“Continuous” Multifunctions in Discrete Spaces with Applications to Fixed Point Theory

Abstract

In this paper, a sound notion of “continuous” multifunctions for discrete spaces is introduced. It turns out that a multifunction is continuous (= both upper and lower semi-continuous) if and only if it is “strong”, or equivalently it takes (original) neighbours into (induced) neighbours w.r.t. Hausdor. metric. Generalizing various previous results, we show that any finite contractible graph has the almost fixed point property (afpp) for strong multifunctions. We then turn to the fixed clique property, or fcp (a desirable strengthening of afpp). We are able to establish this property only under much more explicitly geometric assumptions than in the case of afpp. Specifically, we have to restrict to convex-valued multifunctions, over spaces admitting a suitable notion of convexity; and, for the best results, to the standard n-dimensional digital space with (3ⁿ - 1)-adjacency. In this way we obtain what may be considered (with reservations, discussed in the Conclusion) as an analogue of the classical Kakutani fixed point theorem.

The second author has been partially supported by EPSRC grant on“Digital Topology and Geometry: an Axiomatic Approach with Applications to GIS and Spatial Reasoning”.