scholarly journals Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Poom Kumam ◽  
Thanyarat Jitpeera
Keyword(s):  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Variational Inclusion ◽  
Common Element ◽  
Point Problem ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.

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.

scholarly journals Strong Convergence Theorems for Family of Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-22
Author(s):  
Yekini Shehu
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three sets. We give some applications of our results. Our results extend important recent results.

scholarly journals A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized 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. Our results improve and extend the corresponding results of Marino and Xu (2006), Su et al. (2008), Klin-eam and Suantai (2009), Tan and Chang (2011), and some other authors.

scholarly journals An iterative algorithm for the system of split mixed equilibrium problem

2020 ◽  
Vol 53 (1) ◽  
pp. 309-324
Author(s):  
Ibrahim Karahan ◽  
Lateef Olakunle Jolaoso
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Equilibrium System ◽  
Convergence Conditions ◽  
Mixed Equilibrium Problems ◽  
Split Variational Inequality ◽  
Mixed Equilibrium ◽  
Split Mixed Equilibrium Problem

AbstractIn this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

scholarly journals The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Rabian Wangkeeree ◽  
Panatda Boonman
Keyword(s):  
Hilbert Space ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Solution Set ◽  
Common Element ◽  
Fixed Point Set ◽  
Mixed Equilibrium Problems ◽  
Point Set ◽  
Mixed Equilibrium

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptoticTJmapping and the solution set of SMEP in a real Hilbert space.

scholarly journals The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Kriengsak Wattanawitoon ◽  
Poom Kumam
Keyword(s):  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium ◽  
Modified Block

The propose of this paper is to present a modified block iterative algorithm for finding a common element between the set of solutions of the fixed points of two countable families of asymptotically relatively nonexpansive mappings and the set of solution of the system of generalized mixed equilibrium problems in a uniformly smooth and uniformly convex Banach space. Our results extend many known recent results in the literature.

scholarly journals A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-39 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse-strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

scholarly journals Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Jinhua Zhu ◽  
Shih-Sen Chang ◽  
Min Liu
Keyword(s):  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Countable Family ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The purpose of this paper is first to introduce the concept oftotal quasi-ϕ-asymptotically nonexpansive mappingwhich contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a countable family of total quasi-ϕ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in an uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

scholarly journals A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
General System ◽  
Fixed Point Problem ◽  
Common Element ◽  
Point Problem ◽  
Computational Technique ◽  
Quadratic Minimization

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.

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