## $$CM$$-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

### Citations:

Zbl 0952.54011; Zbl 0887.47040
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