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

Hôǹg Thái Nguyêñ¹ ; Maciej Juniewicz¹ ; Jolanta Ziemińska¹

¹ Institute of Mathematics Szczecin University Wielkopolska 15 70-451 Szczecin, Poland

Studia Mathematica, Tome 144 (2001) no. 2, pp. 135-152

Résumé

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)).

Détail

DOI : 10.4064/sm144-2-3

Keywords: present continuous selection theorem which unifies sense known selection theorems namely prove h upper semicontinuous multivalued map separable metric space lower semicontinuous multivalued map nonconvex decomposable closed values measure space sigma finite measure nonatomic infty bochner lebesgue space functions defined values banach space cap emptyset there exists cm selector pair continuous selector theorem antosiewicz cellina bressan nbsp lojasiewicz generalized fryszkowski bressan nbsp colombo which simultaneously varepsilon approximate continuous selector theorem cellina nbsp colombo fonda bressan nbsp colombo