Explicit Hierarchical Fixed Point Approach to Variational Inequalities

2011 ◽  
Vol 149 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Giuseppe Marino ◽  
Hong-Kun Xu
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Fixed Point Approach ◽  
Hierarchical Fixed Point

scholarly journals ON AN IMPLICIT HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES

2009 ◽  
Vol 80 (1) ◽  
pp. 117-124 ◽  
Author(s):  
FILOMENA CIANCIARUSO ◽  
VITTORIO COLAO ◽  
LUIGI MUGLIA ◽  
HONG-KUN XU
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Fixed Point Theory ◽  
Approximate Solutions ◽  
Fixed Point Problems ◽  
Viscosity Method ◽  
Norm Topology ◽  
Fixed Point Approach ◽  
Hierarchical Fixed Point ◽  
Hierarchical Fixed Point Problems

AbstractMoudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math.13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

Decay solutions for a class of fractional differential variational inequalities

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Tran Dinh Ke ◽  
Nguyen Van Loi ◽  
Valeri Obukhovskii
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Fractional Derivatives ◽  
New Class ◽  
Fixed Point Approach ◽  
Fractional Differential ◽  
Differential Variational Inequalities

AbstractOur aim is to study a new class of differential variational inequalities involving fractional derivatives. Using the fixed point approach, the existence of decay solutions to the mentioned problem is proved.

Viscosity method for hierarchical fixed point and variational inequalities with applications

2011 ◽  
Vol 32 (2) ◽  
pp. 241-250 ◽  
Author(s):  
Shi-sheng Zhang ◽  
Xiong-rui Wang ◽  
H. W. J. Lee ◽  
Chi-kin Chan
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Viscosity Method ◽  
Hierarchical Fixed Point

scholarly journals Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-25
Author(s):  
Lu-Chuan Ceng ◽  
Cheng-Wen Liao ◽  
Chin-Tzong Pang ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Convex Minimization ◽  
Solution Set ◽  
Projection Algorithm ◽  
Common Element ◽  
Point Problem ◽  
Hierarchical Fixed Point ◽  
Mixed Equilibria

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

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