Rolle`s Theorem and Its Generalization Via the Katugampola Fractional Derivative

Abstract

Fractional calculus has extended the classical notions of differentiation and integration to non-integer orders and has become an important tool in many areas of mathematics and applied sciences. Among the modern formulations of fractional operator, The Katugampola fractional derivative provides a unified framework that generalizes several classical fractional derivatives, including the Riemann-Liouville and Hadamard types. In this paper, we review the definitions and some fundamental properties of The Katugampola fractional derivative. We also revisit fractional versions of classical results from differential calculus, particularly Rolle`s theorem and the Mean Value Theorem, within the framework of The Katugampola fractional operator. Furthermore, we present and discuss a general form of Rolle`s theorem in this context. An illustrative example is provided to demonstrate the applicability of this result.