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Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded

Author

Listed:
  • Brian Villegas-Villalpando

    (Centro de Ciencias Básicas, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, Mexico
    These authors contributed equally to this work.)

  • Jorge E. Macías-Díaz

    (Department of Mathematics and Didactics of Mathematics, Tallinn University, 10120 Tallinn, Estonia
    Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, Mexico
    These authors contributed equally to this work.)

  • Qin Sheng

    (Department of Mathematics and Center for Astrophysics, Space Physics and Engineering Research, Baylor University, Waco, TX 76706, USA)

Abstract

Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero measures is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempty sets is a positive extended real number.

Suggested Citation

  • Brian Villegas-Villalpando & Jorge E. Macías-Díaz & Qin Sheng, 2022. "Solution Spaces Associated to Continuous or Numerical Models for Which Integrable Functions Are Bounded," Mathematics, MDPI, vol. 10(11), pages 1-6, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1936-:d:832191
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    References listed on IDEAS

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    1. Iqbal, Zafar & Ahmed, Nauman & Baleanu, Dumitru & Adel, Waleed & Rafiq, Muhammad & Aziz-ur Rehman, Muhammad & Alshomrani, Ali Saleh, 2020. "Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
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