Lyapunov Stability of ψ-Caputo Real-Order Systems with Space-Time Partial Derivatives

Let $\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\in (0,1]$ be the orders and $\psi_{1},\psi_{2},\cdots,\psi_{n}$ be the time-dependent functions that are associated with incommensurate $\widehat{\psi}$-Caputo real-order systems. Is it possible to extend the Lyapunov stability theory for such advanced systems? In this paper, we state and prove new Lyapunov stability theorems that provide efficient, sufficient criteria for the stability of equilibria in such systems when one discovers adequate Lyapunov functions. We also address new conjectures that introduce new perspectives on stability tests with space-time partial derivatives of Lyapunov functions for such systems.