Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces

Firstly, the concept of projective nonexpansive mappings is presented in this paper. The approximate solvability of a generalized system for relaxed cocoercive and involving projective nonexpansive mapping nonlinear variational inequalities in Hilbert spaces is studied, based on the convergence of projection methods. The results presented in this paper extend and improve the main results of many authors.

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Two-Step Systems for G-H-Relaxed Pseudococoercive Nonlinear Variational Problems Based on Projection Methods

Georgian Mathematical Journal ◽

10.1515/gmj.2005.1 ◽

2005 ◽

Vol 12 (1) ◽

pp. 1-10

Author(s):

Ravi P. Agarwal ◽

Donal O'Regan ◽

Ram U. Verma

Keyword(s):

Variational Inequalities ◽

Hilbert Spaces ◽

Variational Problems ◽

Projection Methods ◽

Strongly Monotone ◽

Generalized System ◽

Nonlinear Mappings ◽

Convex Subsets

Abstract The approximation-solvability of a generalized system of nonlinear variational inequalities (SNVI) involving relaxed pseudococoercive mappings, based on the convergence of a system of projection methods, is presented. The class of relaxed pseudococoercive mappings is more general than classes of strongly monotone and relaxed cocoercive mappings. Let 𝐾1 and 𝐾2 be nonempty closed convex subsets of real Hilbert spaces 𝐻1 and 𝐻2, respectively. The two-step SNVI problem considered here is as follows: find an element (𝑥*, 𝑦*) ∈ 𝐻1 × 𝐻2 such that (𝑔(𝑥*), 𝑔(𝑦*)) ∈ 𝐾1 × 𝐾2 and where 𝑆 : 𝐻1 × 𝐻2 → 𝐻1, 𝑇 : 𝐻1 × 𝐻2 → 𝐻2, 𝑔 : 𝐻1 → 𝐻1 and ℎ : 𝐻2 → 𝐻2 are nonlinear mappings.

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Generalized system of relaxed cocoercive variational inequalities governed by nonexpansive mappings

IOP Conference Series Materials Science and Engineering ◽

10.1088/1757-899x/1116/1/012149 ◽

2021 ◽

Vol 1116 (1) ◽

pp. 012149

Author(s):

Sanjeev Gupta

Keyword(s):

Variational Inequalities ◽

Nonexpansive Mappings ◽

Generalized System

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An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Symmetry ◽

10.3390/sym12111915 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1915

Author(s):

Lateef Olakunle Jolaoso ◽

Maggie Aphane

Keyword(s):

Variational Inequalities ◽

Hilbert Spaces ◽

Line Search ◽

Convex Combination ◽

Extragradient Method ◽

Convergence Result ◽

Armijo Line Search ◽

Systems Of Variational Inequalities ◽

Demicontractive Mappings ◽

The Cost

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

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