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# On Some New Impulsive Integral Inequalities

*Journal of Inequalities and Applications*
**volumeÂ 2008**, ArticleÂ number:Â 312395 (2008)

## Abstract

We establish some new impulsive integral inequalities related to certain integral inequalities arising in the theory of differential equalities. The inequalities obtained here can be used as handy tools in the theory of some classes of impulsive differential and integral equations.

## 1. Introduction

Differential and integral inequalities play a fundamental role in global existence, uniqueness, stability, and other properties of the solutions of various nonlinear differential equations; see [1â€“4]. A great deal of attention has been given to differential and integral inequalities; see [1, 2, 5â€“8] and the references given therein. Motivated by the results in [1, 5, 7], the main purpose of this paper is to establish some new impulsive integral inequalities similar to Bihari's inequalities.

Let , , and , then we introduce the following spaces of function:

is continuous for , , and exist, and â€‰

is continuously differentiable for , , and exist, and

To prove our main results, we need the following result (see [1, Theorem 1.4.1]).

Lemma 1.1.

*Assume that*

*the sequence**satisfies*, *with*;

*and**is left-continuous at*, ;

*for*,

*where* , *and* *are constants.*

Then,

## 2. Main Results

In this section, we will state and prove our results.

Theorem 2.1.

*Let* , , *and* *be constants. If*

for , then

for .

Proof.

Define a function by

where is an arbitrary small constant. For , differentiating (2.3) and then using the fact that , we have

and so

For , we have ; thus . Let ; it follows that

From Lemma 1.1, we obtain

Now by using the fact that in (2.7) and then letting , we get the desired inequality in (2.2). This proof is complete.

Theorem 2.2.

*Let*
*and*
*be constants, and let*
*be a nonnegative constant. If*

for , then

for .

Proof.

This proof is similar to that of Theorem 2.1; thus we omit the details here.

Theorem 2.3.

*Let* , *and* *be constants. If*

for , then

for , where

Proof.

Let be an arbitrary small constant, and define a function by

Let ; similar to the proof of Theorem 2.1, we have

Set ; then , and so from (2.14) we get that . Thus, for ,

and for ,

and so . By Lemma 1.1, we have

Let , then we obtain

where is defined in (2.12). Substituting (2.18) into (2.14), we have

Applying Lemma 1.1 again, we obtain

Now using and letting , we get the desired inequality in (2.11).

Theorem 2.4.

*Let* , *and* *be constants. If*

for , then

for .

Proof.

Set

where is an arbitrary small constant; then is nondecreasing. Let , then it follows for that

since is nondecreasing. Also, for , we have . Applying Lemma 1.1, we obtain

Now by using the fact that in (2.25) and letting , we get the inequality (2.22).

Remark 2.5.

If , then (2.1), (2.8), (2.10), and (2.21) have no impulses. In this case, it is clear that Theorems 2.2-2.3 improve the corresponding results of [5, Theorem 1].

Theorem 2.6.

*Let* , *for* , *and* *be constants. Let* *be a nondecreasing function with* , *for* , *and* , *for* ; *here* , *for* . *If*

for , then for

where

Proof.

We first assume that and define a function by the right-hand side of (2.26). Then, , and is nondecreasing. For ,

and for . As , from (2.31) we have

and so

Now assume that for , we have

Then, for , it follows from (2.32) that . Using , we arrive at

From the supposition of , we see that

If , then

Otherwise, we have

This implies, by induction hypothesis, that

Thus, (2.35) and (2.39) yield, for ,

and so

Using (2.41) in , we have the required inequality in (2.27).

If is nonnegative, we carry out the above procedure with instead of , where is an arbitrary small constant, and by letting , we obtain (2.27). The proof is complete.

Remark 2.7.

If , then and the inequality in (2.27) is true for .

An interesting and useful special version of Theorem 2.6 is given in what follows.

Corollary 2.8.

*Let* , *and* *be as in Theorem 2.6 . If*

for , then

for , where is defined by (2.28).

Proof.

Let in Theorem 2.6. Then, (2.26) reduces to (2.42) and

Consequently, by Theorem 2.6, we have

This proof is complete.

## 3. Application

Example 3.1.

Consider the integrodifferential equations

where with and are continuous; is continuous at and exist and ; are constants with . Here, we assume that the solution of (3.1) exists on . Multiplying both sides of (3.1) by and then integrating them from 0 to , we obtain

We assume that

where . From (3.2) and (3.3), we obtain

Now applying Theorem 2.3, we have

where

for all . The inequality (3.5) gives the bound on the solution of (3.1).

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## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 10571050 and 60671066). The project is supported by Scientific Research Fund of Hunan Provincial Education Department (07B041) and Program for Young Excellent Talents at Hunan Normal University.

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Li, J. On Some New Impulsive Integral Inequalities.
*J Inequal Appl* **2008**, 312395 (2008). https://doi.org/10.1155/2008/312395

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DOI: https://doi.org/10.1155/2008/312395