scholarly journals A New General System of Generalized Nonlinear Mixed Composite-Type Equilibria and Fixed Point Problems with an Application to Minimization Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Iterative Scheme ◽  
Real Hilbert Space ◽  
General System ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Bounded Linear ◽  
Minimization Problems ◽  
Strict Pseudocontraction ◽  
Composite Type ◽  
Generalized Equilibrium

We introduce a new general system of generalized nonlinear mixed composite-type equilibria and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend the recent ones announced by many others.

scholarly journals A Hybrid Iterative Scheme for Mixed Equilibrium Problems, General System of Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-23
Author(s):  
S. Imnang ◽  
S. Suantai
Keyword(s):  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
General System ◽  
Iterative Sequence ◽  
Common Element ◽  
Demiclosedness Principle ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Using the demiclosedness principle for nonexpansive mappings, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions, and we also give some examples for mappings which satisfy conditions of the main result.

scholarly journals Hybrid Projection Algorithm for a New General System of Variational Inequalities in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
S. Imnang
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Real Hilbert Space ◽  
General System ◽  
Projection Algorithm ◽  
Common Element ◽  
System Of Variational Inequalities ◽  
Demiclosedness Principle ◽  
Hybrid Projection Algorithm

A new general system of variational inequalities in a real Hilbert space is introduced and studied. The solution of this system is shown to be a fixed point of a nonexpansive mapping. We also introduce a hybrid projection algorithm for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Several strong convergence theorems of the proposed hybrid projection algorithm are established by using the demiclosedness principle. Our results extend and improve recent results announced by many others.

scholarly journals A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

2010 ◽  
Vol 2010 ◽  
pp. 1-27 ◽  
Author(s):  
Jian-Wen Peng ◽  
Soon-Yi Wu ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Infinite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
System Of Equilibrium Problems ◽  
New Iterative Method

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

scholarly journals On Variational Inclusion and Common Fixed Point Problems inq-Uniformly Smooth Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Yanlai Song ◽  
Huiying Hu ◽  
Luchuan Ceng
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Algorithm ◽  
Infinite Family ◽  
General System ◽  
Iterative Sequence ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Uniformly Smooth

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family ofλi-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in aq-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.

scholarly journals A Strong Convergence Theorem for a Common Fixed Point of Two Sequences of Strictly Pseudocontractive Mappings in Hilbert Spaces and Applications

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Kasamsuk Ungchittrakool
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Pseudocontractive Mappings ◽  
Strict Pseudocontraction ◽  
Strictly Pseudocontractive Mappings

We prove a strong convergence theorem for a common fixed point of two sequences of strictly pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main theorem to find a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. The results extend and improve the recent ones announced by Marino and Xu (2007) and others.

scholarly journals Strong convergence algorithm for the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem

2021 ◽  
pp. 1-32
Author(s):  
Shamshad Husain ◽  
Mohd Asad ◽  
Mubashshir Uddin Khairoowala
Keyword(s):  
Fixed Point ◽  
Iterative Scheme ◽  
Fixed Point Problem ◽  
Common Element ◽  
Point Problem ◽  
Variational Inclusions ◽  
Generalized Equilibrium Problem ◽  
Solution Sets ◽  
Generalized Equilibrium ◽  
Split Generalized Equilibrium Problem

The purpose of this paper is to recommend an iterative scheme to approximate a common element of the solution sets of the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem for non-expansive mappings. We prove that the sequences generated by the recommended iterative scheme strongly converge to a common element of solution sets of stated split problems. In the end, we provide a numerical example to support and justify our main result. The result studied in this paper generalizes and extends some widely recognized results in this direction.

scholarly journals An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wanna Sriprad ◽  
Somnuk Srisawat
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Finite Family ◽  
Variational Inequality Problems ◽  
Variational Inclusion ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Convex Minimization Problems

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.

scholarly journals Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Real Hilbert Space ◽  
Monotone Mapping ◽  
Variational Inclusion ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element ◽  
Inclusion Problems ◽  
Minimization Problems ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new general hybrid iterative algorithm for finding a common element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of generalized mixed equilibrium problem, and the set of solution of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Yao and Liou (2010), Tan and Chang (2011), and other authors.

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Adrian Petruşel ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Iterative Process ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Viscosity Approximation ◽  
Inverse Strongly Monotone ◽  
The Common ◽  
Viscosity Approximation Methods

AbstractIn this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.

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