scholarly journals Krasnoselskii–Mann Viscosity Approximation Method for Nonexpansive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1153
Author(s):  
Najla Altwaijry ◽  
Tahani Aldhaban ◽  
Souhail Chebbi ◽  
Hong-Kun Xu
Keyword(s):  
Approximation Method ◽  
Geometric Property ◽  
Nonexpansive Mappings ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Mann Iteration ◽  
Underlying Space ◽  
Uniformly Smooth ◽  
Regularization Parameters ◽  
Uniformly Smooth Banach Spaces

We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods.

scholarly journals Viscosity approximation method for solving variational inequality problem in real Banach spaces

2021 ◽  
pp. 1-20
Author(s):  
Godwin Ugwunnadi
Keyword(s):  
Variational Inequality ◽  
Banach Spaces ◽  
Approximation Method ◽  
Variational Inequality Problem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Inequality Problem ◽  
Mild Conditions ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.

scholarly journals Strong convergence for nonexpansive mappings by viscosity approximation methods in Hadamard manifolds

2015 ◽  
Vol 4 (2) ◽  
pp. 299
Author(s):  
Mandeep Kumari ◽  
Renu Chugh
Keyword(s):  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Approximation Method ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Hadamard Manifolds ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Viscosity Approximation Methods

<p>In 2010, Victoria Martin Marquez studied a nonexpansive mapping in Hadamard manifolds using Viscosity approximation method. Our goal in this paper is to study the strong convergence of the Viscosity approximation method in Hadamard manifolds. Our results improve and extend the recent research in the framework of Hadamard manifolds.</p>

scholarly journals Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Chaichana Jaiboon ◽  
Poom Kumam
Keyword(s):  
Approximation Method ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Variational Inclusions ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation

We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in the literature.

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