Asymptotic properties of solutions to Caputo-Hadamard fractional differential equations

This paper investigates the stability properties of Caputo-Hadamard fractional differential equations. We first establish the asymptotic behavior and rigorously prove a specific convergence rate for these equations. Subsequently, a novel stability criterion termed Logarithmic Mittag-Leffler stability is proposed. Through the construction of an innovative Banach space equipped with a carefully designed weighted norm, and by employing the fixed-point theorem, we demonstrate that when the linearization spectrum of Caputo-Hadamard fractional differential equations lies within a prescribed sector, the equilibrium point of the corresponding nonlinear system exhibits Logarithmic Mittag-Leffler stability. This result leads to a significant extension of Lyapunov's first method to certain classes of Caputo-Hadamard fractional differential equations.