scholarly journals Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 461 ◽  
Author(s):  
Yonghong Yao ◽  
Naseer Shahzad ◽  
Jen-Chih Yao
Keyword(s):  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Convergence Result ◽  
Subgradient Method ◽  
Iterative Sequence ◽  
Common Element ◽  
Lipschitz Continuous ◽  
Subgradient Algorithms ◽  
Continuous Operators

The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature.

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 Construction of a common element for the set of solutions of fixed point problems and generalized equilibrium problems in Hilbert spaces

2016 ◽  
Vol 15 (1) ◽  
pp. 79-96
Author(s):  
Muhammad Aqeel Ahmad Khan
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Finite Family ◽  
Common Element ◽  
Common Solution ◽  
Convergence Results ◽  
Generalized Equilibrium

AbstractIn this paper, we propose and analyse an iterative algorithm for the approximation of a common solution for a finite family of k-strict pseudocontractions and two finite families of generalized equilibrium problems in the setting of Hilbert spaces. Strong convergence results of the proposed iterative algorithm together with some applications to solve the variational inequality problems are established in such setting. Our results generalize and improve various existing results in the current literature.

scholarly journals A Hybrid Extragradient-Like Method for Variational Inequalities, Equilibrium Problems, and an Infinitely Family of Strictly Pseudocontractive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Yaqin Wang ◽  
Hongkun Xu ◽  
Xiaoli Fang
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Sequence ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Mixed Equilibrium Problem ◽  
Pseudocontractive Mappings ◽  
Generalized Mixed Equilibrium ◽  
Strictly Pseudocontractive Mappings

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan (2010) and many others.

scholarly journals Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 503 ◽  
Author(s):  
Habib ur Rehman ◽  
Poom Kumam ◽  
Ioannis K. Argyros ◽  
Wejdan Deebani ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Convergence Result ◽  
Iterative Sequence ◽  
Type Condition ◽  
Mild Conditions ◽  
Slow Moving ◽  
Set Up

In this paper, we propose a new method, which is set up by incorporating an inertial step with the extragradient method for solving a strongly pseudomonotone equilibrium problems. This method had to comply with a strongly pseudomonotone property and a certain Lipschitz-type condition of a bifunction. A strong convergence result is provided under some mild conditions, and an iterative sequence is accomplished without previous knowledge of the Lipschitz-type constants of a cost bifunction. A sufficient explanation is that the method operates with a slow-moving stepsize sequence that converges to zero and non-summable. For numerical explanations, we analyze a well-known equilibrium model to support our well-established convergence result, and we can see that the proposed method seems to have a significant consistent improvement over the performance of the existing methods.

A GENERAL ITERATIVE ALGORITHM FOR EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS IN A HILBERT SPACE

2010 ◽  
Vol 03 (04) ◽  
pp. 685-705
Author(s):  
Tanakit Thianwan
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Finite Family ◽  
Iterative Sequence ◽  
Common Element ◽  
Relaxed Cocoercive Mapping

In this paper, we introduce a general iterative algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. The results obtained in this paper extend and improve the several recent results in this area.

scholarly journals Approximations for Equilibrium Problems and Nonexpansive Semigroups

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Iterative Method ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Nonexpansive Semigroup ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of Zegeye and Shahzad (2013). In the last part of the paper, by the way, we point out that there is a slight flaw in the proof of the main result in Shehu's paper (2012) and perfect the proof.

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 Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Inclusion Problems ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

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