scholarly journals Error bounds for general mixed quasivariational inequalities

2005 ◽  
Vol 2005 (7) ◽  
pp. 1085-1100
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Fixed Point ◽  
Error Bounds ◽  
Fixed Point Problems ◽  
Merit Functions ◽  
Quasivariational Inequalities ◽  
Equivalent Formulation ◽  
Mild Conditions ◽  
Special Cases

It is well known that mixed quasivariational inequalities are equivalent to the implicit fixed-point problems. We use this alternative equivalent formulation to suggest and consider some merit functions for general mixed quasivariational inequalities. We use these merit functions to obtain error bounds for the solution under some mild conditions. Some special cases are also discussed.

scholarly journals Gauss-Seidel type algorithms for a class of variational inequalities

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 395-407
Author(s):  
Muhammad Noor ◽  
Awais Khan ◽  
Khalida Noor ◽  
Amjad Pervez
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Projection Operator ◽  
Fixed Point Problems ◽  
Operator Technique ◽  
Equivalent Formulation ◽  
Nonlinear Operators ◽  
Special Cases ◽  
Projection Operator Technique ◽  
New System

In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation, we propose and analyze Gauss-Seidel type algorithms for solving a system of extended general quasi variational inequalities. Convergence of new method is discussed under some suitable conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems.

scholarly journals Strong Convergence Analysis of Iterative Algorithms for Solving Variational Inclusions and Fixed-Point Problems of Pseudocontractive Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Zhangsong Yao ◽  
Yan-Kuen Wu ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Iterative Algorithms ◽  
Fixed Point Problems ◽  
Variational Inclusions ◽  
Type Method ◽  
Common Solution ◽  
Mild Conditions

Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.

scholarly journals A novel iterative approach for solving common fixed point problems in Geodesic spaces with convergence analysis

2021 ◽  
Vol 37 (2) ◽  
pp. 145-160
Author(s):  
THANATPORN BANTAOJAI ◽  
CHANCHAL GARODIA ◽  
IZHAR UDDIN ◽  
NUTTAPOL PAKKARANANG ◽  
PANU YIMMUANG
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Common Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Analysis ◽  
Nonexpansive Mappings ◽  
Fixed Point Problems ◽  
Iterative Approach ◽  
Mild Conditions ◽  
New Iterative Method

In this paper, we introduce a new iterative method for nonexpansive mappings in CAT(\kappa) spaces. First, the rate of convergence of proposed method and comparison with recently existing method is proved. Second, strong and \Delta-convergence theorems of the proposed method in such spaces under some mild conditions are also proved. Finally, we provide some non-trivial examples to show efficiency and comparison with many previously existing methods.

scholarly journals Damped Algorithms for the Split Fixed Point and Equilibrium Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Li-Jun Zhu ◽  
Minglun Ren
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Equilibrium Problems ◽  
Variational Inequality Problems ◽  
Fixed Point Problems ◽  
Special Cases

The main purpose of this paper is to study the split fixed point and equilibrium problems which includes fixed point problems, equilibrium problems, and variational inequality problems as special cases. A damped algorithm is presented for solving this split common problem. Strong convergence analysis is shown.

scholarly journals Projection Methods and a New System of Extended General Regularized Nonconvex Set-Valued Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Javad Balooee ◽  
Yeol Je Cho ◽  
Mee Kwang Kang
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Projection Methods ◽  
Fixed Point Problems ◽  
Equivalent Formulation ◽  
New System

A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.

scholarly journals Merit functions for absolute value variational inequalities

2021 ◽  
Vol 6 (11) ◽  
pp. 12133-12147
Author(s):  
Safeera Batool ◽  
◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Keyword(s):  
Variational Inequalities ◽  
Error Bounds ◽  
Complementarity Problem ◽  
Merit Functions ◽  
Absolute Value Equations ◽  
Absolute Value ◽  
Special Cases ◽  
The Absolute

<abstract><p>This article deals with a class of variational inequalities known as absolute value variational inequalities. Some new merit functions for the absolute value variational inequalities are established. Using these merit functions, we derive the error bounds for absolute value variational inequalities. Since absolute value variational inequalities contain variational inequalities, absolute value complementarity problem and system of absolute value equations as special cases, the findings presented here recapture various known results in the related domains. The conclusions of this paper are more comprehensive and may provoke futuristic research.</p></abstract>

scholarly journals On New Proximal Point Methods for Solving the Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Proximal Point ◽  
Equivalent Formulation ◽  
Extragradient Methods ◽  
Proximal Point Methods ◽  
New Methods ◽  
Special Cases

It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered under some suitable conditions. Results proved in this paper may stimulate further research in this direction.

scholarly journals Halpern-Subgradient Extragradient Method for Solving Equilibrium and Common Fixed Point Problems in Reflexive Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 743
Author(s):  
Annel Thembinkosi Bokodisa ◽  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Reflexive Banach Space ◽  
Extragradient Method ◽  
Convergence Result ◽  
Fixed Point Problems ◽  
Subgradient Extragradient Method ◽  
Reflexive Banach Spaces ◽  
Mild Conditions ◽  
Lipschitz Constants

In this paper, using the concept of Bregman distance, we introduce a new Bregman subgradient extragradient method for solving equilibrium and common fixed point problems in a real reflexive Banach space. The algorithm is designed, such that the stepsize is chosen without prior knowledge of the Lipschitz constants. We also prove a strong convergence result for the sequence that is generated by our algorithm under mild conditions. We apply our result to solving variational inequality problems, and finally, we give some numerical examples to illustrate the efficiency and accuracy of the algorithm.

scholarly journals Multiple-Sets Split Common Fixed-Point Problems for Demicontractive Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Huanhuan Cui
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Common Fixed Point ◽  
Iterative Methods ◽  
Fixed Point Problems ◽  
Mild Conditions ◽  
Demicontractive Mappings ◽  
Split Common Fixed Point

In this paper, we are concerned with the multiple-sets split common fixed-point problems whenever the involved mappings are demicontractive. We first study several properties of demicontractive mappings and particularly their connection with directed mappings. By making use of these properties, we propose some new iterative methods for solving multiple-sets split common fixed-point problems, as well as multiple-sets spit feasibility problems. Under mild conditions, we establish their weak convergence of the proposed methods.

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