Viscosity Approximations by the Shrinking Projection Method of Quasi-Nonexpansive Mappings for Generalized Equilibrium Problems

We introduce viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of the super hybrid mappings in Hilbert spaces.

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

[2]  Wataru Takahashi,et al.  Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces , 2008 .

[3]  Rabian Wangkeeree.  An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings , 2008 .

[4]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , 2003 .

[5]  Tomonari Suzuki,et al.  Moudafi's viscosity approximations with Meir–Keeler contractions , 2007 .

[6]  Paul-Emile Maingé,et al.  Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems , 2008 .

[7]  Yasunori Kimura,et al.  Viscosity approximations by the shrinking projection method in Hilbert spaces , 2012, Comput. Math. Appl..

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

[9]  F. Browder.  Convergence theorems for sequences of nonlinear operators in Banach spaces , 1967 .

[10]  S. Atsushiba.  VISCOSITY APPROXIMATION METHODS FOR FIXED POINTS PROBLEMS , 2011 .

[11]  高橋 渉.  Introduction to nonlinear and convex analysis , 2009 .

[12]  Rabian Wangkeeree,et al.  An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems , 2009 .

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

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

[15]  Abdellatif Moudafi,et al.  Weak convergence theorems for nonexpansive mappings and equilibrium problems , 2008 .

[16]  Jen-Chih Yao,et al.  A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem , 2010 .

[17]  WEAK AND STRONG CONVERGENCE THEOREMS FOR GENERALIZED HYBRID MAPPINGS IN HILBERT SPACES , 2011 .

[18]  Emmett B. Keeler,et al.  A theorem on contraction mappings , 1969 .

[19]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[20]  U. Mosco.  Convergence of convex sets and of solutions of variational inequalities , 1969 .

[21]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

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

[23]  Yeol Je Cho,et al.  Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions , 2009 .

[24]  Fixed Point Theorems and Weak Convergence Theorems for Generalized Hybrid Mappings in Hilbert Spaces , 2010 .

[25]  M. Tsukada.  Convergence of best approximations in a smooth Banach space , 1984 .

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