Compactness and the fixed point property in ℓ1

In this paper we prove that compactness can be characterized by means of the existence of a fixed point for some classes of mappings defined on convex closed subsets of the space ℓ1ℓ1. Nominally, our result involves nonexpansive mappings, uniformly Lipschitzian mappings and cascading nonexpansive mappings. We also extend the results to some more general classes of Banach spaces.