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

Tom 144 / 2001

Hôǹg Thái Nguyêñ, Maciej Juniewicz, Jolanta Ziemińska 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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