Event № 130
Abstract: Given a closed, convex and bounded subset $C$ of a Banach space $X$, we consider the space of all nonexpansive self-mappings of $C$ equipped with the metric of uniform convergence. We show that the subset of strict contractions is small in the sense that it is sigma-porous. In other words, a typical nonexpansive mapping has Lipschitz constant one. In the case where $X$ is a Hilbert space, this result was proved by F. S. De Blasi and J. Myjak in 1989. If time permits, I also plan to discuss possible generalizations of this result to more general metric spaces. This is joint work with Michael Dymond.