Some new iterative schemes for solving general quasi variational inequalities

Abstract

Several new classes of general quasi variational inequalities involving two arbitrary operators are introduced and considered in this paper. Some important cases are discussed, which can be obtained by choosing suitable and appropriate choice of the operators. It is shown that the implicit obstacle boundary value can be studied via these quasi variational inequalities. Projection technique is applied to establish the equivalent between the general quasi variational inequalities and fixed point problems. This alternative formulation is used to discuss the uniqueness of the solution as well as to propose a wide class of proximal point algorithms. Convergence criteria of the proposed methods is considered. Asymptotic stability of the solution is studied using the first order dynamical system associated with variational inequalities. Second order dynamical systems associated with general quasi variational inequalities are applied to suggest some inertial type methods. Some special cases are discussed as applications of the main results. Several open problems are indicated for future research work.