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Open Access

On Some New Impulsive Integral Inequalities

Journal of Inequalities and Applications20082008:312395

Received: 4 June 2008

Accepted: 21 July 2008

Published: 29 July 2008


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.


Differential EqualityIntegral EquationInduction HypothesisGlobal ExistenceNonlinear Differential Equation

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 [14]. A great deal of attention has been given to differential and integral inequalities; see [1, 2, 58] 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
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.


2. Main Results

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

Theorem 2.1.

Let , , and be constants. If
for , then

for .


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 .


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


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 .


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


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).


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

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



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

Department of Mathematics, Hunan Normal University, Changsha, China


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© Jianli Li. 2008

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