New fractional generalized Gronwall inequalities and Lyapunov theorems with applications

This paper deals with some expressions of the fractional generalized Gronwall inequality when associated with both non-negative and non-positive singular kernels and establishes sharp Mittag- Leffler bounds containing different ingredients. The long-term behavior of non-autonomous fractional order systems by means of modified fractional Lyapunov theorems is analyzed. As an application, we give a few examples that use quadratic Lyapunov functions for typical fractional order systems to predict trajectories that ultimately aim to reach vector 0 as t → ∞.