A synthetic algorithm for families of demicontractive and nonexpansive mappings and equilibrium problems

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 5891-5908
Author(s):  
Ali Abkara ◽  
Mohsen Shekarbaigia
Keyword(s):  
Fixed Points ◽  
Rate Of Convergence ◽  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Common Element ◽  
Numerical Examples ◽  
Demicontractive Mappings

We study the rate of convergence of a new synthetic algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a pair of nonexpansive mappings and two finite families of demicontractive mappings. We then provide some numerical examples to illustrate our main result and the proposed algorithm.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Nimit Nimana
Keyword(s):  
Fixed Points ◽  
Equilibrium Problem ◽  
Projection Method ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Generalized Equilibrium Problem ◽  
Shrinking Projection Method ◽  
Generalized Equilibrium

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.

scholarly journals Convergence Theorem for Equilibrium and Variational Inequality Problems and a Family of Infinitely Nonexpansive Mappings in Hilbert Space

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhou Yinying ◽  
Cao Jiantao ◽  
Wang Yali
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problem ◽  
Convergence Theorem ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Common Element

We introduce a hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the varitional inequality problem and the equilibrium problem in Hilbert space. Under suitable conditions, some strong convergence theorems are obtained. Our results improve and extend the corresponding results in (Chang et al. (2009), Min and Chang (2012), Plubtieng and Punpaeng (2007), S. Takahashi and W. Takahashi (2007), Tada and Takahashi (2007), Gang and Changsong (2009), Ying (2013), Y. Yao and J. C. Yao (2007), and Yong-Cho and Kang (2012)).

scholarly journals Convergence Theorems for BregmanK-Mappings and Mixed Equilibrium Problems in Reflexive Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Bashir Ali ◽  
M. H. Harbau
Keyword(s):  
Fixed Points ◽  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Finite Family ◽  
Iterative Sequence ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of BregmanK-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the BregmanK-mapping is the set of common fixed points of{Ti}i=1N. Using the BregmanK-mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.

scholarly journals A New Hybrid Algorithm for Solving a System of Generalized Mixed Equilibrium Problems, Solving a Family of Quasi--Asymptotically Nonexpansive Mappings, and Obtaining Common Fixed Points in Banach Space

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
J. F. Tan ◽  
S. S. Chang
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Generalized Mixed Equilibrium Problems ◽  
Asymptotically Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi--asymptotically nonexpansive mappings, and null spaces of finite family of -inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors.

scholarly journals Fixed Points Approximation and Solutions of Some Equilibrium and Variational Inequalities Problems

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Bashir Ali
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strong Convergence Theorem ◽  
Points Approximation

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.

scholarly journals Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Prasit Cholamjiak ◽  
Suthep Suantai
Keyword(s):  
Banach Spaces ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Convex Minimization Problems ◽  
The Common ◽  
System Of Equilibrium Problems

We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.

scholarly journals Approximate Equilibrium Problems and Fixed Points

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
H. Mazaheri ◽  
S. A. M. Mohsenalhosseini
Keyword(s):  
Hilbert Space ◽  
Fixed Points ◽  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Common Element ◽  
Approximate Equilibrium

We find a common element of the set of fixed points of a map and the set of solutions of an approximate equilibrium problem in a Hilbert space. Then, we show that one of the sequences weakly converges. Also we obtain some theorems about equilibrium problems and fixed points.

An Iterative Method for Mixed Equilibrium Problems and Fixed Points

2012 ◽  
Vol 263-266 ◽  
pp. 283-286 ◽  
Author(s):  
Qiao Hong Jiang
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Engineering Calculation ◽  
Common Element ◽  
Rounding Errors ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

Fixed point computation plays an important role in the field of engineering calculation. Rounding errors often cause no convergence for iteration sequence or results distortion in many fixed point iterative method. In this paper, we prove the strong convergence of an iterative method for finding a common element of the set of olutions of mixed equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings under some suitable conditions. Result presented in this paper improves and extends the recent known results in this area.

An Iterative Method for Equilibrium Problems and Fixed Point Problems

2014 ◽  
Vol 513-517 ◽  
pp. 382-385
Author(s):  
Chen Min ◽  
Qiao Hong Jiang
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Fixed Point Problems ◽  
Common Element ◽  
Mixed Equilibrium Problems

In this paper, we prove the strong convergence of an iterative method for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings under some suitable conditions. Result presented in this paper improves and extends the recent known results in this area.

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