scholarly journals Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

2022 ◽  
Vol 12 (1) ◽  
pp. 63
Author(s):  
Do Sang Kim ◽  
Nguyen Ngoc Hai ◽  
Bui Van Dinh
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Proximal Point Algorithm ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Common Point ◽  
Hybrid Mapping ◽  
Proximal Point ◽  
Extragradient Algorithm ◽  
Generalized Hybrid Mapping

<p style='text-indent:20px;'>In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.</p>

scholarly journals An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 99 ◽  
Author(s):  
Nopparat Wairojjana ◽  
Habib ur Rehman ◽  
Ioannis K. Argyros ◽  
Nuttapol Pakkaranang
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Numerical Solution ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Suggested Method ◽  
Experimental Results ◽  
Weak Convergence Theorem

Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. This method uses a non-monotonically stepsize technique based on local bifunction values and Lipschitz-type constants. Furthermore, we establish the weak convergence theorem for the suggested method and provide the applications of our results. Finally, several experimental results are reported to see the performance of the proposed method.

scholarly journals A Hybrid Subgradient Algorithm for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudocontraction Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ekkarath Thailert ◽  
Rabian Wangkeeree ◽  
Chanoksuda Khantree
Keyword(s):  
Hilbert Space ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Common Fixed Points ◽  
Common Point ◽  
Strong Convergence Theorem ◽  
Common Solution ◽  
Strict Pseudocontraction ◽  
Subgradient Algorithm ◽  
Convergent Algorithm

We propose a new strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. The strong convergence theorem of proposed algorithms is investigated without the Lipschitz condition for the bifunctions. Our results complement many known recent results in the literature.

scholarly journals A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1165 ◽  
Author(s):  
Pasakorn Yordsorn ◽  
Poom Kumam ◽  
Habib ur Rehman ◽  
Abdulkarim Hassan Ibrahim
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Line Search ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Adaptive Method ◽  
Type Condition ◽  
Computational Performance ◽  
Weak Convergence Theorem

In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others.

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 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 Relatively Inexact Proximal Point Algorithm and Linear Convergence Analysis

2009 ◽  
Vol 2009 ◽  
pp. 1-11
Author(s):  
Ram U. Verma
Keyword(s):  
Convergence Analysis ◽  
Proximal Point Algorithm ◽  
Evolution Equations ◽  
Real Hilbert Space ◽  
Linear Convergence ◽  
Proximal Point ◽  
Evolution Inclusions ◽  
Inclusion Problems ◽  
Set Valued Mapping ◽  
Inexact Proximal

Based on a notion ofrelatively maximal(m)-relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar's theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this new model, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.

scholarly journals A New Iterative Method for Equilibrium Problems and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Abdul Latif ◽  
Mohammad Eslamian
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Common Element ◽  
Continuous Mappings ◽  
New Iterative Method

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

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