Properties of fractional integral operators involving the three-parameters Mittag-Leffler function in the kernels with respect to another function

TL;DR

This paper explores fractional integral operators with three-parameters Mittag-Leffler functions in the kernels, establishing their properties, equivalence of related equations, and deriving solutions using advanced fractional derivatives.

Contribution

It introduces new properties of these operators, proves the equivalence of the Cauchy problem and Volterra integral equation, and provides a solution method using $bla$-Caputo derivatives.

Findings

Established properties of fractional integral operators with Mittag-Leffler kernels

Proved the equivalence between the Cauchy problem and Volterra integral equation

Derived a closed-form solution using successive approximations and $bla$-Caputo fractional derivatives

Abstract

This paper aims to investigate properties associated with fractional integral operators involving the three-parameters Mittag-Leffler function in the kernels with respect to another function. We prove that the Cauchy problem and the Volterra integral equation are equivalent. We find a closed-form to the solution of the Cauchy problem using successive approximations method and $\psi$-Caputo fractional derivative.