An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.

[1]  F. Browder,et al.  Construction of fixed points of nonlinear mappings in Hilbert space , 1967 .

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

[3]  William A. Kirk,et al.  Topics in metric fixed point theory: Minimal displacement , 1990 .

[4]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[5]  L. Ceng,et al.  WEAK CONVERGENCE THEOREM BY A MODIFIED EXTRAGRADIENT METHOD FOR NONEXPANSIVE MAPPINGS AND MONOTONE MAPPINGS , 2009 .

[6]  Hong-Kun Xu,et al.  Viscosity approximation methods for nonexpans mappings , 2004 .

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

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

[9]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[10]  Jian-Wen Peng,et al.  A MODIFIED CQ METHOD FOR EQUILIBRIUM PROBLEMS, FIXED POINTS AND VARIATIONAL INEQUALITY , 2008 .

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

[12]  Viscosity approximation methods for nonexpansive mappings and inverse-strongly monotone mappings in Hilbert spaces , 2008 .

[13]  Tie-Gang Fan,et al.  Viscosity approximation methods for nonexpansive mappings and monotone mappings , 2007 .

[14]  Fengshan Liu,et al.  Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates , 1998 .

[15]  J. Schu,et al.  Weak and strong convergence to fixed points of asymptotically nonexpansive mappings , 1991, Bulletin of the Australian Mathematical Society.

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

[17]  Muhammad Aslam Noor.  Modified projection method for pseudomonotone variational inequalities , 2002, Appl. Math. Lett..

[18]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[19]  Meijuan Shang,et al.  A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings , 2008 .