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 .[2] the typical nonexpansive mapping is not a strict contraction, this behaviour cannot be explained by Banach's fixed point theorem. S. Reich and A. Zaslavski showed in [18,19,21] that, even in more general situations, the typical nonexpansive self mapping is contractive in the sense of Rakotch, i.e. it satisfiesThe mentioned fixed point theorems for single-valued mappings have one thing in common: The fixed point can be attained by iterating the mapping. This tends to be more difficult in the setting of a set-valued mapping, as it is a priori unclear what iterating such a mapping means. The naturally seeming way, to take the union of the point images, leads to lifting a compact-valued mapping to a mapping on the hyperspace of compact sets in C, i.e. liftingwhich was done by Reich and Zaslavski in [17,20,22]. In the mentioned papers the authors have also shown that the typical compact-valued mapping is a contraction in the sense of Rakotch and has a fixed point, i.e. in the example above: there is a set A ∈ K(C) such that A = F (A). Yet there are different approaches, for example the approach of successive approximations: starting with a point x 0 we take a sequence satisfying x n+1 ∈ F (x n ), where we let x n+1 be a minimizer of { x n − y : y ∈ F (x n )}. This was studied already in [15,3], but these papers about successive approximations of set-valued mappings focus on special mappings of the form F = {f, g} with two nonexpansive single valued mappings f, g : C → C.Even though our main interest lies on mappings with compact images, a lot of results will apply to mappings with closed and bounded point images. Set-valued mappings are of interest in various areas, for example Lipschitzian set-valued mappings are being studied in [4]. These set-valued mappings occur in a natural way, for example when computing the subdifferential of a convex mapping at a point. Recently unions of nonexpansive mappings have been considered by M. K. Tam and others; see, for example, [5,24]. Also recently, global convergence and acceleration of fixed point iterations for union upper semicontinuous operators is being investigated by J. H. Alcantara and C.

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We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces. For Banach spaces which are convex and smooth of power type, we exhibit a condition which implies linear convergence of this method. We show these convergence results for iterates of Bregman projections onto closed linear subspaces. Using an intimate connection between the metric projection onto a closed linear subspace and the Bregman projection onto its annihilator, we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated Bregman projection method.

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.

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