scholarly journals Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 270 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Mihai Postolache ◽  
Ching-Feng Wen ◽  
Yonghong Yao
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Variational Inclusions ◽  
Minimization Problems ◽  
Convergence Results ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Implicit And Explicit ◽  
Mixed Equilibria

Multistep composite implicit and explicit extragradient-like schemes are presented for solving the minimization problem with the constraints of variational inclusions and generalized mixed equilibrium problems. Strong convergence results of introduced schemes are given under suitable control conditions.

scholarly journals Algorithms of Common Solutions for Generalized Mixed Equilibria, Variational Inclusions, and Constrained Convex Minimization

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Suliman Al-Homidan
Keyword(s):  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Common Element ◽  
Variational Inclusions ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Fréchet Differentiable ◽  
Implicit And Explicit ◽  
Mixed Equilibria

We introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.

scholarly journals Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Zhaoli Ma ◽  
Lin Wang ◽  
Yunhe Zhao
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Generalized Mixed Equilibrium Problems ◽  
Asymptotically Nonexpansive ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium

We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.

scholarly journals Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

2014 ◽  
Vol 2014 ◽  
pp. 1-26 ◽  
Author(s):  
A. E. Al-Mazrooei ◽  
A. Latif ◽  
J. C. Yao
Keyword(s):  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Generalized Mixed Equilibrium ◽  
Fréchet Differentiable ◽  
Implicit And Explicit ◽  
Mixed Equilibria

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.

scholarly journals Extragradient algorithms for variational inequality, fixed point and generalized mixed equilibrium problems

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1021-1030
Author(s):  
Baohua Guo ◽  
Lijuan Sun
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Generalized Mixed Equilibrium Problems ◽  
Mixed Equilibrium Problems ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

The purpose of this paper is to investigate variational inequalities, fixed point problems and generalized mixed equilibrium problems. Anextragradient iterative algorithm is investigated in the framework of Hilbert spaces. Weak convergence theorems for common solutions are established.

Modified inertial subgradient extragradient method for equilibrium problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Yekini Shehu ◽  
Regina N. Nwokoye
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Extragradient Method ◽  
Subgradient Extragradient Method ◽  
Weak And Strong Convergence ◽  
Extragradient Methods ◽  
Convergence Results ◽  
Inertial Extrapolation ◽  
Special Case

Abstract The subgradient extragradient method with inertial extrapolation step x n + θ n (x n − x n−1) (also known as inertial subgradient extragradient method) has been studied extensively in the literature for solving variational inequalities and equilibrium problems. Most of the inertial subgradient extragradient methods in the literature for both variational inequalities and equilibrium problems have not considered the special case when the inertial factor θ n = 1. The convergence results have always been obtained when the inertial factor θ n is assumed 0 ≤ θ n < 1. This paper considers the relaxed inertial version of subgradient extragradient method for equilibrium problems with 0 ≤ θ n ≤ 1. We give both weak and strong convergence results using this inertial subgradient extragradient method and also give some numerical illustrations.

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