Uniformly bounded composition operators in the Banach space of bounded \((p, k)\)-variation in the sense of Riesz-Popoviciu
Let \([a,b]\) be a compact interval in \(\mathbb R\) and denote by \(RV_{(p,k)}[a,b]\) the vector space of all the functions \(u:[a,b] \to \mathbb R\) such that \(\hat{V}_{(p,k)}^R(u)<+\infty\), that is, such functions which have bounded \((p,k)\) variation in the sense of Riesz-Popoviciu, endowed with the norm \[ \|u\|_{(p,k)}^R=|u(a)|+|u'(a)|+\ldots+|u^{(k-1)}(a)|+(\hat{V}_{(p,k)}^R(u))^{\frac{1}{p}}. \] Let \(p>1\), \(k \geq 2\) a positive integer and let \(f:[a,b]\times\mathbb R \to \mathbb R\) be continuous with respect to the second variable. The main result of this article states that, if the nonautonomous superposition operator \(F\), generated by \(f\), maps the space \(RV_{(p,k)}[a,b]\) into itself and satisfies the inequality \[ \|F(u)-F(v)\|_{(p,k)}^R\leq\gamma (\|u-v\|_{(p,k)}^R),\quad u,v\in RV_{(p,k)}[a,b], \] for some function \(\gamma:[0,+ \infty) \to [0,+ \infty)\), then \(F\) possesses the Matkowski property, that is, there exist functions \(\alpha, \beta \in RV_{(p,k)}[a,b]\) such that \[ f(t,x)= \alpha(t)x+ \beta(t),\; t \in [a,b],\; x \in \mathbb R. \]
scientific article; zbMATH DE number 6137071
scientific article