Some new Gronwall-type inequalities arising in the research of fractional differential equations
© Feng and Meng; licensee Springer. 2013
Received: 7 May 2013
Accepted: 26 August 2013
Published: 11 September 2013
In this paper, some new Gronwall-type inequalities, which can be used as a handy tool in the qualitative and quantitative analysis of the solutions to certain fractional differential equations, are presented. The established results are extensions of some existing Gronwall-type inequalities in the literature. Based on the inequalities established, we investigate the boundedness, uniqueness, and continuous dependence on the initial value and parameter for the solution to a certain fractional differential equation.
In the research of the theory of differential equations, if their solutions are unknown, then it is important to seek for their qualitative and quantitative properties including boundedness, uniqueness, continuous dependence on initial data and so on. It is known that Gronwall’s inequality is very useful in the research of this domain. This inequality reads as follows:
The inequality above has proved to be very effective in the research of boundedness, uniqueness, and continuous dependence on initial data for the solutions to certain differential equations, as it can provide explicit bounds for the unknown function . In the last few decades, motivated by the analysis of solutions to differential equations with more and more complicated forms, various generalizations of this inequality have been presented (see [1–23] for example). But we notice that most of these developed Gronwall-type inequalities are aimed for the research of differential equations of integer order, while less results are concerned with research of fractional differential equations. In order to obtain the desired analysis of the qualitative and quantitative properties of solutions to certain fractional differential equations, it is necessary to further present some new such inequalities suitable for fractional calculus analysis.
In this paper, we establish some new generalized Gronwall-type inequalities suitable for the qualitative and quantitative analysis of the solutions to fractional differential equations. In Section 2, we present the main results, in which new explicit bounds for unknown functions concerned are established. Then, in Section 3, we investigate a certain fractional differential equation, in which the boundedness, uniqueness, and continuous dependence on initial data for the solution to the fractional differential equation are investigated by use of the generalized Gronwall-type inequalities established.
2 Main results
Lemma 1 
where , and is a constant.
Applying Theorem 1 in  to (5), we can get the desired inequality (2). □
which is the desired result. □
Combining (9), (12) and (14), we get the desired result. □
Remark In Theorem 2, if we let , , then Theorem 2 becomes Theorem 1 in .
where , , denotes the Riemann-Liouville fractional derivative defined by .
where is a constant, and T is defined as in Theorem 1.
which is the desired result. □
Theorem 7 If , where L is defined as in Theorem 2, and , then IVP (15)-(16) has a unique solution.
Treating as one independent function, applying Theorem 2 to (22), we obtain , which implies . So, the proof is complete. □
Now we study the continuous dependence on the initial value and parameter for the solution of IVP (15)-(16).
Applying Theorem 2 to (27), after some basic computation, we can get the desired result. □
In this paper, we have established some new generalized Gronwall-type inequalities, which are generalizations of some existing results in the literature. Based on these inequalities, we investigated the boundedness, uniqueness, and continuous dependence on the initial value and parameter for the solution to a certain fractional differential equation. Finally, we note that the presented results in Theorems 2 and 4 can be generalized to Gronwall-type inequalities with more general forms involving arbitrary nonlinear functional terms , and also can be generalized to the 2D case.
The authors would like to thank the referees very much for their valuable suggestions on improving this paper.
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