On asymptotic properties of solutions to fractional differential equations

Abstract

We present some distinct asymptotic properties of solutions to Caputo fractional differential equations (FDEs). First, we show that the non-trivial solutions to a FDE can not converge to the fixed points faster than $t^{-\alpha }$, where $\alpha$ is the order of the FDE. Then, we introduce the notion of Mittag-Leffler stability which is suitable for systems of fractional-order. Next, we use this notion to describe the asymptotical behavior of solutions to FDEs by two approaches: Lyapunov's first method and Lyapunov's second method. Finally, we give a discussion on the relation between Lipschitz condition, stability and speed of decay, separation of trajectories to scalar FDEs.

References in corpus (3)

• Generation of nonlocal fractional dynamical systems by fractional differential equations
• Asymptotic stability of linear fractional systems with constant coefficients and small time dependent perturbations
• An instability theorem for nonlinear fractional differential systems

Cited by in corpus (7)

• Analysis of Nonlinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative
• Stability analysis of multi-term fractional-differential equations with three fractional derivatives
• Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems
• Qualitative analysis of solutions to mixed-order positive linear coupled systems with bounded or unbounded delays
• Implicit Fractional Differential Equations in Banach Spaces via Picard and Weakly Picard Operator Theory
• Hill-Type Stability Analysis of Periodic Solutions of Fractional-Order Differential Equations
• Mittag--Leffler stability of numerical solutions to time fractional ODEs