$CM$-Selectors for pairs of oppositely semicontinuous multivalued maps with ${\Bbb L}_p$-decomposable values
Volume 144 / 2001
Abstract
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)).
