Real Analysis Exchange

On some theorems in fractional calculus for singular functions

Shusheng Fu

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Abstract

In this paper we show that there are close relationships amongst Cantor bar totality, non-integer integral and Hausdorff integral. If a singular function \(f\) has zero Lipschitz (\(1-\nu\))-numbers on a Cantor set \(C\) with \(H\)-dim~\(C=1-\nu\), \(0\lt \nu\lt 1\), then the \(1-\nu\) order fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Moreover, under strong assumptions on the function \(f\) the \(1-\nu\) order derivative of \(f\) exists everywhere. Consequently, the \(\nu\) order fractional integral of \(f\) equals \(f(0)t^{\nu}\). Using the Concept of th Hausdorff derivative we prove that if a singular function has a Hausdorff derivative on \(C\), then the fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Finally, under some assumptions on \(f\) and \(C\), we establish \begin{align*} (D^{-\nu}f)(t)=&\int_0^t\int_0^uv^{\nu-1}f'_H(u-v)\,dv^{1-\nu}\,du\\ =&\int_0^tu^{1-\nu}J_{\nu}(v^{\nu-1}f'_H(u-v))(u)\,du \end{align*} This identity is similar to the identity \[(D^{-\nu}f)(t)=\frac{1}{B_{\nu}}t^{\nu}(J_{\nu}f)(t)\] which professor R. R. Nigmatullin claimed.

Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 122-141.

Dates
First available in Project Euclid: 1 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1338515208

Mathematical Reviews number (MathSciNet)
MR1433601

Zentralblatt MATH identifier
0879.26024

Subjects
Primary: 26A33: Fractional derivatives and integrals 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
Secondary: 26A39: Denjoy and Perron integrals, other special integrals

Keywords
Fractional integral Fractional derivative Hausdorff integral Hausdorff derivative Lipschitz number singular function Cantor type set

Citation

Fu, Shusheng. On some theorems in fractional calculus for singular functions. Real Anal. Exchange 22 (1996), no. 1, 122--141. https://projecteuclid.org/euclid.rae/1338515208


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References

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