On Successive Approximations for Compact-Valued Nonexpansive Mappings

Abstract

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.

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

“…Recall that in [14] similar results to the main results of the current paper were obtained. There the author handles the case of compact-valued nonexpansive mappings, whereas in this paper we regard nonexpansive mappings with values that are compact and convex.…”

“…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]. Lemma 5.1 Let X be a Banach space, C ⊂ X a convex set containing the origin and 0 < r < R. There is a mapping :…”

“…In order to see that this is not a particular case of the situation considered in [14] we show in Proposition 2.1 that the set of convex compact sets is a porous and hence small subset of the hyperspace of compact sets.…”

“…In [4], using different methods, these results have been generalised to the setting of Banach spaces. More generally, in [14] compact-valued nonexpansive mappings on bounded, closed and convex subsets of Banach spaces are considered. The main results of this article establish that the typical compact-valued nonexpansive mapping has a unique sequence of successive approximations for typical initial points in its domain.…”

“…The current article is considered with the same question but for mappings whose point-images are compact and convex. A crucial step in the proof in [14] is to perturb a given nonexpansive mapping F by adding an additional point to F(x) for some x. This point z is required to have positive distance to F(x) rendering F(x) ∪ {z} non-convex.…”