scholarly journals On Variational Inclusion and Common Fixed Point Problems inq-Uniformly Smooth Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Yanlai Song ◽  
Huiying Hu ◽  
Luchuan Ceng
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Algorithm ◽  
Infinite Family ◽  
General System ◽  
Iterative Sequence ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Uniformly Smooth

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family ofλi-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in aq-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.

scholarly journals A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

2010 ◽  
Vol 2010 ◽  
pp. 1-27 ◽  
Author(s):  
Jian-Wen Peng ◽  
Soon-Yi Wu ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Infinite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
System Of Equilibrium Problems ◽  
New Iterative Method

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

scholarly journals Two-Step Viscosity Approximation Scheme for Variational Inequality in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Liping Yang ◽  
Weiming Kong
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Approximation Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Viscosity Approximation ◽  
Convergence Conditions ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

This paper introduces and analyzes a viscosity iterative algorithm for an infinite family of nonexpansive mappings{Ti}i=1∞in the framework of a strictly convex and uniformly smooth Banach space. It is shown that the proposed iterative method converges strongly to a common fixed point of{Ti}i=1∞, which solves specific variational inequalities. Necessary and sufficient convergence conditions of the iterative algorithm for an infinite family of nonexpansive mappings are given. Results shown in this paper represent an extension and refinement of the previously known results in this area.

scholarly journals Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-27 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Infinite Family ◽  
Accretive Operator ◽  
Common Element ◽  
Point Problem ◽  
Fixed Point Set ◽  
Extragradient Methods ◽  
Differentiable Norm ◽  
Implicit And Explicit

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach spaceXwhich has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive inX. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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 Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Pattanapong Tianchai
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Projection Method ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Shrinking Projection Method ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves the corresponding ones due to Kim [Proceedings of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].

scholarly journals A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
General System ◽  
Fixed Point Problem ◽  
Common Element ◽  
Point Problem ◽  
Computational Technique ◽  
Quadratic Minimization

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.

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 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.

Approximation of common fixed points of left Bregman strongly nonexpansive mappings and solutions of equilibrium problems

2015 ◽  
Vol 21 (2) ◽  
Author(s):  
Yekini Shehu ◽  
Ferdinard U. Ogbuisi
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Finite System ◽  
Iterative Sequence ◽  
Common Solution ◽  
System Of Equilibrium Problems

AbstractIn this paper we propose an iterative algorithm based on the hybrid method in mathematical programming for approximating a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which also solves a finite system of equilibrium problems in a reflexive real Banach space. We further prove that our iterative sequence converges strongly to a common fixed point of an infinite family of left Bregman strongly nonexpansive mappings which is also a common solution to a finite system of equilibrium problems. Our result extends many recent and important results in the literature.

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