Geodesic
On the extension and generation of set-valued mappings of bounded variation
V. V. Chistyakov 1   ; A. Rychlewicz 2
1 Department of Mathematics University of Nizhny Novgorod 23 Gagarin Avenue Nizhny Novgorod 603950, Russia
2 Faculty of Mathematics Łódź University Stefana Banacha 22 90-238 Łódź, Poland
Studia Mathematica, Tome 153 (2002) no. 3, pp. 235-247 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We study set-valued mappings of bounded variation of one real variable. First we prove the existence of an extension of a metric space valued mapping from a subset of the reals to the whole set of reals with preservation of properties of the initial mapping: total variation, Lipschitz constant or absolute continuity. Then we show that a set-valued mapping of bounded variation defined on an arbitrary subset of the reals admits a regular selection of bounded variation. We introduce a notion of generated set-valued mappings and show that, under suitable assumptions, set-valued mappings (with arbitrary domains) which are Lipschitzian, of bounded variation or absolutely continuous are generated by certain families of mappings with nice properties. Finally, we prove a Castaing type representation theorem for set-valued mappings of bounded variation.
DOI : 10.4064/sm153-3-2
Keywords: study set valued mappings bounded variation real variable first prove existence extension metric space valued mapping subset reals whole set reals preservation properties initial mapping total variation lipschitz constant absolute continuity set valued mapping bounded variation defined arbitrary subset reals admits regular selection bounded variation introduce notion generated set valued mappings under suitable assumptions set valued mappings arbitrary domains which lipschitzian bounded variation absolutely continuous generated certain families mappings nice properties finally prove castaing type representation theorem set valued mappings bounded variation

V. V. Chistyakov  1   ; A. Rychlewicz  2

1 Department of Mathematics University of Nizhny Novgorod 23 Gagarin Avenue Nizhny Novgorod 603950, Russia
2 Faculty of Mathematics Łódź University Stefana Banacha 22 90-238 Łódź, Poland 
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V. V. Chistyakov; A. Rychlewicz. On the extension and generation of
 set-valued mappings of bounded variation. Studia Mathematica, Tome 153 (2002) no. 3, pp. 235-247. doi: 10.4064/sm153-3-2

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