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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 217641, 7 pages
http://dx.doi.org/10.1155/2013/217641
Research Article

Gronwall-Bellman Type Inequalities and Their Applications to Fractional Differential Equations

1School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
2Department of Mathematics, Jining University, Qufu, Shandong 273155, China

Received 29 May 2013; Accepted 22 July 2013

Academic Editor: Irena Lasiecka

Copyright © 2013 Jing Shao and Fanwei Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which can be used in the qualitative analysis of the solutions to certain fractional differential equations.

1. Introduction

Gronwall-Bellman type integral inequalities play increasingly important roles in the study of quantitative properties of solutions of differential and integral equations, as well as in the modeling of engineering and science problems. The integrals concerning this type of inequalities have regular or continuous kernels, but some problems of theory and practicality require us to solve integral inequalities with singular kernels; see [14] and the references cited therein. For example, Ye and Gao [5] considered the integral inequalities of Henry-Gronwall type and their applications to fractional differential equations with delay; Ma and Pečarić [4] established some weakly singular integral inequalities of Gronwall-Bellman type and used them in the analysis of various problems in the theory of certain classes of differential equations, integral equations, and evolution equations.

In this paper, we study a certain class of nonlinear inequalities of Gronwall-Bellman type, which generalizes some known results and can be used as handy and effective tools in the study of differential equations and integral equations. Furthermore, applications of our results to fractional differential are also involved.

2. Preliminary Knowledge

In this section, we give some inequalities, which will be used in the proof of the main results.

Lemma 1 (Jensen's inequality). Let , and let be nonnegative real numbers. Then, for ,

Lemma 2. Let , , . If , and where . Then, for , one has where .

Proof. Given , for , Define a function , ; then , , is positive and nondecreasing for , and Let , and ; we obtain which implies that And so By the arbitrary of , we obtain the inequality (3). The proof is complete.

Lemma 3. Let , , and . If and where and is a real constant, then where is defined as that in Lemma 2.

Proof. For , we have By Gronwall inequality, we have the inequality (11). We prove that (10) holds for now. Given that and for , we get Define a function , ; then , , is positive and nondecreasing for , and As that in the proof of Lemma 2, we obtain And then By the arbitrary of , we obtain the inequality (10). The proof is complete.

3. Main Results

Now, we are in a position to deal with the integral inequality with weak singular kernels.

Theorem 4. Let . If and where and are constants, then the following assertions hold.(i) Suppose that . Then where , , and .(ii) Suppose that , , and . Then where , and .

Proof. (i) Using the Cauchy-Schwarz inequality, we obtain Using Lemma 1, we obtain Let ; we get Using Lemma 2 and noticing that is nondecreasing, we get by the relationship of and , the first inequality (18) holds.
(ii) By the hypothesis, we get . Using Hölder inequality, we obtain Using Lemma 1, we obtain Let , we get Using Lemma 2 and noticing that is nondecreasing, we get and by the relation of and , (19) holds. The proof is complete.

Theorem 5. Let , and . If with where and are constants, then the following assertions hold.(i) Suppose that . Then where , , and are defined as those in Theorem 4.(ii) Suppose that , , and . Then, where , , and are defined as those in Theorem 4.

Proof. (i) Using the Cauchy-Schwarz inequality by (28), we obtain Using Lemma 1, we obtain Let , we get Using Lemma 3, we get the first inequality of (29) and the second inequality of (29) is easily obtained.
(ii) By the hypothesis, we get . Using Hölder inequality, we obtain Using Lemma 1, we obtain Let ; we get Using Lemma 2, we get the first inequality of (30) and the second inequality of (30) is easily obtained. The proof is complete.

For the case of , this kind of inequalities has been considered by Pachpatte [6] and the case of retarded integral inequalities also has been obtained by Ye and Gao [5, Theorem 2.5]. So, we list only a theorem using different condition and method from Pachpatte [6, Theorem 1.2.4].

Theorem 6. Let , and . If and where , then the following assertions hold.(i) Suppose that . Then where and are defined as those in Theorem 4.(ii) Suppose that , and . Then where and are defined as those in Theorem 4.

Remark 7. In [6, Theorem 1.2.4], is continuously differentiable, but in Theorem 6, is only continuous in the interval , so the methods of [6, Theorem 1.2.4] are invalid for Theorem 6. In [7, Theorem 1], Ye et al. also considered the similar integral inequalities using an iterative method, but we use different methods differing from the previously mentioned two papers.

4. Applications to FDEs

In this section, we present applications of Theorem 4 and Theorem 5 to study certain properties of solutions of fractional differential equations.

Consider the following fractional differential equations: for , , where represents the Caputo fractional derivative of order , , and . The corresponding Volterra fractional integral equation, see [8, Lemma 6.2], becomes

Theorem 8. Suppose that , where , is real number. If is any solution of the initial value problem (40), then the following estimations hold.(i) Suppose that . Then where .(ii) Suppose that , , and . Then where . Notice that , and are the same as those in Theorem 4, .

Proof. By (41), it is easy to derive that Using Theorem 4, we get the desired conclusion. This proves the theorem.

Considering the following fractional differential equations: for , with the given initial condition , , is a given continuously differentiable function on up to order . In this case, we denote , , and , , , and are defined as those in (40).

In [8, Lemma 6.2], the initial value problem (45) is equivalent to the Volterra fractional integral equation:

The next result deals with the upper bounds of solution of (45).

Theorem 9. Suppose that , where , and is real number. If is any solution of the initial value problem (46), then the following estimations hold.(i)Suppose that . Then (ii)Suppose that , and . Then Notice that , and are the same as those in Theorem 8, .

The proof of this theorem is omitted because it is similar to that of Theorem 8.

Acknowledgments

The authors thank the referee for his/her useful comments on this paper. This research was partially supported by the NSF of China (Grants 11171178 and 11271225), Science and Technology Project of High Schools of Shandong Province (Grant J12LI52), and program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.

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