Some comparison principles for fractional differential equations and systems

Abstract

We consider certain classes of fractional differential equations and systems involving weighted fractional derivatives. We establish some comparison principles, and we study the uniqueness and the continuity with respect to the data.

Lyapunov functions for fractional‐order nonlinear systems with Atangana–Baleanu derivative of Riemann–Liouville type

“…Theorem 1 (In a previous study 46 ). Suppose that 𝛼 1 , 𝛼 2 , 𝛼 3 , a and b are arbitrary positive constants, and x = 0 be an equilibrium point for the system (8) and  ⊂ R n be a domain containing the origin. Let V(t, x t ) ∶ [0, ∞) ×  → R be a continuously differentiable function and locally Lipschitz with respect to x such that…”

“…In recent years, fractional differential equations have been employed in nonlinear control theory, signal processing, physical systems modeling, and even in social and biological models. [1][2][3][4][5][6][7][8][9] Considering that, explicit solutions of nonlinear fractional-order systems are difficult to obtain, many stability analysis methods have been developed. In the case of linear systems, some advances have been obtained for commensurate-and incommensurate-order systems.…”