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\(CM\)-Selectors for pairs of oppositely semicontinuous multivalued maps with \({\mathbb L}_p\)-decomposable values. (English) Zbl 0970.54018

The paper provides the proof of the following result: given set-valued maps \(F,G\) defined on a separable metric space \(X\) with the so-called decomposable values in the space \(L^p(T,E)\), \(1\leq p<\infty\), of in the Bochner sense \(p\)-integrable functions \(T\to E\), where \(T\) is a measure space endowed with a \(\sigma\)-finite nonatomic measure and \(E\) is a Banach space, such that \(F(x)\cap G(x)\neq \emptyset\) on \(X\); \(G\) is lower- and \(F\) is upper-semicontinuous, then for any \(\varepsilon>0\), there is a continuous map \(f:X\to L^p(T,E)\) being a selection of \(G\) and an \(\varepsilon\)-graph approximation of \(F\). This result is a ‘decomposable’ version of the ‘convex’ result from [the authors, Z. Anal. Anwend. 19, No. 2, 381-393 (2000; Zbl 0952.54011)]. It is however necessary to underline that the mentioned ‘convex’ version of this result, i.e. concerning set-valued \(F,G:X\to E\) having convex closed values, has appeared first in [H. Ben-El-Mechaiekh and W. Kryszewski, Trans. Am. Math. Soc. 349, No. 10, 4159-4179 (1997; Zbl 0887.47040)]. Some related results are also given.

MSC:

54C65 Selections in general topology
47H04 Set-valued operators
54C60 Set-valued maps in general topology
35R70 PDEs with multivalued right-hand sides
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