Generic properties of nonexpansive mappings on unbounded domains

Abstract

We investigate typical properties of nonexpansive mappings on unbounded, closed and convex subsets of hyperbolic metric spaces. For a metric of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a contraction in the sense of Rakotch on every bounded subset and there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is σ-porous, that is, small in a topological sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.

On generic convergence of successive approximations of mappings with convex and compact point images

“…Note that conditions (1) and (2) are now also satisfied. To see (5), note that since G k (x) = F(x) for an even larger set by (v), we also have this identity for all x ∈ C \ n m=k−1 B(y m , c/2). To end the proof, we confirm (4).…”

“…The aim of this section is to show that we are able to change the value of a set-valued mapping at a given point to a set nearby without increasing the Lipschitz constant too much. For this aim we need the following lemma from [9, Lemma 3.1] or [5,Lemma 5.3], cf. also Lemma 4.2 in [14].…”