On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

隐式形式多值向量均衡问题解的存在性

考虑一类隐式形式多值向量均衡问题的解的存在性,该类问题包含了多值均衡问题、隐式向量均衡问题、多值变分不等式问题、向量变分不等式问题以及向量互补问题作为其特殊情形.利用广义Fan-Browder不动点定理,得到了拓扑向量空间中该类问题解的存要性定理,该结果推广并统一了已有问题解的存在性结果.     

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.     

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.     

Lexicographic variational inequalities with applications

In this article, we consider equivalence properties between various kinds of lexicographic variational inequalities. By employing various concepts of monotonicity, we show that the usual sequential variational inequality is equivalent to the direct lexicographic variational inequality or to the dual lexicographic variational inequality. We establish several existence results for lexicographic variational inequalities. Also, we introduce the lexicographic complementarity problem and establish its equivalence with the lexicographic variational inequality. We illustrate our approach by several examples of applications to vector transportation and vector spatial equilibrium problems.     

局部G-凸空间内的广义矢量拟平衡问题

在局部G-凸空间内引入和研究了几类广义矢量拟平衡问题(GVQEP)．包含了大多数广义矢量平衡问题,广义矢量变分不等式问题,拟平衡问题和拟变分不等式问题作为特殊情形．首先在局部G-凸空间内对一人对策证明了一个平衡存在性定理．作为应用,在非紧局部G-凸空间内对GVQEP的解建立了某些新的存在定理．这些结果和论证方法与最近文献中的结果和论证方法相比较是新的和完全不同的．     

Equilibrium problems and variational inequalities in topological vector spaces

An existence result for the equilibrium problem is proved in a general topological vector space. As applications, existence results are derived for variational inequality problems, vector equilibrium problems and vector variational inequality problems. Our results extend and unify a number of existence theorems in non-compact cases     

Applications of the KKM Property to Coincidence Theorems,Equilibrium Problems,Minimax Inequalities and Variational Relation Problems

In this paper, we establish coincidence-like results in the case when the values of the correspondences are not convex. To do this, we define new type of correspondences, namely, properly quasi-convex-like correspondences. Further, we apply the obtained theorems to solve equilibrium problems and to establish a minimax inequality. In the last part of the paper, we study the existence of solutions for generalized vector variational relation problems. Our analysis is based on the applications of the KKM principle. We establish existence theorems involving new hypothesis and we improve the results of some recent papers.     

Equilibrium Problems under Generalized Convexity and Generalized Monotonicity

Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.     

Equilibrium Problems under Generalized Convexity and Generalized Monotonicity

Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.     

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.     

System of Vector Equilibrium Problems and Its Applications

In this paper, we introduce a system of vector equilibrium problems andprove the existence of a solution. As an application, we derive someexistence results for the system of vector variational inequalities. We alsoestablish some existence results for the system of vector optimizationproblems, which includes the Nash equilibrium problem as a special case.     

Vector complementarity problems with a variable ordering relation

In this paper, we investigate vector complementarity problems with a variable ordering relation. We establish existence results of a solution of a vector complementarity problem under an inclusive type condition. We obtain equivalence results among a vector complementarity problem, a vector variational inequality problem and other related problems.     

Characterizations of Solutions for Vector Equilibrium Problems

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.     

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.     

The Study of KKM Theorems With Applications to Vector Equilibrium Problems and Implict Vector Variational Inequalities Problems

In this paper, we establish some equivalence relations between coincidence theorems and KKM-type theorems. We also obtain some new coincidence theorems and fixed point theorems. Applying the KKM-type theorems we obtain the existence theorems of generalized vector equilibrium problems. From these results, some existence theorems of generalized vector implicit variational inequality problems are established in this paper.     

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.     

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     

Existences theorems of systems of vector quasi-equillibrium problems and mathematical programs with equilibrium constraint

In this paper, we introduce systems of vector quasi-equilibrium problems and prove the existence of their solutions. As applications of our results, we derive the existence theorems for solution of system of vector quasi-saddle point problem, the existences theorems of a solution of system of generalized quasi-minimax inequalities, the mathematical program with equilibrium constraint, semi-infinite and bilevel problems.