On the $\mathbf{\rm\Psi}-$fractional integral and applications

TL;DR

This paper introduces the new $ m ext{ extPsi}$-fractional integral operator, explores its properties, and discusses applications including solutions to population growth models and nonlinear Volterra integral equations.

Contribution

The paper defines a novel $ m extPsi$-fractional integral operator and investigates its properties, including boundedness, with applications to fractional differential equations.

Findings

The $ m extPsi$-fractional integral operator is bounded.

Examples involving Mittag-Leffler functions demonstrate applications.

Discussion on uniqueness of nonlinear $ m extPsi$-fractional Volterra equations.

Abstract

Motivated by the ${\rm \Psi}$-Riemann-Liouville $({\rm \Psi-RL})$ fractional derivative and by the ${\rm \Psi}$-Hilfer $({\rm \Psi-H})$ fractional derivative, we introduced a new fractional operator the so-called $\rm\Psi-$fractional integral. We present some important results by means of theorems and in particular, that the $\rm\Psi-$fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag-Leffler $({\rm M-L})$ function, of paramount importance in the solution of population growth problem, as approached. On the other hand, we realize a brief discussion on the uniqueness of nonlinear $\Psi$-fractional Volterra integral equation (${\rm VIE}$) using $\beta-$distance functions.