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.

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 Fixed Points Approximation and Solutions of Some Equilibrium and Variational Inequalities Problems

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Bashir Ali
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strong Convergence Theorem ◽  
Points Approximation

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.

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 A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2039 ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Adaptive Method ◽  
Convergence Result ◽  
Null Point ◽  
Fixed Point Problems ◽  
Bounded Linear ◽  
Contraction Method ◽  
Projection And Contraction Method ◽  
Self Adaptive

We introduce a new projection and contraction method with inertial and self-adaptive techniques for solving variational inequalities and split common fixed point problems in real Hilbert spaces. The stepsize of the algorithm is selected via a self-adaptive method and does not require prior estimate of norm of the bounded linear operator. More so, the cost operator of the variational inequalities does not necessarily needs to satisfies Lipschitz condition. We prove a strong convergence result under some mild conditions and provide an application of our result to split common null point problems. Some numerical experiments are reported to illustrate the performance of the algorithm and compare with some existing methods.

scholarly journals Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

2015 ◽  
Vol 2015 ◽  
pp. 1-22
Author(s):  
L. C. Ceng ◽  
A. Latif ◽  
C. F. Wen ◽  
A. E. Al-Mazrooei
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Steepest Descent ◽  
Solution Set ◽  
Steepest Descent Method ◽  
Common Element ◽  
Variational Inclusions

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.

scholarly journals Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Prasit Cholamjiak ◽  
Suthep Suantai
Keyword(s):  
Banach Spaces ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Countable Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Convex Minimization Problems ◽  
The Common ◽  
System Of Equilibrium Problems

We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.

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