## Hokkaido Mathematical Journal

### The decompositional structure of certain fractional integral operators

#### 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

https://projecteuclid.org/euclid.hokmj/1573722020

Digital Object Identifier
doi:10.14492/hokmj/1573722020

Mathematical Reviews number (MathSciNet)
MR4031254

Zentralblatt MATH identifier
07145332

#### 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