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AIMS Mathematics

2017, Volume 2, Issue 4: 635-646. doi: 10.3934/Math.2017.4.635
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Research article

The Jordan decomposition of bounded variation functions valued in vector spaces

  • Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
  • Received: 26 September 2017 Accepted: 08 November 2017 Published: 16 November 2017

  • MSC : Primary: 26A45, 26B30; Secondary: 46C05, 06A06

  • In this paper we show the Jordan decomposition for bounded variation functions with values in Riesz spaces. Through an equivalence relation, we prove that this decomposition is satisfied for functions valued in Hilbert spaces. This result is a generalization of the real case. Moreover, we prove that, in general, the Jordan decomposition is not satisfied for vector-valued functions.

    Citation: Francisco J. Mendoza-Torres, Juan A. Escamilla-Reyna, Daniela Rodríguez-Tzompantzi. The Jordan decomposition of bounded variation functions valued in vector spaces[J]. AIMS Mathematics, 2017, 2(4): 635-646. doi: 10.3934/Math.2017.4.635

    Related Papers:

  • In this paper we show the Jordan decomposition for bounded variation functions with values in Riesz spaces. Through an equivalence relation, we prove that this decomposition is satisfied for functions valued in Hilbert spaces. This result is a generalization of the real case. Moreover, we prove that, in general, the Jordan decomposition is not satisfied for vector-valued functions.


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    [3] C. R. Adams, J. A. Clarkson, Properties of functions f (x; y) of bounded variation, Trans. Amer. Math. Soc., 36 (1934), 711-730.
    [4] V. V. Chistyakov, On mappings of bounded variation, J. Dyn. Control Syst., 2 (1997), 261-289.
    [5] V. V. Chistyakov, On the theory of multivalued mappings of bounded variation of one real variable, Sb. Math., 189 (1998), 153-176.
    [6] V. V. Chistyakov, On mappings of bounded variation with values in a metric space, Uspekhi Mat. Nauk, 54 (1999), 189-190.
    [7] V. V. Chistyakov, Metric-valued mappings of bounded variation, J. Math. Sci. (N. Y.), 111 (2002), 3387-3429.
    [8] S. Bianchini, D. Tonon, A decomposition theorem for BV functions, Commun. Pure Appl. Anal., 10 (2011), 1549-1566.
    [9] D. Gou, Y. J. Cho, J. Zhu, Partial Ordering Methods in Nonlinear Problems, New York, NY, USA: Nova Science Publishers, 2004.
    [10] S. Schwabik, Y. Guoju, Topics in Banach Space Integration, Singapore: Real Analysis, vol. 10, World Scientific, 2005.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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