A novel approach to Lyapunov stability of Caputo fractional dynamic equations on time scale using a new generalized derivative

In this work, we introduced a generalized concept of Caputo fractional derivatives, specifically the Caputo fractional delta derivative (Fr $\mathrm{\Delta }$D) and Caputo fractional delta Dini derivative (Fr $\mathrm{\Delta }$DiD) of order $\alpha \in (0,1)$, on an arbitrary time domain $\mathbb{T}$, which was a closed subset of $\mathbb{R}$. By bridging the gap between discrete and continuous time domains, this unified framework enabled a more thorough approach to stability and asymptotic stability analysis on time scales. A key contribution of this work was the new definition of the Caputo Fr $\mathrm{\Delta }$D for a Lyapunov function, which served as the basis for establishing comparison results and stability criteria for Caputo fractional dynamic equations. The proposed framework extended beyond the limitations of traditional integer-order calculus, offering a more flexible and generalizable tool for researchers working with dynamic systems. The inclusion of fractional orders enabled the modeling of more complex dynamics that occur in real-world systems, particularly those involving both continuous and discrete time components. The results presented in this work contributed to the broader understanding of fractional calculus on time scales, enriching the theoretical foundation of dynamic systems analysis. Illustrative examples were included to demonstrate the effectiveness, relevance, and practical applicability of the established stability and asymptotic stability results. These examples highlighted the advantage of our definition of fractional-order derivative over integer-order approaches in capturing the intricacies of dynamic behavior.