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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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