scholarly journals Projection Methods for a System of Nonlinear Mixed Variational Inequalities in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhong-Bao Wang ◽  
Guo-Ji Tang ◽  
Hong-Ling Zhang
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Projection Operator ◽  
Existence And Uniqueness ◽  
Projection Methods ◽  
Uniqueness Of Solution ◽  
Mixed Variational Inequality ◽  
Mixed Variational Inequalities

The existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces is given firstly. A Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces is studied, by using the generalizedf-projection operatorπKf. Our results extend the main results in (Verma (2005); Verma (2001)) from Hilbert spaces to Banach spaces.

scholarly journals On the Hadamard Well-Posedness of Generalized Mixed Variational Inequalities in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Lu-Chuan Ceng ◽  
Yeong-Cheng Liou ◽  
Ching-Feng Wen ◽  
Hui-Ying Hu ◽  
Long He ◽  
...  
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Well Posedness ◽  
Mixed Variational Inequality ◽  
Mixed Variational Inequalities ◽  
Generalized Mixed Variational Inequality ◽  
Generalized Mixed Variational Inequalities

We introduce a new concept of Hadamard well-posedness of a generalized mixed variational inequality in a Banach space. The relations between the Levitin–Polyak well-posedness and Hadamard well-posedness for a generalized mixed variational inequality are studied. The characterizations of Hadamard well-posedness for a generalized mixed variational inequality are established.

scholarly journals Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-38
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Inclusion Problem ◽  
Point Problem ◽  
Well Posedness ◽  
Mixed Variational Inequality ◽  
Mixed Variational Inequalities ◽  
Generalized Mixed Variational Inequality ◽  
Generalized Mixed Variational Inequalities

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

scholarly journals An Interior Projected-Like Subgradient Method for Mixed Variational Inequalities

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Guo-ji Tang ◽  
Xing Wang
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Subgradient Method ◽  
Mixed Variational Inequality ◽  
Convergence Estimate ◽  
Finite Dimensional ◽  
Mixed Variational Inequalities

An interior projected-like subgradient method for mixed variational inequalities is proposed in finite dimensional spaces, which is based on using non-Euclidean projection-like operator. Under suitable assumptions, we prove that the sequence generated by the proposed method converges to a solution of the mixed variational inequality. Moreover, we give the convergence estimate of the method. The results presented in this paper generalize some recent results given in the literatures.

scholarly journals Existence and Algorithm for Solving the System of Mixed Variational Inequalities in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam
Keyword(s):  
Variational Inequalities ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Projection Operator ◽  
Mixed Variational Inequalities

The purpose of this paper is to study the existence and convergence analysis of the solutions of the system of mixed variational inequalities in Banach spaces by using the generalizedfprojection operator. The results presented in this paper improve and extend important recent results of Zhang et al. (2011) and Wu and Huang (2007) and some recent results.

scholarly journals Generalized System for Relaxed Cocoercive Mixed Variational Inequalities and Iterative Algorithms in Hilbert Spaces

2012 ◽  
Vol 20 (3) ◽  
pp. 131-139
Author(s):  
Shuyi Zhang ◽  
Xinqi Guo ◽  
Dan Luan
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Iterative Algorithms ◽  
Operator Technique ◽  
Mixed Variational Inequality ◽  
Generalized System ◽  
Resolvent Operator Technique ◽  
Mixed Variational Inequalities ◽  
The Resolvent Operator

Abstract The approximate solvability of a generalized system for relaxed co- coercive mixed variational inequality is studied by using the resolvent operator technique. The results presented in this paper extend and improve the main results of Chang et al.[1], He and Gu [2] and Verma [3, 4].

scholarly journals The generalized projection methods in countably normed spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Tawfeek ◽  
Nashat Faried ◽  
H. A. El-Sharkawy
Keyword(s):  
Banach Spaces ◽  
Projection Operator ◽  
Metric Projection ◽  
Projection Methods ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Normed Spaces ◽  
Generalized Projection Operator ◽  
Set Valued Mapping ◽  
Uniformly Smooth

AbstractLet E be a Banach space with dual space $E^{*}$ E ∗ , and let K be a nonempty, closed, and convex subset of E. We generalize the concept of generalized projection operator “$\Pi _{K}: E \rightarrow K$ Π K : E → K ” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties. We show the relation between J-orthogonality and generalized projection operator $\Pi _{K}$ Π K and give examples to clarify this relation. We introduce a comparison between the metric projection operator $P_{K}$ P K and the generalized projection operator $\Pi _{K}$ Π K in uniformly convex uniformly smooth complete countably normed spaces, and we give an example explaining how to evaluate the metric projection $P_{K}$ P K and the generalized projection $\Pi _{K}$ Π K in some cases of countably normed spaces, and this example illustrates that the generalized projection operator $\Pi _{K}$ Π K in general is a set-valued mapping. Also we generalize the generalized projection operator “$\pi _{K}: E^{*} \rightarrow K$ π K : E ∗ → K ” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties in these spaces.

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

2021 ◽  
pp. 2150017
Author(s):  
Yinfeng Zhang ◽  
Guolin Yu
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Error Bounds ◽  
Feasible Solution ◽  
Variational Inequality Problem ◽  
Gap Functions ◽  
Strong Monotonicity ◽  
Mixed Variational Inequality ◽  
Related Literature ◽  
Quasi Variational Inequality

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.

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