We introduce the notion of

Lakshmikantham et al. [

In this paper we introduce the notion of

For the basic notions and theorems about fractional calculus, we mainly refer to some books [

We recall the notions of Mittag-Leffler functions which were originally introduced by Mittag-Leffler in 1903 [

Note that the exponential function

We recall briefly the notions and basic properties about fractional integral operators and fractional derivatives of functions [

The Riemann-Liouville fractional integral of order

The Riemann-Liouville fractional derivative of order

If

Note that the Riemann-Liouville fractional derivatives have singularity at

Let

When

When

Hence, we can see that the Caputo derivative is defined for functions for which the Riemann-Liouville derivative exists. Also, we note that the Mittag-Leffler functions

We can obtain the following asymptotic property for

When

If

When

Let

If we set

Let

Let

Let

In the sequential we assume that the solution

Next, we consider the nonhomogeneous linear fractional differential equation with Caputo fractional derivative

If one sets

If

We can obtain the following Caputo fractional differential inequality of Gronwall type by Lemma

Suppose that

There exists a nonnegative function

If we set

We can obtain the following result about fractional integral inequality. It is adapted from the comparison principle regarding nonstrict inequalities in [

Let

Pinto [

We will give the notion of

The zero solution

an

We recall the stability in the sense of Mittag-Leffler [

The zero solution

The zero solution

Note that the Mittag-Leffler stability implies

We can obtain the following result adapted from Theorem 3.4 in [

Suppose that the function

The equation (

Suppose that all conditions of Theorem

We can obtain an upper bound of solutions for Caputo fractional differential equations via fractional Gronwall’s inequality. The following result is adapted from Theorem 5.1 in [

Suppose that

It follows from (

We can obtain the boundedness of solutions for Caputo fractional differential equations via the fractional Lyapunov direct method.

Under the same assumptions of Lemma

Let

We can obtain the following result [

If one sets

In this section we give tow examples which illustrate some results in the previous section.

To illustrate Theorem

The function

We note that the fractional differential equation (

Let

Next, we will give an example to illustrate Theorem

Let

The authors declare no conflict of interests.

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2007585).