Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV2α(I)

  • Aziz, Wadie (Universidad de Los Andes Departamento de Fisica y Matematica) ;
  • Guerrero, Jose Atilio (Universidad Nacional Experimental del Tachira Departamento de Matematica y Fisica) ;
  • Merentes, Nelson (Universidad Central de Venezuela Escuela de Matematicas)
  • Received : 2014.05.01
  • Published : 2015.03.31
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Abstract

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.

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References

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