Continuous selection for set-valued mappings

I. G. Tsar'kov

Izvestiya. Mathematics , Tome 81 (2017) no. 3, pp. 645-669

Résumé

We study properties of set-valued mappings $F$ admitting a continuous selection $f$ which is a continuous $\epsilon$-selection (from the set of $\epsilon$-closest points) for the images $F(x)$ $(x\in X)$. This is interpreted as an $\epsilon$-selection for continuously varying sets in a space with continuously varying norms. We deduce new fixed-point theorems from the results obtained. We also study geometric-topological properties of sets all of whose $r$-neighbourhoods possess a continuous $\epsilon$-selection for every $\epsilon>0$. We obtain a characterization of such sets.

Détail

Keywords: $\epsilon$-selection, continuous selection for set-valued mappings, $\overset{\,_\circ}{B}$-infinite connectedness, $\overset{\,_\circ}{B}$-approximative infinite connectedness, $\overset{\,_\circ}{B}$-neighbourhood infinite connectedness, fixed-point theorems.