HADAMARD-TYPE FRACTIONAL CALCULUS

  • Anatoly A.Kilbas (Department of Mathematics and Mechanics Belarusian State University)
  • Published : 2001.11.01

Abstract

The paper is devoted to the study of fractional integration and differentiation on a finite interval [a, b] of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration $\int_{a}^{x}(t/x)^{\mu}f(t)dt/t$ and the modified differentiation ${\delta}+{\mu}({\delta}=xD,D=d/dx)$ with real $\mu$, being taken n times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space $X^{p}_{c}$(a, b) of Lebesgue measurable functions f on $R_{+}=(0,{\infty})$ such that for c${\in}R=(-{\infty}{\infty})$, in particular in the space $L^{p}(0,{\infty})\;(1{\le}{\le}{\infty})$. The existence almost every where is established for the coorresponding Hadamard-type fractional derivative for a function g(x) such that $x^{p}$g(x) have $\delta$ derivatives up to order n-1 on [a, b] and ${\delta}^{n-1}[x^{\mu}$g(x)] is absolutely continuous on [a, b]. Semigroup and reciprocal properties for the above operators are proved.

Keywords

References

  1. J. Math. Anal. Appl. (to appear) Fractional calculus in the Mellin setting and Hadamard-type fractional integrals P. L. Butzer;A. A. Kilbas;J. Trujillo
  2. J. Math. Anal. Appl. (to appear) Compositions of Hadamard-type fractional integration operators and the semigroup property P. L. Butzer;A. A. Kilbas;J. Trujillo
  3. J. Math. Anal. Appl. (to appear) Mellin transform and integration by parts for Hadamard-type frational integrals P. L. Butzer;A. A. Kilbas;J. Trujillo
  4. Higher Transcendental Functions v.1 A. Erdelyi;W. Magnus;F. Oberhettinger;F. G. Tricomi
  5. J. Math. Pures et Appl. v.4 no.8 Essai sur l'etude des fonctions donnees par leur developpment de Taylor J. Hadamard
  6. Fractional Integrals and Derivatives. Theory and Applications S. G. Samko;A. A. Kilbas;O. I. Marichev