Abstract
In this paper, we introduce and consider a new system of generalized variational inequalities involving five different operators. Using the sunny nonexpansive retraction technique we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.
Highlights
Let (E, · ) be a Banach space and C be a nonempty closed convex subset of E
This paper deals with the problem of convergence of an iterative algorithm for a system of generalized variational inequalities in a Banach space: Find (x∗, y∗) ∈ C × C such that ρA1(y∗) + x∗ − g1(y∗), J(g1(x) − x∗) ≥ 0, ∀x ∈ C, ηA2(x∗) + y∗ − g2(x∗), J(g2(x) − y∗) ≥ 0, ∀x ∈ C, where Ai, gi : C → E are four nonlinear mappings for i = 1, 2, J is the normalized duality mapping and ρ, η > 0 are positive constants
This paper was supported by Dong Eui University Grant (2013AA070). c The Kangwon-Kyungki Mathematical Society, 2013
Summary
Let (E, · ) be a Banach space and C be a nonempty closed convex subset of E. This paper deals with the problem of convergence of an iterative algorithm for a system of generalized variational inequalities in a Banach space: Find (x∗, y∗) ∈ C × C such that (1.1). If x∗ = y∗, problem (1.3) reduces to the following classical variational inequality (VI(A, C)): Find x∗ ∈ C such that (1.4). PC(I − λA), where I is the identity mapping, λ > 0 is a constant and PC is the metric projection of H onto C This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problem. We show that the convergence analysis of the new iterative method under certain mild conditions
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