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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 518646, 9 pages
doi:10.1155/2008/518646
Research Article
On a Generalized Retarded Integral
Inequality with Two Variables
Wu-Sheng Wang
1, 2
and Cai-Xia Shen
1
1
Department of Mathematics, Hechi College, Guangxi, Yizhou 546300, China
2
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Wu-Sheng Wang,
Received 16 November 2007; Accepted 22 April 2008
Recommended by Wing-Sum Cheung
This paper improves Pachpatte’s results on linear integral inequalities with two variables, and gives
an estimation for a general form of nonlinear integral inequality with two variables. This paper does
not require monotonicity of known functions. The result of this paper can be applied to discuss on
boundedness and uniqueness for a integrodifferential equation.
Copyright q 2008 W S. Wang and C X. Shen. 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
Gronwall-Bellman inequality 1, 2 is an important tool in the study of existence, uniqueness,
boundedness, stability, and other qualitative properties of solutions of differential equations
and integral equations. There can be found a lot of its generalizations in various cases from
literature see, e.g., 1–12.In11, Pachpatte obtained an estimation for the integral inequality
ux, y ≤ ax, y
x
0
y
0
fs, t
us, t
s
0
t
0
gs, t, σ, τuσ, τdτdσ
dtds. 1.1
His results were applied to a partial integrodifferential equation:
u
xy
x, yF
x, y,ux, y,
x
0
y
0
h
x, y,τ,σ, ux, y
dτdσ
,
u
x, y
0
αx,u
x
0
,y
βy,
1.2
for boundedness and uniqueness of solutions.
2 Journal of Inequalities and Applications
In this paper, we discuss a more general form of integral inequality:
ψ
ux, y
≤ ax, y
bx
bx
0
cy
cy
0
fx, y,s, t
ϕ
1
us, t
s
bx
0
t
cy
0
gs, t, σ, τϕ
2
uσ, τ
dτdσ
dtds
1.3
for all x, y ∈ x
0
,x
1
× y
0
,y
1
. Obviously, u appears linearly in 1.1,butinour1.3 it
is generalized to nonlinear terms: ϕ
1
us, t and ϕ
2
us, t. Our strategy is to monotonize
functions ϕ
i
s with other two nondecreasing ones such that one has stronger monotonicity than
the other. We apply our estimation to an integrodifferential equation, which looks similar to
1.2 but includes delays, and give boundedness and uniqueness of solutions.
2. Main result
Throughout this paper, x
0
,x
1
,y
0
,y
1
∈ R are given numbers. Let R
:0, ∞,I :x
0
,x
1
,J :
y
0
,y
1
, and Λ : I × J ⊂ R
2
. Consider inequality 1.3, where we suppose that ψ ∈ C
0
R
, R
is strictly increasing such that ψ∞∞, b ∈ C
1
I,I, and c ∈ C
1
J, J are nondecreasing, such
that bx ≤ x and cy ≤ y, a ∈ C
1
Λ, R
,f∈ C
0
Λ
2
, R
,andgx, y,s, t ∈ C
0
Λ
2
, R
are
given, and ϕ
i
∈ C
0
R
, R
i 1, 2 are functions satisfying ϕ
i
00andϕ
i
u > 0 for all
u>0.
Define functions
w
1
s : max
τ∈0,s
ϕ
1
τ
,
w
2
s : max
τ∈0,s
ϕ
2
τ/w
1
τ
w
1
s,
φs : w
2
s/w
1
s.
2.1
Obviously, w
1
,w
2
,andφ in 2.1 are all nondecreasing and nonnegative functions and satisfy
w
i
s ≥ ϕ
i
s,i 1, 2. Let
W
1
u
u
1
ds
w
1
ψ
−1
s
,
2.2
W
2
u
u
1
ds
w
2
ψ
−1
s
,
2.3
Φu
u
W
1
1
ds
φ
ψ
−1
W
−1
1
s
.
2.4
Obviously, W
1
,W
2
,andΦ are strictly increasing in u>0, and therefore the inverses W
−1
1
,W
−1
2
,
and Φ
−1
are well defined, continuous, and increasing. We note that
Φu
u
W
1
1
dx
φ
ψ
−1
W
−1
1
x
u
W
1
1
w
1
ψ
−1
W
−1
1
x
dx
w
2
ψ
−1
W
−1
1
x
W
−1
1
u
1
dx
w
2
ψ
−1
x
W
2
W
−1
1
u
.
2.5
W S. Wang and C X. Shen 3
Furthermore, let
fx, y,s, t : max
τ∈x
0
,x
fτ,y,s,t, which is also nondecreasing in x for each
fixed y, s,andt and satisfies
fx, y,s, t ≥ fx, y, s, t ≥ 0.
Theorem 2.1. If inequality 1.3 holds for the nonnegative function ux, y,then
ux, y ≤ ψ
−1
W
−1
2
Ξx, y
2.6
for all x, y ∈ x
0
,X
1
× y
0
,Y
1
,where
Ξx, y : W
2
W
−1
1
r
2
x, y
bx
bx
0
cy
cy
0
fx, y,s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds,
r
2
x, y : W
1
r
1
x, y
bx
bx
0
cy
cy
0
fx, y,s, tdtds,
r
1
x, y : a
x
0
,y
x
x
0
a
x
s, y
ds,
2.7
and X
1
,Y
1
∈ Λ is arbitrarily given on the boundary of the planar region
R :
x, y ∈ Λ : Ξx, y ∈ Dom
W
−1
2
,r
2
x, y ∈ Dom
W
−1
1
. 2.8
Here Dom denotes the domain of a function.
Proof. By the definition of functions w
i
and
f
i
,from1.3 we get
ψ
ux, y
≤ ax, y
bx
bx
0
cy
cy
0
fx, y,s, t
w
1
us, t
s
bx
0
t
cy
0
g
s, t, σ, τ
w
2
uσ, τ
dτdσ
dtds
2.9
for all x, y ∈ Λ.
Firstly, we discuss the case that ax, y > 0 for all x, y ∈ Λ. It means that r
1
x, y > 0
for all x, y ∈ Λ. In such a circumstance, r
1
x, y is positive and nondecreasing on Λ and
r
1
x, y ≥ a
x
0
,y
x
x
0
a
x
t, ydt. 2.10
Regarding 1.3, we consider the auxiliary inequality
ψ
ux, y
≤ r
1
x, y
bx
bx
0
cy
cy
0
fX, y, s, t
w
1
us, t
s
bx
0
t
cy
0
gs, t, σ, τw
2
uσ, τ
dτdσ
dtds
2.11
4 Journal of Inequalities and Applications
for all x, y ∈ x
0
,X × J,wherex
0
≤ X ≤ X
1
is chosen arbitrarily. We claim that
ux, y ≤ ψ
−1
W
−1
2
W
2
W
−1
1
W
1
r
1
x, y
bx
bx
0
cy
cy
0
fX, y, s, tdtds
bx
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds
2.12
for all x, y ∈ x
0
,X × y
0
,Y
1
,whereY
1
is defined by 2.8.
Let ηx, y denote the right-hand side of 2.11, which is a nonnegative and
nondecreasing function on x
0
,X × J. Then, 2.11 is equivalent to
ux, y ≤ ψ
−1
ηx, y
∀x, y ∈ x
0
,Y × J. 2.13
By the fact that bx ≤ x for x ∈ x
0
,X and the monotonicity of w
i
,ψ,η,andbx,wehave
∂/∂xηx, y
w
1
ψ
−1
ηx, y
≤
∂/∂xr
1
x, y
w
1
ψ
−1
r
1
x, y
b
x
w
1
ψ
−1
ηx, y
×
cy
cy
0
f
1
X, y, bx,t
w
1
u
bx,t
bx
bx
0
t
cy
0
g
bx,t,τ,σ
w
2
uτ,σ
dτdσ
dt
≤
∂/∂xr
1
x, y
w
1
ψ
−1
r
1
x, y
b
x
cy
cy
0
f
1
X, y, bx,tdt
b
x
cy
cy
0
f
1
X, y, bx,t
bx
bx
0
t
cy
0
g
bx,t,τ,σ
φ
uτ,σ
dτdσ
dt
2.14
for all x, y ∈ x
0
,X × J. Integrating the above from x
0
to x,weget
W
1
ηx, y
≤ W
1
r
1
x, y
bx
bx
0
cy
cy
0
f
1
X, y, s, tdtds
bx
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σφ
uτ,σ
dτdσ
dtds
2.15
for all x, y ∈ x
0
,X × J.Let
ψ
ξx, y
: W
1
ηx, y
,
r
2
x, y : W
1
r
1
x, y
bx
bx
0
cy
cy
0
f
1
X, y, s, tdtds.
2.16
W S. Wang and C X. Shen 5
From 2.15, 2.16,weobtain
ψ
ξx, y
≤ r
2
x, y
bx
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σφ
uτ,σ
dτdσ
dtds
2.17
for all x
0
≤ x<X,y
0
≤ y<y
1
.Letβx, y denote the right-hand side of 2.17, which is a
nonnegative and nondecreasing function on x
0
,Y × J. Then, 2.17 is equivalent to
ψ
ξx, y
≤ βx, y ∀x, y ∈ x
0
,Y × J. 2.18
From 2.13, 2.16,and2.18,wehave
ux, y ≤ ψ
−1
ηx, y
ψ
−1
W
−1
1
ψ
ξx, y
≤ ψ
−1
W
−1
1
βx, y
2.19
for all x
0
≤ x<X, y
0
≤ y<Y
1
,whereY
1
is defined by 2.8. By the definitions of φ, ψ,andW
1
,
φψ
−1
W
−1
1
s is continuous and nondecreasing on 0, ∞ and satisfies φψ
−1
W
−1
1
s > 0
for s>0. Let hsψ
−1
W
−1
1
s. Since b
x ≥ 0andbx ≤ x for x ∈ x
0
,X,from2.19 we
have
∂/∂xβx, y
φ
h
βx, y
≤
∂/∂xr
2
x, y
φ
h
r
2
x, y
b
x
φ
h
βx, y
cy
cy
0
f
1
X, y, bx,t
bx
bx
0
t
cy
0
g
bx,t,τ,σ
φ
uτ,σ
dτdσ
dtds
≤
∂/∂xr
2
x, y
φ
h
r
2
x, y
b
x
cy
cy
0
f
1
X, y, bx,t
bx
bx
0
t
cy
0
g
bx,t,τ,σ
dτdσ
dtds
2.20
for all x, y ∈ x
0
,X × y
0
,Y
1
. Integrating the above from x
0
to x,by2.4 we get
Φ
βx, y
≤ Φ
r
2
x, y
bx
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds 2.21
for all x, y ∈ x
0
,X × y
0
,y
1
.By2.19 and the above inequality, we obtain
ux, y
≤ ψ
−1
W
−1
1
Φ
−1
Φ
r
2
x, y
bx
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds
2.22
6 Journal of Inequalities and Applications
for all x, y ∈ x
0
,X × y
0
,Y
1
,whereY
1
is defined by 2.8. It follows from 2.5 that
ux, y ≤ ψ
−1
W
−1
2
W
2
W
−1
1
W
1
r
1
x, y
bx
bx
0
cy
cy
0
f
1
X, y, s, tdtds
bx
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds
,
2.23
which proves the claimed 2.12.
We start from the original inequality 1.3 and see that
ψ
uX, y
≤ r
1
X, y
bX
bx
0
cy
cy
0
fX, y, s, t
ϕ
1
us, t
s
bx
0
t
cy
0
gs, t, σ, τϕ
2
uσ, τ
dτdσ
dtds
2.24
for all y ∈ y
0
,Y
1
; n amely, the auxiliary inequality 2.11 holds for x X, y ∈ y
0
,Y
1
.By
2.12,weget
uX, y ≤ ψ
−1
W
−1
2
W
2
W
−1
1
W
1
r
1
X, y
bX
bx
0
cy
cy
0
f
1
X, y, s, tdtds
bX
bx
0
cy
cy
0
f
1
X, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds
2.25
for all x
0
≤ X ≤ X
1
,y
0
≤ y ≤ Y
1
. This proves 2.6.
The remainder case is that ax, y0 for some x, y ∈ Λ.Let
r
1,ε
x, y : r
1
x, yε, 2.26
where ε>0 is an arbitrary small number. Obviously, r
1,ε
x, y > 0 for all x, y ∈ Λ. Using the
same arguments as above, where r
1
x, y is replaced with r
1,ε
x, y,weget
ux, y ≤ ψ
−1
W
−1
2
W
2
W
−1
1
W
1
r
1,ε
x, y
bx
bx
0
cy
cy
0
f
1
x, y, s, tdtds
bx
bx
0
cy
cy
0
f
1
x, y, s, t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds
2.27
for all x
0
≤ X ≤ X
1
,y
0
≤ y ≤ Y
1
. Letting ε → 0
,weobtain2.6 because of continuity of r
1,ε
in
ε and continuity of ψ
−1
,W
−1
1
,W,W
−1
2
,andW
2
. This completes the proof.
W S. Wang and C X. Shen 7
3. Applications
In 11, the partial integrodifferential equation 1.2 was discussed for boundedness and
uniqueness of the solutions under the assumptions that
Fx, y, u, v
≤ fx, y
|u| |v|
,
h
x, y,s, t, us, t
≤ gx, y,s, t
us, t
,
F
x, y,u
1
,v
1
− F
x, y,u
2
,v
2
≤ fx, y
u
1
− u
2
v
1
− v
2
,
h
x, y,s, t, u
1
− h
x, y,s, t, u
2
≤ gx, y,s, t
u
1
− u
2
,
3.1
respectively. In this section, we further consider the nonlinear delay partial integrodifferential
equation
u
xy
x, yF
x, y,u
bx,cy
,
bx
bbx
0
cy
ccy
0
h
bx,cy,τ,σ,uτ, σ
dτdσ
,
u
x, y
0
αx,u
x
0
,y
βy
3.2
for all x, y ∈ Λ,whereb, c,andu are supposed to be as in Theorem 2.1; h : Λ
2
× R→R,
F : Λ × R
2
→R, α : I→R,andβ : J→R are all continuous functions such that α0β00.
Obviously, the estimation obtained in 11 cannot be applied to 3.2.
We first give an estimation for solutions of 3.2 under the condition
Fx, y, u, v
≤ fx, y
ϕ
1
|u|
|v|
,
h
x, y,s, t, us, t
≤ gx, y,s, t
ϕ
2
us, t
.
3.3
Corollary 3.1. If |αxβy| is nondecreasing in x and y and 3.3 holds, then every solution um, n
of 3.2 satisfies
ux, y ≤ W
−1
2
Ξx, y
∀x, y ∈
x
0
,X
1
×
y
0
,Y
1
, 3.4
where
Ξx, y : W
2
W
−1
1
W
1
αxβy
bx
bx
0
cy
cy
0
f
b
−1
s,c
−1
t
b
b
−1
s
c
c
−1
t
dtds
bx
bx
0
cy
cy
0
f
b
−1
s,c
−1
t
b
b
−1
s
c
c
−1
t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds,
3.5
and W
1
,W
−1
1
,W
2
,W
−1
2
,andX
1
,Y
1
are defined as in Theorem 2.1 .
8 Journal of Inequalities and Applications
Corollary 3.1 actually gives a condition of boundedness for solutions. Concretely, if there
is a positive constant M such that
αxβy
<M,
bx
bx
0
cy
cy
0
f
b
−1
s,c
−1
t
b
b
−1
s
c
c
−1
t
dtds < M,
bx
bx
0
cy
cy
0
f
b
−1
s,c
−1
t
b
b
−1
s
c
c
−1
t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds < M
3.6
on x
0
,X
1
× y
0
,Y
1
, then every solution ux, y of 3.2 is bounded on x
0
,X
1
× y
0
,Y
1
.
Next, we give the condition of the uniqueness of solutions for 3.2.
Corollary 3.2. Suppose
F
x, y,u
1
,v
1
− F
x, y,u
2
,v
2
≤ fx, y
ϕ
1
u
1
− u
2
v
1
− v
2
,
h
x, y,s, t, u
1
− h
x, y,s, t, u
2
≤ gx, y,s, tϕ
2
u
1
− u
2
,
3.7
where f, g, ϕ
1
,ϕ
2
are defined as in Theorem 2.1. There is a positive number M such that
bx
bx
0
cy
cy
0
f
b
−1
s,c
−1
t
b
b
−1
s
c
c
−1
t
dtds < M,
bx
bx
0
cy
cy
0
f
b
−1
s,c
−1
t
b
b
−1
s
c
c
−1
t
s
bx
0
t
cy
0
gs, t, τ, σdτdσ
dtds < M
3.8
on x
0
,X
1
× y
0
,Y
1
.Then,3.2 has at most one solution on x
0
,X
1
× y
0
,Y
1
,whereX
1
,Y
1
are
defined as in Theorem 2.1.
Acknowledgments
This work is supported by the Scientific Research Fund of Guangxi Provincial Education
Department no. 200707MS112, the Natural Science Foundation no. 2006N001, and the
Applied Mathematics Key Discipline Foundation of Hechi College of China.
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