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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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_IEq4_HTML.gif)
is continuous for ,
, and
exist, and
 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_IEq9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equa_HTML.gif)
the sequencesatisfies
, with
;
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equb_HTML.gif)
and
is left-continuous at
,
;
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equc_HTML.gif)
for,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ1_HTML.gif)
where , and
are constants.
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ2_HTML.gif)
2. Main Results
In this section, we will state and prove our results.
Theorem 2.1.
Let ,
, and
be constants. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ3_HTML.gif)
for , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ4_HTML.gif)
for .
Proof.
Define a function by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ5_HTML.gif)
where is an arbitrary small constant. For
, differentiating (2.3) and then using the fact that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ6_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ7_HTML.gif)
For , we have
; thus
. Let
; it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ8_HTML.gif)
From Lemma 1.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ10_HTML.gif)
for , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ12_HTML.gif)
for , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ13_HTML.gif)
for , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ14_HTML.gif)
Proof.
Let be an arbitrary small constant, and define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ15_HTML.gif)
Let ; similar to the proof of Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ16_HTML.gif)
Set ; then
, and so from (2.14) we get that
. Thus, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ17_HTML.gif)
and for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ18_HTML.gif)
and so . By Lemma 1.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ19_HTML.gif)
Let , then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ20_HTML.gif)
where is defined in (2.12). Substituting (2.18) into (2.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ21_HTML.gif)
Applying Lemma 1.1 again, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ22_HTML.gif)
Now using and letting
, we get the desired inequality in (2.11).
Theorem 2.4.
Let , and
be constants. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ23_HTML.gif)
for , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ24_HTML.gif)
for .
Proof.
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ25_HTML.gif)
where is an arbitrary small constant; then
is nondecreasing. Let
, then it follows for
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ26_HTML.gif)
since is nondecreasing. Also, for
, we have
. Applying Lemma 1.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ28_HTML.gif)
for , then for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ29_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ32_HTML.gif)
Proof.
We first assume that and define a function
by the right-hand side of (2.26). Then,
, and
is nondecreasing. For
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ33_HTML.gif)
and for . As
, from (2.31) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ34_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ35_HTML.gif)
Now assume that for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ36_HTML.gif)
Then, for , it follows from (2.32) that
. Using
, we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ37_HTML.gif)
From the supposition of , we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ38_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ39_HTML.gif)
Otherwise, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ40_HTML.gif)
This implies, by induction hypothesis, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ41_HTML.gif)
Thus, (2.35) and (2.39) yield, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ42_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ44_HTML.gif)
for , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ45_HTML.gif)
for , where
is defined by (2.28).
Proof.
Let in Theorem 2.6. Then, (2.26) reduces to (2.42) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ46_HTML.gif)
Consequently, by Theorem 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ47_HTML.gif)
This proof is complete.
3. Application
Example 3.1.
Consider the integrodifferential equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ49_HTML.gif)
We assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ50_HTML.gif)
where . From (3.2) and (3.3), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ51_HTML.gif)
Now applying Theorem 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ52_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F312395/MediaObjects/13660_2008_Article_1797_Equ53_HTML.gif)
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