Mixed Quasi-variational Inequalities Research Articles

The study of error bounds for generalized vector inverse mixed quasi-variational inequalities

In this study, generalized multivalued vector inverse quasi-variational inequality problems are developed, and error bounds are obtained in terms of the residual gap function, the regularized gap function, and the D-gap function. With the help of these constraints, one can effectively estimate the distances between any feasible point and the solution set of problems involving generalized multivalued vector inverse quasi-variational inequality.

Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment

In this paper, we consider a new class of split weak vector mixed quasi-variational inequality problems in fuzzy environments (for short, SWQVIP). Our aim is to establish error bounds for the underlying problem SWQVIP via regularized gap functions. We first propose some regularized gap functions of the problem SWQVIP using the method of nonlinear scalarization functions. Then, error bounds for the problem SWQVIP are investigated based on regularized gap functions with some suitable conditions without monotonicity. Finally, some examples are given to illustrate our results. The main results obtained in this paper are new and extend some corresponding known results in the literature.

Existence and Convergence Results for Generalized Mixed Quasi-Variational Hemivariational Inequality Problem

The main purpose of this paper is threefold. One is to study the existence and convergence problem of solutions for a class of generalized mixed quasi-variational hemivariational inequalities. The second one is to study the existence of optimal control for such kind of generalized mixed quasi-variational hemivariational inequalities under given control u∈U. The third one is to study the relationship between the optimal control and the data for the underlying generalized mixed quasi-variational inequality problems and a class of minimization problem. As an application, we utilize our results to study the elastic frictional problem in a class of Hilbert spaces. The results presented in the paper extend and improve upon some recent results.

Error Bounds for Inverse Mixed Quasi-Variational Inequality via Generalized Residual Gap Functions

In this paper, we investigate error bounds of an inverse mixed quasi variational inequality problem in Hilbert spaces. Under the assumptions of strong monotonicity of function couple, we obtain some results related to error bounds using generalized residual gap functions. Each presented error bound is an effective estimation of the distance between a feasible solution and the exact solution. Because the inverse mixed quasi-variational inequality covers several kinds of variational inequalities, such as quasi-variational inequality, inverse mixed variational inequality and inverse quasi-variational inequality, the results obtained in this paper can be viewed as an extension of the corresponding results in the related literature.

Degree Theory for Generalized Mixed Quasi-variational Inequalities and Its Applications

The present paper is devoted to building degree theory for a generalized mixed quasi-variational inequality in finite dimensional spaces. Then, by employing the obtained results, we prove the existence and stability of solutions to the considered generalized mixed quasi-variational inequality.

Error bounds for mixed set-valued vector inverse quasi-variational inequalities

The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results.

On the lower semicontinuity of the solution map- ping for parametric vector mixed quasivariational inequality problem of the Minty type

In this paper, we study a class of parametric vector mixed quasivariational inequality problem of the Minty type (in short, (MQVIP)). Afterward, we establish some sufficient conditions for the stability properties such as the inner-openness, lower semicontinuity and Hausdorff lower semicontinuity of the solution mapping for this problem. The results presented in this paper is new and wide to the corresponding results in the literature

On the upper semicontinuity of the solution mapping for parametric vector mixed quasivariational inequality

In this paper, we first study a class of parametric generalized vector mixed quasivariational inequality problem of the Minty type in locally convex Hausdorff topological vector spaces, this problem contains many problems as special cases, such as optimization problems, traffic network problems, Nash equilibrium problems, fixed point problems, variational inequality problems and complementarity problems, economic equibrium problems. Then, we establishe the conditions sufficient for stability properties such as: the upper semicontinuity, closedness, outer-continuity, outer-openness of the solution mapping for parametric generalized vector mixed quasivariational inequality problem of the Minty type. The results of the upper semi-continuity and the closeness of the solution mapping for parametric generalized vector mixed quasivariational inequality problem of the Minty type are improve and extend some of the results given by Lalitha and Bhatia. An example is given to demonstrate our results.The results of the outer continuity and the outer-openness of the solution mapping for the parametric generalized vector mixed quasivariational inequality problem of the Minty type are new. We also give some examples to show the relationship between upper semi-continuity, closedness outer continuity and outer-openness.

MIXED QUASI VARIATIONAL INEQUALITIES INVOLVING FOUR NONLINEAR OPERATORS

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the mixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

Solving Mixed Quasi-variational Inequality for Modified Total Generalized Variation for Image Denoising

Image denoising has always been a hot issue in the field of image processing. In order to better reduce the image noise, a modified total generalized variation model is presented. The theory of mixed quasi-variational inequality is introduced to numerically solve the modified model. Compared with several variational models, experiments demonstrate that the proposed method has better performance in edge-preserving while filtering the noise

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

This paper is devoted to investigating a vector inverse mixed quasi-variational inequality (VIMQVI). Our aim is to obtain error bounds for VIMQVI in terms of different gap functions, i.e., the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of VIMQVI. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang. Our results cover and extend similar results for these problems.

Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of p -Laplacian type**Project supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is

The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of p -Laplacian type.

A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.

Existence and stability for a generalized differential mixed quasi-variational inequality

In the present paper, we investigate a generalized differential mixed quasi-variational inequality consisting of a system of an ordinary differential equation and a generalized mixed quasi-variational inequality. By using an important result concerning the measurable selection, we prove the existence of Caratheodory weak solution to ´ the generalized differential mixed quasi-variational inequality. Then, with the existence result, we establish two stability results for the generalized differential mixed quasi-variational inequality under different conditions, i.e., upper semicontinuity and lower semicontinuity of the Caratheodory weak solution with respect to the ´ parameter, which is a perturbation of some mappings in the generalized mixed quasi-variational inequality.

On the well-posedness of differential mixed quasi-variational-inequalities

We discuss the well-posedness and the well-posedness in the generalized sense of differential mixed quasi-variational inequalities ((DMQVIs), for short) in Hilbert spaces. This gives us an outlook to the convergence analysis of approximating sequences of solutions for (DMQVIs). Using these concepts we point out the relation between metric characterizations and well-posedness of (DMQVIs). We also prove that the solution set of (DMQVIs) is compact, if problem (DMQVIs) is well-posed in the generalized sense.

Lagrangian methods for optimal control problems governed by a mixed quasi-variational inequality

This paper studies an optimal control problem where the state of the system is defined by a mixed quasi-variational inequality. Several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem are obtained by using nonlinear Lagrangian methods. Our results are applied to an example where the mixed quasi-variational inequality leads to a bilateral obstacle problem.

The GAP function of a parametric mixed strong vector quasivariational inequality problem

The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational inequality problems, fixed point problems, coincidence point problems, complementary problems etc. There are many authors who have been studied the gap functions for vector variational inequality problem. This problem plays an important role in many fields of applied mathematics, especially theory of optimization. In this paper, we study a parametric gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem (in short, (SQVIP)) in Hausdorff topological vector spaces. (SQVIP) Find
 x ̅ ∈ K(x ̅ ,γ) and z ̅ ∈ T(x ̅ ,γ) such that
 < z ̅ , y-x ̅ >+ f(y, x ̅ ,γ) ∈ Rn+ ∀ y ∈ K(x ̅ ,γ),
 where we denote the nonnegative of Rn by
 Rn+= {t=(t1 ,t2,…,tn )T ∈ Rn |ti >0, i = 1,2, ...,n}.
 Moreover, we also discuss the lower semicontinuity, upper semicontinuity and the continuity for the parametric gap function for this problem. To the best of our knowledge, until now there have not been any paper devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem in Hausdorff topological vector spaces. Hence the results presented in this paper (Theorem 1.3 and Theorem 1.4) are new and different in comparison with some main results in the literature.

Sensitivity Analysis of Solutions for a Class of Generalized Parametric Mixed Quasi-variational Inequalities

In this paper, a class of generalized parametric mixed quasi-variational inequalities is introduced and under the suitable conditions, the sensitivity analysis results of the solutions for the class of generalized parametric mixed quasi-variational inequalities are obtained by using the projective method and the fixed point technique for setvalued contactive mapping. The results presented here improve and extend many known results.

Gap Function and Global Error Bounds for Generalized Mixed Quasi-variational Inequality

Gap Function and Global Error Bounds for Generalized Mixed Quasi-variational Inequality

A Descent Resolvent Method for Mixed Quasi Variational Inequalities

A Descent Resolvent Method for Mixed Quasi Variational Inequalities