Mappings of bounded variation with values in a metric space: Generalizations
The author introduces the notion of the \((\Phi ,\sigma)\)-variation and thoroughly studies properties of functions from a compact interval into a metric space \(X\), which are of bounded variation in this sense. It is shown that such a function \(f\) can be represented as \(f=g\circ \varphi\), where \(g\) is Lipschitzian and \(\varphi\) is increasing and has the same regularity properties as \(f\). Then an analogue of Helly's selection principle is established. For the case when \(X\) is a normed space, differentiability properties of functions of bounded variation are studied. The author derives an integral formula for the total \((\Phi ,\sigma)\)-variation of smooth \(f\). The final part of the paper is devoted to the existence of a regular selection for a compact-valued multifunction of bounded variation. Required selection is a limit point of the suitably constructed sequence of polygonal approximations. Presented theorems extend results obtained for functions of bounded variation in the sense of C. Jordan and F. Riesz (see the previous papers of the author [``On mappings of bounded variation'', J. Dyn. Cont. Syst. 3, No. 2, 261-289 (1997; Zbl 0940.26009); ``On the theory of set-valued maps of bounded variation of one real variable'', Mat. Sb. 189, No. 5, 153-176 (1998); English transl.: Sb. Math. 189, No. 5, 797-819 (1998; Zbl 0933.26014)], and \textit{S. A. Belov} and the author [``A selection principle for mappings of bounded variation'', J. Math. Anal. Appl. 249, No. 2, 351-366 (2000; Zbl 0982.54022)]).
scientific article; zbMATH DE number 1526510