Algorithms for a System of General Variational Inequalities in Banach Spaces

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

[1]  Yair Censor,et al.  The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space , 2011, J. Optim. Theory Appl..

[2]  Tomonari Suzuki Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals , 2005 .

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

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

[5]  Jen-Chih Yao,et al.  Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities , 2008, Math. Methods Oper. Res..

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

[7]  Hong-Kun Xu Inequalities in Banach spaces with applications , 1991 .

[8]  Ram U. Verma,et al.  General convergence analysis for two-step projection methods and applications to variational problems , 2005, Appl. Math. Lett..

[9]  Muhammad Aslam Noor,et al.  Some new unified iteration schemes with errors for nonexpansive mappings and variational inequalities , 2007, Appl. Math. Comput..

[10]  Wataru Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings , 2005 .

[11]  Wataru Takahashi,et al.  Weak and Strong Convergence of Solutions to Accretive Operator Inclusions and Applications , 2000 .

[12]  R. U. Verma,et al.  Projection methods, algorithms, and a new system of nonlinear variational inequalities , 2001 .

[13]  S. Reich Weak convergence theorems for nonexpansive mappings in Banach spaces , 1979 .

[14]  Jen-Chih Yao,et al.  An Extragradient Method for Fixed Point Problems and Variational Inequality Problems , 2007 .

[15]  Yair Censor,et al.  The Split Variational Inequality Problem , 2010 .

[16]  Muhammad Aslam Noor,et al.  On viscosity iterative methods for variational inequalities , 2007 .

[17]  Jen-Chih Yao,et al.  On modified iterative method for nonexpansive mappings and monotone mappings , 2007, Appl. Math. Comput..

[18]  Xiaolong Qin,et al.  Strong Convergence Theorems for Nonexpansive Mapping , 2007, J. Syst. Sci. Complex..

[19]  Shin Min Kang,et al.  Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spac , 2011 .

[20]  Muhammad Aslam Noor,et al.  Strong convergence of three-step relaxed hybrid steepest-descent methods for variational inequalities , 2008, Appl. Math. Comput..

[21]  Wataru Takahashi,et al.  Weak convergence of an iterative sequence for accretive operators in Banach spaces , 2006 .

[22]  Yair Censor,et al.  Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space , 2012 .

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

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

[25]  Muhammad Aslam Noor,et al.  An explicit projection method for a system of nonlinear variational inequalities with different (gamma, r)-cocoercive mappings , 2007, Appl. Math. Comput..

[26]  Wataru Takahashi,et al.  APPROXIMATION OF SOLUTIONS OF VARIATIONAL INEQUALITIES FOR MONOTONE MAPPINGS , 2004 .

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