scholarly journals Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-25 ◽  
Author(s):  
L. C. Ceng ◽  
A. Petruşel ◽  
J. C. Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Approximation Method ◽  
General System ◽  
Solution Set ◽  
Common Element ◽  
Feasibility Problem ◽  
Extragradient Methods ◽  
System Of Variational Inequalities ◽  
Split Feasibility

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.

scholarly journals The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Himanshu Gupta ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
General System ◽  
Solution Set ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Split Feasibility

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution setΞof a general system of variational inequalities, the solution setΓof a split feasibility problem, and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of the Hilbert spaces. These iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge weakly to an element ofFix(S)∩Ξ∩Γunder mild conditions.

scholarly journals Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-32 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Split Feasibility Problem ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Extragradient Methods ◽  
Infinite Dimensional ◽  
Point Set ◽  
Split Feasibility

We consider and study the modified extragradient methods for finding a common element of the solution setΓof a split feasibility problem (SFP) and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element ofFix(S)∩ΓwhereSis strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element ofFix(S)∩Γ. We also propose another extragradient-like algorithm for finding an element ofFix(S)∩ΓwhereS:C→Cis nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element ofFix(S)∩Γ.

scholarly journals Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
S. Imnang
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Real Hilbert Space ◽  
General System ◽  
Projection Algorithm ◽  
Common Element ◽  
System Of Variational Inequalities ◽  
Demiclosedness Principle ◽  
Hybrid Projection Algorithm

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

scholarly journals Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-25
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Convex Minimization ◽  
Solution Set ◽  
Projection Algorithm ◽  
Common Element ◽  
Point Problem ◽  
Hierarchical Fixed Point ◽  
Mixed Equilibria

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and 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 infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just 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 solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

Sign in / Sign up

Export Citation Format

Share Document