Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems

In this paper, a new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Applications to optimization problems are given.

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

[2]  Xiaoming Yuan,et al.  An approximate proximal-extragradient type method for monotone variational inequalities , 2004 .

[3]  Sjur Didrik Flåm,et al.  Equilibrium programming using proximal-like algorithms , 1997, Math. Program..

[4]  Mikhail V. Solodov.  Convergence rate analysis of iteractive algorithms for solving variational inequality problems , 2003, Math. Program..

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

[6]  Tomonari Suzuki,et al.  Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces , 2005 .

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

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

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

[10]  Meijuan Shang,et al.  An iterative method of solution for equilibrium and optimization problems , 2008 .

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

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

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

[14]  Wataru Takahashi,et al.  Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem , 2007 .

[15]  Somyot Plubtieng,et al.  A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings , 2008, Appl. Math. Comput..

[16]  Michael C. Ferris,et al.  Finite termination of the proximal point algorithm , 1991, Math. Program..

[17]  A. Iusem,et al.  Proximal Methods with Penalization Effects in Banach Spaces , 2004 .

[18]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[19]  B. Martinet.  Perturbation des méthodes d'optimisation. Applications , 1978 .

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

[21]  Benar Fux Svaiter,et al.  An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions , 2000, Math. Oper. Res..

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

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

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

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

[26]  Chi Kin Chan,et al.  A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization , 2009 .

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