Application of Quasi-Nonexpansive Operators to an Iterative Method for Variational Inequality

Abstract

In this paper, we prove the convergence in a norm of sequences generated by an iterative process for solving a variational inequality over the subset of fixed points of a quasi-nonexpansive operator $T$ defined on a Hilbert space. The process employs a sequence of quasi-nonexpansive operators for which the subset of common fixed points contains Fix$T$. We prove the convergence under a demi-closedness type condition for the sequence of operators as well as under the assumption that the process is approximately shrinking. We also give examples of methods satisfying these assumptions.