On Superposition Operators in Spaces of Regular and of Bounded Variation Functions

Abstract

For a function $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ we define the superposition operator ${\Psi }_f:{\mathbb{R}}^{[0,1]}\to {\mathbb{R}}^{[0,1]}$ by the formula ${\Psi }_f(\phi )(t)=f(t,\phi (t))$. First we provide necessary and sufficient conditions for $f$ under which the operator ${\Psi }_f$ maps the space $R(0,1)$, of all real regular functions on $[0,1]$, into itself. Next we show that if an operator ${\Psi }_f$ maps the space $BV(0,1)$, of all real functions of bounded variation on $[0,1]$, into itself, then

(1) it maps bounded subsets of $BV(0,1)$ into bounded sets if additionally $f$ is locally bounded,

(2) $f=f_{cr}+f_{dr}$ where the operator ${\Psi }_{f_{cr}}$ maps the space $D(0,1)\cap BV(0,1)$, of all right-continuous functions in $BV(0,1)$, into itself and the operator ${\Psi }_{f_{dr}}$ maps the space $BV(0,1)$ into its subset consisting of functions with countable support

(3) ${\mathrm{limsup}}_{n\to \infty }{n}^{\frac{1}{2}}|f(t_n,x_n)-f(s_n,x_n)|<\infty$ for every bounded sequence $(x_n)\subset \mathbb{R}$ and for every sequence $([s_n,t_n))$ of pairwise disjoint intervals in $[0,1]$ such that the sequence $(|f(t_n,x_n)-f(s_n,x_n)|)$ is decreasing.

Moreover we show that if an operator ${\Psi }_f$ maps the space $D(0,1)\cap BV(0,1)$ into itself, then $f$ is locally Lipschitz in the second variable uniformly with respect to the first variable.