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

Qiao-Luan Li,¹ Xu-Yang Fu,¹ Zhi-Juan Gao,¹ and Wing-Sum Cheung²

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., [1–4]), including many retarded ones (see, e.g., [5–9]).

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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