Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

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

Mathematics ◽

10.3390/math7020187 ◽

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.

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A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

Fixed Point Theory and Applications ◽

10.1155/2009/369215 ◽

2009 ◽

Vol 2009 (1) ◽

pp. 369215 ◽

Cited By ~ 4

Author(s):

Rabian Wangkeeree ◽

Uthai Kamraksa

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Iterative Method ◽

Hilbert Spaces ◽

Variational Inequality Problem ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Fixed Point Problem ◽

Point Problem ◽

General Iterative Method

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Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general system of variational inequalities, and a fixed point problem

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2013-313 ◽

2013 ◽

Vol 2013 (1) ◽

pp. 313 ◽

Cited By ~ 4

Author(s):

Lu-Chuan Ceng ◽

Sy-Ming Guu ◽

Jen-Chih Yao

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Variational Inequalities ◽

General System ◽

Fixed Point Problem ◽

Point Problem ◽

System Of Variational Inequalities ◽

Common Solution ◽

Cq Method

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A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Abstract and Applied Analysis ◽

10.1155/2013/178053 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Abdul Latif ◽

Mohammad Eslamian

Keyword(s):

Hilbert Space ◽

Fixed Point ◽

Variational Inequality ◽

Iterative Method ◽

Equilibrium Problems ◽

Variational Inequality Problem ◽

Real Hilbert Space ◽

Common Element ◽

Continuous Mappings ◽

New Iterative Method

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

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Simultaneous extragradient iterative method to a split equality variational inequality problem and a multiple-sets split equality fixed point problem for multi-valued demicontractive mappings

Advances in Fixed Point Theory ◽

10.28919/afpt/4028 ◽

2019 ◽

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Iterative Method ◽

Variational Inequality Problem ◽

Fixed Point Problem ◽

Point Problem ◽

Inequality Problem ◽

Demicontractive Mappings

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A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

Nonlinear Analysis ◽

10.1016/j.na.2008.04.035 ◽

2009 ◽

Vol 70 (9) ◽

pp. 3307-3319 ◽

Cited By ~ 159

Author(s):

Shih-sen Chang ◽

H.W. Joseph Lee ◽

Chi Kin Chan

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Equilibrium Problem ◽

Variational Inequality Problem ◽

Fixed Point Problem ◽

New Method ◽

Point Problem ◽

Inequality Problem

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Viscosity approximation with weak contractions for fixed point problem, equilibrium problem, and variational inequality problem

Applied Mathematics and Mechanics ◽

10.1007/s10483-010-1360-x ◽

2010 ◽

Vol 31 (10) ◽

pp. 1273-1282 ◽

Cited By ~ 1

Author(s):

Shi-sheng Zhang ◽

Heung-wing Joseph Lee ◽

Chi-kin Chan ◽

Jing-ai Liu

Keyword(s):

Fixed Point ◽

Variational Inequality ◽

Equilibrium Problem ◽

Variational Inequality Problem ◽

Fixed Point Problem ◽

Point Problem ◽

Viscosity Approximation ◽

Inequality Problem

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

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02683-y ◽

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.

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