scholarly journals Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yan Tang
Keyword(s):  
Strong Convergence ◽  
Nonempty Closed Convex Subset ◽  
Finite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Monotone Mappings ◽  
Pseudocontractive Mappings ◽  
Differentiable Norm ◽  
The Common ◽  
Viscosity Approximation Methods

Suppose thatCis a nonempty closed convex subset of a real reflexive Banach spaceEwhich has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.

scholarly journals Viscosity Approximation Methods with Errors and Strong Convergence Theorems for a Common Point of Pseudocontractive and Monotone Mappings: Solutions of Variational Inequality Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yan Tang
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Variational Inequality Problems ◽  
Monotone Mappings ◽  
Convergence Theorems ◽  
Common Solution ◽  
The Common ◽  
Viscosity Approximation Methods

We introduce two proximal iterative algorithms with errors which converge strongly to the common solution of certain variational inequality problems for a finite family of pseudocontractive mappings and a finite family of monotone mappings. The strong convergence theorems are obtained under some mild conditions. Our theorems extend and unify some of the results that have been proposed for this class of nonlinear mappings.

scholarly journals A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

2020 ◽  
Vol 53 (1) ◽  
pp. 152-166 ◽  
Author(s):  
Getahun B. Wega ◽  
Habtu Zegeye ◽  
Oganeditse A. Boikanyo
Keyword(s):  
Strong Convergence ◽  
Approximation Method ◽  
Convergence Theorem ◽  
Finite Family ◽  
Strong Convergence Theorem ◽  
Monotone Mappings ◽  
Important Class ◽  
Numerical Example ◽  
Minimization Problems ◽  
Nonlinear Mappings

AbstractThe purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

scholarly journals A Strong Convergence Theorem for a Common Fixed Point of Two Sequences of Strictly Pseudocontractive Mappings in Hilbert Spaces and Applications

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Kasamsuk Ungchittrakool
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Pseudocontractive Mappings ◽  
Strict Pseudocontraction ◽  
Strictly Pseudocontractive Mappings

We prove a strong convergence theorem for a common fixed point of two sequences of strictly pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main theorem to find a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. The results extend and improve the recent ones announced by Marino and Xu (2007) and others.

scholarly journals Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1525-1536 ◽  
Author(s):  
Habtu Zegeye
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Strong Convergence Theorem ◽  
Monotone Mappings ◽  
Convergence Theorems

In this paper, we study a strong convergence theorem for a common fixed point of a finite family of Bregman strongly nonexpansive mappings in the framework of reflexive real Banach spaces. As a consequence, we prove convergence theorem for a common fixed point of a finite family of Bergman relatively nonexpansive mappings. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding a common zero of a finite family of Bregman inverse strongly monotone mappings and a solution of a finite family of variational inequality problems.

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Adrian Petruşel ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Iterative Process ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Viscosity Approximation ◽  
Inverse Strongly Monotone ◽  
The Common ◽  
Viscosity Approximation Methods

AbstractIn this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.

scholarly journals A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 435
Author(s):  
Wataru Takahashi
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Projection Method ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
Finite Family ◽  
Strong Convergence Theorem ◽  
Monotone Mappings ◽  
Shrinking Projection Method ◽  
Zero Points

In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we get well-known and new strong convergence theorems in a Hilbert space.

scholarly journals A New Iterative Algorithm for the Set of Fixed-Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-24 ◽  
Author(s):  
Atid Kangtunyakarn
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Fixed Point Problems ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Pseudocontractive Mappings

We introduce a new iterative scheme and a new mapping generated by infinite family of nonexpansive mappings and infinite real number. By using both of these ideas, we obtain strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems of infinite family of nonexpansive mappings. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

scholarly journals Strong Convergence Theorems for Solutions of Equilibrium Problems and Common Fixed Points of a Finite Family of Asymptotically Nonextensive Nonself Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Lijuan Zhang ◽  
Hui Tong ◽  
Ying Liu
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Finite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Uniformly Smooth

An iterative algorithm for finding a common element of the set of common fixed points of a finite family of asymptotically nonextensive nonself mappings and the set of solutions for equilibrium problems is discussed. A strong convergence theorem of common element is established in a uniformly smooth and uniformly convex Banach space.

scholarly journals Path Convergence and Approximation of Common Zeroes of a Finite Family ofm-Accretive Mappings in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Yekini Shehu ◽  
Jerry N. Ezeora
Keyword(s):  
Real Banach Space ◽  
Iterative Scheme ◽  
Finite Family ◽  
Duality Mapping ◽  
Strong Convergence Theorem ◽  
Sunny Nonexpansive Retraction ◽  
Mild Conditions ◽  
Pseudocontractive Mappings ◽  
Uniformly Smooth ◽  
Accretive Mappings

LetEbe a real Banach space which is uniformly smooth and uniformly convex. LetKbe a nonempty, closed, and convex sunny nonexpansive retract ofE, whereQis the sunny nonexpansive retraction. IfEadmits weakly sequentially continuous duality mappingj, path convergence is proved for a nonexpansive mappingT:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite family ofm-accretive mappings ofKtoE. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings fromKtoEunder certain mild conditions.

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