Closedness of the solution map for parametric weak vector equilibrium problems

In this paper we introduced a new definition of vector topological pseudomonotonicity to study the parametric weak vector equilibrium problems. The main result gives sufficient conditions for closedness of the solution map defined on the set of parameters.     

Infinite Player Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

In this paper we consider the Nash equilibrium problem for infinite player games with vector payoffs in a topological vector space setting. By employing new concepts of relative (pseudo)monotonicity, we establish several existence results of solutions for usual and normalized vector equilibria. The results strengthen existence results for vector equilibrium problems, which were based on classical pseudomonotonicity concepts. They also extend previous results for vector variational inequalities and finite player games under relative (pseudo)monotonicity.     

Topological pseudomonotonicity and vector equilibrium problems

In this paper, we introduce the topological pseudomonotonicity to vector valued bifunctions, and derive some existence results for vector equilibrium problems with the corresponding bifunctions topologically pseudomonotone. Some applications to vector variational inequalities and existence of Pareto optima are given.     

Upper order-preservation of the solution correspondence to vector equilibrium problems

In this paper, several order-theoretic fixed point theorems are proved on partially ordered topological spaces. Applying these fixed point theorems, we explore the existence and upper order-preservation for parametric vector equilibrium problems. In contrast to the previous results on vector equilibrium problems, the upper order-preservation of solutions is a new subject, which would be useful for predicting the changing trend of solutions to vector equilibrium problems. In addition, neither topological continuity nor convexity of the considered vector-valued bifunction F is required in our results.     

Existence Results for Primal and Dual Generalized Vector Equilibrium Problems With Applications to Generalized Semi-Infinite Programming

In this paper, we introduce several kinds of maximal pseudomonotonicity and establish existence theorems of maximal pseudomonotonicity. From these results we establish the existence theorems of generalized vector equilibrium problems. We establish existence theorems of generalized vector semi-infinite programming, as applications of generalized vector equilibrium problems. This research was supported by the National Science Council of the Republic of China.     

具有控制结构的向量均衡问题与解集的性质

提出一种具有控制结构的向量均衡问题与向量映射的新的伪单调性概念,得到具有控制结构的向量均衡问题解的存在性及其解集的紧凸性．作为应用,得到具有控制结构的向量变分不等式与互补问题的解．     

Generalized Vector Equilibrium Problems for Pseudomonotone Multivalued Bifunctions<Superscript><Emphasis Type="Bold">1</Emphasis></Superscript>

In this paper, we apply a new version of the Brézis, Nirenberg, and Stampacchia theorem; we use pseudomonotonicity and some coercivity conditions to establish some existence result for a solution of generalized vector equilibrium problems for multivalued bifunctions. The proper quasiconvexity of multivalued bifunctions is introduced and existence theorems for generalized vector equilibrium problems related to multivalued mappings with the KKM property are obtained. The new results extend and modify various existence theorems for similar problems.     

Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems

In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems. This research was partially supported by the National Natural Science Foundation of China (Grant 10561007) and the Natural Science Foundation of Jiangxi Province, China.     

Lipschitz continuity of an approximate solution mapping for parametric set-valued vector equilibrium problems

The purpose of this paper is to generalize and improve some topological properties of solutions set to the set-valued vector equilibrium problems by using the scalar characterization method. Moreover, the Lipschitz continuity of an approximate solution mapping for the parametric set-valued vector equilibrium problems is studied.     

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.     

含参广义集值向量均衡问题有效解映射下半连续的最优条件

在实Hausdorff拓扑向量空间中研究一类含参广义集值向量均衡问题弱有效解与有效解映射的下半连续性. 在近似锥-次类凸的条件下, 运用标量化的方法得到弱有效解的标量化结果. 在适当条件下, 得到含参广义集值向量均衡问题弱有效解与有效解映射下半连续性定理.     

Equilibrium and least element problems for multivalued functions

The principal aim of this paper is to extend some recent results which concern problems involving bifunctions to similar generalized problems for multivalued bifunctions. To this end, by using the appropriate notions of strict pseudomonotonicity we establish the relationships between generalized vector equilibrium problems and generalized minimal element problems of feasible sets. Moreover relationships between generalized least element problems of feasible sets and generalized vector equilibrium problems are studied by employing the concept of Z-multibifunctions.     

含参集值向量均衡问题各种有效解映射的下半连续性

在实Hausdorff拓扑向量空间中,引进含参集值向量均衡问题,给出各种有效解的概念.在锥-次类凸的条件下,得到各种有效解的标量化结果.结合集值映射的弱f-性,在适当假设条件下,得到含参集值向量均衡问题各种有效解映射的下半连续性.     

含参弱向量均衡问题的适定性

在实Hausdorff拓扑线性空间中研究了含参弱向量均衡问题的适定性.证明了在适当条件下由近似网定义的含参适定性等价于近似解映射的上半连续性,并给出了所研究问题各种适定的充分性条件.     

On Implicit Vector Variational Inequalities

In this paper, we study the existence of solutions of implicit vector variational inequalities for multifunctions. Generalized pseudomonotonicity concepts are introduced. Our results extend and unify corresponding earlier existence results of many authors for vector variational inequalities under the Hausdorff topological vector space setting.     

Vector Equilibrium Problems,Minimal Element Problems and Least Element Problems

In this paper we introduce some concepts of feasible sets for vector equilibrium problems and some classes of Z-maps for vectorial bifunctions. Under strict pseudomonotonicity assumptions, we investigate the relationship between minimal element problems of feasible sets and vector equilibrium problems. By using Z-maps, we further study the least element problems of feasible sets for vector equilibrium problems. Finally, we prove a generalized sublattice property of feasible sets for vector equilibrium problems associated with Z-maps. This work was supported by the National Natural Science Foundation of China and the Applied Research Project of Sichuan Province (05JY029-009-1). The authors thank Professor Charalambos D. Aliprantis and the referees for valuable comments and suggestions leading to improvements of this paper.     

On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

On nonconvex equilibrium problems

In this paper, we introduce a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We suggest some iterative schemes for solving nonconvex equilibrium problems by using the auxiliary principle technique. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems.     

Multivalued general equilibrium problems

In this paper, we use the auxiliary principle technique to suggest some new classes of iterative algorithms for solving multivalued equilibrium problems. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems. Since multivalued equilibrium problems include equilibrium, variational inequality and complementarity problems as specials cases, our results continue to hold for these problems.     

Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems

In this paper, we consider parametric primal and dual equilibrium problems in locally convex Hausdorff topological vector spaces. Sufficient conditions for the approximate solution maps to be Hausdorff continuous are established. We provide many examples to illustrate the essentialness of the imposed assumptions. As applications of these results, the Hausdorff continuity of the approximate solution maps for optimization problems, variational inequalities and fixed-point problems are derived.