Fractional Differential Equations Involving Caputo Fractional Derivative with Mittag-Leffler Non-Singular Kernel: Comparison Principles and Applications

Abstract

In this paper we study linear and nonlinear fractional differential equations involving the Caputo fractional derivative with Mittag-Leffler non-singular kernel of order $0<\alpha<1.$ 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 obtained sufficient condition determine the initial condition of the associated fractional initial value problem. We then derive comparison principles to the linear fractional equations. We apply these principles to obtain a norm estimate of solutions to the linear equation and to obtain a uniqueness result to the nonlinear equation. We also derive a lower and upper bound of solutions to the nonlinear equation. The applicability of the new results is illustrated through several examples.