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On strong approximations of usc nonconvex-valued mappings. (English) Zbl 1021.54020

Let \(F: X\to Y\) be a multifunction between metric spaces \(X\), \(Y\). A single-valued \(f: X\to Y\) is called \(\varepsilon\)-approximation for \(F\) if the Hausdorff distance between the graphs \(\text{Gr }F\) and \(\text{Gr }f\) in \(X\times Y\) is not greater than \(\varepsilon\). A. Cellina has proved the existence of such approximations for multifunctions with convex values in normed space \(Y\) [cf. C. Olech, Colloq. Math. 19, 285-293 (1968; Zbl 0183.13603); A. Lasota and Z. Opial, Podstawy Sterowania 1, 71-75 (1971; Zbl 0245.54011); G. A. Beer, Rocky Mt. J. Math. 18, 37-47 (1988; Zbl 0648.54015)]. The convexity of values is used only in order to operate with partitions of unity similarly as in the proof of Michael’s selection theorem. Therefore in Cellina’s theorem convexity of values may be replaced by substitutes, for which Michael’s theorem still holds [cf. e.g.,W. Ślȩzak, Probl. Mat. 10, 27-42 (1990; Zbl 0719.54026) for Pasicki’s \(S\)-convexity; similar generalizations are possible for Bielawski’s simplicial convexity, convex structures, etc., and maybe for decomposability under additional assumptions].
In the paper under review the existence of an approximation is proved for usc multifunctions with paraconvex values [see: E. Michael, Math. Scand. 7, 372-376 (1960; Zbl 0093.12001)].
Moreover conditions are given for approximability of usc multifunctions with star-shaped values in comparison with Beers result on \(H\)-continuous multifunctions.
The paper contains 3 open questions.

MSC:

54C60 Set-valued maps in general topology
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
54C65 Selections in general topology
52A01 Axiomatic and generalized convexity
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