The Hybrid Steepest Descent Method for the Variational Inequality Problem Over the Intersection of Fixed Point Sets of Nonexpansive Mappings

2001 ◽  
pp. 473-504 ◽  
Author(s):  
Isao Yamada
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Nonexpansive Mappings ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Inequality Problem ◽  
Hybrid Steepest Descent Method ◽  
Fixed Point Sets

scholarly journals Hybrid Steepest Descent Viscosity Method for Triple Hierarchical Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
L.-C. Ceng ◽  
Q. H. Ansari ◽  
C.-F. Wen
Keyword(s):  
Approximate Solution ◽  
Variational Inequality Problem ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Viscosity Method ◽  
Hybrid Steepest Descent Method ◽  
Triple Hierarchical Variational Inequalities ◽  
Triple Hierarchical Variational Inequality ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (in short, THVIP). By combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP.

scholarly journals A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

2013 ◽  
Vol 2013 ◽  
pp. 1-24 ◽  
Author(s):  
Haitao Che ◽  
Xintian Pan
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Infinite Family ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Common Element ◽  
Hybrid Steepest Descent Method ◽  
Pseudocontractive Mappings ◽  
Strictly Pseudocontractive Mappings

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

scholarly journals Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nguyen Buong
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Descent Method ◽  
Finite Family ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Point Problem ◽  
Lower Semi ◽  
The Family

<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

scholarly journals Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 305-309 ◽  
Author(s):  
Wongvisarut Khuangsatung ◽  
Atid Kangtunyakarn
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Convergence Theorem ◽  
Variational Inequality Problem ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Inequality Problem

The purpose of this article, we give a necessary and sufficient condition for the modified Mann iterative process in order to obtain a strong convergence theorem for finding a common element of the set of fixed point of a finite family of nonexpansive mappings and variational inequality problem in Hilbert space without the conditions ?Ni=1 Fix(Ti)? VI(C,A)??. Moreover, we utilize our main result to fixed point problems of strictly pseudocontractive mappings and the set of solutions of variational inequality problem.

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