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Nonautonomous superposition operators in the spaces of functions of bounded variation. (English) Zbl 1460.47028

The superposition operator (also called composition operator or Nemytskij operator) \(F(x)(t)= f(t,x(t))\) generated by a function \(f: [0,1]\times\mathbb{R}\) has, in spite of its simple form, a rather pathological behaviour in many function spaces, and many problems concerning its properties are still open. For example, it was a long-standing problem to find conditions on \(f\), possibly both necessary and sufficient, under which the operator \(F\) maps the space BV\([0,1]\) into itself. An apparently obvious sufficient condition given by Lyamin in 1986 turned out to be incorrect, as was observed by the first author in 2010 [Math. Comput. Modelling 52, No. 5–6, 791–796 (2010; Zbl 1202.45005)], see also [the reviewer et al., Commun. Appl. Anal. 15, No. 2–4, 153–182 (2011; Zbl 1255.47059)] and confirmed by means of a counterexample by the fourth author [Proc. Am. Math. Soc. 142, No. 5, 1773–1776 (2014; Zbl 1285.47071)].
In this article, which is both a survey of known and a research paper with new results, the authors solve the mentioned problem by giving a necessary and sufficient condition on \(f\) (too technical to be reproduced here) under which \(F\) maps BV\([0,1]\) into itself and is locally bounded. The last additional property is not a severe restriction, since the authors also show that local boundedness of \(F\) implies local boundedness of \(f\), and functions \(f\) which are not locally bounded usually occur very seldom in applications.
Apart from the solution of the mentioned problem, the authors also give other interesting and important contributions to the theory of superposition operators in BV-spaces. For instance, several results are given for functions of the form \(f(t,u)= g(t)h(u)\) which are in a certain sense “intermediate” between the autonomous case \(f=f(u)\) and the nonautonomous case \(f=f(t,u)\). A strong positive feature of this nice paper is that the authors illustrate their abstract results throughout by illuminating examples and counterexamples.

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
26A45 Functions of bounded variation, generalizations
45G10 Other nonlinear integral equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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