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By use of the properties of the modified Riemann-Liouville fractional derivative, some new Gronwall-Bellman-type inequalities are researched. First, we derive some new explicit bounds for the unknown functions lying in these inequalities, which are of different forms from some existing bounds in the literature. Then, we apply the results established to research the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional differential equation.

During the past decades, a lot of integral and difference inequalities have been discovered, which play an important role in the research of the theory of differential, integral, and difference equations. In these inequalities, the Gronwall-Bellman inequality and their generalizations have been paid much attention by many authors (e.g., see [

In this paper, we establish some new Gronwall-Bellman-type inequalities. Based on some basic properties of the modified Riemann-Liouville fractional derivative, we derive explicit bounds for unknown functions concerned in these inequalities. The presented inequalities can be used as a handy tool in the qualitative as well as quantitative analysis of solutions to fractional differential equations.

The modified Riemann-Liouville fractional derivative, defined by Jumarie in [

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 next part of this paper is organized as follows. In Section

Suppose that

Let

In Theorem

Suppose that

Let

Suppose that

Denote

Suppose that

The proof of Theorem

In this section, we will show that the inequalities established above are useful in the research of boundedness, uniqueness, and continuous dependence on the initial value for solutions to fractional differential equations. Consider the following fractional differential equation:

Suppose that

Similar to [

In Theorem

If

Suppose that (

Now we research the continuous dependence on the initial value for the solution of (

Under the conditions of Theorem

Let

Similar to Theorem

We have presented some new Gronwall-Bellman-type inequalities, and based on them we derived explicit bounds for the unknown functions concerned, which are different from the existing bounds in the literature. As one can see, the results established are useful in fulfilling qualitative and quantitative analyses such as the boundedness, uniqueness, and continuous dependence on the initial value and parameter for solutions to certain fractional differential equations. Finally, we note that the method used in Theorems

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank the reviewers very much for their valuable suggestions on improving this paper.