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 Convergence Theorems for Equilibrium Problems and Fixed-Point Problems of an Infinite Family ofki-Strictly Pseudocontractive Mapping in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Haitao Che ◽  
Meixia Li ◽  
Xintian Pan
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Element ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Definition Of ◽  
Strictly Pseudocontractive Mappings

We first extend the definition of Wnfrom an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family ofki-strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

scholarly journals A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-39 ◽  
Author(s):  
Wanpen Chantarangsi ◽  
Chaichana Jaiboon ◽  
Poom Kumam
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Generalized Mixed Equilibrium Problems ◽  
Hybrid Steepest Descent Method ◽  
Mixed Equilibrium Problems ◽  
Relaxed Cocoercive Mapping ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.

scholarly journals The Hybrid Steepest Descent Method for Split Variational Inclusion and Constrained Convex Minimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Jitsupa Deepho ◽  
Poom Kumam
Keyword(s):  
Descent Method ◽  
Convex Minimization ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Constrained Convex Minimization ◽  
Hybrid Steepest Descent Method ◽  
Variational Inclusion Problem

We introduced an implicit and an explicit iteration method based on the hybrid steepest descent method for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem.

scholarly journals Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros ofm-Accretive Operators in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-21
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequalities ◽  
Extragradient Method ◽  
General System ◽  
Accretive Operator ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Descent Methods ◽  
Common Element ◽  
Hybrid Steepest Descent Method

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of anm-accretive operatorAin the setting of real uniformly convex and 2-uniformly smooth Banach spaceX. Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the setA-1(0)of zeros ofAinX. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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