Bulletin of the
Korean Mathematical Society BKMS

The space $BV_\alpha^2(I)$ of all the real functions defined on interval $I=[a,b]\subset\R$, which are of bounded second $\alpha$-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 $h:I\times\R\longrightarrow\R$, and prove, in particular, that if $H$ maps $BV_\alpha^2(I)$ into itself and is globally Lipschitz or uniformly continuous, then $h$ is an affine function with respect to the second variable.

Keywords: variation in the sense of De la Vall\'ee Poussin, uniformly continuous operator, Nemytskii (substitution) operator, Jensen equation

MSC numbers: Primary 47B33; Secondary 26B30