Quadratic Lyapunov Estimates for Variable-Order Fractional Derivatives: Kernel Monotonicity and Definition-Dependence

The quadratic Lyapunov inequality of Aguila-Camacho, Duarte-Mermoud and Gallegos has been extended operator by operator—most recently to the generalised Hattaf mixed operator, which subsumes the Atangana–Baleanu, Hadamard, generalised proportional and tempered ϖ-Caputo cases—each time under hypotheses tailored to the operator at hand. The integration-by-parts technique behind these extensions is not new, and we do not claim it. What we show is that each of those hypotheses is an instance of one structural property, monotonicity of the memory kernel in the integration variable, which also governs the companion “no new maximum” principle. Once the order varies, the property ceases to be automatic. For the history-order operator, in which α depends on the integration variable, the inequality can fail; we give two regimes of failure, a quantitative sufficient condition for validity, and a counterexample whose sign is established analytically rather than computed. The frozen-order (Coimbra-type) operator holds unconditionally, under hypotheses that solutions of fractional differential equations actually satisfy. The practical exposure is limited: within the band used throughout the fractional chaos literature, the inequality survives except in a narrow neighbourhood of the upper endpoint, so existing applications there are substantively safe. As an application, we obtain explicit balls that are positively invariant—not attracting—uniformly in the order function, without a variable-order comparison principle.