Substitution operators in the spaces of functions of bounded variation $BV_alpha^2(I)$

摘要

The space $BV_lpha^2(I)$ of all the real functions defined on interval $I=[a,b]subsetR$, which are of bounded second $lpha$-variation (in the sense De la Vall'{e} Poussin) on $I$ forms a Banach space. In this space we define an operator of substitution $H$ generated by a function $h:IimesRlongrightarrowR$, and prove, in particular, that if $H$ maps $BV_lpha^2(I)$ into itself and is globally Lipschitz or uniformly continuous, then $h$ is an affine function with respect to the second variable.