KKM Theorems on FC-spaces with Their Applications to Variational Inequalities

A new concept of finitely continuous topological spaces （in short, FC-space） without convexity structure and linear structure was introduced. Some KKM type theorems in noncompact FC-spaces were obtained, and from which, some section theorems and wriational inequality theorems were proved under much weak assumptions. Our results improve and generalize the corresponding conclusions in recent literature.     

H-空间的截口定理重合定理相交定理及其应用*

本文的目的是在H-空间上得出几个截口定理、相交定理和重合定理.作为这些结果的应用,我们研究了H-空间中的极大极小不等式和变分不等式解的存在性问题.本文结果改进和发展了引文[1,3,5,6,8,9,12,14,15,17]中的相应结果.     

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.     

广义变分不等式与极大极小定理

本文讨论映Hausdorff拓扑线性空间到Riesz空间[6]的算子变分问题.给出半序空间上变分不等式的绝对形式和相对形式.     

General KKM Theorem with Applications to Minimax and Variational Inequalities

In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general setting. We also establish several existence results for generalized variation inequalities and generalized quasi-variational inequalities.     

The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.     

拓扑空间中的KKM型定理和重合点定理

引入了一类新的有限连续空间(简称, $FC$-空间),并在$FC$-空间中证明了一些新的涉及容许类集值映射和具有紧局部交性质的KKM型定理和重合点定理.作为应用,在$FC$-空间中得到了一些新的不动点定理.这些结果统一和推广了近期文献在抽象凸空间中的一些重要结果.     

广义Minimax不等式及不动点定理

最近,许多作者推广了著名的Ky Fan的minimax不等式.本文提出了T-对角凸凹性条件,发展了这一方面的主要结果.其次,我们讨论了一些不动点问题,推广了Fan-Glicksberg不动点定理.     

一类新的KKM定理及其应用*

本文得到一类新的KKM定理,统一和改进了[2,3,0,7,11]中的结果。作为应用,我们得到了几个匹配定理、重合定理、不动点定理、极大极小不等式定理及截口定理。     

Extragradient Methods for Solving Equilibrium Problems,Variational Inequalities,and Fixed Point Problems

In this paper, we propose the new extragradient algorithms for an α-inverse-strongly monotone operator and a relatively nonexpansive mapping in Banach spaces. We prove convergence theorems by this methods under suitable conditions. Applying our algorithms, we find a zero paint of maximal monotone operators. Using FMINCON optimization toolbox in MATLAB, we give an example to illustrate the usability of our results.     

L-空间中的KKM映射及不动点定理

引入了L-空间和L-空间上的KKM类映射，建立了关于该类映射的一些不动点定理，其中包括Schauder型和Fan-Browder型不动点定理．得到了L-空间中的KyFan匹配定理和叠合点定理．     

拓扑型截口定理及应用*

本文给出一个新型的KKM定理,并用它得到拓扑型截口定理,在第四节至第五节应用此截口定理给出了Browder-Hartman-Stampacchia变分不等式[3].隐变分不等式[8],抽象形式变分不等式[19]的解的存在性定理,和一个集值映射的不动点定理。其结果不仅包含TBrowder[3]中的主要结果为特例,而且,改进和发展了引文[1～19]中的相应结果。     

新型拓扑KKM定理与等价形式及其应用

在不具线性结构拓扑空间—广义区间空间—中得到了新型拓扑KKM定理及其几个等价形式.作为应用,本文研究了广义区间空间中一类抽象形式变分不等式解的存在性问题.     

拓扑有限交性质及极大极小定理*

本文研究了更一般的拓扑有限交的性质。作为应用,利用本文结果给出了更一般的极大极小定理。本文结果推广了[5,6,9]中的主要结果。     

Applications of Equilibrium Problems to a Class of Noncoercive Variational Inequalities

In this paper, we are interested in the existence of solutions for a class of noncoercive variational inequalities involving a p-Laplacian type operator. Our approach is based essentially on equilibrium problems and arguments from recession analysis. Our results are of two types: the first is obtained in a monotone framework; the second is obtained without a monotonicity assumption. The first and the third authors were partially supported by the National Science Council of the Republic of China. The second author was partially supported by NSF, Hunan Province, Grant 04JJY20001. The authors express their sincere thanks to two anonymous referees for careful reading and comments leading to the present version of this paper.     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

广义凸空间内的KKM型定理和极小极大不等式及鞍点定理

本文在非紧G-凸空间内对具有有限闭值和有限开值的G-KKM,广义G-KKM和广义S-KKM映象建立了某些新的KKM型定理。应用这些KKM型定理,在G-凸空间内得到了新的 Ky Fan型极小极大不等式和鞍点定理。这些结论推广了最近文献中的许多已有结果。     

Geometric Properties and Coincidence Theorems with Applications to Generalized Vector Equilibrium Problems

The present paper is divided into two parts. In the first part, we derive a Fan-KKM type theorem and establish some Fan type geometric properties of convex spaces. By applying our results, we also obtain some coincidence theorems and fixed-point theorems in the setting of convex spaces. The second part deals with the applications of our coincidence theorem to establish some existence results for a solution to the generalized vector equilibrium problems.     

抽象凸空间上的弱KKM映射, 相交定理和极大极小不等式

In this paper,we introduce the concept of weakly KKM map on an abstract convex space without any topology and linear structure,and obtain Fan＇s matching theorem and intersection theorem under very weak assumptions on abstract convex spaces.Finally,we give several minimax inequality theorems as applications.These results generalize and improve many known results in recent literature.     

拓扑型极大极小定理和抽象B-H-S变分不等式

本文中证明了区间空间中几个新型的极大极小定理，并研究了抽象形式变分不等式和集值ＢＨＳ变分不等式解的存在性问题．本文的结果包含［１，２，５，７，８］中相应结果为特例．