Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings

We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., (2008) and many others.

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

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

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

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

[5]  P. Kumam,et al.  Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization , 2009 .

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

[7]  Mohammad Asadzadeh.  A RELAXED EXTRAGRADIENT APPROXIMATION METHOD OF TWO INVERSE-STRONGLY MONOTONE MAPPINGS FOR A GENERAL SYSTEM OF VARIATIONAL INEQUALITIES , FIXED POINT AND EQUILIBRIUM PROBLEMS , 2010 .

[8]  József Kolumbán,et al.  Factorization of Minty and Stampacchia variational inequality systems , 2002, Eur. J. Oper. Res..

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

[10]  Chi Kin Chan,et al.  Algorithms of common solutions to quasi variational inclusion and fixed point problems , 2008 .

[11]  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..

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

[13]  Lu-Chuan Zeng,et al.  STRONG CONVERGENCE THEOREMS FOR STRICTLY PSEUDOCONTRACTIVE MAPPINGS OF BROWDER-PETRYSHYN TYPE , 2006 .

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

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

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

[17]  Jen-Chih Yao,et al.  Pseudomonotone Complementarity Problems and Variational Inequalities , 2005 .

[18]  M. O. Osilike,et al.  Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations , 2000 .

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