Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 430512, 18 pages
doi:10.1155/2010/430512
Research Article
A Note on the Integral Inequalities with
Two Dependent Limits
Allaberen Ashyralyev,
1, 2
Emine Misirli,
3
and Ozlem Mogol
3
1
Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey
2
Department of Mathematics, ITTU, 74200 Ashgabat, Turkmenistan
3
Department of Mathematics, Ege University, 35100 Bornova-Izmir, Turkey
Correspondence should be addressed to Emine Misirli,
Received 4 October 2009; Revised 28 April 2010; Accepted 5 July 2010
Academic Editor: Martin Bohner
Copyright q 2010 Allaberen Ashyralyev et al. 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.
The theorem on the Gronwall’s type integral inequalities with two dependent limits is established.
In application, the boundedness of the solutions of nonlinear differential equations is presented.
1. Introduction
Integral inequalities play a significant role in the study of qualitative properties of solutions
of integral, differential and integro-differential equations see, e.g., 1–4 and the references
given therein. One of the most useful inequalities in the development of the theory of
t
0
f
s
ds
1.2
for all t ≥ 0.
2 Journal of Inequalities and Applications
Note that the generalization of this integral inequality and its discrete analogies are
given in papers 5–8. In paper 9 the following useful inequality with two dependent limits
was established.
Lemma 1.2. Let ut be a real-valued nonnegative continuous function defined on −T, T and let c
and a be nonnegative constants. Then the inequality
u
t
≤ c sgn
t
t
−t
au
s
t
t
−t
m
s
u
s
ds
2.1
for t ∈ R, then
u
t
≤ c sgn
t
t
−t
m
u
p
s
h
s
u
s
ds
2.3
Journal of Inequalities and Applications 3
for t ∈ R, then
u
t
≤
a
t
b
× sgn
t
t
−t
a
s
g
s
1
p
h
s
p − 1
p
h
dr
ds
1/p
2.4
for all t ∈ R.
(iii) Let ct be a real-valued positive continuous and nondecreasing function defined on R and
p>1 be a real constant. If
u
p
t
≤ c
p
t
b
t
sgn
t
t
−t
1 b
t
exp
sgn
t
t
−t
b
r
g
r
h
r
c
1−p
−sgn
s
s
−s
b
r
g
r
h
r
c
1−p
r
p
dr
ds
g
s
u
p
s
h
s
u
s
ds
2.7
for t ∈ R, then
u
t
≤
a
p
h
r
dr
×
sgn
t
t
0
sgn
s
s
−s
∂
∂s
k
s, r
s
s
−s
k
r, r
b
r
g
r
1
p
h
r
dr
4 Journal of Inequalities and Applications
1
p
h
y
dy dr
ds
sgn
t
t
−t
k
s, s
exp
−sgn
s
g
s
1
p
h
s
p − 1
p
h
s
B
k
t, s
ds
k−
t, s
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
exp
t
s
r
−r
∂
∂r
k
2.10
B
k
t, s
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
exp
t
s
r
−r
∂
∂r
, −t ≤ s ≤ 0.
2.11
Proof. i Define a function vt by
v
t
c
2
2sgn
t
t
−t
m
s
u
s
ds.
2.12
Note that vt is a nonnegative function and v0c
2
. Then 2.1 can be rewritten as
2
t
−t
m
s
u
s
ds.
2.14
Differentiating 2.14 and using 2.13,weget
v
t
≤ 2m
t
v
t
2m
−t
.
2.16
Integrating the last inequality from 0 to t,weget
v
t
≤ c
t
0
m
s
ds
t
0
m
−s
ds c sgn
t
Differentiating 2.18 and using 2.13,weget
−v
t
≤ 2m
t
v
t
2m
−t
v
t
.
2.19
Dividing both sides of 2.19 by 2
vt,weget
−
t
m
s
ds
0
t
m
−s
ds c sgn
t
t
−t
m
s
ds.
2.21
6 Journal of Inequalities and Applications
Finally, using 2.17 and 2.21,weobtain
v
g
s
u
p
s
h
s
u
s
ds.
2.23
It is evident that vt is an even and n onnegative function. We have that
u
p
t
≤ a
t
u
t
≤
a
t
b
t
v
t
p
p − 1
p
.
2.25
Let t ≥ 0. Then
v
t
u
p
t
h
t
u
t
g
−t
u
p
−t
h
−t
u
−t
b
−t
g
−t
1
p
h
−t
a
t
g
t
1
p
h
−t
.
2.28
Denoting
B
t
b
t
g
t
1
p
h
t
,A
t
t
− v
t
B
t
B
−t
≤ A
t
A
−t
. 2.30
Journal of Inequalities and Applications 7
From that it follows that
exp
t
≤ exp
t
s
B
r
B
−r
dr
A
s
A
−s
2.31
for any s ≤ t. Integrating the last inequality from 0 to t and using v00, we get
v
ds. 2.32
It is easy to see that
t
s
B
r
B
−r
dr
t
−t
B
r
dr −
s
−s
B
r
exp
−
s
−s
B
r
dr
ds. 2.34
Since 0 ≤ s ≤ t, we have that
v
t
≤ exp
sgn
t
t
−t
B
s
exp
−
−s
s
B
r
dr
ds
exp
sgn
t
t
−t
B
r
A
s
exp
−sgn
s
s
−s
B
r
dr
ds
exp
sgn
t
t
−t
dr
.
2.35
8 Journal of Inequalities and Applications
Applying 2.24,weobtain
u
t
≤
a
t
b
t
exp
sgn
t
t
−t
B
ds
1/p
.
2.36
From 2.36,and2.29 it follows 2.4 for t ≥ 0. Let t ≤ 0; then
v
t
−
t
−t
g
s
u
p
s
h
s
u
u
p
−t
h
−t
u
−t
.
2.37
Using 2.24 and 2.25,weget
−v
t
≤ v
t
B
t
v
s
v
s
B
s
B
−s
≤ exp
s
t
B
r
B
−s
exp
s
t
B
r
B
−r
dr
ds.
2.40
It is easy to see that
s
t
B
r
−t
t
B
r
dr
0
t
A
s
A
−s
exp
−
−s
s
B
r
s
A
−s
exp
−
−s
s
B
r
dr
ds
exp
sgn
t
t
−t
B
−s
exp
s
−s
B
r
dr
ds
exp
sgn
t
t
−t
B
r
dr
−s
s
B
r
dr
ds
exp
sgn
t
t
−t
B
r
dr
−t
t
A
dr
sgn
t
t
−t
A
s
exp
−sgn
s
s
−s
B
r
dr
ds.
p
h
s
c
1−p
s
u
s
c
s
ds.
2.44
Now the application of the inequality proven in ii yields the desired result in 2.6.
iv We define a function vt by
v
t
sgn
t
We have that
u
p
t
≤ a
t
b
t
v
t
,u
t
≤
a
t
b
s
h
s
u
s
ds.
2.47
Differentiating 2.47,weget
v
t
k
t, t
g
t
u
u
−t
t
−t
∂
∂t
k
t, s
g
s
u
p
s
h
s
1
p
h
t
k
t, −t
b
−t
g
−t
1
p
h
−t
t, t
g
t
a
t
h
t
1
p
a
t
p − 1
p
k
t, −t
∂t
k
t, s
g
s
a
s
h
s
1
p
a
s
p − 1
p
ds.
t
−t
∂
∂t
k
t, s
B
s
ds
k
t, t
A
t
k
t, −t
A
−t
A
s
k
s, −s
A
−s
s
−s
∂
∂s
k
s, r
A
r
dr
× exp
B
y
dy
dr
.
2.51
Journal of Inequalities and Applications 11
Since kt, sk−t, s, we have that
v
t
≤
t
0
k
s, s
A
s
k
r, r
B
r
k
−r, −r
B
−r
r
−r
∂
∂r
k
r, y
B
y
dy
s
−s
∂
∂s
k
s, r
A
r
dr
× exp
−
s
−s
k
r, r
B
r
dr
exp
−
s
−s
k
r, r
B
r
dr
t
s
r
−r
∂
∂r
k
r, y
B
y
dr ds
t
s
r
−r
∂
∂r
k
r, y
B
y
dy dr
ds
.
2.53
Since 0 ≤ s ≤ t, we have that
v
t
≤ exp
s
−s
∂
∂s
k
s, r
A
r
dr
× exp
−
s
−s
k
r, r
B
r
dr
exp
−
s
−s
k
r, r
B
r
dr
t
s
r
−r
∂
∂r
k
r, y
B
y
dr
t
−s
r
−r
∂
∂r
k
r, y
B
y
dy dr
ds
.
2.54
12 Journal of Inequalities and Applications
Using 2.9 and 2.11,weget
v
t
≤ exp
s
−s
∂
∂s
k
s, r
A
r
dr
× exp
−sgn
s
s
−s
k
r, r
B
t
0
k
s, s
A
s
exp
−sgn
s
s
−s
k
r, r
B
r
dr
B
r
dr
B
k
t, s
ds
exp
sgn
t
t
−t
k
r, r
B
r
× exp
−sgn
s
s
−s
k
r, r
B
r
dr sgn
t
t
s
sgn
r
r
s
−s
k
r, r
B
r
dr
A
s
B
k
t, s
ds
.
2.55
Let t ≤ 0. Then
v
2.56
Differentiating 2.56,weget
−v
t
k
t, t
g
t
u
p
t
h
t
u
t
t, s
g
s
u
p
s
h
s
u
s
ds.
2.57
Journal of Inequalities and Applications 13
Using 2.46 and Young’s inequality, we obtain that
−v
t
b
−t
g
−t
1
p
h
−t
t
−t
∂
∂t
k
t, s
b
h
t
1
p
a
t
p − 1
p
k
t, −t
g
−t
a
−t
h
s
h
s
1
p
h
s
p − 1
p
ds.
2.58
Using 2.29,weget
−v
t
≤ v
t
ds
k
t, t
A
t
k
t, −t
A
−t
−t
t
∂
∂t
k
t, s
A
−s
s
∂
∂s
k
s, r
A
r
dr
× exp
s
t
k
r, r
B
r
t
≤
0
t
k
s, s
A
s
k
−s, −s
A
−s
−s
s
∂
∂s
−r
−r
r
∂
∂r
k
r, y
B
y
dy
dr
ds.
2.61
14 Journal of Inequalities and Applications
Using 2.41,weget
v
t
≤ exp
dr
× exp
−
−s
s
k
r, r
B
r
dr
s
t
−r
r
∂
∂r
k
r, y
B
r
dr
s
t
−r
r
∂
∂r
k
r, y
B
y
dy dr
ds
0
t
k
−s, −s
r, y
B
y
dy dr ds
.
2.62
Since t ≤ s ≤ 0, we have that
v
t
≤ exp
sgn
t
t
−t
k
r, r
B
dr
× exp
−sgn
s
s
−s
k
r, r
B
r
dr
s
t
−r
r
∂
∂r
k
B
r
dr
s
t
−r
r
∂
∂r
k
r, y
B
y
dy dr
ds
−t
0
k
k
r, y
B
y
dy dr
ds
.
2.63
Using 2.9 and 2.10,weget
v
t
≤ exp
sgn
t
t
−t
k
A
r
dr
× exp
−sgn
s
s
−s
k
r, r
B
r
dr sgn
t
t
s
sgn
exp
−sgn
s
s
−s
k
r, r
B
r
dr
B
k−
t, s
ds
−t
0
t, s
ds
Journal of Inequalities and Applications 15
exp
sgn
t
t
−t
k
r, r
B
r
dr
×
sgn
t
k
r, r
B
r
dr sgn
t
t
s
sgn
r
×
r
−r
∂
∂r
k
r, y
B
B
r
dr
A
s
B
k
t, s
ds
.
2.64
The inequality 2.8 follows from 2.29, 2.55,and2.64. Theorem 2.1 is proved.
3. An Application
In this section, we indicate an application of Theorem 2.1 part ii to obtain the explicit
bound on the solution of the following boundary value problem for one dimensional partial
differential equations:
v
p
tt
t, x
v
t, l
,v
x
t, 0
v
x
t, l
,t∈ R,
v
0,x
ϕ
x
,v
t
0,x
ψ
2p
t, x
dx
1/2
h
t
l
0
v
2
t, x
dx
1/2
3.2
for all t ∈ R. Here gt and ht are real-valued nonnegative continuous functions defined on
R.
This allows us to reduce the nonlocal boundary-value 3.1 to the initial-value problem
v
p
tt
the formula Aux−axu
x
x
x
δux, with the domain DA{ux : u
x ∈
L
2
0,l,u0ul,u
0u
l} see, e.g., 15, 16.
Let us give a corollary of Theorem 2.1.
Theorem 3.1. The solution of problem 3.1 satisfies the estimates
l
0
v
2p
t, x
dx
1/2p
≤
dr
× sgn
t
t
−t
M
g
s
1
p
l
1−1/p
h
s
p − 1
p
r
dr
ds
1/p
3.4
for all t ∈ R. Here M
l
0
ϕ
2p
xdx
1/2
p/δ
l
0
ϕ
2p−1
xψ
2
xdx
1/2
.
Proof. It is known thatthe formula see, e.g., 15, 16
t − s
F
s, v
s
ds
3.5
gives a solution of problem 3.3. Here
c
t
e
itA
1/2
e
−itA
1/2
2
,s
t
A
−1/2
−1/2
H →H
≤
1
√
δ
,
3.7
we get
v
p
t
H
≤
v
p
0
H
1
√
δ
s
vs
H
ds.
3.8
Journal of Inequalities and Applications 17
Since
v
p
0
H
1
√
δ
v
p
0
x
dx
1/2
,
vs
H
≤ l
1−1/p
v
p
s
1/p
H
3.9
we have that
v
p
t
H
≤ M
1
√
1/p
H
ds.
3.10
Denote that utv
p
t
1/p
H
. Then
u
p
t
≤ M
1
√
δ
sgn
t
t
−t
g
δ
exp
sgn
t
t
−t
1
√
δ
g
r
1
p
l
1−1/p
h
r
dr
s
× exp
−sgn
s
s
−s
1
√
δ
g
r
1
p
l
1−1/p
h
r
dx
1/2p
.
3.13
Therefore, the inequality 3.4 follows from the last inequality. Theorem 3.1 is proved.
Acknowledgments
The authors thank professor O. Celebi Turkey,professorR.P.AgarwalUSA,and
anonymous reviewers for their valuable comments.
18 Journal of Inequalities and Applications
References
1 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, vol. 155 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.
2 E. F. Beckenbach and R. Bellman, Inequalities, Springer, New York, NY, USA, 1965.
3 D. S. Mitrinovi
´
c, Analytic Inequalities, Springer, New York, NY, USA, 1970.
4 T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of
differential equations,” Annals of Mathematics II, vol. 20, no. 4, pp. 292–296, 1919.
5 B. G. Pachpatte, “Some new finite difference inequalities,” Computers & Mathematics with Applications,
vol. 28, no. 1–3, pp. 227–241, 1994.
6 E. Kurpınar, “On inequalities in the theory of differential equations and their discrete analogues,”
Pan-American Mathematical Journal, vol. 9, no. 4, pp. 55–67, 1999.
7 B. G. Pachpatte, “On the discrete generalizations of Gronwall’s inequality,” Journal of Indian
Mathematical Society, vol. 37, pp. 147–156, 1973.
8 B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197 of Mathematics in Science and
Engineering, Academic Press, New York, NY, USA, 1998.
9 Y. Dj. Mamedov and S. Ashirov, “A Volterra type integral equation,” Ukrainian Mathematical Journal,
vol. 40, no. 4, pp. 510–515, 1988.
Copyright: Tài liệu đại học ©