Substitution operators in the spaces of functions of bounded variation $BV_\alpha^2(I)$
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 649-659
https://doi.org/10.4134/BKMS.2015.52.2.649
Published online March 31, 2015
Wadie Aziz, Jos\'e Atilio Guerrero, and Nelson Merentes
Universidad de Los Andes, Universidad Nacional Experimental del T\'achira, Universidad Central de Venezuela
Abstract : 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
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