Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

Abstract

In this paper, we introduce and analyze implicit and explicit iteration methods for solving a variational inequality problem over the set of common fixed points of an infinite family of nonexpansive mappings on a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Strong convergence results are given. Our results improve and extend the corresponding results in the literature.

Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces

“…Finally, under very mild conditions, we prove the strong convergence of the proposed methods by using V-mappings instead of W-ones. Our results improve and extend the corresponding results announced by some others, e.g., Ceng et al [7] and Buong and Phuong [5].…”

“…Assume that λ ∈ (0, α κ 2 ) and µ ∈ (0, β κ 2 ) where κ is the 2-uniformly smooth constant of X (see Lemma 2.3). Very recently, in order to solve GSVI (1.1), Ceng et al [7] introduced an implicit algorithm of Mann's type. Algorithm 2.7 ([7, Algorithm 3.6]).…”

“…It is worth mentioning that the system of variational inequalities plays an important role in game theory and economics. Namely, the Nash equilibrium problem can be formulated in the form of variational inequality; see e.g., [1,7] and the references therein. Existing results.…”

Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities

“…Assume that λ ∈ (0, α κ 2 ) and µ ∈ (0, β κ 2 ) where κ is the 2-uniformly smooth constant of X (see Lemma 2.2). Very recently, in order to solve GSVI (1.1), Ceng et al [4] introduced an implicit algorithm of Mann's type. Algorithm 2.6 ([4, Algorithm 3.6]).…”

“…It was proven in [4] that the net {x t } converges in norm, as t → 0 + , to the unique solution x * ∈ GSVI(C, A, B) to the following VI:…”