scholarly journals Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family ofki-Strictly Pseudocontractive Mapping in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Haitao Che ◽  
Meixia Li ◽  
Xintian Pan
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Element ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Definition Of ◽  
Strictly Pseudocontractive Mappings

We first extend the definition of Wnfrom an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family ofki-strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

scholarly journals A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

2010 ◽  
Vol 2010 ◽  
pp. 1-27 ◽  
Author(s):  
Jian-Wen Peng ◽  
Soon-Yi Wu ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Infinite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
System Of Equilibrium Problems ◽  
New Iterative Method

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

scholarly journals General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Peichao Duan ◽  
Aihong Wang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Iterative Methods ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Convergence Theorems ◽  
Pseudocontractive Mappings

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

scholarly journals A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Haitao Che ◽  
Xintian Pan
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Infinite Family ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Common Element ◽  
Hybrid Steepest Descent Method ◽  
Pseudocontractive Mappings ◽  
Strictly Pseudocontractive Mappings

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

scholarly journals An Iterative Method for Solving the Generalized System of Relaxed Cocoercive Quasivariational Inclusions and Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings

2010 ◽  
Vol 2010 ◽  
pp. 1-23
Author(s):  
Pattanapong Tianchai ◽  
Rabian Wangkeeree
Keyword(s):  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Strong Convergence Theorem ◽  
Viscosity Approximation ◽  
Generalized System ◽  
Pseudocontractive Mappings ◽  
Quasivariational Inclusions ◽  
Strictly Pseudocontractive Mappings

We introduce an iterative scheme by the viscosity approximation to find the set of solutions of the generalized system of relaxed cocoercive quasivariational inclusions and the set of common fixed points of an infinite family of strictly pseudocontractive mappings problems in Hilbert spaces. We suggest and analyze an iterative scheme under some appropriate conditions imposed on the parameters; we prove that another strong convergence theorem for the above two sets is obtained. The results presented in this paper improve and extend the main results of Li and Wu (2010) and many others.

scholarly journals Viscosity Approximation Methods for Equilibrium Problems, Variational Inequality Problems of Infinitely Strict Pseudocontractions in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Aihong Wang
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Approximation Methods ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Pseudocontractive Mappings ◽  
Viscosity Approximation Methods

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

scholarly journals On Common Solutions for Fixed-Point Problems of Two Infinite Families of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-32
Author(s):  
Pattanapong Tianchai
Keyword(s):  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Common Fixed Points ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Pseudocontractive Mappings ◽  
Quasivariational Inclusions ◽  
Strictly Pseudocontractive Mappings

This paper is concerned with a common element of the set of common fixed points for two infinite families of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a novel general iterative scheme based on the viscosity approximation method, and applicability of the results has shown difference with the results of many others existing in the current literature.

scholarly journals A Hybrid Extragradient-Like Method for Variational Inequalities, Equilibrium Problems, and an Infinitely Family of Strictly Pseudocontractive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Yaqin Wang ◽  
Hongkun Xu ◽  
Xiaoli Fang
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Sequence ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Mixed Equilibrium Problem ◽  
Pseudocontractive Mappings ◽  
Generalized Mixed Equilibrium ◽  
Strictly Pseudocontractive Mappings

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan (2010) and many others.

scholarly journals Shrinking Projection Method for Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Pattanapong Tianchai
Keyword(s):  
Projection Method ◽  
Hilbert Spaces ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Shrinking Projection Method ◽  
Pseudocontractive Mappings ◽  
Quasivariational Inclusions ◽  
Strictly Pseudocontractive Mappings

This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

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