A Multiplicative Extension of Weighted Hadamard Fractional Integrals

Abstract

This paper presents an innovative category of weighted Hadamard fractional integral operators situated within the paradigm of multiplicative calculus. The operator introduced herein broadens the traditional weighted Hadamard fractional integral by integrating the multiplicative framework of non-Newtonian calculus, thereby establishing a cohesive linkage between weighted fractional analysis and multiplicative fractional calculus. We delineate the formulation of both the left- and right-sided multiplicative weighted Hadamard fractional integral operators and explore their core analytical characteristics. Specifically, a multiplicative linearity property is demonstrated, the continuity of the logarithmic representation of the operator is affirmed, and boundedness results are extracted in weighted Lebesgue-type spaces through the application of Hölder's inequality. Furthermore, we establish that the proposed operators comply with a semigroup property, which ensures alignment with the classical framework of fractional integration. These findings illustrate that the introduced operators maintain crucial structural attributes of weighted fractional integrals while seamlessly extending them to the multiplicative context. The theoretical framework developed herein lays a foundational basis for subsequent explorations in multiplicative fractional calculus, fractional integral inequalities, and associated applications within the realm of mathematical analysis.