Superposition operators in the algebra of functions of two variables with finite total variation
The author studies the Nemytskij operator \((Hf)(x) = h(x,f(x))\) in the space of functions \(f : [a,b]\times [c,d]\to\mathbb{R}\) of bounded variation. In particular, he shows that a Lipschitz condition for \(H\) in the norm of this space leads to a (mild or strong) degeneracy of the generating function \(h\). This extends previous results of the author [J. Appl. Anal. 6, 173--186 (2000; Zbl 0997.47051), Positivity 5, 323--358 (2001; Zbl 1027.47046), and J. Math. Sci., New York 111, 3387--3429 (2002; Zbl 1033.26014)]{} and by many other authors.
scientific article; zbMATH DE number 1837389