Fanky极大极小不等式的进一步推广及对变分不等式的应用

本文引出广义KKM映象的概念,其包含著名的KKM映象为特例.借助于这一映象,我们给出著名的KKM定理和Fan Ky极大极小不等式以一种新的推广形式.作为应用,我们考虑了变分不等式解的存在性问题.本文结果是引文[1～6]中相应结果的改进和发展.     

Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces

In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed – monotone mappings in reflexive Banach spaces. We present also the solvability of variational-like inequalities with relaxed – semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve the corresponding results of Refs. 1–6.     

Generalized KKM-type theorems for weakly generalized KKM mapping with some applications

In this paper, we establish some new generalized KKM-type theorems based on weakly generalized KKM mapping without any convexity structure in topological spaces. As applications, some minimax inequalities and an existence theorem of equilibrium points for an abstract generalized vector equilibrium problem are proved in topological spaces. The results presented in this paper unify and generalize some known results in recent literature.     

Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings

In this paper, we consider the implicit quasi-variational inequality without continuity assumptions of data mappings. Our approach here is completely different from the one based on KKM theorem in the literature. Interesting applications to generalized quasi-variational inequalities for both discontinuous mappings and discontinuous fuzzy mappings are given.This paper was partially supported by the NSC grant no. 86-2115-M-110-004.     

自反Banach空间中广义混合双线性变分不等式解的存在性

介绍了自反B anach空间中一类广义混合双线性型变分不等式,利用极大极小不等式和辅助变分原理技术证明了这类混合双线性型变分不等式解的存在性与唯一性.     

一般化凸空间上极大极小不等式及其应用

我们首先改进了已有的KKM型定理,然后根据改进后的KKM型定理得到了一般化凸空间上的KyFan型极大极小不等式,最后利用它给出了Fan-Kneser型不等式族及其一些应用.我们的结论对文献中的相应结果进行了改进和一般化。     

On the Existence of Solutions of Quasivariational Inclusion Problems

Quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.     

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.     

Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces

In this paper, we introduce and investigate a new class of mixed quasi-variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to Ding–Tan and a lemma due to Chang, we establish some existence and uniqueness results of solution for the mixed quasi-variational-like inequality. Next, by using a KKM theorem due to Fan and an auxiliary principle technique due to Cohen, we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms. Our results extend, improve and unify several known results in the literature.     

广义拟凹映射和向量极小极大不等式

引入z-广义c-拟凹向量映射,研究它和广义KKM映射的等价性,应用这些结果改进了-Ky Fan向量极小极大不等式,并且获得了一强向量极小极大不等式．     

Generalized 2-KKM theorems and their applications in hyperconvex metric spaces

In this work, we first define the 2-KKM mapping and the generalized 2-KKM mapping on a metric space, and then we apply the property of the hyperconvex metric space to get a KKM theorem and a fixed point theorem without a compactness assumption. Next, by using this KKM theorem, we get some variational inequality theorems and minimax inequality theorems.     

Generalized Variational-Like Inequalities with Nonmonotone Set-Valued Mappings

In this paper, we introduce a new class of generalized variational-like inequalities involving a nonmonotone-type set-valued mapping which is more general than those in the known literature. By applying a KKM-type theorem established by the author, some existence theorems for the solutions to the class of generalized variational-like inequalities are proved in Hausdorff locally convex topological vector spaces.     

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.     

On Nondifferentiable and Nonconvex Vector Optimization Problems

In this paper, we prove the equivalence among the Minty vector variational-like inequality, Stampacchia vector variational-like inequality, and a nondifferentiable and nonconvex vector optimization problem. By using a fixed-point theorem, we establish also an existence theorem for generalized weakly efficient solutions to the vector optimization problem for nondifferentiable and nonconvex functions.     

一般化凸空间上截口定理及其对择一不等式的应用

我们根据一般化凸空间上的KKM型定理得到了截口定理,然后作为它的应用讨论了若干个择一不等式.最后,引进了一个具体的一般化凸空间并在该空间上讨论了择一不等式解的存在性问题.     

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.     

New generalizations of basic theorems in the KKM theory

In the present paper, we obtain a new KKM type theorem for intersectionally closed-valued KKM maps and some useful new basic consequences. Typical examples of them are abstract forms of Fan’s matching theorem, Fan’s geometric lemma, the Fan-Browder fixed point theorem, maximal element theorems, Fan’s minimax inequality, variational inequalities, and others.     

拓扑空间上的KKM定理,不动点定理,广义变分不等式

引进没有任何凸结构的拓扑空间上的广义R-KKM映射的定义并利用古典的KKM原理得到一般拓扑空间上的KKM型定理以及若干个变形结果,然后作为应用给出了非紧的拓扑空间上不动点定理和广义变分不等式解的存在性定理.     

Generalized Vector Variational-Like Inequalities

Using a generalized Fan’s KKM theorem, some existence results for generalized vector variational-like inequalities in noncompact settings are established. Some applications to vector optimization problems are given. The results presented in this paper extend and unify corresponding results of other authors.     

Stability of solutions for variational relation problems with applications

In this paper, we prove that most of problems in variational relations (in the sense of Baire category) are essential and that, for any problem in variational relations, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan’s minimax inequality theorem, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games, the existence of essential component of the set of solutions for vector Ky Fan’s minimax inequality, the existence of essential components of the set of KKM points and the existence of essential components of the set of solutions for Ky Fan’s section theorem.