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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 562691, 9 pages

http://dx.doi.org/10.1155/2014/562691
Research Article

Some Nonlinear Gronwall-Bellman-Gamidov Integral Inequalities and Their Weakly Singular Analogues with Applications

1School of Science, Southwest University of Science and Technology, Mianyang 621010, China

2School of Business, Sichuan University, Chengdu 610064, China

3School of Preparatory Education, Southwest University for Nationalities, Chengdu 610041, China

Received 17 March 2014; Accepted 23 April 2014; Published 13 May 2014

Academic Editor: Robert A. Van Gorder

Copyright © 2014 Kelong Cheng et al. 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 Gronwall-Bellman-Gamidov type integral inequalities with power nonlinearity and their weakly singular analogues are established, which can give the explicit bound on solution of a class of nonlinear fractional integral equations. An example is presented to show the application for the qualitative study of solutions of a fractional integral equation with the Riemann-Liouville fractional operator.

1. Introduction

Integral inequalities, which provide explicit bounds on unknown functions, play a fundamental role in the development of the theory of linear and nonlinear differential equation and integral equation. One of the best known and widely used inequalities is the so-called Gronwall-Bellman integral inequality. In view of the important applications of the Gronwall-Bellman inequality, in the past few years, Pachpatte [13] established a number of new generalizations of such inequality which can be used as powerful tools in the study of certain new classes of differential and integral equations. Meanwhile, many authors have researched various generalizations of the Gronwall-Bellman inequality; for example, we refer the reader to [412].

In [6], Baǐnov and Simeonov discussed the following useful integral inequality: which came from the study of the boundary value problem for higher order differential equations by Gamidov [13] and was extended by Pachpatte [2] as follows:

Remark 1. It should be noted that the derived result in [2] is not right. In the proof, it involves the definition of , which was treated as a constant by mistake. For example, consider the following integral equation: in which , , , and . We can obtain the solution of the equation above; that is, . According to the formula of its upper bound reported in [2], one gets that for . Clearly, the result does not hold for . Hence, a revised one will be provided in later section.

On the other hand, Zheng [14] also established a weakly singular version of the Gronwall-Bellman-Gamidov inequality as follows: and discussed its application in a fractional integral equation with the modified Riemann-Liouville derivative. As for weakly singular inequalities and their applications, more results can be found (e.g., see [1523] and the references therein).

In this paper, motivated by the work in [2, 6, 14], we consider a Gronwall-Bellman-Gamidov integral inequality with power nonlinearity, and its weakly singular analogue where , , and ( ) is the ordered parameter group. The presented inequalities can be used as a handy tool in the qualitative as well as quantitative analysis of solutions of certain fractional differential equation and integral equation. Furthermore, an application of our result to certain fractional integral equation with the Riemann-Liouville (R-L) fractional operator is also involved.

2. Nonlinear Gronwall-Bellman-Gamidov Inequalities

Throughout this paper, denotes the set of real numbers, , , and ( is a constant). denotes the collection of continuous functions from to .

We firstly give some lemmas, which will be used in the proof of the main results.

Lemma 2 (see [10]). Let , , and . Then for any .

Lemma 3. Suppose , , , and . If then for , provided that .

Proof. Let . Obviously, is a constant. It follows from (8) that which yields

Integrating (11) with respect to from 0 to , we have

It is easy to observe that

provided that . Substituting the inequality above into (10), we get (9).

Lemma 4. Suppose , , , , , and . If , , and are nondecreasing and satisfies then for , provided that

Proof. Fix any , ; then for , we have since , , and are nondecreasing. Define by the right side of (17); then , for . From (19), we have

Letting in (20) and integrating it with respect to from 0 to , we get

Since is arbitrary, from (21) with replaced by and , we have where

According to (22), it follows from (23) that

Applying Lemma 3, we have

Substituting the inequality above into (20), we can get (13). The proof is complete.

Remark 5. Even if , , and are not nondecreasing, the result also holds, since we can replace it by , , and .

Remark 6. Pachpatte [24] also discussed inequality (14) and derived a slightly complicated bound, but the formula of bound of in our lemma is quite simple and can be extended easily.

Theorem 7. Let , , , , and be defined as in Lemma 4. Suppose that satisfies (5). If then

for , where , , , , and are constants, and for any ( ).

Proof. Letting from (5), we have or

Applying Lemma 2, for any ( ), we get Substituting (32) into (29), we get which is similar to (14), where , , , , and are defined as in (28). Clearly, , , , , and , , are nondecreasing since , , are nondecreasing, respectively.

Applying Lemma 4 to (33), we have From (31) and (34), we get (27).

When , in Theorem 7, a Gronwall-Bellman-Pachpatte-Ou-Iang type inequality is obtained as follows.

Corollary 8. Let , , , , and be defined as in Lemma 4. Suppose that satisfies If then for , where for any ( ).

When , in Theorem 7, we can also get an interesting result as follows.

Corollary 9. Let , , , , and be defined as in Lemma 4. Suppose that satisfies If then for , where and for any .

3. Nonlinear Weakly Singular Integral Inequalities

Lemma 10 (discrete Jensen inequality). Let be nonnegative real numbers and a real number. Then

Lemma 11 (see [16]). Let , , , and be positive constants. Then where ( , ) is the well-known -function and .

Assume that for the parameter group , , and such that , ( , ).

Theorem 12. Under assumption , let , , , , and be defined as in Lemma 4. Suppose that satisfies (6). If then for , where , , , , , , and are constants, , and for any ( ).

Remark 13. When , inequality (6) can be reduced to the case discussed by Ma and Pečarić [12]. But their result is based on the assumption that the ordered parameter group ( ) obeys distribution I or II (for details, see [16]), which leads up to slightly complicated formula of bound on solutions.

Proof. From assumption , using the Hölder inequality with indices , to (6), we get

Applying Lemma 10 to (48), we have where , ( ) are given in (47).

Letting , we have which is similar to inequality (5), where , , and are also given in (47).

An application of Theorem 7 to the inequality above gives that holds for , where , , , , and are also given in (47). Since , we can get (46).

Similarly, if we take , in (5), the following result is obtained.

Corollary 14. Under assumption , let , , , , and be defined as in Lemma 4. Suppose that satisfies If then for , where , and are constants, , , , are defined as in Theorem 12, and for any .

Remark 15. If we take , , similar to Corollary 8, we can get a general Ou-Iang type singular inequality of (6). Here we leave the details to the reader.

When we take , in (6), we also get the following result.

Corollary 16. Let , , , , and be defined as in Lemma 4. Suppose that satisfies If then for , where , , , , , , , , , , , are defined as in Theorem 12, ( ), the choice of satisfies that and , and , are replaced by

Remark 17. If we take , , , , and , inequality (6) becomes inequality (4). So, Zheng’s result [14] is the special case of our result.

Furthermore, if we take , , and , the weakly singular case of the Gronwall-Bellman-Gamidov-Ou-Iang type inequality also can be obtained.

Corollary 18. Let , , , , and be defined as in Lemma 4. Suppose that satisfies If then for , where , , and are defined as in Theorem 12, , , , , are defined as in Corollary 16, and for any ( ).

4. Applications

In this section, we give some applications of our result in the study of the boundedness of solutions of a fractional integral equation with the Riemann-Liouville (R-L) fractional operator.

Definition 19 (see [25]). The R-L fractional integral of order is defined by the following expression:

Consider the following fractional integral equation: where and .

Theorem 20. If is nondecreasing, and , where . Under the condition , the following estimate holds, where , , , , and and , , and are defined as in Corollary 18.

Proof. According to Definition 19, from (65), we have for . Hence Letting , , and , and applying Corollary 18, we have From (62), we get the desired estimate (66) which implies that in (65) is bounded.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their helpful comments and valuable suggestions. This work is supported by the Doctoral Program Research Funds of Southwest University of Science and Technology (no. 11zx7129) and the Fundamental Research Funds for the Central Universities (no. skqy201324).

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