New viscosity method for hierarchical fixed point approach to variational inequalities

Abstract

A new viscosity method for hierarchically approximating some common fixed point of an infinite family of nonexpansive mappings is presented; and some strong convergence theorems for solving variational inequality problems and hierarchical fixed point problems are obtained without the aid of the convex linear combination of a countable family of nonexpansive mappings. Solutions are sought in the set of fixed points of another nonexpansive mapping. The results improve those of the authors with the related interest. MSC: 47H09; 47H10; 65J15; 47J25

The common solution for a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem

“…It is known that the hierarchical fixed point problem (1.3) is related with some monotone variational inequalities and convex programming problems; see [7][8][9][10]. Hence, various methods to solve the hierarchical fixed point problem have been studied by many authors; see, for example [8,[11][12][13][14] and the references therein. Now, we give some definitions of nonlinear mappings which are used in the next sections.…”

Convergence Theorems for Hierarchical Fixed Point Problems and Variational Inequalities

“…The existence problem of hierarchical fixed points for a nonlinear mapping and approximation problem has been studied by several authors (see [5][6][7][8][9][10][11][12]). In 2006, Marino and Xu [13] introduced the following viscosity iterative method:…”

Triple-hierarchical problems with variational inequality

“…If C is bounded closed convex and T is a nonexpansive mapping of C into itself, then F ix(T ) is nonempty [15]. We discuss the following variational inequality problem over the fixed point set of a nonexpansive mapping, which is called hierarchical problem denoted by HP see [1,3,4,5,7,18,19,21,22,23]. Let A : C → H be a monotone continuous mapping and T : C → H a nonexpansive mapping.…”