# On Some New Impulsive Integral Inequalities

- Jianli Li
^{1}Email author

**2008**:312395

**DOI: **10.1155/2008/312395

© Jianli Li. 2008

**Received: **4 June 2008

**Accepted: **21 July 2008

**Published: **29 July 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 [1–4]. A great deal of attention has been given to differential and integral inequalities; see [1, 2, 5–8] 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:

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.*

## 2. Main Results

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

Theorem 2.1.

*Let*, ,

*and*

*be constants. If*

for .

Proof.

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 .

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*

Proof.

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

Theorem 2.4.

*Let*,

*and*

*be constants. If*

for .

Proof.

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*

Proof.

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 , where is defined by (2.28).

Proof.

This proof is complete.

## 3. Application

Example 3.1.

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

## Declarations

### 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.

## Authors’ Affiliations

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## Copyright

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