集值映射的广义对称向量拟平衡问题系统

In this paper, a system of generalized symmetric vector quasi-equilibrium problems for set-valued mappings is introduced. By using a scalarization method and a fixed-point theorem, the existence result for its solution is established. The main result extends the corresponding results in Fu （J. Math. Anal. Appl. 285, 708-713, 2003） and Zhang, Chen and Li （OR Transactions 10, 24-32, 2006）.     

锥凸对称向量拟均衡问题解集的通有稳定性

在拓扑向量空间中,利用Ky Fan截口定理得到一个锥凸向量拟均衡问题弱Pareto解的存在性结果.作为该结果的应用,得到了一个对称向量拟均衡问题在支付映射为锥凸条件下弱Pareto解的存在性定理.该定理在较弱的条件下回答了Fu在文献[1]中提出的第二个问题,即在支付映射为锥凸且连续的条件下对称向量拟均衡问题的弱Pareto解是否存在.最后在赋范线性空间中研究了锥凸对称向量拟均衡问题弱Pareto解集的通有稳定性.     

The Study of the System of Generalized Vector Quasi-equilibrium Problems

In this paper, we study the system of generalized vector quasi-equilibrium problems, which includes as special cases the system of vector quasi-equilibrium problems and the system of generalized vector equilibrium problems, and establish the existence and essential components of the solution set under perturbations of its best-reply map. Moreover, we also derive a new existence theorem of Ky Fan’s points for a set-valued map.     

局部FC-一致空间内具有联立广义矢量拟平衡组约束的数学规划(英文)

在局部FC-一致空间内引入和研究了某些新的联立广义矢量拟平衡问题组和具有联立广义矢量拟平衡组约束的数学规划.应用作者在局部FC-一致空间得到的Himmelberg型不动点定理,首先在局部FC-一致空间内对联立广义矢量拟平衡问题组的解证明了某些新的存在性定理.作为应用,对具有联立广义矢量拟平衡组约束的数学规划的解得到了某些新的存在性定理.     

The existence of solutions for the system of generalized vector quasi-equilibrium problems

In this paper, we study the system of generalized vector quasi-equilibrium problems which includes as special cases the system of generalized vector equilibrium problems and the system of vector quasi-equilibrium problems, and establish the existence result for its solutions by means of the Kakutani–Fan–Glicksberg fixed points theorem. As special cases, we also derive the existence results for weakly Pareto–Nash equilibrium points for multiobjective generalized game problems and multiobjective game problems.     

System of Generalized Vector Quasi-Equilibrium Problems with Applications to Fixed Point Theorems for a Family of Nonexpansive Multivalued Mappings

In this paper, we establish the existence theorems of the generalized vector quasi-equilibrium problems. From some existence theorem, we establish fixed point theorems for a family of lower semicontinuous or nonexpansive multivalued mappings. We also obtain the existence theorems of system of mixed generalized vector variational-like inequalities and existence theorems of the Debreu vector equilibrium problems and the Nash vector equilibrium problems.     

The Stability of Set of Solutions for Symmetric Vector Quasi-Equilibrium Problems

In this paper, the stability of the set of solutions for symmetric vector quasi-equilibrium problems is discussed. Then, we prove a generic stability theorem and give an existence theorem for essentially connected components of the set of solutions for symmetric vector quasi-equilibrium problems. Finally, we apply these results to vector weak saddle point problems with constraints. This research was partially supported by the Natural Science Foundation of China (Grant number: 10561007) and Natural Science Foundation of Jiangxi Province, China.     

New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces

Motivated by the Suzuki’s type fixed point theorems, we give several new existence theorems for scalar quasi-equilibrium problems, and vector quasi-equilibrium problem on complete metric spaces. We give important examples for our results. Note that the solution of quasi-equilibrium problem (resp. vector quasi-equilibrium problem) is unique under suitable conditions, and we can find the unique solution by the Picard iteration. Besides, we also give a new coincidence theorem on complete metric spaces. Finally, we give a new minimax theorem on complete metric spaces. Note that the solution of minimax theorem is unique under suitable conditions, and we can find the unique solution by the Picard iteration.     

Gap Functions and Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems

This paper deals with generalized vector quasi-equilibrium problems. By virtue of a nonlinear scalarization function, the gap functions for two classes of generalized vector quasi-equilibrium problems are obtained. Then, from an existence theorem for a generalized quasi-equilibrium problem and a minimax inequality, existence theorems for two classes of generalized vector quasi-equilibrium problems are established. This research is partially supported by the Postdoctoral Fellowship Scheme of The Hong Kong Polytechnic University and the National Natural Science Foundation of China.     

乘积FC-空间中广义混合向量拟平衡问题系统

本文在乘积FC-空间中引入并研究一类新的广义混合向量拟平衡问题系统,利用极大元定理,得到了关于广义混合向量拟平衡问题系统解的一些存在性定理,改进和推广了前人已有的某些结果．     

对称向量拟均衡问题有效解的存在性

利用标量化方法建立对称向量拟均衡问题有效解的存在性定理。作为标量化方法的应用,利用这一方法得到向量变分不等式和拟向量变分不等式有效解的存在性定理。     

广义强向量拟均衡问题组解的存在性

本文研究了一类集值广义强向量拟均衡问题组解的存在性问题.利用集值映射的自然拟C-凸性和集值映射的下(-C)-连续性的定义和Kakutani-Fan-Glicksberg不动点定理,在不要求锥C的对偶锥C~*具有弱*紧基的情况下,建立了该类集值广义强向量拟均衡问题组解的存在性定理.所得结果推广了该领域的相关结果.     

向量拟平衡问题的本质解及解集的本质连通区

本文研究向量拟平衡问题,得到了向量拟平衡问题解的一个存在性结果,证明了在满足一定的连续性和凸性条件的问题构成的空间Y中,大多数(在Baire分类意义下)问题的解集是稳定的,并证明Y的某子集中,每个向量拟平衡问题的解集中至少存在一个本质连通区。作为应用,我们导出了多目标广义对策弱Pareto-Nash平衡点的存在性,证明了在满足一定的连续性和凸性条件的多目标广义对策构成的空间P中,大多数对策的弱Pareto-Nash平衡点是稳定的,并证明了P中的每个对策的弱Pareto-Nash平衡点集中至少有一个本质连通区。     

On the existence of solutions to generalized quasi-equilibrium problems

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.     

局部FC-一致空间内凝聚映象的极大元和广义对策及应用(Ⅱ)

在局部FC-一致空间内,引入和研究了某些新的涉及凝聚集值映象的广义矢量拟平衡问题组．应用作者在文(Ⅰ)中对局部FC-一致空间内的凝聚集值映象得到的极大元存在性定理,在局部FC-一致空间内,对这些广义矢量拟平衡问题组的解,证明了某些新的存在性定理．这些结果改进和推广了文献中某些已知结果到局部FC-一致空间．     

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.     

On generalized operator quasi-equilibrium problems

In this paper, we will introduce the generalized operator equilibrium problem and generalized operator quasi-equilibrium problem which generalize the operator equilibrium problem due to Kazmi and Raouf [K.R. Kazmi, A. Raouf, A class of operator equilibrium problems, J. Math. Anal. Appl. 308 (2005) 554-564] into multi-valued and quasi-equilibrium problems. Using a Fan-Browder type fixed point theorem in [S. Park, Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31 (1994) 493-519] and an existence theorem of equilibrium for 1-person game in [X.-P. Ding, W.K. Kim, K.-K. Tan, Equilibria of non-compact generalized games with L^∗-majorized preferences, J. Math. Anal. Appl. 164 (1992) 508-517] as basic tools, we prove new existence theorems on generalized operator equilibrium problem and generalized operator quasi-equilibrium problem which includes operator equilibrium problems.     

Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces

In this paper we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally G-convex uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally G-convex uniform spaces. These theorems are new and generalize some known results in the literature.     

System of vector quasi-variational inclusions with some applications

In this paper, we consider systems of vector quasi-variational inclusions which include systems of vector quasi-equilibrium problems for multivalued maps, systems of vector optimization problems and several other systems as special cases. We establish existence results for solutions of these systems. As applications of our results, we derive the existence results for solutions of system vector optimization problems, mathematical programs with systems of vector variational inclusion constraints and bilevel problems. Another application of our results provides the common fixed point theorem for a family of lower semicontinuous multivalued maps. Further applications of our results for existence of solutions of systems of vector quasi-variational inclusions are given to prove the existence of solutions of systems of Minty type and Stampacchia type generalized implicit quasi-variational inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.