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Some new Gronwall-type inequalities arising in the research of fractional differential equations
Journal of Inequalities and Applications volume 2013, Article number: 429 (2013)
Abstract
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.
MSC:26D10.
1 Introduction
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:
Gronwall’s inequality Suppose u, a, b are continuous functions with . Then
implies
Furthermore, if a is nondecreasing, then we have
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 [24]
Assume that , with . Then, for any , we have
Theorem 2 Suppose that , are constants, with for , where T is the Lipschitz constant, u, a, h are nonnegative functions locally integrable on with h nondecreasing and bounded by M, where M is a positive constant. If the following inequality is satisfied:
then we have the following explicit estimate for u:
where , and is a constant.
Proof Denote the right-hand side of (1) by . Then we have
Furthermore,
By use of Lemma 1, we obtain that
Applying Theorem 1 in [25] to (5), we can get the desired inequality (2). □
Corollary 3 Under the conditions of Theorem 2, furthermore, assume that a is nondecreasing. Then we have the following estimate:
Proof Since a is nondecreasing, then is also nondecreasing, and from (2) we obtain
which is the desired result. □
Theorem 4 Suppose that α, u, a, h are defined as in Theorem 2, b is a nonnegative function locally integrable on , and p, q are constants with . If a is nondecreasing and the following inequality is satisfied:
then we have
where
Proof Denote the right-hand side of (1) by . Then we have
Furthermore, an application of Lemma 1 yields that
Let
Then we have
Since a is nondecreasing, then z is also nondecreasing, and by use of Gronwall’s inequality, we get that
Moreover,
Since the structure of (13) is the same as that of (5), following in a similar manner to the proof in Theorem 2, we get that
Combining (9), (12) and (14), we get the desired result. □
Corollary 5 For Theorem 4, similar to the proof of Corollary 3, we can obtain the following estimate for u:
Remark In Theorem 2, if we let , , then Theorem 2 becomes Theorem 1 in [25].
3 Applications
In this section, we show that the inequalities established above are useful in the research of boundedness, uniqueness, continuous dependence on the initial value and parameter for the solutions to fractional differential equations. Consider the following IVP for a certain fractional differential equation:
with the initial condition
where , , denotes the Riemann-Liouville fractional derivative defined by .
Theorem 6 For IVP (15)-(16), if , where L is defined as in Theorem 2, then we have the following estimate:
where is a constant, and T is defined as in Theorem 1.
Proof The equivalent integral form of IVP (15)-(16) is denoted as follows:
So,
Then a suitable application of Theorem 2 (with ) to (18) yields
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.
Proof Suppose that IVP (15)-(16) has two solutions , . Then we have
Furthermore,
which implies
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).
Theorem 8 Let u be the solution of IVP (15)-(16), and let be the solution of the following IVP:
If , where ε is arbitrarily small, and , where L is defined as in Theorem 2, and , then we have
Proof The equivalent integral form of IVP (23) is denoted as follows:
So, we have
Furthermore,
Applying Theorem 2 to (27), after some basic computation, we can get the desired result. □
4 Conclusions
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.
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The authors would like to thank the referees very much for their valuable suggestions on improving this paper.
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QF carried out the main part of this article. All authors read and approved the final manuscript.
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Feng, Q., Meng, F. Some new Gronwall-type inequalities arising in the research of fractional differential equations. J Inequal Appl 2013, 429 (2013). https://doi.org/10.1186/1029-242X-2013-429
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DOI: https://doi.org/10.1186/1029-242X-2013-429