Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

一类积集上的广义向量变分不等式

研究一类积集上具某种权向量的广义向量变分不等式组及其广义向量变分不等式的有关问题,刻画它们之间解的相互关系.在映射的次连续性和关于某向量广义单调性的条件下,利用集值映射的不动点定理,对所讨论的几种类型的广义向量变分不等式给出解的存在性.     

Systems of generalized variational inequalities and their applications ∗

In this paper, we first introduce the system of generalized implicit variational inequalities and prove the existence of its solution. Then we derive existence results for systems of generalized variational and variational like inequalities and system of variational inequalities. As applications, we establish some existence results for a solution to the system of optimization problems which includes the Nash equilibrium problem as a special case     

Generalized Vector Variational Inequalities

In this paper, we introduce a generalized vector variational inequality problem (GVVIP) which extends and unifies vector variational inequalities as well as classical variational inequalities in the literature. The concepts of generalized C-pseudomonotone and generalized hemicontinuous operators are introduced. Some existence results for GVVIP are obtained with the assumptions of generalized C-pseudomonotonicity and generalized hemicontinuity. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained.     

Existence of Solutions for Generalized Vector Variational-like Inequalities

In this paper, we consider a generalized vector variational-like inequality problem (for short, GVVLIP), which includes generalized vector variational inequalities, vector variational inequalities and classical variational inequalities as special cases. The concepts of generalized C-pseudomonotone-like and generalized H-hemicontinuous-like operators are introduced. Some existence results for GVVLIP are obtained under the assumptions of generalized C-pseudomonotone-like property and generalized H-hemicontinuous-like property. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained. In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The third author was partially supported by Grant NSC 94-2213-E-110-035.     

An existence result for generalized vector equilibrium problems without pseudomonotonicity

In this work, we introduce and study a class of generalized vector equilibrium problems for multifunctions which includes a number of generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. By using the KKM–Fan theorem and Nadler’s result, we prove an existence theorem for solutions for this class of generalized vector equilibrium problems in Banach spaces. Applications to generalized vector variational-like inequalities are given.     

System of nonsmooth variational inequalities with applications

In this paper, we consider a system of vector variational inequalities and a system of nonsmooth variational inequalities defined by means of Clarke directional derivative. We also consider the Nash equilibrium problem with vector pay-offs and its scalarized form. We present some relations among these systems and problems. The existence results for a solution of system of nonsmooth variational inequalities are given. As a consequence, we derive an existence result for a solution of Nash equilibrium problem with vector pay-offs.     

Generalized Vector Variational Inequalities over Countable Product of Sets

In this paper, we consider vector variational inequalities with set-valued mappings over countable product sets in a real Banach space setting. By employing concepts of relative pseudomonotonicity, we establish several existence results for generalized vector variational inequalities and for systems of generalized vector variational inequalities. These results strengthen previous existence results which were based on the usual monotonicity type assumptions     

Generalized Vector Equilibrium Problems in Generalized Convex Spaces

In this paper, we introduce and study a class of abstract generalized vector equilibrium problems (AGVEP) in generalized convex spaces which includes most vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems, and generalized vector variational inequality problems as special cases. By using the generalized GKKM and generalized SKKM type theorems due to the first author, some new existence results of equilibrium points for the AGVEP are established in noncompact generalized convex spaces. As consequences, some recent results in the literature are obtained under much weaker assumptions.     

Existence of Solutions to Implicit Vector Variational Inequalities

In this paper, we study a class of implicit vector variational inequalities which contain implicit variational inequalities and generalized quasivariational inequalities as special cases. By employing the Fan–Kakutani fixed-point theorem and the Oettli scalarization procedure, respectively, we establish several existence results for implicit vector variational inequalities.     

Some generalized vector variational inequalities and complementarity problems for multivalued mappings

In this paper, we introduce and study a new class of generalized vector variational inequalities and complementarity problems for multivalued mappings. We prove the existence of solutions for this kind of vector variational inequality and discuss the relations between the solutions of the generalized vector variational inequalities and the solutions of generalized vector complementarity problems in Hausdorff topological vector spaces. Our results extend and improve some results in this field.     

Exceptional Families and Existence Theorems for Variational Inequality Problems

This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provides a new powerful tool for the study of the existence theory for variational inequalities.     

SYSTEM OF GENERALIZED VECTOR VARIATIONAL INEQUALITIES

1IntroductionA vector variational inequality(for short,VVI),as an important generalization of the clas-sical variational inequality,was first introduced and studied by Giannessi[6]in finite dimensionalEuclidean spaces.Later on,a VVI was studied and genera…     

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques. The first two authors were supported by SABIC and Fast Track Research Grants SAB-2006-05. They are grateful to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities.     

Generalized Vector Variational and Quasi-Variational Inequalities with Operator Solutions

In a recent paper, Domokos and Kolumbán introduced variational inequalities with operator solutions to provide a suitable unified approach to several kinds of variational inequality and vector variational inequality in Banach spaces. Inspired by their work, in this paper, we further develop the new scheme of vector variational inequalities with operator solutions from the single-valued case into the multi-valued one. We prove the existence of solutions of generalized vector variational inequalities with operator solutions and generalized quasi-vector variational inequalities with operator solutions. Some applications to generalized vector variational inequalities and generalized quasi-vector variational inequalities in a normed space are also provided.     

Equilibrium Problems in the Quasimonotone Case

Existence results for quasimonotone vector equilibrium problems and quasimonotone vector variational inequalities are obtained starting from an existence result for a scalar equilibrium problem involving two quasimonotone bifunctions. These results are established under weaker conditions than in previous works.     

Exceptional Families of Elements for Variational Inequalities in Banach Spaces

In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in more general spaces and under considerably weaker assumptions on the defining map.     

Existence of Solutions to Generalized Vector Quasi-variational-like Inequalities with Set-valued Mappings

In this paper, we introduce and study a class of generalized vector quasivariational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space.     

Existence of Solutions to Generalized Vector Quasi-Variational-Like Inequalities with Set-Valued Mappings

In this paper, we introduce and study a class of generalized vector quasi-variational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space.     

Generalized Vector Quasi-Variational Inequality Problems Over Product Sets

In this paper we consider vector quasi-variational inequality problems over product sets (in short, VQVIP). Moreover we study generalizations of this model, namely problems of a system of vector quasi-variational inequalities (in short, SVQVIP), generalized vector quasi-variational inequality problems over product sets (in short, GVQVIP) and problems of a system of generalized vector quasi-variational inequalities (in short, SGVQVIP). We show that every solution of (VQVIP) (respectively, (GVQVIP)) is a solution of (SVQVIP) (respectively, (SGVQVIP)). By defining relatively pseudomonotone and relatively maximal pseudomonotone maps and by employing a known fixed point theorem, we establish the existence of a solution of (VQVIP) and (SVQVIP). These existence results are then used to derive the existence of a solution of (GVQVIP) and (SGVQVIP), respectively, The results of this paper extend recent results in the literature. They are obtained in a more general setting.