Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 240790, 21 pages
doi:10.1155/2010/240790
Research Article
Nonlinear Retarded Integral Inequalities with Two
Variables and Applications
Wu-Sheng Wang,1, 2 Zizun Li,1 Yong Li,3 and Yong Huang3
1
School of Mathematics and Computing Science, Guilin University of Electronic Technology,
Guilin 541004, China
2
Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China
3
Department of Mathematics and Computer Information Engineering, Baise College,
Guangxi Baise 533000, China
Correspondence should be addressed to Wu-Sheng Wang, wang4896@126.com
Received 8 June 2010; Revised 27 August 2010; Accepted 6 October 2010
Academic Editor: Sever Silverstru Dragomir
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.
We consider some new nonlinear retarded integral inequalities with two variables, which extend
the results in the work of W.-S. Wang 2007, and the one in the work of Y.-H. kim 2009.
These inequalities include not only a nonconstant term outside the integrals but also more
than one distinct nonlinear integrals without assumption of monotonicity. Finally, we give some
applications to the boundary value problem of a partial differential equation for boundedness and
uniqueness.
1. Introduction
Integral inequalities that give explicit bounds on unknown functions provide a very useful
and important device in the study of many qualitative as well as quantitative properties
of solutions of partial differential equations, integral equations, and integrodifferential
equation. One of the best known and widely used inequalities in the study of nonlinear
differential equations is Gronwall inequality 1, which states that if u and f are nonnegative
continuous functions on the interval a, b satisfying
ut ≤ c t
fsusds,
t ∈ a, b,
1.1
α
where c is a nonnegative constant, then we have
t
ut ≤ c exp
fsds ,
α
t ∈ a, b.
1.2
2
Journal of Inequalities and Applications
Since the inequality 1.2 provides an explicit bound of the unknown function u it furnishes
a handy tool in the study of various properties of solutions of differential equations. Because
of its fundamental importance, several generalizations and analogous results of Gronwall
inequality 1, 2 and its applications have attracted great interests of many mathematicians
e.g., 3–5. Some recent works can be found, for example, in 6–17 and some references
therein. In 2005, Agarwal et al. 6 investigated the inequality
ut ≤ at n bi t
bi t0 i1
gi t, swi usds,
t0 ≤ t < t1 .
1.3
In 2006, Cheung 9 studied the inequality
up x, y ≤ a b1 x c1 y
p
p−q
p
p−q
c1 y 0 b1 x0 b2 x c2 y
b2 x0 c2 y 0 g1 s, tuq s, tdt ds
1.4
g2 s, tuq s, tψus, tdt ds,
for all x, y ∈ x0 , X × y0 , Y , where a is a constant.
In 2007, Wang 16 discussed the retarded integral inequality
bi x ci y
k
u x, y ≤ a x, y p
bi x0 i1
ci y 0 fi x, y, s, t ϕi us, tds dt,
1.5
for all x, y ∈ x0 , x1 × y0 , y1 .
In 2008, Agarwal et al. 7 discussed the retarded integral inequality
ϕut ≤ c n αi t
i1
αi t0 uq s fi sϕus gi s ds,
1.6
for all t ∈ t0 , T , where c is a constant.
In 2009, Kim 12 obtained the explicit bound of the unknown function of the following
inequality:
n
ψ u x, y ≤ a x, y c x, y
i1
n
c x, y
i1
for all x, y ∈ x0 , X × y0 , Y .
αi x βi y
αi x0 αi x βi y
αi x0 βi y 0 βi y0 uq s, tgi s, t ds dt
uq s, tfi s, tϕus, tds dt,
1.7
Journal of Inequalities and Applications
3
The purpose of the present paper is to establish some new nonlinear retarded
integral inequalities of Gronwall-Bellman type with two variables. We can demonstrate that
inequalities 1.4, 1.5, and 1.7, considered in 9, 12, 16, respectively, can also be solved
with our results. We also apply our results to study the boundedness and uniqueness of the
solutions of the boundary value problem of a partial differential equation.
2. Main Result
Throughout this paper, R denotes the set of real numbers, and x0 , y0 , x1 , y1 ∈ R are given
numbers. R : 0, ∞, I : x0 , x1 , J : y0 , y1 are the subsets of R and Λ : I × J ⊂ R2 . For
any s, t ∈ Λ, let Λs, t denote the subset x0 , s × y0 , t ∩ Λ of Λ. C1 U, V denotes the set of
continuous differentiable functions of U into V .
Consider the following inequality:
ψ u x, y
α x β y
n
i
i
≤ a x, y uq s, tgi x, y, s, t ds dt
αi x0 βi y0 i1
δi x γi y
δi x0 γi y0 u s, tfi x, y, s, t ϕi us, tds dt ,
q
∀ x, y ∈ Λ.
2.1
Our inequality 2.1 not only includes a nonconstant term outside the integrals but also more
than one distinct nonlinear integral without assumption of monotonicity. When fi x, y, s, t cx, yfi s, t, gi x, y, s, t cx, ygi s, t, αi x δi x, βi y γi y, and ϕi u ϕu,
our inequality 2.1 reduces to 1.7 studied in 12. When uq s, t 1, gi x, y, s, t 0, and
ψu uq , our inequality 2.1 reduces to 1.5 studied in 16.
Suppose that
H1 ψ is a strictly increasing continuous function on R , ψ0 0;
H2 all ϕi , i 1, 2, . . . , n are continuous functions on R and positive on 0, ∞;
H3 ax, y ≥ 0 on Λ, and a is nondecreasing in each variable;
H4 αi , δi ∈ C1 I, I and βi , γi ∈ C1 J, Ji 1, 2, . . . , n are nondecreasing such that
αi x ≤ x and δi x ≤ x on I, βi y ≤ y and γi y ≤ y on J;
H5 q > 0 is a constant;
H6 all fi , gi i 1, 2, . . . , n are nonnegative functions on Λ × Λ.
Firstly, we technically consider a sequence of functions wi s, which can be calculated
recursively by
w1 u : max ϕ1 τ,
τ∈0, u
wi1 u : max
τ∈0, u
ϕi1 τ
wi u,
wi τ
u > 0,
2.2
u > 0, i 1, 2, . . . , n − 1.
4
Journal of Inequalities and Applications
Moreover, we define the following functions:
Ψu :
u
0
Wi u :
u
0
wi
ψ −1
ds
q ,
ψ −1 s
ds
−1 ,
Ψ s
u ≥ 0,
2.3
u ≥ 0, i 1, 2, . . . , n.
2.4
Obviously, both Ψ and Wi are strictly increasing and continuous functions. Letting Ψ−1 , Wi−1
denote Ψ, Wi inverse function, respectively, then both Ψ−1 and Wi−1 are also continuous and
increasing functions.
Let
fi x, y, s, t :
gi x, y, s, t :
fi τ, ξ, s, t,
2.5
gi τ, ξ, s, t, i 1, 2, . . . , n.
2.6
max
τ, ξ∈x0 , x×y0 , y max
τ, ξ∈x0 , x×y0 , y Then fi x, y, s, t and gi x, y, s, t are nonnegative and nondecreasing in x, y for each fixed
s, t and satisfy fi x, y, s, t ≥ fi x, y, s, t, gi x, y, s, t ≥ gi x, y, s, t, i 1, 2, . . . , n.
Theorem 2.1. Suppose that H1 –H6 hold and ux, y is a nonnegative function on Λ satisfying
2.1. Then
δn x γn y
−1
−1
−1
Ψ Wn Wn Ξn x, y u x, y ≤ ψ
,
fn x, y, s, t ds dt
δn x0 γn y0 2.7
for x, y ∈ ΛX1 , Y1 , where
n
Ξ1 x, y : Ψ a x, y αi x βi y
i1
Ξi x, y :
−1
Wi−1
αi x0 Wi−1 Ξi−1 x, y
βi y0 gi x, y, s, t ds dt,
δi−1 x γi−1 y
δi−1 x0 γi−1 y0 fi−1 x, y, s, t dsdt ,
i 2, 3, . . . , n.
2.8
Journal of Inequalities and Applications
5
X1 , Y1 ∈ Λ is arbitrarily given on the boundary of the planar region
R1 :
x, y ∈ Λ : Wi Ξi x, y ∞
≤
≤
Wn Ξn x, y
∞
δn x γn y
δn x0 γn y0 2.9
fn x, y, s, t ds dt
ds
ψ −1 s
0
γi y0 δi x0 fi x, y, s, t ds dt
ds
, i 1, 2, . . . , n,
wi ψ −1 Ψ−1 s
0
Wn−1
δi x γi y
q .
Corollary 2.2. Let u, a, fi , gi , αi , βi , δi , γi and ϕi u i 1, 2, . . . , n be as defined in Theorem 2.1.
Suppose that p > q > 0 are constants. If
n
u x, y ≤ a x, y p
α x β y
i
i
αi x0 i1
βi y 0 uq s, tgi x, y, s, t ds dt
δi x γi y
δi x0 γi y0 2.10
uq s, tfi x, y, s, t ϕi us, tds dt ,
for all x, y ∈ Λ, then
u x, y ≤
G−1
n
p − q
Gn Bn x, y p
δn x γn y
δn x0 γn y0 fn X, Y, s, t
1/p−q
,
2.11
for all x, y ∈ ΛX2 ,Y2 , where fi and gi are defined by 2.5 and 2.6, and
n
p−q/p p − q B1 x, y : a x, y
p i1
Bi x, y :
G−1
i−1
Gi−1 Bi−1 x, y
p−q
p
αi x βi y
αi x0 gi x, y, s, t ds dt,
βi y 0
δi−1 x γi−1 y
δi−1 x0 γi−1 y0 fi−1 x, y, s, t ds dt ,
i 2, 3, . . . , n,
Gi x, y :
u
0
ds
,
wi s1/p−q
u ≥ 0, i 1, 2, . . . , n.
2.12
6
Journal of Inequalities and Applications
G−1
i denotes the inverse function of Gi , and X2 , Y2 ∈ Λ lies on the boundary of the planar region
R2 :
p − q
x, y Λ : Gi Bi x, y p
≤
∞
0
δi x γi y
γi y0 δi x0 fi x, y, s, t ds dt
2.13
ds
, i 1, 2, . . . , n .
wi s1/p−q
Corollary 2.3. Let u, a, q, fi , gi , δi , γi and ϕi ui 1, 2, . . . , n be as defined in Theorem 2.1.
Supposing that
n
ψ u x, y ≤ a x, y δi x γi y
i1
γi y0 δi x0 uq s, tfi x, y, s, t ϕi us, tds dt,
2.14
for all x, y ∈ Λ, then
δn x γn y
−1
−1
−1
Ψ Wn Wn Kn x, y ,
u x, y ≤ ψ
fn x, y, s, t ds dt
δn x0 γn y0 2.15
for all x, y ∈ ΛX3 ,Y3 , where
K1 x, y : Ψ a x, y ,
Ki x, y :
−1
Wi−1
Wi−1 Ki−1 x, y
δi−1 x γi−1 y
δi−1 x0 γi−1 y0 fi−1 x, y, s, t ds dt ,
2.16
i 2, 3, . . . , n.
fi is defined by 2.5, Ψ, Wi are as defined in 2.3 and 2.4, respectively, and Ψ−1 Wi−1 denote the
inverse functions of Ψ and Wi . X3 , Y3 ∈ Λ lies on the boundary of the planar region
R3 : x, y ∈ Λ : Wi Ki x, y ≤
∞
0
wi
ψ −1
Wn−1 Wn En x, y ∞
0
ds
q
ψ −1 s
δi x0 γi y0 fi x, y, s, t ds dt
ds
−1 , i 1, 2, . . . , n,
Ψ s
≤
δi x γi y
.
δn x γn y
δn x0 γn y0 fn x, y, s, t ds dt
2.17
Journal of Inequalities and Applications
7
Theorem 2.4. Suppose that H1 –H6 hold and ux, yis a nonnegative function on Λ satisfying
α x β y
n
i
i
uq s, tgi x, y, s, t ds dt
ψ u x, y ≤ a x, y αi x0 βi y0 i1
αi x βi y
βi y 0 αi x0 u s, tfi x, y, s, t ϕi us, tds dt ,
q
2.18
∀ x, y ∈ Λ.
Then
αn x βn y
−1
−1
−1
Ψ Wn Wn Hn x, y u x, y ≤ ψ
,
fn x, y, s, t ds dt
αn x0 βn y0 2.19
for x, y ∈ ΛX4 , Y4 , where
n
H1 x, y : Ψ a x, y αi x βi y
i1
−1
Wi−1 Hi−1 x, y
Hi x, y : Wi−1
βi y0 αi x0 gi x, y, s, t ds dt,
αi−1 x βi−1 y
αi−1 x0 βi−1 y0 2.20
fi−1 x, y, s, t ds dt ,
for i 2, 3, . . . , n, and X4 , Y4 ∈ Λ is arbitrarily given on the boundary of the planar region
R4 :
x, y ∈ Λ : Wi Hi x, y
≤
∞
0
wi
≤
0
αi x0 βi y 0 ds
−1 , i 1, 2, . . . , n
Ψ s
ds
αi x βi y
ψ −1
Wn−1 Wn Hn x, y
∞
ψ −1 s
q .
αn x βn y
αn x0 βn y 0 fi x, y, s, t ds dt
fn x, y, s, t ds dt
2.21
8
Journal of Inequalities and Applications
Corollary 2.5. Suppose that H1 –H5 hold and ux, yis a nonnegative function on Λ satisfying
n
ψ u x, y ≤ a x, y α x β y
i
i
βi y 0 αi x0 i1
uq s, tgi s, tds dt
αi x βi y
αi x0 βi y0 u s, tfi s, tϕi us, tds dt ,
q
∀ x, y ∈ Λ,
2.22
where fi , gi i 2, 3, . . . , n are nonnegative functions on Λ. Then
αn x βn y
−1
−1
−1
Ψ Wn Wn Ln x, y u x, y ≤ ψ
fn s, tds dt
,
αn x0 βn y0 2.23
for x, y ∈ ΛX5 , Y5 , where
n
L1 x, y : Ψ a x, y αi x βi y
i1
−1
Wi−1 Li−1 x, y
Li x, y : Wi−1
αi x0 βi y 0 gi s, tds dt,
αi−1 x βi−1 y
αi−1 x0 βi−1 y0 2.24
fi−1 s, tds dt ,
for i 2, 3, . . . , n, and X5 , Y5 ∈ Λ is arbitrarily given on the boundary of the planar region
R5 :
x, y ∈ Λ : Wi Li x, y ≤
∞
wi
0
Wn−1
≤
∞
0
ψ −1
Wn Ln x, y q .
ψ −1 s
αi x0 βi y 0 fi s, tds dt
ds
−1 , i 1, 2, . . . , n
Ψ s
ds
αi x βi y
αn x βn y
αn x0 βn y0 fn s, tds dt
2.25
Journal of Inequalities and Applications
9
3. Proofs and Remarks
Proof of Theorem 2.1. Obviously, the sequence wi s defined by ϕi s in 2.2 is nondecreasing
nonnegative functions and satisfies wi s ≥ ϕi s, i 1, 2, . . . , n. Moreover, the ratios
wi1 s/wi s, i 1, 2, . . . , n − 1 are all nondecreasing. From 2.1, 2.2 and 2.5, 2.6, we
have
α x β y
n
i
i
uq s, t
gi x, y, s, t ds dt
ψ u x, y ≤ a x, y αi x0 βi y0 i1
δi x γi y
δi x0 γi y0 u s, tfi x, y, s, t wi us, tds dt ,
q
3.1
∀ x, y ∈ Λ.
We first discuss the case that ax, y > 0 for all x, y ∈ Λ. Consider the auxiliary inequality
α x β y
n
i
i
ψ u x, y ≤ aX, Y uq s, t
gi X, Y, s, tds dt
αi x0 βi y0 i1
δi x γi y
δi x0 γi y0 3.2
uq s, tfi X, Y, s, twi us, tds dt ,
for all x, y ∈ ΛX, Y , where x0 ≤ X ≤ X1 and y0 ≤ Y ≤ Y1 are chosen arbitrarily. Let z1 x, y
denote the function on the right-hand side of 3.2, which is a nonnegative and nondecreasing
function on ΛX, Y and z1 x0 , y aX, Y . Then, we get the equivalent form of 3.2
u x, y ≤ ψ −1 z1 x, y , ∀ x, y ∈ ΛX, Y .
3.3
Since wi is nondecreasing and satisfies wi u > 0 for u > 0. By the definition of z1 , hypothesis
H4 , the monotonicity of ψ −1 and z1 , and 3.3, we have
βi y
n
∂z1 x, y
αi x
uq αi x, t
gi X, Y, αi x, tdt
∂x
β
y
i 0
i1
δi x
γi y
γi y0 u δi x, tfi X, Y, δi x, twi uδi x, tdt
q
10
Journal of Inequalities and Applications
≤
n
αi x
i1
βi y βi y 0 δi x
ψ −1 z1 αi x, t
γi y γi y0 q
gi X, Y, αi x, tdt
ψ −1 z1 δi x, t
q
fi X, Y, δi x, t
−1
×wi ψ z1 δi x, t dt
βi y
n
q αi x
gi X, Y, αi x, tdt
≤ ψ z1 x, y
βi y0 i1
γi y
−1
δi x
fi X, Y, δi x, twi ψ z1 δi x, t dt .
γi y0 3.4
−1
From 3.4, we have
βi y
n
∂z1 x, y /∂x
αi x
gi X, Y, αi x, tdt
−1 q ≤
ψ z1 x, y
βi y 0 i1
δi x
γi y
γi y0 fi X, Y, δi x, t
3.5
−1
×wi ψ z1 δi x, t dt .
Keeping y fixed in 3.5, setting s x, integrating both sides of 3.5 with respect to s from x0
to x, and using the definition of Ψ in 2.3, we have
Ψ z1 x, y
n
≤ Ψ z1 x0 , y x αi s
x δi s
x0
α X β Y n
i
i
i1
αi x0 ≤ An X, Y βi y 0 δi x0 δi x0 γi y0 γi y
γi y0 gi X, Y, αi s, tdt ds
fi X, Y, δi s, twi ψ z1 δi s, t dt ds
−1
gi X, Y, s, tds dt
δi x γi y
n δi x γi y
i1
βi y0 x0
i1
≤ ΨaX, Y βi y
γi y0 fi X, Y, s, twi ψ −1 z1 s, t ds dt
fi X, Y, s, twi ψ −1 z1 s, t ds dt,
3.6
Journal of Inequalities and Applications
11
for all x, y ∈ ΛX, Y , where
An X, Y ΨaX, Y n αi X βi Y αi x0 i1
βi y 0 gi X, Y, s, tds dt.
3.7
Let
v x, y Ψ z1 x, y .
3.8
From 3.6, we have
δi x γi y
n
v x, y ≤ An X, Y δi x0 i1
γi y0 fi X, Y, s, twi ψ −1 Ψ−1 vs, t ds dt,
3.9
for all x, y ∈ ΛX, Y . We claim that the unknown function v in 3.9 satisfies
v x, y ≤
Wn−1
Wn En X, Y δn X γn Y δn x0 γn y0 fn X, Y, s, tds dt ,
3.10
for all x, y ∈ ΛX, Y , where
3.11
E1 X, Y : An X, Y ,
Ei X, Y :
−1
Wi−1
Wi−1 Ei−1 X, Y δi−1 X γi−1 Y δi−1 x0 γi−1 y0 fi−1 X, Y, s, tdsdt , i 2, 3, . . . , n
3.12
Now, we prove 3.10 by induction. For n 1, let z2 x, y denote the function on the righthand side of 3.9, which is a nonnegative and nondecreasing function on ΛX, Y , z2 x0 , y A1 X, Y and vx, y ≤ z2 x, y. Then we have
γ1 y
∂z2 x, y
δ1 x
f1 X, Y, δ1 x, tw1 ψ −1 Ψ−1 vδ1 x, t dt
∂x
γ1 y0 γ1 y
−1
−1
≤ w1 ψ Ψ z2 x, y
δ1 x
f1 X, Y, δ1 x, tdt ,
γ1 y0 3.13
for all x, y ∈ ΛX, Y . From 3.13, we have
γ1 y
∂z2 x, y /∂x
f1 X, Y, δ1 x, tdt,
≤ δ1 x
w1 ψ −1 Ψ−1 z2 x, y
γ1 y0 ∀ x, y ∈ ΛX, Y .
3.14
12
Journal of Inequalities and Applications
Keeping y fixed in 3.14, setting s x, integrating both sides of 3.14 with respect to s from
x0 to x, and using the definition of Wi in 2.4, we have
W1 z2 x, y ≤ W1 z2 x0 , y x x0
W1 A1 X, Y ≤ W1 A1 X, Y δ1 s
γ1 y
γ1 y0 δ1 x γ1 y
δ1 x0 γ1 y0 δ1 X γ1 Y δ1 x0 γ1 y0 f1 X, Y, δ1 s, tdt ds
f1 X, Y, s, tds dt
3.15
f1 X, Y, s, tds dt,
for all x, y ∈ ΛX, Y . Using vx, y ≤ z2 x, y, from 3.15, we obtain
v x, y ≤ z2 x, y ≤
W1−1
W1 A1 X, Y δ1 X γ1 Y δ1 x0 γ1 y0 f1 X, Y, s, tds dt ,
3.16
for all x, y ∈ ΛX, Y . This proves that 3.10 is true for n 1.
Next, we make the inductive assumption that 3.10 is true for n k. Now, we consider
v x, y ≤ Ak1 X, Y k1 δi x γi y
i1
γi y0 δi x0 fi X, Y, s, twi ψ −1 Ψ−1 vs, t ds dt,
3.17
for all x, y ∈ ΛX, Y . Let z3 x, y denote the nonnegative and nondecreasing function on the
right-hand side of 3.17. Then z3 x0 , y Ak1 X, Y and
v x, y ≤ z3 x, y .
3.18
Let
φi u :
wi1 u
,
w1 u
i 1, 2, . . . , k.
3.19
Journal of Inequalities and Applications
13
By 2.2, we see that each φi , i 1, 2, . . . , k, is a nondecreasing function. Then, we have
γi y
−1 −1
k1 Ψ vδi x, t dt
∂z3 x, y /∂x
i1 δi x γi y0 fi X, Y, δi x, twi ψ
w1 ψ −1 Ψ−1 z3 x, y
w1 ψ −1 Ψ−1 z3 x, y
k1
≤
γi y
δi x
i1
≤ δ1 x
γi y0 γ1 y
γ1 y0 fi X, Y, δi x, twi ψ −1 Ψ−1 z3 δi x, t dt
w1 ψ −1 Ψ−1 z3 x, y
f1 X, Y, δ1 x, tdt
γi1 y
k
fi1 X, Y, δi1 x, t
δi1 x
γi1 y0 i1
× φi ψ −1 Ψ−1 z3 δi1 x, t dt,
3.20
for all x, y ∈ ΛX, Y . Keeping y fixed in 3.20, setting s x, integrating both sides of 3.20
with respect to s from x0 to x, and using the definition of Wi in 2.4, we have
W1 z3 x, y ≤ W1 Ak1 X, Y δ1 X γ1 Y δ1 x0 k δi1 x γi1 y
i1
δi1 x0 γi1 y0 γ1 y0 f1 X, Y, s, tds dt
fi1 X, Y, s, tφi ψ −1 Ψ−1 z3 s, t
3.21
ds dt,
for all x, y ∈ ΛX, Y . Let
η x, y : W1 z3 x, y ,
θ1 X, Y : W1 Ak1 X, Y δ1 X γ1 Y δ1 x0 γ1 y 0 3.22
f1 X, Y, s, tds dt.
3.23
Using 3.22 and 3.23, from 3.21, we have
η x, y ≤ θ1 X, Y k δi1 x γi1 y
i1
δi1 x0 γi1 y0 ds dt,
fi1 X, Y, s, tφi ψ −1 Ψ−1 W1−1 ηs, t
∀ x, y ∈ ΛX, Y .
3.24
14
Journal of Inequalities and Applications
It has the same form as 3.9. Let ρs : ψ −1 Ψ−1 W1−1 s. Since ψ −1 , Ψ−1 , W1−1 , and
φi are continuous, nondecreasing, and positive on 0, ∞, each φi ρs is continuous,
nondecreasing, and positive on 0, ∞. Moreover,
wi1 ρs
φi1 ρs
ϕi1 τ
,
max
wi τ
τ∈0, ρs
φi ρs
wi ρs
i 1, 2, . . . , k − 1,
3.25
which are also continuous, nondecreasing, and positive on 0, ∞. Therefore, the inductive
assumption for 3.9 can be used to 3.24, and then we have
δk1 X γk1 Y −1
η x, y ≤ Φk Φk θk X, Y fk1 X, Y, s, tds dt ,
δk1 x0 γk1 y0 3.26
for all x, y ∈ ΛX, Y , where
Φi u :
u
0
φi
ψ −1
ds
−1 ,
W1 s
Ψ−1
θi1 X, Y :
Φ−1
i
Φi θi X, Y u > 0, i 1, 2, . . . , k,
δi1 X γi1 Y δi1 x0 γi1 y0 fi1 X, Y, s, tds dt ,
3.27
3.28
i 1, 2, . . . , k − 1
We note that
Φi u u
0
w1 ψ −1 Ψ−1 W1−1 s ds
wi1 ψ −1 Ψ−1 W1−1 s
W −1 u
1
wi1
0
3.29
ds
−1 Ψ s
ψ −1
Wi1 W1−1 u ,
i 1, 2, . . . , k.
Thus, from 3.18, 3.22, 3.26, and 3.29, we have
v x, y ≤ z3 x, y W1−1 η x, y
≤
W1−1
−1
Wk1
Φ−1
k
Φk θk X, Y δk1 X γk1 Y Wk1 W1−1 θk X, Y δk1 x0 γk1 y0 fk1 X, Y, s, tds dt
δk1 X γk1 Y δk1 x0 γk1 y0 fk1 X, Y, s, tds dt ,
3.30
Journal of Inequalities and Applications
15
for all x, y ∈ ΛX, Y . We can prove that the term of W1−1 θk X, Y in 3.30 is just the same
as Ek1 X, Y defined in 3.12. Let θi X, Y : W1−1 θi X, Y . By 3.23, we have
θ1 X, Y W1−1 θ1 X, Y W1−1
W1 Ak1 X, Y δ1 X γ1 Y γ1 y 0 δ1 x0 f1 X, Y, s, tds dt
3.31
E2 X, Y .
Then using 3.28 and 3.29, we get
δi X γi Y −1
−1
fi X, Y, s, tds dt
θi X, Y W1 Φi−1 Φi−1 θi−1 X, Y δi x0 γi y0 δi X γi Y −1
−1
Wi W1 θi−1 X, Y Wi
fi X, Y, s, tds dt
δi x0 γi y0 δi X γi Y −1
Wi θi−1 X, Y Wi
fi X, Y, s, tds dt
δi x0 γi y0 δi X γi Y −1
Wi Ei X, Y Wi
fi X, Y, s, tds dt
δi x0 γi y0 3.32
i 2, 3, . . . , k.
Ei1 X, Y ,
This proves that W1−1 θk X, Y in 3.30 is just the same as Ek1 X, Y defined in 3.12.
Therefore, from 3.29, 3.30, and 3.32, we obtain
Φi θi X, Y δi1 X γi1 Y δi1 x0 γi1 y0 Wi1 Ei1 X, Y ≤
∞
0
fi1 X, Y, s, tds dt
δi1 X γi1 Y δi1 x0 ds
−1 −1 wi1 ψ Ψ s
γi1 y0 fi1 X, Y, s, tds dt
W1 ∞
0
φi
ψ −1
3.33
ds
−1 ,
W1 s
Ψ−1
i 1, 2, . . . , k. The relations of 3.33 imply that in 3.26 and 3.28
Φi θi X, Y δi1 X γi1 Y δi1 x0 γi1 y0 ,
fi1 X, Y, s, tds dt ∈ Dom Φ−1
i
3.34
16
Journal of Inequalities and Applications
i 1, 2, . . . , k. Hence, 3.30 can be equivalently written as
δk1 X γk1 Y −1
v x, y ≤ Wk1 Wk1 Ek1 X, Y fi1 X, Y, s, tds dt,
δk1 x0 γk1 y0 3.35
for all x, y ∈ ΛX, Y . The claim in 3.10 is proved by induction.
Therefore, by 3.3, 3.8, and 3.10, we have
u x, y ≤ ψ −1 z1 x, y ≤ ψ −1 Ψ−1 v x, y
≤ψ
−1
Ψ
−1
Wn−1
Wn En X, Y δn X γn Y δn x0 γn y0 fn X, Y, s, tds dt
3.36
,
for all x, y ∈ ΛX, Y . Hence, we obtain the estimation of the unknown function u in the
auxiliary inequality 3.2.
Letting x X, y Y , from 3.36, we have
uX, Y ≤ ψ
−1
Ψ
−1
Wn−1
Wn En X, Y δn X γn Y δn x0 γn y0 fn X, Y, s, tds dt
,
3.37
for all x0 ≤ X ≤ X1 , y0 ≤ Y ≤ Y1 . Since Ξi X, Y Ei X, Y and X and Y are arbitrarily chosen,
this proves 2.7.
The remainder case is that ax, y 0 for some x, y ∈ Λ. Let
aε x, y a x, y ε,
3.38
where ε > 0 is an arbitrary small number. Obviously, aε x, y > 0, for all x, y ∈ Λ. Using the
same arguments as above, where ax, y is replaced with aε x, y, we get
δn x γn y
−1
−1
−1
u x, y ≤ ψ
,
Ψ
Wn Wn En, ε x, y fn X, Y, s, tds dt
δn x0 γn y0 3.39
for all x, y ∈ ΛX1 , Y1 . Letting ε → 0 , we obtain 2.7 because of continuity of aε in ε and
continuity of ψ −1 , Ψ−1 , Wi , and Wi−1 for i 1, 2, . . . , n. This completes the proof.
The proofs of Corollary 2.2, 2.3, 2.5 and Theorem 2.4 are similar to the argument in the
proofs of Theorem 2.1 with appropriate modification. We omit the details here.
Remark 3.1. When ax, y a, n 1 and gx, y, s, t p/p − qg1 s, t, fx, y, s, t p/p −
qg2 s, t, Corollary 2.2 reduces to Theorem 2.4 in 9.
Remark 3.2. When ψu up and q 0, Corollary 2.3 reduces to Theorem 1 of Wang 16.
Journal of Inequalities and Applications
17
Remark 3.3. When fi x, y, s, t cx, yfi s, t, gi x, y, s, t cc, ygi s, t, Theorem 2.4
reduces to Theorem 2.1 of Kim 12.
4. Applications
In this section, we apply our results to study the boundedness and uniqueness of the
solutions of boundary value problem to a partial differential equation. We consider the partial
differential equation with the initial boundary conditions:
∂2 ψ z x, y
F x, y, ϕ1 z δ1 x, γ1 y , . . . , ϕn z δn x, γn y
,
∂x∂y
ψ z x, y0 a1 x, ψ z x0 , y a2 y , a1 x0 a2 y0 0,
4.1
4.2
for all x, y ∈ Λ, where Λ I × J is defined as in Section 2, δi , γi are defined in H4 , ψ is a
continuous and strictly increasing odd function on R, satisfying ψ0 0, ψu > 0 for u > 0,
F : Λ × Rn → R, a1 : I → R, a2 : J → R, and ϕi : R → R are nondecreasing continuous
functions, and the ratio ϕi1 /ϕi is also nondecreasing, and ϕi u > 0 for u > 0 i 1, 2, . . . , n.
In the following corollary, we firstly apply our result to discuss boundedness on the
solution of problem 4.1.
Corollary 4.1. Assume that F : Λ×Rn → R is a continuous function for which there exist a constant
q > 0, nonnegative functions fi x, y ∈ CΛ, R , ϕi ∈ CR , R , i 1, 2, . . . , n, such that
n
F x, y, ϕ1 u1 , . . . , ϕn un ≤
|ui |q fi x, y ϕi |ui |,
4.3
i1
a1 x a2 y ≤ a x, y ,
4.4
for all x, y ∈ Λ, and a : Λ → R is nondecreasing in each variable. If zx, y is any solution of
problem 4.1 with condition 4.2, then
x y
z x, y ≤ ψ −1 Ψ−1 Wn−1 Wn K
n x, y fn s, tds dt
,
x0
4.5
y0
for all x, y ∈ ΛX6 ,Y6 , where
1 x, y : Ψ a x, y ,
K
i x, y :
K
−1
Wi−1
i−1 x, y
Wi−1 K
x y
x0
fi−1 s, tds dt ,
y0
i 2, 3, . . . , n,
4.6
18
Journal of Inequalities and Applications
and Ψ, Ψ−1 , W, W −1 are as defined in Theorem 2.1. X6 , Y6 ∈ Λ lies on the boundary of the planar
region
R6 :
i x, y x, y ∈ Λ : Wi K
x y
x0
≤
∞
ϕi
0
ψ −1
Wn−1
Wn
fi s, tds dt
y0
ds
−1 , i 1, 2, . . . , n,
Ψ s
n x, y K
x y
x0
4.7
fn s, tds dt
y0
≤
∞
0
ds
q .
ψ −1 s
Proof. It is easy to see that the solution zx, y of 4.1 satisfies the following equivalent
integral equation:
ψ z x, y a1 x a2 y x y
x0
F s, t, ϕ1 z δ1 s, γ1 t , . . . , ϕn z δn s, γn t ds dt.
y0
4.8
By 4.3,4.4, and 4.8, we have
ψ z x, y a1 x a2 y x y
x0
F s, t, ϕ1 z δ1 s, γ1 t , . . . , ϕn z δn s, γn t ds dt
y0
n
≤ a x, y i1
n
≤ a x, y i1
x y
x0
z δi s, γi t q fi s, tϕi z δi s, γi t ds dt
4.9
y0
fi δi−1 s, γi−1 t
q
−1 −1 |zs, t| ϕi |zs, t|ds dt.
δ
γ
δ
γ
s
t
γi y0 i
i
i i
δi x γi y
δi x0 Since |ψzx, y| ψ|zx, y|, 4.9 is the form of 2.14. Applying Corollary 2.3 to
inequality 4.9, using the relation
x y
fi δi−1 s, γi−1 t
fi s, tds dt,
−1 −1 ds dt γi y0 δi δi s γi γi t
x0 y 0
δi x γi y
δi x0 we obtain the estimation of zx, y as given in 4.5.
4.10
Journal of Inequalities and Applications
19
Corollary 4.1 gives a condition of boundedness for solutions, concretely. If there is a
M > 0,
i x, y < M,
K
x y
x0
fi s, tds dt < M,
i 1, 2, . . . , n,
y0
4.11
for all x, y ∈ Λ. Then every solution zx, y of 4.1 is bounded on Λ.
Next, we discuss the uniqueness of the solutions of 4.1.
Corollary 4.2. Additionally, assume that
F s, t, ϕ1 u11 , . . . , ϕn u1n − F s, t, ϕ1 u21 , . . . , ϕn u2n ≤
n
q fi s, tψu1i − ψu2i ϕi ψu1i − ψu2i ,
4.12
i1
for u1i , u2i ∈ R, i 1, 2, . . . , n, and x, y ∈ Λ, where Λ is defined as in Section 2, 0 < q < 1 is
nondecreasing with the
a constant, fi ∈ CΛ, R , ϕi : R → R , i 1, 2, . . . , n are continuous
∞
nondecreasing ratio ϕi1 /ϕi such that ϕi u > 0 for all u > 0, and 0 ds/ϕi s ∞, i 1, 2, . . . , n,
and ψ : R → R is a strictly increasing odd function satisfying ψu > 0, for all u > 0. Then, 4.1
has at most one solution on Λ
Proof. Let zx, y and zx, y be two solutions of 4.1. By 4.8 and 4.12, we have
ψ z x, y − ψ z x, y ≤ε
n x y
i1
x0
q
fi s, tψ z δi s, γi t − ψ z δi s, γi t y0
× ϕi ψ z δi s, γi t − ψ z δi s, γi t ds dt
ε
4.13
q
fi δi−1 s, γi−1 t zs, t
−1 −1 ψzs, t − ψ
γi y0 δi δi s γi γi t
n δi x γi y
i1
δi x0 zs, t ds dt
× ϕi ψzs, t − ψ
for all x, y ∈ Λ, which is an inequality of the form 2.14, where ε > 0 is an arbitrary
small number. Applying Corollary 2.2, we obtain an estimation of the difference |ψzx, y −
ψ
zx, y| in the form 4.5. Namely,
1/1−q
x y
ψ z x, y − ψ z x, y ≤ Ω−1
Ωn Pn x, y 1 − q
fn s, tds dt
,
n
x0
y0
4.14
20
Journal of Inequalities and Applications
for all x, y ∈ Λ, where
P1 x, y : ε1−q ,
Pi x, y :
Ω−1
i−1
Ωi−1 Pi−1 x, y
1−q
x y
x0
Ωi u :
u
1
ds
,
ϕi s1/1−q
fi−1 s, tdsdt , i 2, 3, . . . , n,
y0
u ≥ 0,
4.15
i 1, 2, . . . , n,
and Ω−1
i denotes the inverse function of Ωi .
Furthermore, by the definition of Ωi , we conclude that
lim Ωi u −∞,
u→0
lim Ω−1
i u 0,
i 1, 2, . . . , n.
u → −∞
4.16
Letting ε → 0, it follows that
Ωi Pi x, y 1 − q
x y
x0
Ω−1
i
Ωi Pi x, y 1 − q
x y
x0
fi s, tds dt −∞,
i 1, 2, . . . , n,
y0
4.17
fi s, tds dt 0,
i 2, . . . , n.
y0
Thus, from 4.5, we deduce that |ψzx, y − ψ
zx, y| ≤ 0, implying that zx, y zx, y,
for all x, y ∈ Λ, since ψ is strictly increasing. The uniqueness is proved.
Acknowledgments
The authors are very grateful to the editor and the referees for their helpful comments
and valuable suggestions. This work is supported by the Natural Science Foundation of
Guangxi Autonomous Region 0991265, the Scientfic Research Foundation of the Education
Department of Guangxi Autonomous Region 200707MS112, the Key Discipline of Applied
Mathematics 200725 and the Key Project 2009YAZ-N001 of Hechi University of China.
References
1 T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of
differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919.
2 R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol.
10, pp. 643–647, 1943.
3 D. Baı̆nov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its
Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
4 I. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of
differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 7, pp. 81–94, 1956.
5 B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197 of Mathematics in Science and
Engineering, Academic Press, San Diego, Calif, USA, 1998.
Journal of Inequalities and Applications
21
6 R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its
applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005.
7 R. P. Agarwal, Y.-H. Kim, and S. K. Sen, “New retarded integral inequalities with applications,” Journal
of Inequalities and Applications, vol. 2008, Article ID 908784, 15 pages, 2008.
8 C.-J. Chen, W.-S. Cheung, and D. Zhao, “Gronwall-Bellman-type integral inequalities and applications to BVPs,” Journal of Inequalities and Applications, vol. 2009, Article ID 258569, 15 pages, 2009.
9 W.-S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 9, pp. 2112–2128, 2006.
10 S. K. Choi, S. Deng, N. J. Koo, and W. Zhang, “Nonlinear integral inequalities of Bihari-type without
class H,” Mathematical Inequalities & Applications, vol. 8, no. 4, pp. 643–654, 2005.
11 S. S. Dragomir and Y.-H. Kim, “Some integral inequalities for functions of two variables,” Electronic
Journal of Differential Equations, vol. 2003, no. 10, pp. 1–13, 2003.
12 Y.-H. Kim, “Gronwall, Bellman and Pachpatte type integral inequalities with applications,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2641–e2656, 2009.
13 O. Lipovan, “Integral inequalities for retarded Volterra equations,” Journal of Mathematical Analysis
and Applications, vol. 322, no. 1, pp. 349–358, 2006.
14 Q.-H. Ma and E.-H. Yang, “On some new nonlinear delay integral inequalities,” Journal of Mathematical
Analysis and Applications, vol. 252, no. 2, pp. 864–878, 2000.
15 Q.-H. Ma and E.-H. Yang, “Some new Gronwall-Bellman-Bihari type integral inequalities with delay,”
Periodica Mathematica Hungarica, vol. 44, no. 2, pp. 225–238, 2002.
16 W.-S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to
BVP,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 144–154, 2007.
17 W. Zhang and S. Deng, “Projected Gronwall-Bellman’s inequality for integrable functions,”
Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 393–402, 2001.