This paper introduces a new two-parameter stability framework for fractional differential systems based on a pair of constants that model decay and robustness in memory-driven dynamics. Classical stability and inequality methods in fractional calculus rely almost exclusively on single-parameter Lipschitz-type bounds, which provide no mechanism for quantifying disturbance tolerance or practical asymptotic behavior. To address this gap, we develop a comprehensive K–R stability theory for Caputo-type fractional differential equations that simultaneously captures Mittag–Leffler decay rates and residual perturbation radii. Three main analytical results are obtained. First, we establish a K–R fractional stability inequality that generalizes exponential-type bounds to Mittag–Leffler decay while incorporating a robustness term . Second, we derive a new K–R fractional Grönwall inequality, which extends the classical and fractional Grönwall–Bellman inequalities by embedding both contraction and disturbance effects. This result provides sharper estimates for existence, uniqueness, and sensitivity analysis in nonlinear fractional systems. Third, we formulate a K–R Lyapunov stability framework, yielding practical Mittag–Leffler stability conditions and explicit stability radii for nonlinear systems under persistent disturbances. The theoretical contributions are complemented by numerical simulations based on the fractional Adams–Bashforth–Moulton scheme, verifying the accuracy and sharpness of the proposed bounds. Applications are demonstrated for fractional control systems, viscoelastic models, and neural dynamics, showing how the K–R constants naturally encode system memory, damping behavior, and robustness to uncertainty. Overall, the K–R framework unifies and extends several foundational tools in fractional calculus, offering a versatile and analytically tractable approach to stability, robustness, and inequality theory. The results open new pathways for modeling, analysis, and control of complex systems governed by fractional-order dynamics.
Ramakrishna Rao Pasupuleti (Fri,) studied this question.