scholarly journals Strong Convergence to a Solution of a Variational Inequality Problem in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yasunori Kimura ◽  
Kazuhide Nakajo
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Proximal Point ◽  
Inequality Problem ◽  
Common Solution ◽  
Differentiable Norm

We consider the variational inequality problem for a family of operators of a nonempty closed convex subset of a 2-uniformly convex Banach space with a uniformly Gâteaux differentiable norm, into its dual space. We assume some properties for the operators and get strong convergence to a common solution to the variational inequality problem by the hybrid method proposed by Haugazeau. Using these results, we obtain several results for the variational inequality problem and the proximal point algorithm.

scholarly journals An Inertial Iterative Algorithm to Find Common Solution of a Split Generalized Equilibrium and a Variational Inequality Problem in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Mohammad Farid ◽  
Rehan Ali ◽  
Watcharaporn Cholamjiak
Keyword(s):  
Variational Inequality ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Variational Inequality Problem ◽  
Field Of Study ◽  
Inequality Problem ◽  
Numerical Example ◽  
Common Solution ◽  
Generalized Equilibrium ◽  
The Common

In this paper, we introduce and study an iterative algorithm via inertial and viscosity techniques to find a common solution of a split generalized equilibrium and a variational inequality problem in Hilbert spaces. Further, we prove that the sequence generated by the proposed theorem converges strongly to the common solution of our problem. Furthermore, we list some consequences of our established algorithm. Finally, we construct a numerical example to demonstrate the applicability of the theorem. We emphasize that the result accounted in the manuscript unifies and extends various results in this field of study.

INERTIAL RELAXED TSENG METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEM IN HILBERT SPACE

2021 ◽  
Vol 10 (12) ◽  
pp. 3597-3623
Author(s):  
F. Akusah ◽  
A.A. Mebawondu ◽  
H.A. Abass ◽  
M.O. Aibinu ◽  
O.K. Narain
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extrapolation Method ◽  
Minimum Norm ◽  
Minimum Norm Solution ◽  
Inequality Problem ◽  
Weaker Conditions

The research efforts of this paper is to present a new inertial relaxed Tseng extrapolation method with weaker conditions for approximating the solution of a variational inequality problem, where the underlying operator is only required to be pseudomonotone. The strongly pseudomonotonicity and inverse strongly monotonicity assumptions which the existing literature used are successfully weakened. The strong convergence of the proposed method to a minimum-norm solution of a variational inequality problem are established. Furthermore, we present an application and some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature.

scholarly journals A Generalized Strong Convergence Algorithm in the Presence of Errors for Variational Inequality Problems in Hilbert Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Mostafa Ghadampour ◽  
Donal O’Regan ◽  
Ebrahim Soori ◽  
Ravi P. Agarwal
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Variational Inequality Problem ◽  
Numerical Algorithms ◽  
Variational Inequality Problems ◽  
Numerical Examples ◽  
Inequality Problem ◽  
Convergence Of An Algorithm ◽  
Computational Errors

In this paper, we study the strong convergence of an algorithm to solve the variational inequality problem which extends a recent paper (Thong et al., Numerical Algorithms. 78, 1045-1060 (2018)). We reduce and refine some of their algorithm conditions and we prove the convergence of the algorithm in the presence of some computational errors. Then, using the MATLAB software, the result will be illustrated with some numerical examples. Also, we compare our algorithm with some other well-known algorithms.

scholarly journals Algorithms for Solving the Variational Inequality Problem over the Triple Hierarchical Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Point Set ◽  
Monotone Variational Inequality

This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions.

scholarly journals An Iterative Method for a Common Solution of a Combination of the Split Equilibrium Problem, a Finite Family of Nonexpansive Mapping and a Combination of Variational Inequality Problem

2021 ◽  
Vol 52 ◽  
Author(s):  
Ihssane Hay ◽  
Abdellah Bnouhachem ◽  
Themistocles M. Rassias
Keyword(s):  
Variational Inequality ◽  
Iterative Method ◽  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Variational Inequality Problem ◽  
Convergence Result ◽  
Finite Family ◽  
Split Equilibrium Problem ◽  
Inequality Problem ◽  
Common Solution

The present paper aims to deal with an iterative algorithm for finding common solution of the combination of the split equilibrium problem and a finite family of non-expansive mappings and the combination of variational inequality problem in the setting of real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution to these problems. A numerical example is presented to illustrate the proposed method and convergence result. The results and method presented in this paper generalize, extend and unify some known results in the literatures.

scholarly journals A Cyclic Method for Solutions of a Class of Split Variational Inequality Problem in Banach Space

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Bashir Ali ◽  
Aisha A. Adam ◽  
Yusuf Ibrahim
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Variational Inequality Problem ◽  
Strong Convergence Theorem ◽  
Inequality Problem ◽  
Cyclic Algorithm ◽  
Split Variational Inequality ◽  
Split Variational Inequality Problem

In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, unify, and generalize certain recent results in the literature.

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