On Lyapunov Stability of Caputo Fractional Dynamic Equations on Time Scale using a New Generalized Derivative

Abstract

In this work, we introduce a new and generalized concept (herein referred to as Caputo fractional delta derivative and Caputo fractional delta dini derivative of order $\alpha\in(0,1)$) for Caputo fractional derivatives on an arbitrary time domain $\mathbb{T}$ which is a closed subset of $\mathbb{R}$. Combining the continuous and discrete time domains, we create a unified framework for stability analysis on time scales. Comparison results and stability criteria for the considered Caputo fractional dynamic equations are presented based on a new definition for Caputo fractional delta derivative of a Lyapunov function, contributing to the broader understanding of fractional calculus on time scales. The work also incorporates an illustrative example to demonstrate the relevance, effectiveness and applicability of the established stability results over that of the integer order.