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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 980396, 7 pages
http://dx.doi.org/10.1155/2014/980396
Research Article

On Retarded Integral Inequalities for Dynamic Systems on Time Scales

1College of Mathematics & Information Science, Hebei Normal University, Shijiazhuang 050024, China
2Department of Mathematics, The University of Hong Kong, Hong Kong

Received 13 September 2013; Accepted 16 January 2014; Published 20 February 2014

Academic Editor: Jaeyoung Chung

Copyright © 2014 Qiao-Luan Li 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

The object of this paper is to establish some nonlinear retarded inequalities on time scales which can be used as handy tools in the theory of integral equations with time delays.

1. Introduction

Integral inequalities play an important role in the qualitative analysis of differential and integral equations. The well-known Gronwall inequality provides explicit bounds for solutions of many differential and integral equations. On the basis of various initiatives, this inequality has been extended and applied to various contexts (see, e.g., [14]), including many retarded ones (see, e.g., [59]).

Recently, Ye and Gao [7] obtained the following.

Theorem A. Let , , , , and with where . Then, the following assertions hold.
Suppose that . Then, where , , , , and If, in addition, and are nondecreasing -functions, then
Suppose that . Then, where , , , , , and If, in addition, and are nondecreasing -functions, then

In this paper, we will further investigate functions satisfying the following more general inequalities: where is any time scale, , , , , and are real-valued nonnegative rd-continuous functions defined on , and are positive constants, , , , and .

First, we make a preliminary definition.

Definition 1. We say that a function is regressive provided that holds, where is graininess function; that is, . The set of all regressive and rd-continuous functions will be denoted by .

2. Main Results

For convenience, we first cite the following lemma.

Lemma 2 (see [10]). Let , , ; then for any .

Lemma 3. Let , , , , for all , is rd-continuous on , and and are real constants. If is rd-continuous and then for and for .
Furthermore, if and are nondecreasing with , then where .

Proof. Let . Then, , and is positive, nondecreasing for . By Lemma 2, we get for . Multiplying (16) by , we get Integrating both sides from to , we obtain For , , so Using (18) and (19), we get for .
Noting that , inequalities (13) and (14) follow.
Finally, if and are nondecreasing, then for , by (14), we have If , by (13), The proof is complete.

Theorem 4. Assume that satisfies condition (8), , , , ; then where , , and + .
If, in addition, and are nondecreasing, and , then where

Proof. The second inequality in (23) is obvious. Next, we will prove the first inequality in (23). For , using Hölder’s inequality with indices and , we obtain from (8) By Jensen’s inequality , we get For the first integral in (26), we have the estimate Hence, and so Let ; then we have For , we have ; that is, . By Lemma 3, we get Hence, the first inequality in (23) follows.
Finally, if and are nondecreasing, and , by Lemma 3, we have The proof is complete.

Lemma 5. Let , , , , , and and let be rd-continuous on , where and are real constants. If is rd-continuous and then for and for .
Furthermore, if and are nondecreasing with , then

Proof. Let . Then, , , is positive and nondecreasing for . Further, we have For , using Lemma 2, we have Integrating both sides from to , we obtain For , Hence, we get Integrating both sides from to , we obtain Using , we get inequalities (34) and (35).
Finally, if and are nondecreasing, then, by (35), for . Furthermore, by (34), for . The proof is complete.

Theorem 6. Assume that satisfies condition (9), , , , , , .
If, in addition, and are nondecreasing, and , then where .

Proof. For , using Hölder’s inequality with indices and , we obtain from (9) that By Jensen’s inequality , we get So, Let , ; we have for . For , we have ; that is, . By Lemma 5, we get The proof is complete.

The following is a simple consequence of Theorem 4.

Corollary 7. Suppose that , then where , , .
If  , then the conclusion reduces to that of Theorem  A for .

Conflict of Interests

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

Acknowledgments

The first author’s research was supported by NNSF of China (11071054), Natural Science Foundation of Hebei Province (A2011205012). The corresponding author’s research was partially supported by an HKU URG grant.

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