scholarly journals Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yeong-Cheng Liou ◽  
Yonghong Yao ◽  
Chun-Wei Tseng ◽  
Hui-To Lin ◽  
Pei-Xia Yang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Nonlinear Operator ◽  
Iterative Algorithms ◽  
Fixed Point Problem ◽  
Solution Set ◽  
Point Problem ◽  
General Variational Inequality

We consider a general variational inequality and fixed point problem, which is to find a pointx*with the property that (GVF):x*∈GVI(C,A)andg(x*)∈Fix(S)whereGVI(C,A)is the solution set of some variational inequalityFix(S)is the fixed points set of nonexpansive mappingS, andgis a nonlinear operator. Assume the solution setΩof (GVF) is nonempty. For solving (GVF), we suggest the following methodg(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n≥0. It is shown that the sequence{xn}converges strongly tox*∈Ωwhich is the unique solution of the variational inequality〈F(x*)-g(x*),g(x)-g(x*)〉≤0, for allx∈Ω.

scholarly journals Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 187
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
General System ◽  
Inequality Constraint ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Common Fixed Point Problem ◽  
Monotone Variational Inequality

In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions.

scholarly journals Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1502
Author(s):  
Sun Young Cho
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Strict Pseudocontraction ◽  
Adaptive Step

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

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.

scholarly journals General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Nopparat Wairojjana ◽  
Poom Kumam
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Bounded Operator ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Linear Bounded Operator ◽  
New Methods

This paper deals with new methods for approximating a solution to the fixed point problem; findx̃∈F(T), whereHis a Hilbert space,Cis a closed convex subset ofH,fis aρ-contraction fromCintoH,0<ρ<1,Ais a strongly positive linear-bounded operator with coefficientγ̅>0,0<γ<γ̅/ρ,Tis a nonexpansive mapping onC,andPF(T)denotes the metric projection on the set of fixed point ofT. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality〈(A-γf)x̃+τ(I-S)x̃,x-x̃〉≥0forx∈F(T), whereτ∈[0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

scholarly journals Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1189 ◽  
Author(s):  
Yonghong Yao ◽  
Mihai Postolache ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included.

Sign in / Sign up

Export Citation Format

Share Document