On Successive Approximations for Compact-Valued Nonexpansive Mappings

We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping F has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of F . In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have a unique trajectory.