Vector Quasivariational Inequality Research Articles

A topological approach for vector quasi-variational inequalities with set-valued functions

A topological approach for vector quasi-variational inequalities with set-valued functions

Existence conditions for symmetric strong vector quasi-equilibrium problems

In the following paper, the symmetric strong vector quasi-equilibrium problems will be studied thoroughly. Afterward, the existence conditions of solution sets for these problems has been established. The results which are presented in this paper improve and extend the main results mentioned in the literature. Our results can be illustrated by some interesting examples. In 1994, Noor and Oettli introduced the following the symmetric scalar quasi-equilibrium problem. This problem is one of the generalization of the symmetric scalar quasi-equilibrium problem which is presented by Noor and Oettli. Since then, the symmetric vector quasi-equilibrium problem has been investigated by a huge number of authors in different ways. The research works mentioned above are one of our motivation to improve and extend the problem. So, in this paper, we will introduce the vector quasi-equilibrium problems. Afterward, some existence conditions of solution sets for these problems will be established. The symmetric vector quasi-equilibrium problems consist of many optimization - related models namely symmetric vector quasi-variational inequality problems, fixed point problems, coincidence-point problems and complementarity problems, etc. In recent years, a lot of results for existence of solutions for symmetric vector quasi-equilibrium problems, vector quasi-equilibrium problems, vector quasivariational inequality problems and optimization problems have been established by many authors in different ways. We will present our work in the following steps. In the first section of our paper, we will introduce the model of symmetric vector quasi-equilibrium problems. In the following section, we recall definitions, lemmas which can be used for the main results. In the last section, we will establish some conditions for existence and closedness of the solutions set by applying fixed-point theorem for symmetric vector quasi-equilibrium problems. The results presented in this paper improve and extend the main results in the literature. Some examples are given to illustrate our results. Hence our results, Theorem 3.1 and Theorem 3.6 have significant improvements.

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

Continuity of the solution set mappings for parametric unified weak quasivariational inequalities via free-disposal set mappings

ABSTRACT In this study, we mainly study the stability for parametric unified weak quasivariational inequality (PUWQVI) under free-disposal set mappings. Firstly, by using a nonlinear scalarization method, we establish the semicontinuity of gap functions and solution mappings for (PUWQVI). Then, under the free-disposal set mapping assumption, we obtain some sufficient conditions of the outer-continuity and the outer-openness for (PUWQVI), and the inner-openness for parametric unified set-valued strong vector quasivariational inequality, respectively. Moreover, examples are given to illustrate our results, our results are new and improve the corresponding ones in the literature.

Regularized gap functions and error bounds for generalized mixed weak vector quasivariational inequality problems in fuzzy environments

In this paper, we establish gap functions for a new class of generalized mixed weak vector quasivariational inequality problems in fuzzy environments (in short, (MQVIPF)). These functions are constructed through terms of regularized gap functions using the method of nonlinear scalarization functions. Moreover, error bounds are provided for (MQVIPF) via regularized gap functions under suitable assumptions. The main results obtained in this paper improve and extend some corresponding known results. Some examples are given to illustrate our results.

Inverse problems for vector variational and vector quasi-variational inequalities

Inverse problems for vector variational and vector quasi-variational inequalities

Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

In this paper, we consider vector quasi-equilibrium problems under perturbation in terms of suitable asymptotically solving sequences, not embedding given problems into a parameterized family. By employing some types of convergences for mapping and set sequences, we obtain the Painleve–Kuratowski upper convergence of solution sets for the reference problems. Then, using nonlinear scalarization functions, we propose gap functions for such problems, and later employing these functions, we study necessary and sufficient conditions for Painleve–Kuratowski lower convergence and Painleve–Kuratowski convergence. As an application, we discuss the special case of vector quasi-variational inequality.

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.

On the stability of solution mappings parametric generalized vector quasivariational inequality problems of the Minty type

In this paper, we study two parametric weak and strong vector quasivariational inequality problems of the Minty type. The stability properties of the exact solution sets and approximate solution sets for these problems such as the upper semicontinuity, the lower semicontinuity, the Hausdorff lower semicontinuity, the continuity and the Hausdorff continuity are obtained. The results presented in the paper improve and extend the main results in the literature.

Scalarized system of nonsmooth vector quasi-variational inequalities with applications to Debreu type vector equilibrium problems

In this work, we utilize a scalarization method to introduce a system of nonsmooth vector quasi-variational inequalities. We also study their relationship to Debreu type vector equilibrium problems. Then we establish some existence results for solutions of these systems by using maximal element theorems for a family of set-valued maps.

Well-posedness for parametric generalized vector quasivariational inequality problems of the Minty type

In this paper, we introduce the concepts of well-posedness, and of well-posedness in the generalized sense for parametric generalized vector quasivariational inequality problems of the Minty type. The necessary and sufficient conditions for the various kinds of well-posedness of these problems are obtained. Our results are different from some main results in the literature and extend them.MSC:90C31, 49J53, 49J40, 49J45.

Efficiency in vector quasi-equilibrium problems and applications

In this paper, we give sufficient conditions for the existence of efficient solutions of a generalized vector quasi-equilibrium problem in topological vector spaces. The motivations for introducing this problem come from practical problems in traffic networks and the optimal control theory for discrete-time dynamical systems. The main results of the paper are proven with the help of a strongly monotonic function which can be constructed from the data of the problem under consideration. Some notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, are also used in our study. As applications, we obtain existence results in vector quasi-optimization problems, Stampacchia set-valued vector quasi-variational inequality problems and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.

Generalized vector quasi-equilibrium problems on Hadamard manifolds

In this paper, a generalized vector quasi-equilibrium problem (GVQEP) is introduced and studied on Hadamard manifolds. An existence theorem of solutions for the GVQEP is established under some suitable conditions. Some applications to a generalized vector quasi-variational inequality, a generalized vector variational-like inequality and a vector optimization problem are also presented on Hadamard manifolds.

Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces

This paper is devoted to the semicontinuity of solutions of a parametric generalized Minty vector quasivariational inequality problem with set-valued mappings [(in short (PGMVQVI)] in Hausdorff topological vector spaces, when the mapping and the constraint sets are perturbed by different parameters. The upper (lower) semicontinuity and closedness of the solution set mapping for (PGMVQVI) are established under some appropriate assumptions. The sufficient and necessary conditions of the Hausdorff lower semicontinuity and Hausdorff continuity of the solution set mapping for (PGMVQVI) are also derived without monotonicity. As an application, we discuss the upper semicontinuity for the solution set mapping of a special case of the (PGMVQVI).

Nonsmooth multiobjective optimization problems and weak vector quasi-variational inequalities

In this article, we investigate a nonsmooth multiobjective optimization problem (MOP) under generalized invexity. First, the Kuhn–Tucker type optimality conditions for MOP are obtained. Furthermore, the relationships between weakly efficient solutions of MOP and vector valued saddle points of its Lagrange function are established. Last but not the least, the relations between weakly efficient solutions of MOP and solutions of Hartman–Stampacchia weak vector quasi-variational inequalities and Hartman–Stampacchia nonlinear weak vector quasi-variational inequalities are also derived under some suitable assumptions. These results extend and improve some known results in the literature.

An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

AN EXTENSION OF MULTI-VALUED QUASI-GENERALIZED SYSTEM

Recently, Kazmi and Khan [7] introduced a kind of equilibrium problem called generalized system (GS) with a single-valued bi-operator F. Next, in [10], the first author considered a generalization of (GS) into a multi-valued circumstance called the multi-valued quasi-generalized system (in short, MQGS). In the current work, we provide an extension of (MQGS) into a system of (MQGS) in general settings. This system is called the generalized multi-valued quasi-generalized system (in short, GMQGS). Using the existence theorem for abstract economy by Kim [8], we prove the existence of solutions for (GMQGS) in the framework of Hausdorff topological vector spaces. As an application, an existence result of a system of generalized vector quasi-variational inequalities is derived.

Well‐Posedness of Generalized Vector Quasivariational Inequality Problems

We introduce several types of the Levitin‐Polyak well‐posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. Criteria and characterizations of these types of the Levitin‐Polyak well‐posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper unify, generalize, and extend some known results in the literature.

Well-posedness for parametric strong vector quasi-equilibrium problems with applications

In this article, we generalize the concept of well-posedness to the parametric strong vector quasi-equilibrium problem. Under some suitable conditions, we establish some characterizations of well-posedness for parametric strong vector quasi-equilibrium problems. The corresponding concept of well-posedness in the generalized sense is also investigated for the parametric strong vector quasi-equilibrium problem. As applications, we investigate the well-posedness for strong vector quasi-variational inequality problems and strong vector quasi-optimization problems.2000 Mathematics subject classification: 49K40; 90C31.

On the solution semicontinuity to a parametric generalized vector quasivariational inequality

In this paper, the semicontinuities of the solution set map are investigated for a parametric generalized vector quasivariational inequality in locally convex Hausdorff topological vector spaces. The upper semicontinuity and closedness of the solution set map are obtained. A parametric gap function is proposed by using a nonlinear scalarization function. By virtue of the parametric gap function and a key assumption, the Hausdorff lower semicontinuity of the solution set map is established.