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A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces. (English) Zbl 1101.54025
Let $X$ be a paracompact topological space, $Y$ a hyperconvex metric space and $F:X\to Y$ a multifunction with sub-admissible values. A subset $B\subset Y$ is called sub-admissible if $coC\subset B$ for each finite $C\subset B$, where $coC$ denotes the intersection of all closed balls in $Y$ containing $C$. It is proved that if $F$ is quasi-lsc (i.e., for each $x\in X$ and $\varepsilon>0$ there is a point $y\in F(x)$ and a neighbourhood $U(x)$ of $x$ such that for each $t\in U(x)$, $F(t)\cap B(y,\varepsilon)\ne\emptyset)$, then $F$ admits a continuous selection. Two fixed-point theorems are deduced. Remark: It would be interesting to locate the generalized convexity used in the paper under review in the framework of Bielawski’s simplicial convexity, cf. {\it R. Bielawski} [J. Math. Anal. Appl, 127, 155--171 (1987; Zbl 0638.52002)].

MSC:
54C65Continuous selections
26E25Set-valued real functions
52A01Axiomatic and generalized geometric convexity
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References:
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