Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 462163, 9 pages
doi:10.1155/2010/462163
Research Article
A New Nonlinear Retarded Integral Inequality and
Its Application
Wu-Sheng Wang,
1
Ri-Cai Luo,
1
and Zizun Li
2
1
Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China
2
School of Mathematics and Computing Science, Guilin University of Electronic Technology,
Guilin 541004, China
Correspondence should be addressed to Wu-Sheng Wang, [email protected]
Received 28 April 2010; Revised 9 July 2010; Accepted 15 August 2010
Academic Editor: L
´
aszl
´
o Losonczi
Copyright q 2010 Wu-Sheng Wang 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 main objective of this paper is to establish a new retarded nonlinear integral inequality with
two variables, which provide explicit bound on unknown function. This inequality given here can
be used as tool in the study of integral equations.
t, s
w
i
u
s
ds, t
0
≤ t <t
1
.
1.1
Agarwal et al. 2 obtained the explicit bound to the unknown function of the following
retarded integral inequality
ϕ
u
t
≤ c
n
i1
α
i
ϕ
2
log
u
s
ds.
1.2
2 Journal of Inequalities and Applications
Cheung 3 investigated the inequality in two variables
u
p
x, y
≤ a
p
p − q
b
1
x
b
1
x
b
2
x
0
c
2
y
c
2
y
0
g
2
s, t
u
q
s, t
ψ
u
w
u
s, t
dt ds
γx
γ
x
0
δy
δ
y
0
f
s, t
w
u
s
ds
c
2
2
2
t
0
h
s
u
s
ds
. 1.5
Pachpatte 9 firstly got the estimation of the unknown function of the following inequality:
u
t
≤
ds
, 1.6
then, the estimation was used to study the boundedness, asymptotic behavior, slowly growth
of the solutions of the integral equation
u
t
k
c
1
t
−
t
0
f
1
t − s
u
s
ds
u
x, y
≤
c
1
x, y
α
1
x
α
1
x
0
β
1
y
β
1
2
x
0
β
2
y
β
2
y
0
f
2
s, t
ϕ
2
u
s, t
dt ds
.
:0, ∞. For functions hx,gx, y, h
x denotes the
derivative of hx,andg
x
x, y denotes the partial derivative gx, y on x. Consider 1.8,
and suppose that
H
1
ψ ∈ CR
, R
is a strictly increasing function with ψ00andψt →∞as
t →∞;
H
2
c
1
,c
2
: Δ → 0, ∞ are nondecreasing in each variable;
H
3
ϕ
i
∈ CR
, R
:
r
0
ds
ϕ
ψ
−1
s
,
Ψ
r
:
r
0
ds
ϕ
ψ
−1
Φ
on Δ satisfying 1.8. Then one has
u
x, y
≤ ψ
−1
Φ
−1
Ψ
−1
E
x, y
, 2.2
4 Journal of Inequalities and Applications
for all x, y ∈ x
0
,X
1
× y
0
,Y
1
,where
β
i
y
0
f
i
s, t
dt
×
α
3−i
s
α
3−i
x
0
β
3−i
y
β
3−i
i1
α
i
x
α
i
x
0
β
i
y
β
i
y
0
c
3−i
s, t
f
i
, Ψ
−1
E
x, y
∈ Dom
Φ
−1
. 2.4
Proof. From the inequality 1.8, for all x, y ∈ x
0
,X × J, we have
ψ
u
x, y
≤
c
1
X, y
s, t
dt ds
×
c
2
X, y
α
2
x
α
2
x
0
β
2
y
β
2
y
c
1
X, y
α
1
x
α
1
x
0
β
1
y
β
1
y
0
f
1
β
2
y
β
2
y
0
f
2
s, t
ϕ
2
u
s, t
dt ds
.
2.6
By the assumptions H
2
–H
5
i
y
β
i
y
0
f
i
α
i
x
,t
ϕ
i
u
α
i
x
,t
f
3−i
s, t
ϕ
3−i
u
s, t
dt ds
≤ ϕ
ψ
−1
θ
x, y
2
i1
α
X, y
α
3−i
x
α
3−i
x
0
β
3−i
y
β
3−i
y
0
f
3−i
s, t
ϕ
≤
2
i1
α
i
x
β
i
y
β
i
y
0
f
i
α
i
x
0
f
3−i
s, t
ϕ
3−i
ψ
−1
θ
s, t
dt ds
.
2.8
By taking s x in 2.8 and then integrating it from x
0
to x,weget
Φ
θ
x, y
c
3−i
X, y
f
i
s, t
dt ds
2
i1
α
i
x
α
i
x
0
β
i
3−i
y
0
f
3−i
σ, t
ϕ
3−i
ψ
−1
θ
σ, t
dt dσ
ds
≤ Φ
c
1
X, y
X, y
f
i
s, t
dt ds
2
i1
α
i
x
α
i
x
0
β
i
y
β
i
f
3−i
σ, t
ϕ
3−i
ψ
−1
θ
σ, t
dt dσ
ds,
2.9
for all x, y ∈ x
0
,X × y
0
,y
1
, where using the definition of Φ in 2.1. Similarly to the
above statement, we define a function ωx, y by the right-hand side of 2.9, then ωx, y is
6 Journal of Inequalities and Applications
a positive and nondecreasing function in each variable, θx, y ≤ Φ
c
3−i
X, yf
i
s, tdt ds.Differentiating ωx, y for x,by
the relation among ϕ and ϕ
1
,ϕ
2
, we have
ω
x
x, y
2
i1
α
i
x
β
i
y
β
3−i
y
β
3−i
y
0
f
3−i
σ, t
ϕ
3−i
ψ
−1
θ
σ, t
dt dσ
≤
2
i1
α
3−i
x
α
3−i
x
0
β
3−i
y
β
3−i
y
0
f
3−i
σ, t
ϕ
ψ
−1
β
i
y
β
i
y
0
f
i
α
i
x
,t
dt
×
α
3−i
x
α
×
y
0
,Y
1
,
2.10
where Y
1
is defined by 2.4.From2.10 , we have
ω
x
x, y
ϕ
ψ
−1
Φ
−1
ω
x, y
≤
dt
×
α
3−i
x
α
3−i
x
0
β
3−i
y
β
3−i
y
0
f
3−i
σ, t
dt dσ,
2.11
for all x, y ∈ x
0
α
i
x
0
β
i
y
β
i
y
0
f
i
s, t
dt
×
α
3−i
s
α
3−i
c
2
X, y
2
i1
α
i
X
α
i
x
0
β
i
y
β
i
y
0
y
β
i
y
0
f
i
s, t
dt
×
α
3−i
s
α
3−i
x
0
β
3−i
y
β
≤ ψ
−1
Φ
−1
ω
x, y
≤ ψ
−1
Φ
−1
Ψ
−1
Ψ
Φ
c
1
X, y
3−i
X, y
f
i
s, t
dt ds
2
i1
α
i
x
α
i
x
0
β
i
y
y
0
f
3−i
σ, t
dt dσ
ds
.
2.13
Let x X,from2.13we observe that
u
X, y
≤ ψ
−1
Φ
−1
Ψ
−1
Ψ
y
β
i
y
0
c
3−i
X, y
f
i
s, t
dt ds
2
i1
α
i
X
α
i
x
0
β
3−i
y
β
3−i
y
0
f
3−i
σ, t
dt dσ
ds
.
2.14
Since X ∈ x
0
,X
1
is arbitrary, from 2.14, we get the required estimation 2.2.
3. Applications
In this section, we present an application of our result to obtain bound of the solution of a
1
y
0
g
1
x − s, t
ϕ
1
z
s, t
dt ds
×
a
2
x, y
α
2
, ∀
x, y
∈ Δ,
3.1
where ψ : R → R is a strictly increasing function with ψ00, |ψr| ψ|r| > 0, and
ψt →∞as t →∞, k is a given positive constant, |a
1
|, |a
2
| : Δ → R
are bounded functions
and nondecreasing in each variable, functions α
i
and β
i
satisfy hypothesis H
4
,i1,2, g
i
,z ∈
C
0
Δ, R and ϕ
i
∈ C
0
. Then all solutions zx, y of 3.1 have the estimate
z
x, y
≤ ψ
−1
Φ
−1
Ψ
−1
H
x, y
, 3.2
for all x, y ∈ x
0
,X
2
× y
β
i
y
β
i
y
0
f
i
s, t
dt
×
α
3−i
s
α
3−i
x
0
β
a
2
x, y
2
i1
α
i
x
α
i
x
0
β
i
y
β
boundary of the planar region
R :
x, y
∈ Δ : H
x, y
∈ Dom
Ψ
−1
, Ψ
−1
H
x, y
∈ Dom
Φ
−1
. 3.4
Proof. From the integral equation 3.1, we have
ψ
0
β
1
y
β
1
y
0
g
1
x − s, t
ϕ
1
|
z
s, t
|
2
y
0
g
2
x − s, t
ϕ
2
|
z
s, t
|
dt ds
≤
s, t
ϕ
1
|
z
s, t
|
dt ds
×
a
2
x, y
α
2
|
dt ds
, ∀
x, y
∈ Δ.
3.5
Clearly, inequality 3.5 is in the form of 1.8. T hus, the estimate 3.2 of the solution zx, y
in this corollary is obtained immediately by our Theorem 2.1.
Journal of Inequalities and Applications 9
Acknowledgments
The authors are very grateful to the editor and the referees for their helpful comments
and valuable suggestions. This paper is supported by the Natural Science Foundation of
Guangxi Autonomous Region 0991265, the Scientific Research Foundation of the Education
Department of Guangxi Autonomous Region 200707MS112, and the Key Discipline of
Applied Mathematics of Hechi University of China 200725.
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