Geodesic
Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces
Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 835-851

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The Michael selection theorem is extended to the case of set-valued mappings with not necessarily convex values. Classical approximation problems on cone-spaces with symmetric and asymmetric seminorms are considered. In particular, conditions for existence of continuous selections for convex subsets of asymmetric spaces are studied. The problem of existence of a Chebyshev centre for a bounded set is solved in a semilinear space consisting of bounded convex sets with Hausdorff semimetric.
Keywords: selection of a set-valued mapping, Michael's selection theorem, fixed point, asymmetric space, Chebyshev centre, convex set, $\varepsilon$-selection. 
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     author = {I. G. Tsar'kov},
     title = {Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces},
     journal = {Izvestiya. Mathematics },
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     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a6/}
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I. G. Tsar'kov. Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces. Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 835-851. http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a6/