Gronwall inequalities are common tools in studying differential and integral equations analytically. Existence, uniqueness and stability results can be obtained using these inequalities. In this paper, we provide new versions of the Gronwall inequality to the normalized fractional integrals with exponential and Mittag-Leffler kernels. The obtained inequalities are used to establish existence and uniqueness results to the fractional Cauchy problem with the normalized derivative of Mittag-Leffler kernel. Comparison principles are derived based on an estimate of the normalized derivative of a function at its extreme points. These comparison principles are then used to obtain a pre-norm estimates of solutions for related linear fractional differential equations. Two examples are presented to illustrate the efficiency of the obtained re-sults. Further, a numerical example is studied to illustrate the solutions of a non-homogeneous normalized system in the Mittag-Leffler kernel case.
Abdeljawad et al. (Wed,) studied this question.