#### 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.