Gradient-type projection methods for quasi-variational inequalities

Abstract

We study methods for solving quasi-variational inequalities which are a notable generalization of the variational inequalities. Solving quasi-variational inequality requires that the corresponding variational inequality be solved concurrently with the calculation of a fixed point of the set-valued mapping. For this reason, the literature on quasi-variational inequalities is not very extensive in what concerns solution methods. In this paper we suggest and analyze a new continuous and iterative variants of some generalizations of the gradient-type projection method for solving quasi-variational inequalities. Using the technique of Noor, we also propose a new two-step iterative scheme. We also establish sufficient conditions for the convergence of the proposed methods and estimate the rates of convergence.