scholarly journals Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Lu-Chuan Ceng ◽  
Abdul Latif ◽  
Saleh A. Al-Mezel
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
General System ◽  
Variational Inclusion ◽  
Uniformly Convex ◽  
Extragradient Methods ◽  
Extragradient Algorithm

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

scholarly journals Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-44
Author(s):  
L. C. Ceng ◽  
A. E. Al-Mazrooei ◽  
A. A. N. Abdou ◽  
A. Latif
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
General System ◽  
Countable Family ◽  
Uniformly Convex ◽  
Mann Iteration ◽  
Differentiable Norm ◽  
Iteration Methods

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

scholarly journals Modified Relaxed Extragradient Method for a General System of Variational Inequalities and Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yuanheng Wang ◽  
Liu Yang
Keyword(s):  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
General System ◽  
Iterative Sequence ◽  
Monotone Mappings ◽  
System Of Variational Inequalities ◽  
Common Solution ◽  
Demiclosedness Principle

The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.

scholarly journals Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6267-6281
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
General System ◽  
Inequality Constraint ◽  
Implicit And Explicit

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

scholarly journals Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium Problems, a General System of Variational Inequalities and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Qiao-Li Dong ◽  
Yan-Ni Guo ◽  
Fang Su
Keyword(s):  
Iterative Method ◽  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
General System ◽  
Countable Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
System Of Variational Inequalities

Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others.

scholarly journals Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-18
Author(s):  
Lu-Chuan Ceng ◽  
Saleh A. Al-Mezel ◽  
Abdul Latif
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Steepest Descent Method ◽  
Point Problem ◽  
Hierarchical Fixed Point Problem ◽  
Hierarchical Fixed Point

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings inX, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

scholarly journals Stable Perturbed Iterative Algorithms for Solving New General Systems of Nonlinear Generalized Variational Inclusion in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Ting-jian Xiong ◽  
Heng-you Lan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Iterative Algorithms ◽  
General System ◽  
Variational Inclusion ◽  
Operator Technique ◽  
Accretive Mappings ◽  
General Systems ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator

We introduce and study a new general system of nonlinear variational inclusions involving generalizedm-accretive mappings in Banach space. By using the resolvent operator technique associated with generalizedm-accretive mappings due to Huang and Fang, we prove the existence theorem of the solution for this variational inclusion system in uniformly smooth Banach space, and discuss convergence and stability of a class of new perturbed iterative algorithms for solving the inclusion system in Banach spaces. Our results presented in this paper may be viewed as an refinement and improvement of the previously known results.

scholarly journals Convergence results of a faster iterative scheme including multi-valued mappings in Banach spaces

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1359-1368
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Muhammad Khan ◽  
Naseer Muhammad
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Current Literature ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Efficiency ◽  
Convergence Results

In this paper, we study M-iterative scheme in the new context of multi-valued generalized ?-nonexpansive mappings. A uniformly convex Banach space is used as underlying setting for our approach. We also provide a new example of generalized ?-nonexpasive mappings. We connect M iterative scheme and other well known schemes with this example, to show the numerical efficiency of our results. Our results improve and extend many existing results in the current literature.

scholarly journals Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-18
Author(s):  
Lu-Chuan Ceng ◽  
Abdul Latif ◽  
Abdullah E. Al-Mazrooei
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Uniformly Convex ◽  
Viscosity Approximation ◽  
Uniformly Smooth ◽  
Viscosity Approximation Methods

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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