scholarly journals Strong convergence of a selection of ishikawa-reich-sabach-type algorithm

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1505-1516
Author(s):  
Obiageli Isiogugu ◽  
Paranjothi Pillay ◽  
Osuo-Siseken Uzoma-Oguguo
Keyword(s):  
Strong Convergence ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Contemporary Literature ◽  
Iteration Scheme ◽  
Common Elements ◽  
Type Mapping ◽  
Type Algorithm ◽  
The Common ◽  
Selection Of

We establish the strong convergence of a selection of an Ishikawa-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points F(T) of a multi-valued (or singlevalued) pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a contribution to the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of the sequence {Kn}?n=1 of closed convex subsets of H from an arbitrary x0 ? H and the sequence {xn}?n=1 of the metric projections of x0 into Kn. The results obtained are contributions to the resolution of the controversy over the computability and applicability of such algorithms in the contemporary literature.

scholarly journals On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
F. O. Isiogugu ◽  
P. Pillay ◽  
P. U. Nwokoro
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Contemporary Literature ◽  
Iteration Scheme ◽  
Common Elements ◽  
Type Mapping ◽  
The Common ◽  
Selection Of

We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points F(T) of a multivalued (or single-valued) k-strictly pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence {Kn}n=1∞ of closed convex subsets of H from an arbitrary x0∈H and a sequence {xn}n=1∞ of the metric projections of x0 into Kn. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.

scholarly journals Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 480
Author(s):  
Manatchanok Khonchaliew ◽  
Ali Farajzadeh ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Constraint Qualifications ◽  
Solution Sets ◽  
New Algorithms

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm.

scholarly journals Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Bin-Chao Deng ◽  
Tong Chen ◽  
Zhi-Fang Li
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Cyclic Algorithm ◽  
The Common

Let{Ti}i=1NbeNstrictly pseudononspreading mappings defined on closed convex subsetCof a real Hilbert spaceH. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality〈(γf-μB)x*,v-x*〉≤0,∀v∈⋂i=1NFix(Ti).

scholarly journals Strong Convergence Theorems for Equilibrium Problems andk-Strict Pseudocontractions in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Dao-Jun Wen
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Finite Family ◽  
Common Element ◽  
Convergence Theorems

We introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of a finite family ofk-strictly pseudo-contractive nonself-mappings. Strong convergence theorems are established in a real Hilbert space under some suitable conditions. Our theorems presented in this paper improve and extend the corresponding results announced by many others.

scholarly journals Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1504
Author(s):  
Li Wei ◽  
Xin-Wang Shen ◽  
Ravi P. Agarwal
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Iterative Design ◽  
Common Solution ◽  
Monotone Inclusions ◽  
New Ideas ◽  
The Common

Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under mild restrictions, which extend and complement some already existing work.

scholarly journals Strong Convergence to Common Fixed Points of a Countable Family of Asymptotically Strictly Quasi-ϕ-Pseudocontractions

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Wei-Qi Deng
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Iterative Solution ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Countable Family ◽  
Strong Convergence Theorem ◽  
System Of Equilibrium Problems

Based on an original idea, namely, a specific way of choosing the indexes of the involved mappings, we propose a new hybrid shrinking iteration scheme for approximating some common fixed points of a countable family of asymptotically strictly quasi-ϕ-pseudocontractions and obtain a strong convergence theorem in the framework of Banach space. Our result extends other authors, related results existing in the current literature. As application, an iterative solution to a system of equilibrium problems is provided.

scholarly journals Strong Convergence Theorems for Family of Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-22
Author(s):  
Yekini Shehu
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of nonexpansive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three sets. We give some applications of our results. Our results extend important recent results.

scholarly journals Shrinking Projection Methods for Accelerating Relaxed Inertial Tseng-Type Algorithm with Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Hasanen A. Hammad ◽  
Habib ur Rehman ◽  
Manuel De la Sen
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Minimization Problem ◽  
Numerical Experiments ◽  
Real Hilbert Space ◽  
Projection Methods ◽  
Novel Structure ◽  
Convergence Results ◽  
Type Algorithm ◽  
Shrinking Projection Methods

Our main goal in this manuscript is to accelerate the relaxed inertial Tseng-type (RITT) algorithm by adding a shrinking projection (SP) term to the algorithm. Hence, strong convergence results were obtained in a real Hilbert space (RHS). A novel structure was used to solve an inclusion and a minimization problem under proper hypotheses. Finally, numerical experiments to elucidate the applications, performance, quickness, and effectiveness of our procedure are discussed.

scholarly journals A New Iterative Scheme for Solving the Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Pakkapon Preechasilp
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Solution

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

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