scholarly journals Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria

2014 ◽  
Vol 2014 ◽  
pp. 1-25
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen ◽  
Zhao-Rong Kong
Keyword(s):  
Weak Convergence ◽  
Iterative Algorithm ◽  
Projection Method ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Generalized Mixed Equilibrium Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Convergence Criteria ◽  
Shrinking Projection Method

We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

scholarly journals Multistep Hybrid Iterations for Systems of Generalized Equilibria with Constraints of Several Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-27
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Iterative Algorithm ◽  
Projection Method ◽  
Real Hilbert Space ◽  
Fixed Point Problem ◽  
Generalized Mixed Equilibrium Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Shrinking Projection Method ◽  
Generalized Mixed Equilibrium ◽  
Fréchet Differentiable

We first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: the generalized mixed equilibrium problem, finitely many variational inclusions, the minimization problem for a convex and continuously Fréchet differentiable functional, and the fixed-point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions.

scholarly journals Convex Minimization with Constraints of Systems of Variational Inequalities, Mixed Equilibrium, Variational Inequality, and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-28
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Projection Method ◽  
Real Hilbert Space ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Shrinking Projection Method ◽  
Mixed Equilibrium

We introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

scholarly journals Iterative Schemes for Convex Minimization Problems with Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Strong Convergence ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Generalized Mixed Equilibrium Problem ◽  
Strong Convergence Theorem ◽  
Generalized Mixed Equilibrium ◽  
Generalized Equilibrium ◽  
Fréchet Differentiable ◽  
Implicit Iterative Algorithm

We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

scholarly journals Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Poom Kumam ◽  
Thanyarat Jitpeera
Keyword(s):  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Variational Inclusion ◽  
Common Element ◽  
Point Problem ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.

scholarly journals A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
General System ◽  
Fixed Point Problem ◽  
Common Element ◽  
Point Problem ◽  
Computational Technique ◽  
Quadratic Minimization

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.

scholarly journals Iterative Algorithms for Systems of Generalized Equilibrium Problems with the Constraints of Variational Inclusion and Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Abdul Latif ◽  
Abdullah E. Al-Mazrooei
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Generalized Mixed Equilibrium Problem ◽  
Monotone Mappings ◽  
Point Problem ◽  
Common Solution ◽  
Generalized Equilibrium

We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.

scholarly journals A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Mei Yuan ◽  
Xi Li ◽  
Xue-song Li ◽  
John J. Liu
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Projection Method ◽  
Convergence Theorem ◽  
Projection Operator ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalizedf-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

Hybrid projection method for a system of unrelated generalized mixed variational-like inequality problems

2019 ◽  
Vol 26 (1) ◽  
pp. 63-78
Author(s):  
Kaleem Raza Kazmi ◽  
Rehan Ali
Keyword(s):  
Banach Space ◽  
Projection Method ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Solution Set ◽  
Common Element ◽  
Monotone Mappings ◽  
Point Problem ◽  
Existence Of A Solution ◽  
Hybrid Projection Method

Abstract The aim of this paper is to consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems. Further, we prove some properties of a solution set of generalized mixed variational-like inequality problems. Furthermore, we use a hybrid projection method to find a common element of a solution set of a system of unrelated generalized mixed variational-like inequality problems for generalized relaxed α-monotone mappings and the set of fixed points of a common fixed point problem for a family of generalized asymptotically quasi-ϕ-nonexpansive mappings in a reflexive, uniformly smooth and strictly convex Banach space.

scholarly journals Hybrid Iterative Scheme for Triple Hierarchical Variational Inequalities with Mixed Equilibrium, Variational Inclusion, and Minimization Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Solution Set ◽  
Steepest Descent Method ◽  
Projection Algorithm ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Variational Inclusions ◽  
Mixed Equilibrium

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

Sign in / Sign up

Export Citation Format

Share Document