KKM Mapping Research Articles

On a Generalized Class of Minimax Inequalities

Some results on a generalized class of minimax inequalities based on the r– I– G– H-KKM mapping theorems in a G– H-space setting are presented. The r– I– G– H-KKM mappings represent a new class of KKM mappings in G– H-spaces as well as in the interval spaces.

Relative KKM type selection theorems and their applications

First some nonempty finite intersection theorems involving relative KKM mappings are presented, and then an application to a new class of minimax equalities is given.

The Characterization of Generalized Metric KKM Mappings with Open Values in Hyperconvex Metric Spaces and Some Applications

In this note, the characterization for a set-valued mapping with finitely metrically open values being a generalized metric KKM mapping in hyperconvex metric spaces is established. This result could be regarded as a dual form of corresponding results for the Fan–KKM principle in hyperconvex metric spaces obtained recently by M. A. Khamsi [J. Math. Anal. Appl.204 (1996) 298–306] and W. A. Kirk, B. Sims, and X. Z. Yuan [Nonlinear Analysis (1999) (in press)]. Then we show that the finite intersection property of generalized metric KKM mappings with finitely metrically open values indeed is equivalent to the finite intersection property of generalized metric KKM mappings with finitely metrically closed values in hyperconvex spaces. As applications, we first obtain Ky Fan type matching theorems for both closed and open covers in hyperconvex spaces, which, in turn, are used to establish fixed point theorems for set-valued mappings in hyperconvex metric spaces.

A Generalization of H-Space and Some Results on Multivalued Mappings Without Convexity

In this paper we introduce the concept of G-H-space which is a generalization of H-space and a G-H-KKM map which unifies H-KKM maps and generalized KKM maps, and obtain intersection and fixed point theorems in G-H-space. We also give applications to the existence of greatest and maximal elements for preference relations in topological spaces.

Generalized KKM theorem and variational inequalities

The purpose of this paper is to introduce the concept of generalized KKM mapping and to obtain some general versions of the famous KKM theorem, Ky Fan's minimax inequality, and the Browder-Hartman-Stampacchia variational inequality theorem. As applications we utilize these results to study the saddle point problem and the existence problem of solutions for a class of quasi-variational inequalities. The results presented in this paper extend and improve some recent results of other authors.

Some further generalizations of Ky Fan's minimax inequality and its applications to variational inequalities

The purpose of this paper is to introduce the concept of generalized KKM mapping and to obtain some general version of the famous KKM theorem and Ky Fan's minimax inequality. As applications, we utilize the results presented in this paper to study the saddle point problem and the existence problem of solutions for a class of quasi-variational inequalities. The results obtained in this paper extend and improve some recent results of [1–6].