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SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV<sup>2</sup><sub>α</sub>(I)
Aziz, Wadie,Guerrero, Jose Atilio,Merentes, Nelson Korean Mathematical Society 2015 대한수학회보 Vol.52 No.2
The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.
Glazowska, Dorota,Guerrero, Jose Atilio,Matkowski, Janusz,Merentes, Nelson Korean Mathematical Society 2013 대한수학회보 Vol.50 No.2
We prove, under some general assumptions, that a generator of any uniformly bounded Nemytskij operator, mapping a subset of space of functions of bounded variation in the sense of Wiener-Young into another space of this type, must be an affine function with respect to the second variable.
SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BVα2 (I)
Wadie Aziz,Jos´e Atilio Guerrero,Nelson Merentes 대한수학회 2015 대한수학회보 Vol.52 No.2
The space BVα2 (I) of all the real functions defined on interval I = [a, b] ⊂ R, which are of bounded second α-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 × R → R, and prove, in particular, that if H maps BVα2 (I) into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.
Dorota G lazowska,Jos\'e Atilio Guerrero,Janusz Matkowski,Nelson Merentes 대한수학회 2013 대한수학회보 Vol.50 No.2
We prove, under some general assumptions, that a generator of any uniformly bounded Nemytskij operator, mapping a subset of space of functions of bounded variation in the sense of Wiener-Young into another space of this type, must be an affine function with respect to the second variable.