Uniformly continuous composition operators in the space of bounded \(\varphi \)-variation functions
The authors write: ``We prove that, under some general assumptions, a generator of any uniformly continuous Nemytskii operator, mapping a subset of [the] space of bounded variation functions in the sense of Wiener into another space of this type, must be an affine function.'' An important ingredient in the proof is a result about left regularization by \textit{V. V. Chistyakov} [J. Math. Sci., 110, No.~2, 2455--2466 (2002; Zbl 1033.26013)]. The authors show that the left regularization \(h^-\) of the generating function \(h\) satisfies the Jensen functional equation and by modifying the standard argument [see \textit{M. Kuczma}, ``An introduction to the theory of functional equations and inequalities'' (Prace Naukowe Uniwersytetu Ślaskiego w Katowicach 489; Warszawa-Kraków-Katowice: Państwowe Wydawnictwo Naukowe) (1985; Zbl 0555.39004)], they obtain that \(h^-\) is an affine function. Reviewer's remarks. In the statement of their Theorem 3.1, the authors should have spelt out that the space \(Y\) is not only a normed space, but a Banach space. Also, the proof of the theorem seems to have a gap, consequently their Remark 3.5 appears to be incorrect. Fortunately, the gap can easily be filled if, instead of using the functions \(\eta_{\alpha,\beta}\), more general piecewise linear functions are used.
\(\varphi\)-variation in the sense of Wiener
scientific article; zbMATH DE number 5669299
scientific article