Comparison principles for differential equations involving Caputo fractional derivative with Mittag-Leffler non-singular kernel - Mohammed Al-Refai (Electronic Journal of Differential Equations, 2018)
In this article we study linear and nonlinear differential equations involving the Caputo fractional derivative with Mittag-Leffler non-singular kernel of order $0<α<1$. We first obtain a new estimate of the fractional derivative of a function at its extreme points and derive a necessary condition for the existence of a solution to the linear fractional equation. The condition obtained determines the initial condition of the associated fractional initial-value problem. Then we derive comparison principles for the linear fractional equations, and apply these principles for obtaining norm estimates of solutions and to obtain a uniqueness results. We also derive lower and upper bounds of solutions. The applicability of the new results is illustrated through several examples.