scholarly journals A Viscosity Approximation Method with a Weakly Contractive Mapping of General Iterative Processes for Nonexpansive Semigroups in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Pongsakorn Sunthrayuth ◽  
Chanan Sudsukh ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Approximation Method ◽  
Contractive Mapping ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Iterative Processes ◽  
Weakly Contractive Mapping ◽  
Weakly Contractive

We introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for finding common fixed point of nonexpansive semigroups {T(t):t∈ℝ+} in the framework of Banach spaces. We proved that under some mild conditions these iterative processes converge strongly to the common fixed point of {T(t):t∈ℝ+}, which is the unique solution of some variational inequality. The results obtained in this paper extend and improve on the recent results of Li et al. (2009), Chen and He (2007), and many others as special cases.

scholarly journals A New Modified Hybrid Steepest-Descent by Using a Viscosity Approximation Method with a Weakly Contractive Mapping for a System of Equilibrium Problems and Fixed Point Problems with Minimization Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Uamporn Witthayarat ◽  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Approximation Method ◽  
Contractive Mapping ◽  
Steepest Descent ◽  
Common Element ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Weakly Contractive Mapping ◽  
Minimization Problems ◽  
Weakly Contractive

The purpose of this paper is to consider a modified hybrid steepest-descent method by using a viscosity approximation method with a weakly contractive mapping for finding the common element of the set of a common fixed point for an infinite family of nonexpansive mappings and the set of solutions of a system of an equilibrium problem. The sequence is generated from an arbitrary initial point which converges in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. The results presented in this paper generalize and improve the results of Moudafi (2000), Marino and Xu (2006), Tian (2010), Saeidi (2010), and some others. Finally, we give an application to minimization problems and a numerical example which support our main theorem in the last part.

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.

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