scholarly journals On a General Extragradient Implicit Method and Its Applications to Optimization

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 124
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Extragradient Method ◽  
General System ◽  
Zero Point ◽  
Fixed Point Problem ◽  
Implicit Method ◽  
Point Problem ◽  
Viscosity Approximation ◽  
Uniformly Smooth ◽  
Common Fixed Point Problem ◽  
Mann’S Iteration

Let X be a Banach space with both q-uniformly smooth and uniformly convex structures. This article introduces and considers a general extragradient implicit method for solving a general system of variational inequalities (GSVI) with the constraints of a common fixed point problem (CFPP) of a countable family of nonlinear mappings { S n } n = 0 ∞ and a monotone variational inclusion, zero-point, problem. Here, the constraints are symmetrical and the general extragradient implicit method is based on Korpelevich’s extragradient method, implicit viscosity approximation method, Mann’s iteration method, and the W-mappings constructed by { S n } n = 0 ∞ .

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 System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 5
Author(s):  
Lu-Chuan Ceng ◽  
Mihai Postolache ◽  
Xiaolong Qin ◽  
Yonghong Yao
Keyword(s):  
Banach Spaces ◽  
Infinite Family ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Variational Inclusions ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Mild Conditions ◽  
Pseudocontractive Mappings

The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point problem, we construct an implicit multiple-viscosity approximation method for solving the GSVI. Under very mild conditions, we prove the strong convergence of the proposed method to a solution of the GSVI with the HVI constraint, for infinitely many pseudocontractions.

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 Systems of Variational Inequalities with Nonlinear Operators

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 338 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
General System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Nonlinear Operators ◽  
Pseudocontractive Mappings ◽  
Common Fixed Point Problem

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces.

scholarly journals Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Jing Zhao ◽  
Songnian He
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Viscosity Approximation ◽  
Approximation Process ◽  
Common Fixed Point Problem ◽  
Split Common Fixed Point

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixed-point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

scholarly journals Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 933 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Meijuan Shang
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Optimization System ◽  
Common Fixed Point Problem ◽  
Implicit Rule ◽  
Viscosity Iterative Method ◽  
Mann’S Iteration

In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where 1 < q ≤ 2 . We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonexpansive mappings in X. The generalized Mann-like viscosity implicit rule investigated in this work is based on the Korpelevich’s extragradient technique, the implicit viscosity iterative method and the Mann’s iteration method. We show that the iterative sequences governed by our generalized Mann-like viscosity implicit rule converges strongly to a solution of the general optimization system.

scholarly journals A Parallel Hybrid Bregman Subgradient Extragradient Method for a System of Pseudomonotone Equilibrium and Fixed Point Problems

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 216
Author(s):  
Annel Thembinkosi Bokodisa ◽  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Convergence Result ◽  
Point Problem ◽  
Subgradient Extragradient Method ◽  
Reflexive Banach Spaces ◽  
Common Fixed Point Problem ◽  
Parallel Hybrid

We introduce a new parallel hybrid subgradient extragradient method for solving the system of the pseudomonotone equilibrium problem and common fixed point problem in real reflexive Banach spaces. The algorithm is designed such that its convergence does not require prior estimation of the Lipschitz-like constants of the finite bifunctions underlying the equilibrium problems. Moreover, a strong convergence result is proven without imposing strong conditions on the control sequences. We further provide some numerical experiments to illustrate the performance of the proposed algorithm and compare with some existing methods.

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