Hokkaido Mathematical Journal

The decompositional structure of certain fractional integral operators

Min-Jie LUO and Ravinder Krishna RAINA

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Abstract

The aim of this paper is to investigate the decompositional structure of generalized fractional integral operators whose kernels are the generalized hypergeometric functions of certain type. By using the Mellin transform theory proposed by Butzer and Jansche [J. Fourier Anal. 3 (1997), 325-376], we prove that these operators can be decomposed in terms of Laplace and inverse Laplace transforms. As applications, we derive two very general results involving the $H$-function. We also show that these fractional integral operators when being understood as integral equations possess the $\mathcal{L}$ and $\mathcal{L}^{-1}$ solutions. We also consider the applications of the decompositional structures of the fractional integral operators to some specific integral equations and one of such integral equations is shown to possess a solution in terms of an Aleph $(\aleph)$-function.

Article information

Source
Hokkaido Math. J., Volume 48, Number 3 (2019), 611-650.

Dates
First available in Project Euclid: 14 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1573722020

Digital Object Identifier
doi:10.14492/hokmj/1573722020

Mathematical Reviews number (MathSciNet)
MR4031254

Zentralblatt MATH identifier
07145332

Subjects
Primary: 26A33: Fractional derivatives and integrals 33C60: Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions) 44A10: Laplace transform 44A15: Special transforms (Legendre, Hilbert, etc.) 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) [See also 47B35]

Keywords
Fractional integral operator generalized hypergeometric function integral equation $H$-function Laplace transform Mellin transform

Citation

LUO, Min-Jie; RAINA, Ravinder Krishna. The decompositional structure of certain fractional integral operators. Hokkaido Math. J. 48 (2019), no. 3, 611--650. doi:10.14492/hokmj/1573722020. https://projecteuclid.org/euclid.hokmj/1573722020


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