New Approach To A Generalized Fractional Integral

TL;DR

This paper introduces a new generalized fractional integral formula that unifies existing types, establishes conditions for boundedness, proves a semigroup property, and offers a broad definition of fractional derivatives.

Contribution

It presents a novel unified fractional integral formula, extends the theory with boundedness conditions, and defines fractional derivatives in a general framework.

Findings

Unified fractional integral formula encompassing Riemann-Liouville and Hadamard types

Proved semigroup property for the new operator

Established boundedness conditions in Lebesgue spaces

Abstract

The paper presents a new formula for the fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as special cases. Conditions are given for such a generalized fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. Finally, we give a general definition of the Fractional derivatives.