Abstract
We first study the asymptotic behavior of infinite products of nonexpansive self-mappings of a star-shaped complete metric space which is not necessarily bounded. Using the notion of contractivity and the Baire category approach, we show that a typical (generic) sequence of nonexpansive mappings is contractive and generates infinite products which have the same asymptotic behavior. This result is then applied to certain classes of nonexpansive set-valued self-mappings of a closed and star-shaped subset of a Banach space. It is shown that in these classes most (in the sense of Baire category) mappings are contractive. In particular, we show that this fact is true for a certain class of compact-valued mappings. It follows that a generic mapping in this class has a fixed point.
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