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 Levitin-Polyak well-posedness of completely generalized mixed variational inequalities in reflexive banach spaces

2017 ◽  
Vol 48 (1) ◽  
pp. 95-121
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Fixed Point Problem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Well Posedness ◽  
Mixed Variational Inequality ◽  
Mixed Variational Inequalities ◽  
Generalized Mixed Variational Inequality ◽  
Generalized Mixed Variational Inequalities ◽  
Well Posed

Let $X$ be a real reflexive Banach space. In this paper, we first introduce the concept of Levitin-Polyak well-posedness of a completely generalized mixed variational inequality in $X$, and establish some characterizations of its Levitin-Polyak well-posedness. Under suitable conditions, we prove that the Levitin-Polyak well-posedness of a completely generalized mixed variational inequality is equivalent both to the Levitin-Polyak well-posedness of a corresponding inclusion problem and to the Levitin-Polyak well-posedness of a corresponding fixed point problem. We also derive some conditions under which a completely generalized mixed variational inequality in $X$ is Levitin-Polyak well-posed. Our results improve, extend and develop the early and recent ones in the literature.

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 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.

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