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.

scholarly journals Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1189 ◽  
Author(s):  
Yonghong Yao ◽  
Mihai Postolache ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included.

scholarly journals Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 187
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
General System ◽  
Inequality Constraint ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Common Fixed Point Problem ◽  
Monotone Variational Inequality

In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions.

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 Iterations for systems of variational inequalities, fixed points of pseudocontractions and zeros of accretive operators

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4769-4784
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
General System ◽  
Accretive Operator ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Accretive Operators ◽  
Common Solution ◽  
Pseudocontractive Mappings ◽  
Systems Of Variational Inequalities

In this paper, we introduce implicit composite three-step Mann iterations for finding a common solution of a general system of variational inequalities, a fixed point problem of a countable family of pseudocontractive mappings and a zero problem of an accretive operator in Banach spaces. Strong convergence of the suggested iterations are given.

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 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 Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 860 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petruşel ◽  
Ching-Feng Wen ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Pseudomonotone Operator ◽  
Strictly Pseudocontractive Mapping ◽  
Common Solution ◽  
Convergence Results ◽  
Asymptotically Nonexpansive

Let VIP indicate the variational inequality problem with Lipschitzian and pseudomonotone operator and let CFPP denote the common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space. Our object in this article is to establish strong convergence results for solving the VIP and CFPP by utilizing an inertial-like gradient-like extragradient method with line-search process. Via suitable assumptions, it is shown that the sequences generated by such a method converge strongly to a common solution of the VIP and CFPP, which also solves a hierarchical variational inequality (HVI).

scholarly journals Convex Minimization with Constraints of Systems of Variational Inequalities, Mixed Equilibrium, Variational Inequality, and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Projection Method ◽  
Real Hilbert Space ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Shrinking Projection Method ◽  
Mixed Equilibrium

We introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

scholarly journals The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Himanshu Gupta ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
General System ◽  
Solution Set ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Split Feasibility

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution setΞof a general system of variational inequalities, the solution setΓof a split feasibility problem, and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of the Hilbert spaces. These iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge weakly to an element ofFix(S)∩Ξ∩Γunder mild conditions.

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