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 Strong Convergence Iterative Algorithms for Equilibrium Problems and Fixed Point Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Shenghua Wang ◽  
Shin Min Kang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Iterative Algorithms ◽  
Common Fixed Points ◽  
Common Element ◽  
Fixed Point Set ◽  
Point Set

We first introduce the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.

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 Extragradient methods with CQ technique for fixed point problems and equilibrium problems

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4783-4793
Author(s):  
Zhangsong Yao ◽  
Yeong-Cheng Liou ◽  
Li-Jun Zhu
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Algorithms ◽  
Convergence Result ◽  
Fixed Point Problems ◽  
Common Element ◽  
Extragradient Methods ◽  
Extragradient Algorithm

In this paper, we study iterative algorithms for solving fixed point problems and equilibrium problems in Hilbert spaces. We present an extragradient algorithm with CQ technique for finding a common element of the fixed points of pseudocontractive operators and the solutions of pseudomonotone equilibrium problems. Strong convergence result of the proposed algorithm is proved.

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 Fixed Point Problems for Nonexpansive Mappings and Zero Point Problems for Accretive Operators Using Viscosity Implicit Midpoint Rules in Banach Spaces

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 257 ◽  
Author(s):  
Huancheng Zhang ◽  
Yunhua Qu ◽  
Yongfu Su
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Zero Point ◽  
Accretive Operator ◽  
Accretive Operators ◽  
Fixed Point Set ◽  
Implicit Midpoint Rule ◽  
Convergence Results ◽  
Point Set

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results of the proposed algorithm and improves the relevant results of researchers in this field. In the end, this paper gives numerical examples to support the main results.

scholarly journals Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-32 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Split Feasibility Problem ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Extragradient Methods ◽  
Infinite Dimensional ◽  
Point Set ◽  
Split Feasibility

We consider and study the modified extragradient methods for finding a common element of the solution setΓof a split feasibility problem (SFP) and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element ofFix(S)∩ΓwhereSis strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element ofFix(S)∩Γ. We also propose another extragradient-like algorithm for finding an element ofFix(S)∩ΓwhereS:C→Cis nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element ofFix(S)∩Γ.

scholarly journals Common Solution for a Finite Family of Equilibrium Problems, Quasi-Variational Inclusion Problems and Fixed Points on Hadamard Manifolds

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1161
Author(s):  
Jinhua Zhu ◽  
Jinfang Tang ◽  
Shih-sen Chang ◽  
Min Liu ◽  
Liangcai Zhao
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Finite Family ◽  
Variational Inclusion ◽  
Point Problem ◽  
Hadamard Manifolds ◽  
Inclusion Problems ◽  
Common Solution

In this paper, we introduce an iterative algorithm for finding a common solution of a finite family of the equilibrium problems, quasi-variational inclusion problems and fixed point problem on Hadamard manifolds. Under suitable conditions, some strong convergence theorems are proved. Our results extend some recent results in literature.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

scholarly journals Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yuanheng Wang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

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