Strong convergence of a hybrid method for pseudomonotone variational inequalities and fixed point problems

Abstract

In this paper, we suggest a hybrid method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. We derive a necessary and sufficient condition for the strong convergence of the sequences generated by the proposed method.

A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space

“…Equilibrium problems are generalized by several problems such as: optimization problems, variational inequalities, etc. In recent years, several methods have been proposed for finding a solution of equilibrium problem (2) in Hilbert space [5,7,15,19,18]. In 2010, for finding a common element of the set of fixed points of nonexpansive mappings, the set of the solutions of variational inequalities for αinverse strongly monotone operators, and the set of the solutions of equilibrium problems in Hilbert space, Saeidi [12] proposed the following iterative method:…”

A Parallel Hybrid Method for Equilibrium Problems, Variational Inequalities and Nonexpansive Mappings in Hilbert Space

“…Equilibrium problems are generalized by several problems such as: optimization problems, variational inequalities, etc. In recent years, several methods have been proposed for finding a solution of equilibrium problem (2) in Hilbert space [5,7,16,18,19]. In 2010, for finding a common element of the set of fixed points of nonexpansive mappings, the set of the solutions of variational inequalities for α-inverse strongly monotone operators, and the set of the solutions of equilibrium problems in Hilbert space, Saeidi [12] proposed the following iterative method:…”