Isotonicity of the proximity operator and mixed variational inequalities in Hilbert spaces

2020 ◽  
Vol 114 (4) ◽  
Author(s):  
Dezhou Kong ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Proximity Operator ◽  
Mixed Variational Inequalities

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 Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1915
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Line Search ◽  
Convex Combination ◽  
Extragradient Method ◽  
Convergence Result ◽  
Armijo Line Search ◽  
Systems Of Variational Inequalities ◽  
Demicontractive Mappings ◽  
The Cost

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

scholarly journals Some Proximal Methods for Solving Mixed Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Muhammad Aslam Noor ◽  
Zhenyu Huang
Keyword(s):  
Variational Inequalities ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Proximal Methods ◽  
Proximal Point ◽  
Equivalent Formulation ◽  
New Methods ◽  
Special Cases ◽  
Mixed Variational Inequalities

It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest some new proximal point methods for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered under some suitable conditions. Our method of constructing these iterative methods is very simple. Results proved in this paper may stimulate further research in this direction.

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