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

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.

Generalized System for Relaxed Cocoercive and Involving Projective Nonexpansive Mapping Variational Inequalities

2011 ◽  
Vol 393-395 ◽  
pp. 792-795
Author(s):  
Guang Hui Gu ◽  
Yong Fu Su
Keyword(s):  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Projection Methods ◽  
Generalized System

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.

scholarly journals Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Monica Gabriela Cojocaru ◽  
Stephane Pia
Keyword(s):  
Dynamical Systems ◽  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Dual Space ◽  
Variational Problems ◽  
Base Space ◽  
Projected Dynamical Systems

This paper presents a generalization of the concept and uses of projected dynamical systems to the case of nonpivot Hilbert spaces. These are Hilbert spaces in which the topological dual space is not identified with the base space. The generalization consists of showing the existence of such systems and their relation to variational problems, such as variational inequalities. In the case of usual Hilbert spaces these systems have been extensively studied, and, as in previous works, this new generalization has been motivated by applications, as shown below.

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