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A NEW ℱα-TYPE INTEGRABLE FUNCTIONS AND RETARDED FRACTAL DIFFERENTIAL EQUATIONS.

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Title: A NEW ℱα-TYPE INTEGRABLE FUNCTIONS AND RETARDED FRACTAL DIFFERENTIAL EQUATIONS.
Authors: SAJID, MOHAMMAD1 (AUTHOR) msajd@qu.edu.sa, KALITA, HEMANTA2 (AUTHOR) hemanta30kalita@gmail.com, DAS, ABHISHIKTA3 (AUTHOR) abhishikta.math@gmail.com
Source: Fractals. 2026, Vol. 34 Issue 8, p1-13. 13p.
Subjects: Integrable functions, Delay differential equations, Riemann integral, Fractional differential equations, Fractals, Integrals
Abstract: The gauge integral, also known as the Henstock–Kurzweil integral, is a generalization of the Riemann integral. Functions that are not integrable due to singularities in the contexts of Lebesgue or Riemann integration can still be classified as gauge integrable. In this paper, vector-valued gauge integral type ℱ α -integral is introduced. Several properties of the newly introduced integral have been discussed. A measure of noncompactness has been established for fractal sets. In applications, along with some additional conditions and the newly introduced fractal measure of noncompactness, we prove an existence theorem for the fractal retarded differential equation D ℱ α ℏ (∂) = ♭ (∂ , ℏ ∂) whenever ♭ is a vector-valued ℱ α -integrable and D ℱ α is fractal differentiation. [ABSTRACT FROM AUTHOR]
Copyright of Fractals is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 194725823
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PubType: Academic Journal
PubTypeId: academicJournal
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  Data: A NEW ℱα-TYPE INTEGRABLE FUNCTIONS AND RETARDED FRACTAL DIFFERENTIAL EQUATIONS.
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  Data: <searchLink fieldCode="JN" term="%22Fractals%22">Fractals</searchLink>. 2026, Vol. 34 Issue 8, p1-13. 13p.
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  Data: <searchLink fieldCode="DE" term="%22Integrable+functions%22">Integrable functions</searchLink><br /><searchLink fieldCode="DE" term="%22Delay+differential+equations%22">Delay differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Riemann+integral%22">Riemann integral</searchLink><br /><searchLink fieldCode="DE" term="%22Fractional+differential+equations%22">Fractional differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Fractals%22">Fractals</searchLink><br /><searchLink fieldCode="DE" term="%22Integrals%22">Integrals</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: The gauge integral, also known as the Henstock–Kurzweil integral, is a generalization of the Riemann integral. Functions that are not integrable due to singularities in the contexts of Lebesgue or Riemann integration can still be classified as gauge integrable. In this paper, vector-valued gauge integral type ℱ α -integral is introduced. Several properties of the newly introduced integral have been discussed. A measure of noncompactness has been established for fractal sets. In applications, along with some additional conditions and the newly introduced fractal measure of noncompactness, we prove an existence theorem for the fractal retarded differential equation D ℱ α ℏ (∂) = ♭ (∂ , ℏ ∂) whenever ♭ is a vector-valued ℱ α -integrable and D ℱ α is fractal differentiation. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Fractals is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1142/S0218348X26400104
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 13
        StartPage: 1
    Subjects:
      – SubjectFull: Integrable functions
        Type: general
      – SubjectFull: Delay differential equations
        Type: general
      – SubjectFull: Riemann integral
        Type: general
      – SubjectFull: Fractional differential equations
        Type: general
      – SubjectFull: Fractals
        Type: general
      – SubjectFull: Integrals
        Type: general
    Titles:
      – TitleFull: A NEW ℱα-TYPE INTEGRABLE FUNCTIONS AND RETARDED FRACTAL DIFFERENTIAL EQUATIONS.
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            NameFull: SAJID, MOHAMMAD
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            NameFull: KALITA, HEMANTA
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            NameFull: DAS, ABHISHIKTA
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          Dates:
            – D: 01
              M: 09
              Text: 2026
              Type: published
              Y: 2026
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              Value: 0218348X
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              Value: 34
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              Value: 8
          Titles:
            – TitleFull: Fractals
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