPROXIMATION METHODS FOR NONEXPANSIVE MAPPINGS , 2004 .

[15]  Paul-Emile Maingé,et al.  Strong convergence of an iterative method for hierarchical fixed point problems , 2007 .

[16]  Abdellatif Moudafi,et al.  Weak convergence theorems for nonexpansive mappings and equilibrium problems , 2008 .

[17]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[18]  J. Reinermann,et al.  Über Fixpunkte kontrahierender Abbildungen und schwach konvergente Toeplitz-Verfahren , 1969 .

[19]  Jen-Chih Yao,et al.  A hybrid iterative scheme for mixed equilibrium problems and fixed point problems , 2008 .

[20]  Giuseppe Marino,et al.  A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem , 2009 .

[21]  Jian-Wen Peng,et al.  A NEW HYBRID-EXTRAGRADIENT METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS, FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS , 2008 .

[22]  Yonghong Yao.  A general iterative method for a finite family of nonexpansive mappings , 2007 .

[23]  Hong-Kun Xu.  An Iterative Approach to Quadratic Optimization , 2003 .