scholarly journals Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
D. R. Sahu ◽  
Shin Min Kang ◽  
Vidya Sagar
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strong Convergence Theorem

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

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 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 Hybrid subgradient algorithm for equilibrium and fixed point problems by approximation of nonexpansive mapping

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1721-1729
Author(s):  
Seyed Aleomraninejad ◽  
Kanokwan Sitthithakerngkiet ◽  
Poom Kumam
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Expansive Mapping ◽  
Subgradient Algorithm

In this paper anew algorithm considered on a real Hilbert space for finding acommonpoint in the solution set of a class of pseudomonotone equilibrium problem and the set of fixed points of nonexpansive mappings. We produce this algorithm by mappings Tk that are approximations of non-expansive mapping T. The strong convergence theorem of the proposed algorithms is investigated. Our results generalize some recent results in the literature.

scholarly journals A New Iterative Scheme for Solving the Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Pakkapon Preechasilp
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Solution

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

scholarly journals A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Watcharaporn Cholamjiak ◽  
Suthep Suantai
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Multivalued Maps

We introduce a new monotone hybrid iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established.

scholarly journals Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Songnian He ◽  
Caiping Yang
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Functions ◽  
Level Sets ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Original Domain

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0, for allx∈C, whereCis the intersection of finite level sets of convex functions defined on a real Hilbert spaceHandF:H→His anL-Lipschitzian andη-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution ofVI(C,F). Since our algorithm avoids calculating the projectionPC(calculatingPCby computing several sequences of projections onto half-spaces containing the original domainC) directly and has no need to know any information of the constantsLandη, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

scholarly journals Convergence Theorems of Iterative Schemes For Nonexpansive Mappings

2017 ◽  
Vol 12 (12) ◽  
pp. 6845-6851
Author(s):  
Inaam Mohammed Ali Hadi ◽  
Dr. salwa Salman Abd
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Common Fixed Point ◽  
Convex Analysis ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Iterative Schemes

In this paper, we give a type of iterative scheme for sequence of nonexpansive mappings and we study the strongly convergence of these schemes in real Hilbert space to common fixed point which is also a solution of a variational inequality. Also there are some consequent of this results in convex analysis

scholarly journals A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-Nonexpansive Mappings and System of Equilibrium Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Yekini Shehu
Keyword(s):  
Strong Convergence ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Hybrid Methods ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Uniformly Smooth ◽  
Convex Minimization Problems ◽  
System Of Equilibrium Problems

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalizedf-projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization problems in Banach spaces. Our results extend many known recent results in the literature.

scholarly journals An Alternative Regularization Method for Equilibrium Problems and Fixed Point of Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Shuo Sun
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Regularization Method ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Sunny Nonexpansive Retraction

We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.

scholarly journals Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-25 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam ◽  
Kriengsak Wattanawitoon
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Projection Method ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Hybrid Projection Method ◽  
Generalized Equilibrium

The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

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