A hybrid extragradient method extended to fixed point problems and equilibrium problems

In this article, we present a new hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone and Lipschitz-type continuous bifunction. We obtain strongly convergent theorems for the sequences generated by these processes in a real Hilbert space.

[1]  B. Halpern.  Fixed points of nonexpanding maps , 1967 .

[2]  Stella Dafermos,et al.  Exchange price equilibria and variational inequalities , 1990, Math. Program..

[3]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[4]  R. Wittmann.  Approximation of fixed points of nonexpansive mappings , 1992 .

[5]  W. Oettli,et al.  From optimization and variational inequalities to equilibrium problems , 1994 .

[6]  A. Moudafi.  Proximal point algorithm extended to equilibrium problems , 1999 .

[7]  I. Konnov.  Application of the Proximal Point Method to Nonmonotone Equilibrium Problems , 2003 .

[8]  Antonino Maugeri,et al.  Equilibrium problems and variational models , 2003 .

[9]  Giandomenico Mastroeni,et al.  Gap Functions for Equilibrium Problems , 2003, J. Glob. Optim..

[10]  Hong-Kun Xu,et al.  Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities , 2003 .

[11]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[12]  Pham Ngoc Anh,et al.  Using the Banach Contraction Principle to Implement the Proximal Point Method for Multivalued Monotone Variational Inequalities , 2005 .

[13]  Wataru Takahashi,et al.  Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings , 2006 .

[14]  Lu-Chuan Zeng,et al.  STRONG CONVERGENCE THEOREM BY AN EXTRAGRADIENT METHOD FOR FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS , 2006 .

[15]  Wataru Takahashi,et al.  Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces , 2007 .

[16]  T. D. Quoc,et al.  Extragradient algorithms extended to equilibrium problems , 2008 .

[17]  J. Strodiot,et al.  GENERALIZED PROJECTION METHOD FOR NON-LIPSCHITZ MULTIVALUED MONOTONE VARIATIONAL INEQUALITIES , 2009 .

[18]  Y. Liou,et al.  An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems , 2009 .

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

[20]  Shenghua Wang,et al.  New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces , 2010, J. Comput. Appl. Math..

[21]  Pham Ngoc Anh,et al.  Outer approximation algorithms for pseudomonotone equilibrium problems , 2011, Comput. Math. Appl..