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 Wang1, 2 and Cai-Xia Shen1
1
2
Department of Mathematics, Hechi College, Guangxi, Yizhou 546300, China
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Wu-Sheng Wang, wang4896@126.com
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. In 11, Pachpatte obtained an estimation for the integral inequality
ux, y ≤ ax, y x y
0
fs, t us, t 0
s t
0
gs, t, σ, τuσ, τdτdσ dtds.
1.1
0
His results were applied to a partial integrodifferential equation:
x y
uxy x, y F x, y, ux, y,
h x, y, τ, σ, ux, y dτdσ ,
u x, y0 αx,
0
for boundedness and uniqueness of solutions.
0
u x0 , y βy,
1.2
2
Journal of Inequalities and Applications
In this paper, we discuss a more general form of integral inequality:
ψ ux, y
bx cy
t
s
≤ ax, y fx, y, s, t ϕ1 us, t gs, t, σ, τϕ2 uσ, τ dτdσ dtds
bx0 cy0 bx0 cy0 1.3
for all x, y ∈ x0 , x1 × y0 , y1 . Obviously, u appears linearly in 1.1, but in our 1.3 it
is generalized to nonlinear terms: ϕ1 us, t and ϕ2 us, t. Our strategy is to monotonize
functions ϕi 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, x0 , x1 , y0 , y1 ∈ R are given numbers. Let R : 0, ∞, I : x0 , x1 , J :
y0 , y1 , and Λ : I × J ⊂ R2 . Consider inequality 1.3, where we suppose that ψ ∈ C0 R , R is strictly increasing such that ψ∞ ∞, b ∈ C1 I, I, and c ∈ C1 J, J are nondecreasing, such
that bx ≤ x and cy ≤ y, a ∈ C1 Λ, R , f ∈ C0 Λ2 , R , and gx, y, s, t ∈ C0 Λ2 , R are
given, and ϕi ∈ C0 R , R i 1, 2 are functions satisfying ϕi 0 0 and ϕi u > 0 for all
u > 0.
Define functions
w1 s : max ϕ1 τ ,
τ∈0,s
w2 s : max ϕ2 τ/w1 τ w1 s,
τ∈0,s
2.1
φs : w2 s/w1 s.
Obviously, w1 , w2 , and φ in 2.1 are all nondecreasing and nonnegative functions and satisfy
wi s ≥ ϕi s, i 1, 2. Let
u
ds
W1 u −1 ,
2.2
1 w1 ψ s
u
ds
W2 u −1 ,
2.3
w
s
ψ
2
1
u
ds
Φu −1 −1 .
2.4
W1 s
W 1 1 φ ψ
Obviously, W1 , W2 , and Φ are strictly increasing in u > 0, and therefore the inverses W1−1 , W2−1 ,
and Φ−1 are well defined, continuous, and increasing. We note that
u
dx
Φu −1 −1 W1 x
W1 1 φ ψ
w1 ψ −1 W1−1 x dx
−1 −1 W1 x
W1 1 w2 ψ
u
W −1 u
1
1
dx
W2 W1−1 u .
w2 ψ −1 x
2.5
W.-S. Wang and C.-X. Shen
3
y, s, t : maxτ∈x ,x fτ, y, s, t, which is also nondecreasing in x for each
Furthermore, let fx,
0
y, s, t ≥ fx, y, s, t ≥ 0.
fixed y, s, and t and satisfies fx,
Theorem 2.1. If inequality 1.3 holds for the nonnegative function ux, y, then
ux, y ≤ ψ −1 W2−1 Ξx, y
2.6
for all x, y ∈ x0 , X1 × y0 , Y1 , where
Ξx, y :
W2 W1−1
r2 x, y
r2 x, y : W1 r1 x, y r1 x, y : a x0 , y x
bx cy
bx0 cy0 bx cy
bx0 cy0 t
s
fx, y, s, t
bx0 cy0 gs, t, τ, σdτdσ dtds,
y, s, tdtds,
fx,
2.7
ax s, yds,
x0
and X1 , Y1 ∈ Λ is arbitrarily given on the boundary of the planar region
R : x, y ∈ Λ : Ξx, y ∈ Dom W2−1 , r2 x, y ∈ Dom W1−1 .
2.8
Here Dom denotes the domain of a function.
Proof. By the definition of functions wi and fi , from 1.3 we get
ψ ux, y
≤ ax, y
bx cy
bx0 cy0 t
s
y, s, t w1 us, t fx,
bx0 cy0 g s, t, σ, τ w2 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 r1 x, y > 0
for all x, y ∈ Λ. In such a circumstance, r1 x, y is positive and nondecreasing on Λ and
r1 x, y ≥ a x0 , y x
ax t, ydt.
2.10
x0
Regarding 1.3, we consider the auxiliary inequality
ψ ux, y
≤ r1 x, y
bx cy
bx0 cy0 s
y, s, t w1 us, t fX,
t
bx0 cy0 gs, t, σ, τw2 uσ, τ dτdσ dtds
2.11
4
Journal of Inequalities and Applications
for all x, y ∈ x0 , X × J, where x0 ≤ X ≤ X1 is chosen arbitrarily. We claim that
bx cy
y, s, tdtds
ux, y ≤ ψ −1 W2−1 W2 W1−1 W1 r1 x, y fX,
bx0 cy0 bx cy
bx0 cy0 s
f1 X, y, s, t
t
bx0 cy0 2.12
gs, t, τ, σdτdσ dtds
for all x, y ∈ x0 , X × y0 , Y1 , where Y1 is defined by 2.8.
Let ηx, y denote the right-hand side of 2.11, which is a nonnegative and
nondecreasing function on x0 , X × J. Then, 2.11 is equivalent to
ux, y ≤ ψ −1 ηx, y
∀x, y ∈ x0 , Y × J.
2.13
By the fact that bx ≤ x for x ∈ x0 , X and the monotonicity of wi , ψ, η, and bx, we have
∂/∂xηx, y
w1 ψ −1 ηx, y
≤
∂/∂xr1 x, y
b x
−1 −1 w1 ψ r1 x, y
w1 ψ ηx, y
×
≤
cy
cy0 bx t
f1 X, y, bx, t w1 u bx, t bx0 cy0 cy
∂/∂xr1 x, y
b x
w1 ψ −1 r1 x, y
cy0 g bx, t, τ, σ w2 uτ, σ dτdσ dt
f1 X, y, bx, tdt
cy
bx t
b x
f1 X, y, bx, t
cy0 bx0 cy0 g bx, t, τ, σ φ uτ, σ dτdσ dt
2.14
for all x, y ∈ x0 , X × J. Integrating the above from x0 to x, we get
W1 ηx, y ≤ W1 r1 x, y bx cy
bx0 cy0 bx cy
bx0 cy0 f1 X, y, s, tdtds
f1 X, y, s, t
s
t
bx0 cy0 2.15
gs, t, τ, σφ uτ, σ dτdσ dtds
for all x, y ∈ x0 , X × J. Let
ψ ξx, y : W1 ηx, y ,
bx cy
f1 X, y, s, tdtds.
r2 x, y : W1 r1 x, y bx0 cy0 2.16
W.-S. Wang and C.-X. Shen
5
From 2.15, 2.16, we obtain
ψ ξx, y
≤ r2 x, y bx cy
bx0 cy0 s
f1 X, y, s, t
t
bx0 cy0 gs, t, τ, σφ uτ, σ dτdσ dtds
2.17
for all x0 ≤ x < X, y0 ≤ y < y1 . Let βx, y denote the right-hand side of 2.17, which is a
nonnegative and nondecreasing function on x0 , Y × J. Then, 2.17 is equivalent to
ψ ξx, y ≤ βx, y
∀x, y ∈ x0 , Y × J.
2.18
From 2.13, 2.16, and 2.18, we have
ux, y ≤ ψ −1 ηx, y ψ −1 W1−1 ψ ξx, y ≤ ψ −1 W1−1 βx, y
2.19
for all x0 ≤ x < X, y0 ≤ y < Y1 , where Y1 is defined by 2.8. By the definitions of φ, ψ, and W1 ,
φψ −1 W1−1 s is continuous and nondecreasing on 0, ∞ and satisfies φψ −1 W1−1 s > 0
for s > 0. Let hs ψ −1 W1−1 s. Since b x ≥ 0 and bx ≤ x for x ∈ x0 , X, from 2.19 we
have
∂/∂xβx, y
φ h βx, y
≤
∂/∂x
r2 x, y
φ h r2 x, y
b x
φ h βx, y
≤
cy
cy0 bx t
f1 X, y, bx, t
g bx, t, τ, σ φ uτ, σ dτdσ dtds
bx0 cy0 cy
bx t
∂/∂x
r2 x, y
g bx, t, τ, σ dτdσ dtds
f1 X, y, bx, t
b x
φ h r2 x, y
cy0 bx0 cy0 2.20
for all x, y ∈ x0 , X × y0 , Y1 . Integrating the above from x0 to x, by 2.4 we get
Φ βx, y ≤ Φ r2 x, y bx cy
bx0 cy0 f1 X, y, s, t
s
t
bx0 cy0 gs, t, τ, σdτdσ dtds
2.21
for all x, y ∈ x0 , X × y0 , y1 . By 2.19 and the above inequality, we obtain
ux, y
s
bx cy
f1 X, y, s, t
≤ ψ −1 W1−1 Φ−1 Φ r2 x, y bx0 cy0 t
bx0 cy0 gs, t, τ, σdτdσ dtds
2.22
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Journal of Inequalities and Applications
for all x, y ∈ x0 , X × y0 , Y1 , where Y1 is defined by 2.8. It follows from 2.5 that
bx cy
ux, y ≤ ψ −1 W2−1 W2 W1−1 W1 r1 x, y f1 X, y, s, tdtds
bx0 cy0 bx cy
bx0 cy0 t
s
f1 X, y, s, t
bx0 cy0 2.23
gs, t, τ, σdτdσ dtds
,
which proves the claimed 2.12.
We start from the original inequality 1.3 and see that
ψ uX, y
≤ r1 X, y
bX cy
bx0 cy0 s
y, s, t ϕ1 us, t fX,
t
bx0 cy0 gs, t, σ, τϕ2 uσ, τ dτdσ dtds
2.24
for all y ∈ y0 , Y1 ; namely, the auxiliary inequality 2.11 holds for x X, y ∈ y0 , Y1 . By
2.12, we get
bX cy
uX, y ≤ ψ −1 W2−1 W2 W1−1 W1 r1 X, y f1 X, y, s, tdtds
bx0 cy0 s
bX cy
bx0 cy0 f1 X, y, s, t
t
bx0 cy0 2.25
gs, t, τ, σdτdσ dtds
for all x0 ≤ X ≤ X1 , y0 ≤ y ≤ Y1 . This proves 2.6.
The remainder case is that ax, y 0 for some x, y ∈ Λ. Let
r1,ε x, y : r1 x, y ε,
2.26
where ε > 0 is an arbitrary small number. Obviously, r1,ε x, y > 0 for all x, y ∈ Λ. Using the
same arguments as above, where r1 x, y is replaced with r1,ε x, y, we get
bx cy
−1
−1
ux, y ≤ ψ
W2 W2 W1 W1 r1,ε x, y f1 x, y, s, tdtds
−1
bx0 cy0 bx cy
bx0 cy0 f1 x, y, s, t
s
t
bx0 cy0 2.27
gs, t, τ, σdτdσ dtds
for all x0 ≤ X ≤ X1 , y0 ≤ y ≤ Y1 . Letting ε → 0 , we obtain 2.6 because of continuity of r1,ε in
ε and continuity of ψ −1 , W1−1 , W, W2−1 , and W2 . 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, tus, t,
F x, y, u1 , v1 − F x, y, u2 , v2 ≤ fx, y u1 − u2 v1 − v2 ,
h x, y, s, t, u1 − h x, y, s, t, u2 ≤ gx, y, s, tu1 − u2 ,
3.1
respectively. In this section, we further consider the nonlinear delay partial integrodifferential
equation
bx
uxy x, y F x, y, u bx, cy ,
cy
bbx0 ccy0 u x, y0 αx,
h bx, cy, τ, σ, uτ, σ dτdσ ,
u x0 , y βy
3.2
for all x, y ∈ Λ, where b, c, and u are supposed to be as in Theorem 2.1; h : Λ2 × R→R,
F : Λ × R2 →R, α : I→R, and β : J→R are all continuous functions such that α0 β0 0.
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 ϕ1 |u| |v| ,
h x, y, s, t, us, t ≤ gx, y, s, tϕ2 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 ≤ W2−1 Ξx, y ∀x, y ∈ x0 , X1 × y0 , Y1 ,
3.4
where
bx cy
f b−1 s, c−1 t
−1
Ξx, y : W2 W1 W1 αx βy dtds
−1
−1
bx0 cy0 b b s c c t
s t
f b−1 s, c−1 t
gs, t, τ, σdτdσ dtds,
−1 −1 cy0 b b s c c t
bx0 cy0 bx cy
bx0 and W1 , W1−1 , W2 , W2−1 , and X1 , Y1 are defined as in Theorem 2.1 .
3.5
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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
αx βy < M,
f b−1 s, c−1 t
−1 −1 dtds < M,
cy0 b b s c c t
bx cy
bx0 s t
f b−1 s, c−1 t
gs, t, τ, σdτdσ dtds < M
−1 −1 cy0 b b s c c t
bx0 cy0 bx cy
bx0 3.6
on x0 , X1 × y0 , Y1 , then every solution ux, y of 3.2 is bounded on x0 , X1 × y0 , Y1 .
Next, we give the condition of the uniqueness of solutions for 3.2.
Corollary 3.2. Suppose
F x, y, u1 , v1 − F x, y, u2 , v2 ≤ fx, y ϕ1 u1 − u2 v1 − v2 ,
h x, y, s, t, u1 − h x, y, s, t, u2 ≤ gx, y, s, tϕ2 u1 − u2 ,
3.7
where f, g, ϕ1 , ϕ2 are defined as in Theorem 2.1. There is a positive number M such that
f b−1 s, c−1 t
−1 −1 dtds < M,
cy0 b b s c c t
bx cy
bx0 s t
f b−1 s, c−1 t
gs,
t,
τ,
σdτdσ
dtds < M
−1
−1
cy0 b b s c c t
bx0 cy0 bx cy
bx0 3.8
on x0 , X1 × y0 , Y1 . Then, 3.2 has at most one solution on x0 , X1 × y0 , Y1 , where X1 , Y1 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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