Iterative Methods for Solving Systems of Variational Inequalities in Reflexive Banach Spaces

Abstract

We prove strong convergence theorems for three iterative algorithms which approximate solutions to systems of variational inequalities for mappings of monotone type. All the theorems are set in reflexive Banach spaces and take into account possible computational errors.

An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces

“…Many authors have considered the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of a variational inequality problem for γ −inverse strongly monotone mapping (see, e.g., [13,15,20,23,26,34]). …”

A composition projection method for feasibility problems and applications to equilibrium problems

“…Indeed, since the distance d(A, B) = 0 and E = Fix(P A P B ) = ∅. (ii) The sequence (x n ) generated by Algorithm (2.1) can be guaranteed the strong convergence under the assumptions of Theorem 2.1 and Corollaries 2.2 and 2.3; (iii) Compared with the Algorithms 6.1 and 6.2 of Kassay, Reich and Sabach [13] and Algorithm 6.1 of Sabach [17], the step Q n+1 = {z ∈ A B :…”

“…It is worth noting that many authors studied the common element for variational inequalities, equilibrium problem, maximal monotone operators and fixed points of nonlinear operators which can be considered as special cases of the problem (FP) (see, [2,3,11,13,14,15,17]). In many practical problems, the set A B is empty.…”

Hybrid iterative algorithm for an infinite families of closed, uniformly asymptotic regular and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings in Banach spaces

“…x in E. Hence, D(·, ·) is reduced to the usual map φ(·, ·) as D(x, y) = ||x|| 2 − 2 x, Jy + ||y|| 2 = φ(x, y) for all x, y ∈ E. The Bregman projection P roj g C (x) is reduced to the generalized projection Π C (x) (see [6][7][8][9][10][11][12][13][14][15][16]), which is defined by…”