Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

It is well known that the Gronwall-Bellman inequality [

On the other hand, recently, Jumarie presented a new definition for the fractional derivative named the modified Riemann-Liouville fractional derivative (see [

The modified Riemann-Liouville derivative of order

The Riemann-Liouville fractional integral of order

Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows (the interval concerned below is always defined by

The modified Riemann-Liouville derivative for a constant is zero.

The modified Riemann-Liouville derivative has many excellent characters in handling many fractional calculus problems. Many authors have investigated various applications of the modified Riemann-Liouville fractional derivative. For example, in [

Motivated by the wide applications of the modified Riemann-Liouville fractional derivative, in this paper, we use this type of fractional derivative to establish some fractional Gronwall-Bellman type inequalities. Based on these inequalities and some basic properties of the modified Riemann-Liouville fractional derivative, we derive explicit bounds for unknown functions concerned in these inequalities. As for applications, we apply these inequalities to research qualitative properties such as the boundedness, uniqueness, and continuous dependence on initial data for solutions to some certain fractional differential and integral equations.

We organize the rest of this paper as follows. In Section

Assume that

Let

By the properties

Substituting

The desired result can be obtained subsequently.

Suppose

where

Fix

Then we have

Since

By Lemma

Letting

Combining (

In Lemma

We note that if we take

Suppose that

then we have

Denote

Since

Furthermore,

where

Suppose

then we have the following explicit estimate for

Fix

Since

which implies

That is,

Substituting

which implies

and furthermore,

Letting

Since

Suppose

then we have the following explicit estimate for

Let

Then we have

Since

So

Since

Noticing that the structure of (

Combining (

In this section, we apply the inequalities established above to research of boundedness, uniqueness, continuous dependence on the initial value for solutions to certain fractional differential and integral equations. Let us first consider the following IVP of fractional differential equation:

Suppose that

Similar to [

So

Then a suitable application of Lemma

At the end of the proof of Theorem

In Theorem

If

Suppose that the IVP (

Furthermore,

which implies

Treating

Now we research the continuous dependence on the initial value for the IVP (

Suppose that

where

For the IVP (

So by (

Furthermore,

A suitable application of Lemma

Theorem

Consider the following fractional integral equation:

Suppose that

From (

Then a suitable application of Theorem

For the fractional integral equation (

Let

A combination of (

Furthermore, we have

Then treating

So the continuous dependence on the initial value

From the two examples presented above, one can see that the main results established in Section

This work was partially supported by the Natural Science Foundation of Shandong Province (in China) (Grant no. ZR2013AQ009) and the Doctoral Initializing Foundation of Shandong University of Technology (in China) (Grant no. 4041-413030).