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 w J. Math. Anal. Appl. 204 1996.298]306x and W. A. Kirk, B. Sims, and X. Z. Yuan wNonlinear Analysis 1999. in press.x. 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.