- Research Article
- Open access
- Published:
On Some New Impulsive Integral Inequalities
Journal of Inequalities and Applications volume 2008, Article number: 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 [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:
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 sequencesatisfies, with;
andis 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).
References
Lakshmikantham V, BaÄnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Singapore; 1989:xii+273.
BaÄnov DD, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications (East European Series). Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.
Liu X, Wang Q: The method of Lyapunov functionals and exponential stability of impulsive systems with time delay. Nonlinear Analysis: Theory, Methods & Applications 2007,66(7):1465–1484. 10.1016/j.na.2006.02.004
Li J, Shen J: Periodic boundary value problems for delay differential equations with impulses. Journal of Computational and Applied Mathematics 2006,193(2):563–573. 10.1016/j.cam.2005.05.037
Pachpatte BG: On some new inequalities related to certain inequalities in the theory of differential equations. Journal of Mathematical Analysis and Applications 1995,189(1):128–144. 10.1006/jmaa.1995.1008
Pachpatte BG: On some new inequalities related to a certain inequality arising in the theory of differential equations. Journal of Mathematical Analysis and Applications 2000,251(2):736–751. 10.1006/jmaa.2000.7044
Pachpatte BG: Integral inequalities of the Bihari type. Mathematical Inequalities & Applications 2002,5(4):649–657.
Tatar N-E: An impulsive nonlinear singular version of the Gronwall-Bihari inequality. Journal of Inequalities and Applications 2006, 2006:-12.
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, J. On Some New Impulsive Integral Inequalities. J Inequal Appl 2008, 312395 (2008). https://doi.org/10.1155/2008/312395
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/312395