General Quasi-variational Inequalities Research Articles

Recent Developments in General Quasi Variational Inequalities

In this paper, we present a number of new and known numerical techniques for solving general quasi variational inequalities, introduced by Noor [34] in 1988, using various techniques including projection, Wiener-Hopf equations, auxiliary principle, dynamical systems coupled with finite difference approach and sensitivity analysis. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis is also investigated. Some special cases are discussed as applications of the main results. Several open problems are suggested for future research.

Iterative Methods for Random Generalized Quasi Variational Inequalities With Applications

Iterative Methods for Random Generalized Quasi Variational Inequalities With Applications

A Unified Inertial Iterative Approach for General Quasi Variational Inequality with Application

In this paper, we design two inertial iterative methods involving one and two inertial steps for investigating a general quasi-variational inequality in a real Hilbert space. We establish an existence result and a non-trivial example is furnished to substantiate our theoretical findings. We discuss the convergence of the inertial iterative algorithms to approximate the solution of a general quasi-variational inequality. Finally, we apply an inertial iterative scheme with two inertial steps to investigate a delay differential equation. The results presented herein can be seen as substantial generalizations of some known results.

Some new classes of general quasi variational inequalities

In this paper, we introduce and consider some new classes of general quasi variational inequalities, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We propose some new inertial projection methods for solving the general quasi variational inequalities and related problems. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inequalities include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inequalities for further research activities.

Inertial projection methods for solving general quasi-variational inequalities

In this paper, we consider a new class of quasi-variational inequalities, which is called the general quasi-variational inequality. Using the projection operator technique, we establish the equivalence between the general quasi-variational inequalities and the fixed point problems. We use this alternate formulation to propose some new inertial iterative schemes for solving the general quasi-variational inequalities. The convergence criteria of the new inertial projection methods under some appropriate conditions is investigated. Since the general quasi-variational inequalities include the quasi-variational inequalities, variational inequalities, complementarity problems and the related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare the efficiency of the proposed methods with other known methods.

Quasi-Variational Inequality Problems over Product Sets with Quasi-monotone Operators

Quasi-variational inequalities are variational inequalities in which the constraint map depends on the current point. Due to this characteristic, specific proofs have been built to prove adapted existence results. Semicontinuity and generalized monotonicity are assumed and many efforts have been made in the last decades to use the weakest concepts. In the case of quasi-variational inequalities defined on a product of spaces, the existence statements in the literature require pseudomonotonicity of the operator, a hypothesis that is too strong for many applications, in particular in economics. On the other hand, the current minimal hypotheses for existence results for general quasi-variational inequalities are quasi-monotonicity and local upper sign-continuity. But since the product of quasi-monotone (respectively, locally upper sign-continuous) operators is not in general quasi-monotone (respectively, locally upper sign-continuous), it is thus quite difficult to use these general-type existence result in the quasi-variational inequalities defined on a product of spaces. In this work we prove, in an infinite-dimensional setting, several existence results for product-type quasi-variational inequalities by only assuming the quasi-monotonicity and local upper sign-continuity of the component operators. Our technique of proof is strongly based on an innovative stability result and on the new concept of net-lower sign-continuity.

A Perturbed Iterative Algorithm for Split General Mixed Variational Inequality Problem

In this paper, we introduce a split general mixed variational inequality problem, which is a natural extension of a split variational inequality problem, split general quasi-variational inequality problem in Hilbert spaces. Using the resolvent operator technique, we propose a perturbed iterative algorithm for the split general mixed variational inequality problem. Further, we discuss the convergence criteria of the iterative algorithm. The results presented in this paper extend and improve many previously known results in this area.

Split General Strong Nonlinear Quasi-Variational Inequality Problem

We introduce a split general strong nonlinear quasi-variational inequality problem which is a natural extension of a split general quasi-variational inequality problem, split variational inequality problem, and quasi-variational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for the split general strongly nonlinear quasi-variational inequality problem and discuss the convergence criteria of the iterative algorithm. The results presented here generalized, unify, and improve many previously known results for quasi-variational and variational inequality problems.

Existence of General Competitive Equilibria: A Variational Approach

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities.

Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge's fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

Split general quasi-variational inequality problem

Abstract In this paper, we introduce a split general quasi-variational inequality problem which is a natural extension of a split variational inequality problem, quasivariational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for a split general quasi-variational inequality problem and discuss some special cases. Further, we discuss the convergence criteria of these iterative algorithms. The results presented in this paper generalize, unify and improve many previously known results for quasi-variational and variational inequality problems.

Generalized Quasi-Variational Inequalities for Pseudo-Monotone Type III and Strongly Pseudo-Monotone Type III Operators on Non-Compact Sets

In this paper, the authors prove some existence results of solutions for a new class of generalized quasi-variational inequalities (GQVI) for pseudo-monotone type III operators and strongly pseudo-monotone type III operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GQVI for pseudo-monotone type III operators, we shall use Chowdhury and Tan’s generalized version [1] of Ky Fan’s minimax inequality [2] as the main tool.

Three step iterative algorithms for solving a class of quasi variational inequalities

In this paper, we discuss the existence of a solution of extended general quasi variational inequalities using projection technique. Using the technique of Noor, we suggest and analyze a new class of three-step iterative schemes for solving extended general quasi variational inequalities. Under some certain conditions on operators, we also discuss the convergence of the proposed iterative scheme. Results proved in this paper continue to hold for previously known and new classes of quasi variational inequalities. Ideas and techniques of this paper may stimulate further research in this field. The implementation and comparison of the proposed new algorithms is an interesting problem for further research.

Parallel schemes for solving a system of extended general quasi variational inequalities

In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that the system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation, we propose and analyze some parallel schemes for solving a system of extended general quasi variational inequalities. The convergence of these new schemes is discussed under some mild conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems. The ideas and techniques of this paper may stimulate further research in this dynamic field.

On a New Class of Systems of Generalized Quasi-Variational Inequalities

We introduce new types of systems of generalized quasi-variational inequalities and we prove the existence of the solutions by using results of pair equilibrium existence for free abstract economies. We consider the fuzzy models and we also introduce the random free abstract economy and the random equilibrium pair. The existence of the solutions for the systems of quasi-variational inequalities comes as consequences of the existence of equilibrium pairs for the considered free abstract economies.

Some Iterative Schemes for Solving Extended General Quasi Variational Inequalities

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems. This equivalence is used to suggest and analyze some iterative methods for solving the extended general quasi variational inequalities. Convergence analysis is also considered. We have also shown that the extended general quasi variational inequalities are equivalent to the extended general implicit Wiener-Hopf equations. This alternative formulation is used to suggest and analyze some iterative methods. The convergence analysis of these new methods under some suitable conditions is investigated. Several special cases are discussed. Since the extended general quasi variational inequalities include general variational inequalities, quasi variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. Results of this paper may stimulate further research in this fascinating area.

Gap functions and error bounds for inverse quasi-variational inequality problems

This paper is dedicated to inverse quasi-variational inequalities which correspond to a mixture of quasi-variational inequalities and inverse variational inequalities. Our aim is to obtain local/global error bounds for inverse quasi-variational inequality problems in terms of different gap functions/merit functions i.e. the residual gap function, the regularized gap function and the D-gap function. These bounds provide effective estimated distances between a specific point and the exact solution of the inverse quasi-variational inequality problem. Since the class of inverse quasi-variational inequalities includes the classes of general quasi-variational inequalities and variational inequalities, our results cover and extend similar results for these problems.

Solving quasi-variational inequalities via their KKT conditions

We propose to solve a general quasi-variational inequality by using its Karush–Kuhn–Tucker conditions. To this end we use a globally convergent algorithm based on a potential reduction approach. We establish global convergence results for many interesting instances of quasi-variational inequalities, vastly broadening the class of problems that can be solved with theoretical guarantees. Our numerical testings are very promising and show the practical viability of the approach.

Existence Results for General Mixed Quasivariational Inequalities

We consider and study a new class of variational inequality, which is called the general mixed quasivariational inequality. We use the auxiliary principle technique to study the existence of a solution of the general mixed quasivariational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

On general quasi-variational inequalities

A new class of general quasi-variational inequalities involving two operators is introduced and studied. Using essentially the projection operator technique, we establish the equivalence between the general quasi-variational inequalities and the fixed-point problem and the Wiener–Hopf equations. These alternative equivalent formulations have been used to suggest and analyze several iterative methods for solving the general quasi-variational inequalities. We also discuss the convergence criteria of these iterative methods under some suitable conditions. Several special cases are also discussed.