A conjecture on the Schauder fixed point theorem and the degree of non-densifiability

Abstract

We revisit a conjecture posed in 1972 by Nussbaum on a generalization of the Schauder fixed point theorem. Instead of requiring compactness of iterates, we introduce the notion of n-DND-condensing mappings, based on the degree of non-densifiability (DND). Roughly speaking, the DND measures, in the specified sense, the distance from a bounded subset of a metric space to the class of Peano continua contained in it. Our main result shows that under suitable additional conditions such mappings have a fixed point. Also, through some examples, we show that the new result works in cases where other known results do not.