scholarly journals Iterative Algorithms for General Multivalued Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yonghong Yao ◽  
Muhammad Aslam Noor ◽  
Yeong-Cheng Liou ◽  
Shin Min Kang
Keyword(s):  
Variational Inequalities ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Convergence Result ◽  
Common Element ◽  
Projection Technique ◽  
Mild Conditions ◽  
Special Cases ◽  
The Common ◽  
Multivalued Variational Inequalities

We introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

scholarly journals On a class of multivalued variational inequalities

1998 ◽  
Vol 11 (1) ◽  
pp. 79-93 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Optimization Problems ◽  
Iterative Algorithms ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Projection Techniques ◽  
Multivalued Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener-Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

A Unified Iteration Method with Error for Inclution Problem

2011 ◽  
Vol 403-408 ◽  
pp. 1584-1587
Author(s):  
Han Jun Chen ◽  
Yang Xue ◽  
Hong Zhao
Keyword(s):  
Iteration Method ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Inclusion Problem ◽  
Operator Technique ◽  
Convergence Criteria ◽  
Special Cases ◽  
Resolvent Operator Technique ◽  
The Common ◽  
The Resolvent Operator

In this paper, we suggest and analyze a unified iteration method with error for finding the common element of the set of fixed points of nonexpansive mappings and the set of the solutions of the inclusion problem using the resolvent operator technique. We also study the convergence criteria of the unified iteration method with error under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement the previously known results.

scholarly journals Extension of Extragradient Techniques for Variational Inequalities

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 111
Author(s):  
Yonghong Yao ◽  
Ke Wang ◽  
Xiaowei Qin ◽  
Li-Jun Zhu
Keyword(s):  
Variational Inequalities ◽  
Convergence Result ◽  
Type Method ◽  
Mild Conditions ◽  
The Common

An extragradient type method for finding the common solutions of two variational inequalities has been proposed. The convergence result of the algorithm is given under mild conditions on the algorithm parameters.

scholarly journals Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Rabian Wangkeeree ◽  
Uraiwan Boonkong
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Common Fixed Points ◽  
Common Element ◽  
Fixed Point Set ◽  
Generalized Equilibrium ◽  
Point Set ◽  
The Common

We introduce new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the systems of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in Hilbert spaces.

scholarly journals On certain classes of variational inequalities and related iterative algorithms

1996 ◽  
Vol 9 (1) ◽  
pp. 43-56 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Approximate Solution ◽  
Variational Inequalities ◽  
Unique Solution ◽  
Iterative Algorithm ◽  
Iterative Algorithms ◽  
Projection Technique ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Special Cases

In this paper, we introduce and study some new classes of variational inequalities and Wiener-Hopf equations. Essentially using the projection technique, we establish the equivalence between the multivalued general quasi-variational inequalities and the multivalued implicit Wiener-Hopf equations. This equivalence enables us to suggest and analyze a number of iterative algorithms for solving multivalued general quasi-variational inequalities. We also consider the auxiliary principle technique to prove the existence of a unique solution of the variational-like inequalities. This technique is used to suggest a general and unified iterative algorithm for computing the approximate solution. Several special cases which can be obtained from our main results are also discussed. The results proved in this paper represent a significant refinement and improvement of the previously known results.

scholarly journals Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Yekini Shehu ◽  
Yeol Je Cho
Keyword(s):  
Variational Inequalities ◽  
Bandwidth Allocation ◽  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Adaptive Method ◽  
Convergence Result ◽  
Common Fixed Points ◽  
Common Element ◽  
Generalized Nash Equilibrium Problem ◽  
Nash Equilibrium Problem

AbstractIn this paper, using a Bregman distance technique, we introduce a new single projection process for approximating a common element in the set of solutions of variational inequalities involving a pseudo-monotone operator and the set of common fixed points of a finite family of Bregman quasi-nonexpansive mappings in a real reflexive Banach space. The stepsize of our algorithm is determined by a self-adaptive method, and we prove a strong convergence result under certain mild conditions. We further give some applications of our result to a generalized Nash equilibrium problem and bandwidth allocation problems. We also provide some numerical experiments to illustrate the performance of our proposed algorithm using various convex functions and compare this algorithm with other algorithms in the literature.

Iterative algorithm for a split equilibrium problem and fixed problem for finite asymptotically nonexpansive mappings in Hlbert space

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1423-1434 ◽  
Author(s):  
Sheng Wang ◽  
Min Chen
Keyword(s):  
Equilibrium Problem ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Fixed Point Set ◽  
Split Equilibrium Problem ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Point Set ◽  
The Common

In this paper, we propose an iterative algorithm for finding the common element of solution set of a split equilibrium problem and common fixed point set of a finite family of asymptotically nonexpansive mappings in Hilbert space. The strong convergence of this algorithm is proved.

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.

scholarly journals Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Kyung Soo Kim
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Nonempty Closed Convex Subset ◽  
Common Element ◽  
Mild Conditions ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Reversible Semigroup ◽  
Invariant Means

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI={T(s):s∈S}on a nonempty closed convex subsetCof a Banach space with respect to a sequence of asymptotically left invariant means{μn}defined on an appropriate invariant subspace ofl∞(S), whereSis a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed pointsF(I), whereF(I)=⋂{F(T(s)):s∈S}.

scholarly journals Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6267-6281
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
General System ◽  
Inequality Constraint ◽  
Implicit And Explicit

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

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