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## $CM$-Selectors for pairs of oppositely semicontinuous multivalued maps with ${\Bbb L}_p$-decomposable values

### Tom 144 / 2001

Studia Mathematica 144 (2001), 135-152 MSC: Primary 54C65, 54C60; Secondary 47H04, 35R70. DOI: 10.4064/sm144-2-3

#### Streszczenie

We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if $F$ is an $H$-upper semicontinuous multivalued map on a separable metric space $X$, $G$ is a lower semicontinuous multivalued map on $X$, both $F$ and $G$ take nonconvex $L_p(T, E)$-decomposable closed values, the measure space $T$ with a $\sigma$-finite measure $\mu$ is nonatomic, $1\le p< \infty$, $L_p(T, E)$ is the Bochner–Lebesgue space of functions defined on $T$ with values in a Banach space $E$, $F(x) \cap G(x)\not = \emptyset$ for all $x \in X$, then there exists a $CM$-selector for the pair $(F,G)$, i.e. a continuous selector for $G$ (as in the theorem of H. Antosiewicz and A. Cellina (1975), A. Bressan (1980), S. /Lojasiewicz, Jr. (1982), generalized by A. Fryszkowski (1983), A. Bressan and G. Colombo (1988)) which is simultaneously an $\varepsilon$-approximate continuous selector for $F$ (as in the theorem of A. Cellina, G. Colombo and A. Fonda (1986), A. Bressan and G. Colombo (1988)).

#### Autorzy

• Hôǹg Thái NguyêñInstitute of Mathematics
Szczecin University
Wielkopolska 15
70-451 Szczecin, Poland
e-mail
• Maciej JuniewiczInstitute of Mathematics
Szczecin University
Wielkopolska 15
70-451 Szczecin, Poland
e-mail
• Jolanta ZiemińskaInstitute of Mathematics
Szczecin University
Wielkopolska 15
70-451 Szczecin, Poland
e-mail

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