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.

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 Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Long He ◽  
Yun-Ling Cui ◽  
Lu-Chuan Ceng ◽  
Tu-Yan Zhao ◽  
Dan-Qiong Wang ◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Implicit Rule

AbstractIn a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

scholarly journals Iterative Scheme for Split Variational Inclusion and a Fixed-Point Problem of a Finite Collection of Nonexpansive Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
M. Dilshad ◽  
A. F. Aljohani ◽  
M. Akram
Keyword(s):  
Fixed Point ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Common Element ◽  
Finite Collection ◽  
Point Problem ◽  
Common Solution ◽  
The Common

This article is aimed at introducing an iterative scheme to approximate the common solution of split variational inclusion and a fixed-point problem of a finite collection of nonexpansive mappings. It is proven that under some suitable assumptions, the sequences achieved by the proposed iterative scheme converge strongly to a common element of the solution sets of these problems. Some consequences of the main theorem are also given. Finally, the convergence analysis of the sequences achieved from the iterative scheme is illustrated with the help of a numerical example.

scholarly journals Some Results for Split Equality Equilibrium Problems in Banach Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 194
Author(s):  
Zhaoli Ma ◽  
Lin Wang ◽  
Yeol Cho
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Banach Spaces ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Common Element ◽  
Point Problem ◽  
Convex Minimization Problem ◽  
Strong And Weak Convergence

In this paper, a new algorithm for finding a common element of a split equality fixed point problem for nonexpansive mappings and split equality equilibrium problem in three Banach spaces is introduced. Also, some strong and weak convergence theorems for the proposed algorithm are proved. Finally, the main results obtained in this paper are applied to solve the split equality convex minimization problem.

scholarly journals Variational inequalities, variational inclusions and common fixed point problems in Banach spaces

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2939-2951
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Uniformly Smooth ◽  
Common Fixed Point Problem

In this paper, let X be a uniformly convex and q-uniformly smooth Banach space with 1 < q ? 2. We introduce and study modified implicit extragradient iterations for treating a common solution of a common fixed-point problem of a countable family of nonexpansive mappings, a general system of variational inequalities, and a variational inclusion in X.

scholarly journals Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 305-309 ◽  
Author(s):  
Wongvisarut Khuangsatung ◽  
Atid Kangtunyakarn
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Convergence Theorem ◽  
Variational Inequality Problem ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Inequality Problem

The purpose of this article, we give a necessary and sufficient condition for the modified Mann iterative process in order to obtain a strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and variational inequality problem in Hilbert space without the conditions ?Ni=1 Fix(Ti)? VI(C,A)??. Moreover, we utilize our main result to fixed point problems of strictly pseudocontractive mappings and the set of solutions of variational inequality problem.

scholarly journals Hybrid viscosity approximation methods for systems of variational inequalities and hierarchical fixed point problems

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1927-1947
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Variational Inequalities ◽  
Nonexpansive Mappings ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
System Of Variational Inequalities ◽  
Hierarchical Fixed Point Problem ◽  
Hierarchical Fixed Point

In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities with a hierarchical fixed point problem constraint for an infinite family of nonexpansive mappings. We show that the proposed algorithms converge strongly to a solution of the general system of variational inequalities, which is a unique solution of the hierarchical fixed point problem.

scholarly journals An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Shamshad Husain ◽  
Nisha Singh
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Fixed Point Problem ◽  
Common Fixed Points ◽  
Finite Family ◽  
Common Element ◽  
Point Problem

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

scholarly journals Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed-Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-32
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
General System ◽  
Point Problem ◽  
Strictly Pseudocontractive Mapping ◽  
Common Solution

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed-point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi's hybrid-extragradient method, we propose and analyze new hybrid-extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings.

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