Abstract
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.
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