scholarly journals A Viscosity Approximation Scheme for Finding Common Solutions of Mixed Equilibrium Problems, a Finite Family of Variational Inclusions, and Fixed Point Problems in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Bin-Chao Deng ◽  
Tong Chen ◽  
Baogui Xin
Keyword(s):  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Finite Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Mixed Equilibrium Problems ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings ◽  
Mixed Equilibrium

We introduce an iterative method for finding a common element of set of fixed points of nonexpansive mappings, the set of solutions of a finite family of variational inclusion with set-valued maximal monotone mappings and inverse strongly monotone mappings, and the set of solutions of a mixed equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of Plubtemg and Sripard and many others.

scholarly journals S-iteration process of Halpern-type for common solutions of nonexpansive mappings and monotone variational inequalities

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1727-1746 ◽  
Author(s):  
D.R. Sahu ◽  
Ajeet Kumar ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Common Element ◽  
Monotone Mappings ◽  
Monotone Variational Inequalities ◽  
Numerical Examples ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings

This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.

scholarly journals A New Iterative Method for the Set of Solutions of Equilibrium Problems and of Operator Equations with Inverse-Strongly Monotone Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jong Kyu Kim ◽  
Nguyen Buong ◽  
Jae Yull Sim
Keyword(s):  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Finite Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Operator Equations ◽  
Strongly Monotone ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings ◽  
New Iterative Method

The purpose of the paper is to present a new iteration method for finding a common element for the set of solutions of equilibrium problems and of operator equations with a finite family ofλi-inverse-strongly monotone mappings in Hilbert spaces.

scholarly journals Viscosity Iterative Schemes for Finding Split Common Solutions of Variational Inequalities and Fixed Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Zhenhua He ◽  
Wei-Shih Du
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Monotone Mappings ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Iterative Schemes ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings

We introduce some new iterative schemes based on viscosity approximation method for finding a split common element of the solution set of a pair of simultaneous variational inequalities for inverse strongly monotone mappings in real Hilbert spaces with a family of infinitely nonexpansive mappings. Some strong convergence theorems are also given. Our results generalize and improve some well-known results in the literature and references therein.

scholarly journals Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Wiyada Kumam ◽  
Prapairat Junlouchai ◽  
Poom Kumam
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Maximal Monotone ◽  
Iterative Sequence ◽  
Common Element ◽  
Monotone Mappings ◽  
Strongly Monotone ◽  
Special Cases ◽  
Inverse Strongly Monotone ◽  
Strongly Monotone Mappings

We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., (2008) and many others.

scholarly journals A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

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

We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions 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), Su et al. (2008), Klin-eam and Suantai (2009), Tan and Chang (2011), and some other authors.

scholarly journals Relaxed Iterative Algorithms for Generalized Mixed Equilibrium Problems with Constraints of Variational Inequalities and Variational Inclusions

2014 ◽  
Vol 2014 ◽  
pp. 1-25 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Chi-Ming Chen ◽  
Ching-Feng Wen ◽  
Chin-Tzong Pang
Keyword(s):  
Variational Inequalities ◽  
Finite Family ◽  
Generalized Mixed Equilibrium Problem ◽  
Monotone Mappings ◽  
Variational Inclusions ◽  
Strongly Monotone ◽  
Inverse Strongly Monotone ◽  
Generalized Mixed Equilibrium ◽  
Strongly Monotone Mappings ◽  
Mixed Equilibrium

We introduce and analyze a relaxed extragradient-like viscosity iterative algorithm for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems which also solves a variational inequality problem.

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 Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions 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. Furthermore, we give a numerical example which supports our main theorem in the last part.

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