scholarly journals MULTI-STEP ALGORITHMS FOR SOLVING EQUILIBRIUM PROBLEMS

2018 ◽  
Vol 23 (3) ◽  
pp. 453-472 ◽  
Author(s):  
Pham Ngoc Anh ◽  
Dang Van Hieu
Keyword(s):  
Hilbert Space ◽  
Prior Knowledge ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Numerical Examples

The paper introduces and analysizes the convergence of two multi-step proximal-like algorithms for pseudomonotone and Lipschitz-type continuous equilibrium problems in a real Hilbert space. The algorithms are combinations between the multi-step proximal-like method and Mann or Halpern iterations. The weakly and strongly convergent theorems are established with the prior knowledge of two Lipschitz-type continuous constants. Moreover, by choosing two sequences of suitable stepsizes, we also show that the multi-step proximal-like algorithm for strongly pseudomonotone and Lipschitz-type continuous equilibrium problems where the construction of solution approximations and the establishing of its convergence do not require the prior knowledge of strongly pseudomonotone and Lipschitz-type continuous constants of bifunctions. Finally, several numerical examples are reported to illustrate the convergence and the performance of the proposed algorithms over classical extragradient-like algorithms.

scholarly journals The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 463 ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Wiyada Kumam ◽  
Meshal Shutaywi ◽  
Wachirapong Jirakitpuwapat
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Gradient Method ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Considerable Influence ◽  
Numerical Examples ◽  
Convergence Results ◽  
Monotone Variational Inequality ◽  
Number Of Iterations

In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations.

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 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.

scholarly journals New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1677-1693 ◽  
Author(s):  
Shenghua Wang ◽  
Yifan Zhang ◽  
Ping Ping ◽  
Yeol Cho ◽  
Haichao Guo
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Fixed Point Theorems ◽  
Equilibrium Problems ◽  
Convex Combination ◽  
Projection Methods ◽  
Extragradient Methods ◽  
Numerical Examples

In the literature, the most authors modify the viscosity methods or hybrid projection methods to construct the strong convergence algorithms for solving the pseudomonotone equilibrium problems. In this paper, we introduce some new extragradient methods with non-convex combination to solve the pseudomonotone equilibrium problems in Hilbert space and prove the strong convergence for the constructed algorithms. Our algorithms are very different with the existing ones in the literatures. As the application, the fixed point theorems for strict pseudo-contraction are considered. Finally, some numerical examples are given to show the effectiveness of the algorithms.

New hybrid algorithms for global minimization of common best proximity points of some generalized nonexpansive mappings

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2381-2391
Author(s):  
Jenwit Puangpee ◽  
Suthep Suantai
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Best Proximity Point ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Hybrid Algorithms ◽  
Global Minimization ◽  
Numerical Examples ◽  
Common Best Proximity Point ◽  
Two Hybrid

In this paper, we introduce two hybrid algorithms for finding a common best proximity point of two best proximally nonexpansive mappings. We establish strong convergence theorems of the proposed algorithms under some control conditions in a real Hilbert space. Moreover, some numerical examples are given for supporting our main theorems.

scholarly journals Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications

2021 ◽  
Vol 6 (2) ◽  
pp. 1538-1560
Author(s):  
Habib ur Rehman ◽  
◽  
Poom Kumam ◽  
Kanokwan Sitthithakerngkiet ◽  
◽  
...  
Keyword(s):  
Hilbert Space ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Type Method

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 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.

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