scholarly journals Iterative methods for a fixed point of hemicontractive-type mapping and a solution of a variational inequality problem

2016 ◽  
Vol 25 (2) ◽  
pp. 183-196
Author(s):  
TESFALEM HADUSH MECHE ◽  
◽  
MENGISTU GOA SANGAGO ◽  
HABTU ZEGEYE ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Process ◽  
Variational Inequality Problem ◽  
Monotone Mapping ◽  
Common Point ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Type Mapping ◽  
Point Set

In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a Lipschitz hemicontractive-type multi-valued mapping and the solution set of a variational inequality problem for a monotone mapping. Our results improve and extend most of the results that have been proved for this class of nonlinear mappings.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

scholarly journals Algorithms for Solving the Variational Inequality Problem over the Triple Hierarchical Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Point Set ◽  
Monotone Variational Inequality

This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions.

scholarly journals Strong Convergence of an Iterative Algorithm for Hierarchical Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Poom Kumam ◽  
Thanyarat Jitpeera
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Variational Inequality Problem ◽  
Fixed Point Set ◽  
Mild Conditions ◽  
Hierarchical Problems ◽  
Point Set

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

scholarly journals An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
H. Zegeye ◽  
N. Shahzad
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Variational Inequality Problem ◽  
Monotone Mapping ◽  
Common Point ◽  
Finite Family ◽  
Monotone Mappings ◽  
Nonlinear Operators ◽  
Asymptotically Nonexpansive

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

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