Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 312395, 8 pages
doi:10.1155/2008/312395
Research Article
On Some New Impulsive Integral Inequalities
Jianli Li
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Correspondence should be addressed to Jianli Li, ljianli@sina.com
Received 4 June 2008; Accepted 21 July 2008
Recommended by Wing-Sum Cheung
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.
Copyright q 2008 Jianli Li. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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 0 ≤ t0 < t1 < t2 < · · · , limk→∞ tk ∞, R 0, ∞, and I ⊂ R, then we introduce
the following spaces of function:
tk , u0 , utk , and ut−k exist, and
P CR , I {u : R → I, u is continuous for t /
−
utk utk , k 1, 2, . . .},
tk , u 0 , u tk ,
P C1 R , I {u ∈ P CR , I : u is continuously differentiable for t /
−
−
and u tk exist, and u tk u tk , k 1, 2, . . .}.
To prove our main results, we need the following result see 1, Theorem 1.4.1.
Lemma 1.1. Assume that
A0 the sequence {tk } satisfies 0 ≤ t0 < t1 < t2 < · · · , with limk→∞ tk ∞;
A1 m ∈ P C1 R , R and mt is left-continuous at tk , k 1, 2, . . .;
A2 for k 1, 2, . . . , t ≥ t0 ,
m t ≤ ptmt qt, t / tk ,
m tk ≤ dk mtk bk ,
where q, p ∈ P CR , R, dk ≥ 0, and bk are constants.
1.1
2
Journal of Inequalities and Applications
Then,
mt ≤ mt0 t
dk exp
t0
t0 <tk <t
t0 <tk <t
t t
psds dk exp
pσdσ qsds
dj
t0 s<tk <t
t
psds bk ,
exp
s
1.2
t ≥ t0 .
tk
tk <tj <t
2. Main results
In this section, we will state and prove our results.
Theorem 2.1. Let u, f ∈ P CR , R , bk ≥ 1, and c ≥ 0 be constants. If
t
2
2
u t ≤ c 2 fsusds bk2 − 1 u2 tk ,
0
for t ∈ R , then
ut ≤ c
2.1
0<tk <t
bk
0<tk <t
t 0
bk fsds,
2.2
0<tk <t
for t ∈ R .
Proof. Define a function zt by
2
zt c ε 2
t
0
fsusds bk2 − 1 u2 tk ,
2.3
0<tk <t
where ε > 0 is an
tk , differentiating 2.3 and then using the
arbitrary small constant. For t /
fact that ut ≤ zt, we have
z t 2ftut ≤ 2ft zt,
2.4
and so
d
zt
z t
≤ ft.
dt
2 zt
2.5
For t tk , we have ztk − ztk bk2 − 1u2 tk ≤ bk2 − 1ztk ; thus ztk ≤ bk2 ztk . Let
zt xt; it follows that
tk , t ≥ 0,
x t ≤ ft, t /
x tk ≤ bk xtk , k 1, 2 . . . .
2.6
From Lemma 1.1, we obtain
t t bk bk fsds ≤ c ε
bk bk fsds.
xt ≤ x0
0<tk <t
0
s<tk <t
0<tk <t
0
s<tk <t
2.7
Now by using the fact that ut ≤ zt xt in 2.7 and then letting ε → 0, we get the
desired inequality in 2.2. This proof is complete.
Jianli Li
3
Theorem 2.2. Let u, f ∈ P CR , R and bk ≥ 1 be constants, and let c be a nonnegative constant.
If
u t ≤ c 2
2
2
t
0
fsu2 s hsus ds bk2 − 1 u2 tk ,
2.8
0<tk <t
for t ∈ R , then
ut ≤ c
t t
t
bk exp
fsds bk exp
fτdτ hsds,
0
0<tk <t
0
s<tk <t
2.9
s
for t ∈ R .
Proof. This proof is similar to that of Theorem 2.1; thus we omit the details here.
Theorem 2.3. Let u, f, g, h ∈ P CR , R , c ≥ 0, and bk ≥ 1 be constants. If
u t ≤ c 2
2
2
t 0
s
fsus us gτuτdτ hsus ds bk2 − 1 u2 tk ,
0
0<tk <t
2.10
for t ∈ R , then
ut ≤ c
bk
0<tk <t
t 0
bk fsas hsds,
2.11
s<tk <t
for t ∈ R , where
at c
t t
t
bk exp
fτ gτdτ bk exp
fτ gτdτ hsds.
0
0<tk <t
0
s<tk <t
s
2.12
Proof. Let ε > 0 be an arbitrary small constant, and define a function zt by
t s
zt c ε 2
fsus us gτuτdτ hsus ds bk2 − 1 u2 tk .
2
0
Let
0
0<tk <t
2.13
zt xt; similar to the proof of Theorem 2.1, we have
t
x t ≤ ft xt gsxsds ht,
t/
tk ,
0
x tk ≤ bk xtk ,
2.14
k 1, 2, . . . .
t
Set vt xt 0 gsxsds; then vt ≥ xt, and so from 2.14 we get that x t ≤ ftvt
ht. Thus, for t /
tk ,
v t x t gtxt ≤ ftvt ht gtxt ≤ ft gtvt ht,
2.15
4
Journal of Inequalities and Applications
and for t tk ,
v tk − vtk x tk − xtk ≤ bk − 1xtk ≤ bk − 1vtk ,
2.16
and so vtk ≤ bk vtk . By Lemma 1.1, we have
vt ≤ cε
t
t
t bk exp
fτgτdτ bk exp
fτgτdτ hsds.
0
0<tk <t
0
s
s<tk <t
2.17
Let ε → 0, then we obtain
vt ≤ at,
2.18
where at is defined in 2.12. Substituting 2.18 into 2.14, we have
x t ≤ ftat ht, t / tk ,
x tk ≤ bk xtk , k 1, 2, . . . .
2.19
Applying Lemma 1.1 again, we obtain
xt ≤ c ε
bk
t 0
0<tk <t
bk fsas hsds.
2.20
s<tk <t
Now using ut ≤ xt and letting ε → 0, we get the desired inequality in 2.11.
Theorem 2.4. Let u, f, g, h ∈ P CR , R , c ≥ 0, and bk ≥ 1 be constants. If
t s
gτuτdτ hsus ds bk2 − 1u2 tk ,
u2 t ≤ c2 2
fsus
0
0
2.21
0<tk <t
for t ∈ R , then
ut ≤ c
t
bk exp
0<tk <t
t 0
s
fs
0
0
t
τ
bk exp
fτ
s
s<tk <t
gτdτ ds
2.22
gωdω dτ hsds,
0
for t ∈ R .
Proof. Set
zt c ε2 2
t s
0
gτuτdτ
fsus
0
hsus ds bk2 − 1 u2 tk , 2.23
0<tk <t
where ε is an arbitrary small constant; then zt is nondecreasing. Let xt follows for t /
tk that
t
t
x t ≤ ft gsxsds ht ≤ ft gsds xt ht
0
0
zt, then it
2.24
Jianli Li
5
since xt is nondecreasing. Also, for t tk , we have xtk ≤ bk xtk . Applying Lemma 1.1,
we obtain
xt ≤ c εc
s
t
bk exp
fs
gτdτ ds
0
0<tk <t
t 0
0
τ
t
bk exp
fτ
s
s<tk <t
gωdω dτ hsds.
2.25
0
Now by using the fact that ut ≤ xt in 2.25 and letting ε → 0, we get the inequality
2.22.
Remark 2.5. If bk ≡ 1, 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 u, f ∈ P CR , R , ht, s ∈ CR2 , R , for 0 ≤ s ≤ t < ∞, c ≥ 0, bk ≥ 1, and
p > 1 be constants. Let g ∈ P CR , R be a nondecreasing function with gu > 0, for u > 0, and
gλu ≥ μλgu, for λ > 0, u ∈ R; here μλ > 0, for λ > 0. If
up t ≤ c t fsgus s
0
hs, σguσdσ ds bk − 1up tk ,
0
2.26
0<tk <t
for t ∈ R , then for 0 ≤ t < T,
1/p
t bk
psds
bk ,
ut ≤ G−1 G c
1/p 0 s<tk <t μ b
0<tk <t
k
2.27
where
pt ft t
2.28
ht, σdσ,
0
r
ds
1/p for r ≥ r0 > 0,
g
s
r0
t bk
−1
psds
∈
dom
G
.
bk T sup t ≥ 0 : G c
1/p 0 s<tk <t μ b
0<tk <t
k
Gr 2.29
2.30
Proof. We first assume that c > 0 and define a function zt by the right-hand side of 2.26.
tk ,
Then, zt > 0, z0 c, ut ≤ zt1/p , and zt is nondecreasing. For t /
z t ftgut t
ht, σguσdσ
0
≤ ftg zt1/p ht, σg zσ1/p dσ
0
1/p ≤ g zt
t
ft t
ht, σdσ ,
0
2.31
6
Journal of Inequalities and Applications
and for t tk , ztk ≤ bk ztk . As t ∈ 0, t1 , from 2.31 we have
zt
t
ds
Gzt − Gz0 1/p ≤ psds,
z0 g s
0
and so
2.32
t
zt ≤ G−1 Gc psds .
2.33
0
Now assume that for 0 ≤ t ≤ tn , we have
t bk
−1
psds
.
2.34
bk G c
zt ≤ G
1/p 0 0<tk <t μ b
0<tk <t
k
t
Then, for t ∈ tn , tn1 , it follows from 2.32 that Gzt ≤ Gztn tn psds. Using ztk ≤
bk ztk , we arrive at
t
psds.
2.35
Gzt ≤ Gbn ztn tn
From the supposition of g, we see that
λu
λv
ds
ds
λ
Gλu − Gλv 1/p −
1/p ≤ 1/p Gu − Gv,
μ λ
0 g s
0 g s
If Gztn ≤ Gc
for u ≥ v, λ > 0.
n−1
2.36
k1 bk ,
then
t
t n
bk
Gzt ≤ Gbn ztn psds ≤ G c bk 1/p psds.
tn
0 s<tk <t μ b
k1
k
2.37
Otherwise, we have
n−1
bn
Gbn ztn − G c
bk ≤ 1/p Gztn − G c bk .
μ bn
0<tk <t
k1
2.38
This implies, by induction hypothesis, that
tn tn bk
bk
bn
Gbn ztn − G c
psds
bk ≤ 1/p 1/p 1/p psds.
0 s<tk <tn μ b
0 s<tk <t μ b
μ bn
0<tk <t
k
k
2.39
Thus, 2.35 and 2.39 yield, for 0 < t ≤ tn1 ,
t bk
Gzt ≤ G c
bk 1/p psds,
0 s<tk <t μ b
0<tk <t
k
and so
t bk
zt ≤ G−1 G c
psds
.
bk 1/p 0 s<tk <t μ b
0<tk <t
k
2.40
2.41
Using 2.41 in ut ≤ zt1/p , we have the required inequality in 2.27.
If c is nonnegative, we carry out the above procedure with c ε instead of c, where
ε > 0 is an arbitrary small constant, and by letting ε → 0, we obtain 2.27. The proof is
complete.
Jianli Li
Remark 2.7. If
t ∈ R .
7
∞
r0
ds/gs1/p ∞, then G∞ ∞ 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 u, f, h, c, p, and bk be as in Theorem 2.6. If
t s
bk − 1up tk ,
fsus hs, σuσdσ ds up t ≤ c 0
0
2.42
0<tk <t
for t ∈ R , then
p/p−1
p−1/p
p − 1 t p−1/p
bk
bk
psds
,
ut ≤
c
p
0 s<tk <t
0<tk <t
2.43
for t ∈ R , where pt is defined by 2.28.
Proof. Let gu u in Theorem 2.6. Then, 2.26 reduces to 2.42 and
p
p−1/p
r p−1/p − r0
,
p−1
p/p−1
p−1
p−1/p
G−1 r r r0
.
p
Gr 2.44
Consequently, by Theorem 2.6, we have
ut ≤
p/p−1
p−1/p
p − 1 t p−1/p
bk
bk
psds
.
c
p
0 s<tk <t
0<tk <t
2.45
This proof is complete.
3. Application
Example 3.1. Consider the integrodifferential equations
t
x t − F t, xt, Kt, s, xsds ht,
0
x tk bk xtk ,
3.1
k 1, 2, . . . ,
x0 x0 ,
where 0 t0 < t1 < t2 < · · · with limk→∞ tk ∞; h : R → R and K : R2 × R → R are
tk ; limt→tk Ft, ·, · and limt→t−k Ft, ·, · exist
continuous; F : R × R2 → R is continuous at t /
and limt→t−k Ft, ·, · Ft, ·, ·; bk are constants with |bk | ≥ 1 k 1, 2, . . .. Here, we assume
that the solution xt of 3.1 exists on R . Multiplying both sides of 3.1 by xt and then
integrating them from 0 to t, we obtain
t s
xsF s, xs, Ks, τ, xτdτ hsxs ds bk2 − 1 x2 tk .
x2 t x02 2
0
0
0<tk <t
3.2
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Journal of Inequalities and Applications
We assume that
|Kt, s, xs| ≤ ftgs|xs|,
|Ft, xt, v| ≤ ft|xt| |v|,
3.3
where f, g ∈ CR , R . From 3.2 and 3.3, we obtain
2
|xt| ≤ |x0 | 2
t
2
s
fs|xs| |xs| gτ|xτ|dτ |hs||xs| ds
|bk |2 −1 |xtk |2 .
0
0
0<tk <t
3.4
Now applying Theorem 2.3, we have
|xt| ≤ |x0 |
|bk | 0<tk <t
t 0
|bk | fsas hsds,
3.5
s<tk <t
where
at |x0 |
t
t t
|bk | exp
fτgτdτ |bk | exp
fτ gτdτ hsds,
0<tk <t
0
0
s<tk <t
s
3.6
for all t ∈ R . The inequality 3.5 gives the bound on the solution xt of 3.1.
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.
References
1 V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.
2 D. D. Baı̆nov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its
Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
3 X. Liu and Q. Wang, “The method of Lyapunov functionals and exponential stability of impulsive
systems with time delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 7, pp. 1465–
1484, 2007.
4 J. Li and J. Shen, “Periodic boundary value problems for delay differential equations with impulses,”
Journal of Computational and Applied Mathematics, vol. 193, no. 2, pp. 563–573, 2006.
5 B. G. Pachpatte, “On some new inequalities related to certain inequalities in the theory of differential
equations,” Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 128–144, 1995.
6 B. G. Pachpatte, “On some new inequalities related to a certain inequality arising in the theory of
differential equations,” Journal of Mathematical Analysis and Applications, vol. 251, no. 2, pp. 736–751,
2000.
7 B. G. Pachpatte, “Integral inequalities of the Bihari type,” Mathematical Inequalities & Applications, vol.
5, no. 4, pp. 649–657, 2002.
8 N.-E. Tatar, “An impulsive nonlinear singular version of the Gronwall-Bihari inequality,” Journal of
Inequalities and Applications, vol. 2006, Article ID 84561, 12 pages, 2006.