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On the theory of set-valued maps of bounded variation of one real variable. (English. Russian original) Zbl 0933.26014
Sb. Math. 189, No. 5, 797-819 (1998); translation from Mat. Sb. 189, No. 5, 153-176 (1998).
The author considers (set-valued) maps of bounded variation taking values in metric or normed spaces. The following structure theorem is proved: A map \(f: [a,b]\to X\) has bounded variation iff there is a nondecreasing bounded function \(\varphi: [a,b]\to \mathbb{R}\) and a map \(g\) satisfying \(V(g,[c,d])= d-c\) for every \(a\leq c\leq d\leq b\) such that \(f= g\circ\varphi\). It is shown that every compact set-valued map into a Banach space that is of bounded variation (or Lipschitz, or absolutely continuous) has a selection of bounded variation (Lipschitz or absolutely continuous, respectively).

MSC:
26E25 Set-valued functions
26A16 Lipschitz (Hölder) classes
26A45 Functions of bounded variation, generalizations
54C65 Selections in general topology
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