scholarly journals Algorithms for a System of General Variational Inequalities in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Jin-Hua Zhu ◽  
Shih-Sen Chang ◽  
Min Liu
Keyword(s):  
Variational Inequality ◽  
Banach Spaces ◽  
Existence Theorems ◽  
Extragradient Method ◽  
Finite Family ◽  
Existence Problem ◽  
Special Cases ◽  
General Variational Inequalities ◽  
Solvability Problem ◽  
Nonlinear Variational Inequality

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

scholarly journals Common Fixed Points of Multistep Noor Iterations with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
S. Imnang ◽  
S. Suantai
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Finite Family ◽  
Special Cases

We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

scholarly journals A Generalized System of Nonlinear Variational Inequalities in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Prapairat Junlouchai ◽  
Anchalee Kaewcharoen ◽  
Somyot Plubtieng
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Generalized System ◽  
Uniformly Smooth ◽  
Strictly Convex Banach Spaces ◽  
Generalized Projection Method ◽  
Nonlinear Variational Inequality

We introduce a new generalized system of nonlinear variational inequality problems (GSNVIP) by using the generalized projection method. Moreover, we introduce an iterative scheme for finding a solution to this problem. Moreover, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces under suitable conditions. The results presented in the paper improve and extend some recent results.

scholarly journals Strong convergence theorems for variational inequalities and fixed point problems in Banach spaces

2021 ◽  
Vol 37 (3) ◽  
pp. 477-487
Author(s):  
MONDAY OGUDU NNAKWE ◽  
◽  
" JERRY N." EZEORA ◽  
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Metric Projection ◽  
Finite Family ◽  
Common Solution ◽  
Monotone Type

In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel’skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of η-strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized non-expansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature.

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.

scholarly journals Sensitivity analysis for a new class of generalized parametric nonlinear ordered variational inequality problems in ordered Banach spaces

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Kottakkaran Sooppy Nisar ◽  
Mohd. Sarfaraz ◽  
Ahmed Morsy ◽  
Md. Kalimuddin Ahmad
Keyword(s):  
Sensitivity Analysis ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Existence Result ◽  
Variational Inequality Problems ◽  
New Class ◽  
Special Cases ◽  
Ordered Banach Spaces

Abstract In this study, we introduce a class of new generalized parametric nonlinear ordered variational inequality problems and discuss its existence result. Also, we prove the sensitivity of the solution for the parametric inequality class with the help of B-restricted-accretive method in ordered Banach spaces. Some special cases of the main results are also discussed.

scholarly journals Existence theorems for vector variational inequalities

1996 ◽  
Vol 54 (3) ◽  
pp. 473-481 ◽  
Author(s):  
Aris Daniilidis ◽  
Nicolas Hadjisavvas
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Convex Subset ◽  
Variational Inequality Problem ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Multivalued Operator ◽  
Inequality Problem ◽  
Vector Variational Inequality Problem

Given two real Banach spaces X and Y, a closed convex subset K in X, a cone with nonempty interior C in Y and a multivalued operator from K to 2L(x, y), we prove theorems concerning the existence of solutions for the corresponding vector variational inequality problem, that is the existence of some x0 ∈ K such that for every x ∈ K we have A(x − x0) ∉ − int C for some A ∈ Tx0. These results correct previously published ones.

ON A NEW NUMERICAL METHOD FOR SOLVING GENERAL VARIATIONAL INEQUALITIES

2011 ◽  
Vol 25 (32) ◽  
pp. 4443-4455 ◽  
Author(s):  
ABDELLAH BNOUHACHEM ◽  
MUHAMMAD ASLAM NOOR ◽  
MOHAMED KHALFAOUI ◽  
ZHAOHAN SHENG
Keyword(s):  
Lower Bound ◽  
Global Convergence ◽  
Variational Inequalities ◽  
Complementarity Problems ◽  
Extragradient Method ◽  
Mild Conditions ◽  
Modified Method ◽  
Adaptive Technique ◽  
Special Cases ◽  
General Variational Inequalities

In this paper, we suggest and analyze a new extragradient method for solving the general variational inequalities involving two operators. We also prove the global convergence of the proposed modified method under certain mild conditions. We used a self-adaptive technique to adjust parameter ρ at each iteration. It is proved theoretically that the lower-bound of the progress obtained by the proposed method is greater than that by the extragradient method. An example is given to illustrate the efficiency and its comparison with the extragradient method. Since the general variational inequalities include the classical variational inequalities and complementarity problems as special cases, our results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this field.

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