scholarly journals An Iterative Method for Finding Common Solution of the Fixed Point Problem of a Finite Family of Nonexpansive Mappings and a Finite Family of Variational Inequality Problems in Hilbert Space

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Shamshad Husain ◽  
Nisha Singh
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Fixed Point Problem ◽  
Common Fixed Points ◽  
Finite Family ◽  
Common Element ◽  
Point Problem

In this paper, a hybrid iterative algorithm is proposed for finding a common element of the set of common fixed points of finite family of nonexpansive mappings and the set of common solutions of the variational inequality for an inverse strongly monotone mapping on the real Hilbert space. We establish the strong convergence of the proposed method for approximating a common element of the above defined sets under some suitable conditions. The results presented in this paper extend and improve some well-known corresponding results in the earlier and recent literature.

A GENERAL ITERATIVE ALGORITHM FOR EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS IN A HILBERT SPACE

2010 ◽  
Vol 03 (04) ◽  
pp. 685-705
Author(s):  
Tanakit Thianwan
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Finite Family ◽  
Iterative Sequence ◽  
Common Element ◽  
Relaxed Cocoercive Mapping

In this paper, we introduce a general iterative algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. The results obtained in this paper extend and improve the several recent results in this area.

scholarly journals Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1502
Author(s):  
Sun Young Cho
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Strict Pseudocontraction ◽  
Adaptive Step

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

scholarly journals The System of Mixed Equilibrium Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Rabian Wangkeeree ◽  
Panatda Boonman
Keyword(s):  
Hilbert Space ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Solution Set ◽  
Common Element ◽  
Fixed Point Set ◽  
Mixed Equilibrium Problems ◽  
Point Set ◽  
Mixed Equilibrium

We first introduce the iterative procedure to approximate a common element of the fixed-point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptoticTJmapping and the solution set of SMEP in a real Hilbert space.

scholarly journals Approximation of Common Fixed Points of a Sequence of Nearly Nonexpansive Mappings and Solutions of Variational Inequality Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
D. R. Sahu ◽  
Shin Min Kang ◽  
Vidya Sagar
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Variational Inequality Problem ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strong Convergence Theorem

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings.

scholarly journals A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Pongsakorn Sunthrayuth ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
General System ◽  
Fixed Point Problem ◽  
Common Element ◽  
Point Problem ◽  
Computational Technique ◽  
Quadratic Minimization

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.

scholarly journals Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems, Variational Inequality Problems and Common Fixed Point Problems, of Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Jing Zhao
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Common Element ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for aγ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Poom Kumam ◽  
Somyot Plubtieng
Keyword(s):  
Variational Inequality ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Inverse Strongly Monotone Mapping ◽  
Inverse Strongly Monotone ◽  
Viscosity Approximation Methods

AbstractWe use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P C is a metric projection. We consider the iteration process {x n} of C defined by x 1 = x ∈ C is arbitrary and $$ x_{n + 1} = \alpha _n f(x_n ) + (1 - \alpha _n )S_n P_C (x_n + \lambda _n Ax_n ) $$ where f is a contraction on C, {S n} is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x n} converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.

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.

scholarly journals Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 881 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Xiaolong Qin ◽  
Yekini Shehu ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Common Solution ◽  
Asymptotically Nonexpansive

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

scholarly journals Convergence Theorem for Equilibrium and Variational Inequality Problems and a Family of Infinitely Nonexpansive Mappings in Hilbert Space

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhou Yinying ◽  
Cao Jiantao ◽  
Wang Yali
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problem ◽  
Convergence Theorem ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Common Element

We introduce a hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the varitional inequality problem and the equilibrium problem in Hilbert space. Under suitable conditions, some strong convergence theorems are obtained. Our results improve and extend the corresponding results in (Chang et al. (2009), Min and Chang (2012), Plubtieng and Punpaeng (2007), S. Takahashi and W. Takahashi (2007), Tada and Takahashi (2007), Gang and Changsong (2009), Ying (2013), Y. Yao and J. C. Yao (2007), and Yong-Cho and Kang (2012)).

Sign in / Sign up

Export Citation Format

Share Document