Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 258569, 15 pages
doi:10.1155/2009/258569
Research Article
Gronwall-Bellman-Type Integral Inequalities and
Applications to BVPs
Chur-Jen Chen, Wing-Sum Cheung, and Dandan Zhao
Department of Mathematics, Tunghai University, Taichung, Taiwan
Correspondence should be addressed to Wing-Sum Cheung, wscheung@hku.hk
Received 9 May 2008; Revised 26 November 2008; Accepted 29 January 2009
Recommended by Alexander Domoshnitsky
We establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two
variables. These inequalities generalize former results and can be used as handy tools to study the
qualitative as well as the quantitative properties of solutions of differential equations. Example of
applying these inequalities to derive the properties of BVPs is also given.
Copyright q 2009 Chur-Jen Chen 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.
1. Introduction
The Gronwall-Bellman inequality states that if u and f are nonnegative continuous functions
on an interval a, b satisfying
ut ≤ C t
fsusds
1.1
a
for some constant C ≥ 0, then
ut ≤ C exp
t
fsds ,
t ∈ a, b.
1.2
a
Inequality 1.2 provides an explicit bound to the unknown function and hence
furnishes a handy tool in the study of quantitative and qualitative properties of solutions
of differential and integral equations. Because of its fundamental importance over the years
many generalizations and analogous results of 1.2 have been established see, e.g., 1–20.
Among various of Gronwall-Bellman-type inequalities, a very useful one is the following.
2
Journal of Inequalities and Applications
Theorem 1.1 see 21. If u and f are nonnegative continuous functions defined on 0, ∞ such
that
x
u2 x ≤ k2 2 fsusds
1.3
0
for all x ∈ 0, ∞, where k ≥ 0 is a constant, then
ux ≤ k x
1.4
fsds
0
for all x ∈ 0, ∞.
Inequality 1.4 is called Ou-Iang’s inequality, which was established by Ou-Iang
during his study of the boundedness of certain kinds of second-order differential equations.
Recently, Pachpatte established the following generalization of Ou-Iang-type inequality.
Theorem 1.2 see 20. Let u, f, g be nonnegative continuous functions defined on 0, ∞, and let
ϕ be a continuous nondecreasing function on 0, ∞ with ϕr > 0 for r > 0. If
x
fsus gsusϕ us ds
u2 x ≤ k2 2
1.5
0
for all x ∈ 0, ∞, where k ≥ 0 is a constant, then
x
x
ux ≤ Φ−1 Φ k fsds gsds
0
1.6
0
r
for all x ∈ 0, x1 , where Φ−1 is the inverse function
of Φ, Φr : 1 ds/ϕs, r > 0, and x1 ∈
x
x
0, ∞ is chosen such that Φk 0 fsds 0 gsds ∈ DomΦ−1 for all x ∈ 0, x1 .
Bainov-Simeonov and Lipovan gave the following Gronwall-Bellman-type inequalities, which are useful in the study of global existence of solutions of certain integral equations
and functional differential equations.
Theorem 1.3 see 1. Let I 0, a, J 0, b, where, for the sake of covenience, we allow a, b to
be ∞ (in this case we mean interval 0, ∞). Let c ≥ 0 be a constant, let ϕ ∈ C0, ∞, 0, ∞ be
nondecreasing with ϕr > 0 for r > 0, and b ∈ CI × J, 0, ∞. If u ∈ CI × J, 0, ∞ satisfies
x y
bs, tϕ us, t dt ds
1.7
x y
bs, tdt ds
ux, y ≤ Φ Φc 1.8
ux, y ≤ c 0
0
for all x, y ∈ I × J, then
−1
0
0
Journal of Inequalities and Applications
3
for all x, y ∈ 0, x1 × 0, y1 , where Φ is defined as in Theorem 1.2 and x1 , y1 ∈ I × J is chosen
x y
such that Φc 0 0 bs, tdt ds ∈ DomΦ−1 for all x, y ∈ 0, x1 × 0, y1 .
Theorem 1.4 see 13. Suppose u, f are nonnegative continuous functions defined on xo , X,
ϕ ∈ C0, ∞ with ϕr > 0 for r > 0, and α ∈ C1 xo , X, xo , X with αx ≤ x on xo , X are
nondecreasing functions. If
ux ≤ k αx
αxo fsϕ us ds
1.9
for all x ∈ xo , X, where k ≥ 0 is a constant, then
αx
ux ≤ Φ−1 Φk fsds
1.10
αxo for all x ∈ xo , x1 , where Φ is defined as in Theorem 1.2, and x1 ∈ xo , X is chosen such that
αx
Φk αxo fsds ∈ DomΦ−1 for all x ∈ xo , x1 .
Very recently, the above results have been further generalized by Cheung to the
following.
Theorem 1.5 see 7. Let k ≥ 0 and p > q > 0 be constants. Let a, b ∈ Cxo , X ×
yo , Y , 0, ∞, α, γ ∈ C1 xo , X, xo , X, β, δ ∈ C1 yo , Y , yo , Y , and ϕ, h ∈
C0, ∞, 0, ∞ be functions satisfying
i α, β, γ, δ are nondecreasing and α, γ ≤ idI , β, δ ≤ idJ ;
ii ϕ is nondecreasing with ϕr > 0 for r > 0.
If u ∈ Cxo , X × yo , Y , 0, ∞ satisfies
p
u x, y ≤ k p−q
p
p
p−q
αx βy
αxo βyo γx δy
γxo δyo as, tuq s, tdt ds
bs, tuq s, tϕ us, t dt ds
1.11
for all x, y ∈ xo , X × yo , Y , then
1/p−q
1−q/p
Ax, y Bx, y
ux, y ≤ Φ−1
p−q Φp−q k
1.12
for all x, y ∈ xo , x1 × yo , y1 , where
Ax, y αx βy
αxo βyo as, tdt ds,
Bx, y γx δy
γxo δyo bs, tdt ds,
1.13
4
Journal of Inequalities and Applications
r
1/p−q
∈ C0, ∞, 0, ∞, and
Φ−1
p−q is the inverse function of Φp−q , Φp−q r : 1 ds/ϕs
x1 , y1 ∈ xo , X × yo , Y is chosen such that Φp−q k1−q/p Ax, y Bx, y ∈ DomΦ−1
p−q for
all x, y ∈ xo , x1 × yo , y1 .
In this paper, we intend to establish some new nonlinear Gronwall-Bellman-Ou-Iang
type integral inequalities with two variables. The setup is basically along the line of 7 but
this is by all means a nontrivial improvement of the results there. Examples are also given
to illustrate the usefulness of these inequalities in the study of qualitative as well as the
quantitative properties of solutions of BVPs.
2. Main Results
Throughout this paper, xo , yo ∈ R are two fixed numbers. Let I xo , X ⊂ R, J yo , Y ⊂ R,
and Δ I × J ⊂ R2 , here we allow X or Y to be ∞. We denote by Ci U, V the set of all
i-times continuously differentiable functions of U into V , and Co U, V CU, V . Partial
derivatives of a function zx, y are denoted by zx , zy , zxy , and so forth. The identity function
will be denoted as id and so, in particular, idU is the identity function of U onto itself.
Let R 0, ∞ and for any ϕ, ψ, h ∈ CR , R , define
Φh r :
r
ds
−1 ,
ϕ
h
s
1
Ψh r :
Φh 0 : lim Φh r,
r
ds
−1 ,
ψ
h
s
1
r > 0,
2.1
Ψh 0 : lim Ψh r.
r →0
r →0
Note that we allow Φh 0 and Ψh 0 to be −∞ here.
Theorem 2.1. Suppose u ∈ CΔ, R . Let c ≥ 0 be a constant. If γ ∈ C1 I, I, δ ∈ C1 J, J,
b ∈ CΔ, R , and ϕ, h ∈ CR , R are functions satisfying
i γ, δ are nondecreasing and γ ≤ idI , δ ≤ idJ ;
ii ϕ is nondecreasing with ϕr > 0 for r > 0;
iii h is strictly increasing with h0 0 and ht → ∞ as t → ∞;
iv for any x, y ∈ Δ,
h ux, y ≤ c γx δy
γxo δyo bs, tϕ us, t dt ds,
2.2
then we have
ux, y ≤ h−1 Φ−1
h Φh c Bx, y
2.3
for all x, y ∈ xo , x1 × yo , y1 , where
Bx, y γx δy
γxo δyo bs, tdt ds,
2.4
Journal of Inequalities and Applications
5
is the inverse function of Φh , and x1 , y1 ∈ Δ is chosen such that Φh c Bx, y ∈ DomΦ−1
Φ−1
h
h
for all x, y ∈ xo , x1 × yo , y1 .
Proof. It suffices to only consider the case c > 0, since the case c 0 can then be arrived at
by continuity argument. If we let gx, y denote the right-hand side of 2.2, then we have
g > 0, u ≤ h−1 g on Δ, and g is nondecreasing in each variable. Hence for any x, y ∈ Δ,
gx x, y γ x
≤ γ x
δy
δyo δy
δyo b γx, t ϕ u γx, t dt
b γx, t ϕ h−1 g γx, t dt
δy
≤ γ xϕ h−1 g γx, δy
δyo 2.5
b γx, t dt
−1 δy b γx, t dt.
≤ γ xϕ h gx, y
δyo By the definition of Φh ,
Φh ◦ gx x, y dΦh · gx x, y
dr gx,y
δy
1
· γ xϕ h−1 gx, y
−1 ϕ h gx, y
δy
b γx, t dt .
≤ γ x
≤
δyo b γx, t dt
2.6
δyo Integrating with respect to x over xo , x, we have
Φh gx, y − Φh gxo , y ≤
x
γ ξ
δy
xo
≤
γx1 δy
γxo δyo δyo b γξ, t dt dξ
2.7
bs, tdt ds,
that is,
Φh gx, y ≤ Φh gxo , y Bx, y.
2.8
ux, y ≤ h−1 g ≤ h−1 Φ−1
h Φh c Bx, y
2.9
Therefore,
for all x, y ∈ xo , x1 × yo , y1 .
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Journal of Inequalities and Applications
Corollary 2.2. Suppose u ∈ CΔ, R . Let c ≥ 0 be a constant. If γ ∈ C1 I, I, δ ∈ C1 J, J,
b ∈ CΔ, R , and h ∈ CR , R are functions satisfying
i γ, δ are nondecreasing and γ ≤ idI , δ ≤ idJ ;
ii h is strictly increasing with h0 0 and ht → ∞ as t → ∞;
iii for any x, y ∈ Δ,
h ux, y ≤ c γx δy
γxo δyo bs, th us, t dt ds,
2.10
then we have
ux, y ≤ h−1 ceBx,y
2.11
for all x, y ∈ Δ, where
Bx, y γx δy
γxo δyo 2.12
bs, tdt ds.
Proof. Let ϕ h, then Φh r ln r. The corollary now follows immediately from Theorem 2.1.
Remark 2.3. It is easily seen that Theorem 2.1 generalizes Theorems 1.3 and 1.4.
Theorem 2.4. Suppose u ∈ CΔ, R . Let c ≥ 0 be a constant. If α, γ ∈ C1 I, I, β, δ ∈ C1 J, J,
a, b ∈ CΔ, R , and ϕ, h ∈ CR , R are functions satisfying
i α, β, γ, δ are nondecreasing and α, γ ≤ idI , β, δ ≤ idJ ;
ii ϕ is nondecreasing with ϕr > 0 for r > 0;
iii ht and Ht : ht/t, t > 0, are strictly increasing with Ht → ∞ as t → ∞;
iv for any x, y ∈ Δ,
h ux, y ≤ c αx βy
αxo βyo as, tus, tdt ds γx δy
γxo δyo bs, tus, tϕ us, t dt ds,
2.13
then we have
ux, y ≤ H −1 Φ−1
Φ
H
H
c
Ax,
y
Bx,
y
h−1 c
2.14
for all x, y ∈ xo , x1 × yo , y1 , where
Ax, y αx βy
αxo βyo as, tdt ds,
Bx, y γx δy
γxo δyo bs, tdt ds,
2.15
Journal of Inequalities and Applications
7
−1
c Ax, y Φ−1
H is the inverse function of ΦH , and x1 , y1 ∈ Δ is chosen such that ΦH c/h
−1
Bx, y ∈ DomΦH for all x, y ∈ xo , x1 × yo , y1 .
Proof. It suffices to consider the case c > 0. If we let gx, y denote the right-hand side of
2.13, then we have g > 0, u ≤ h−1 g on Δ, and g is nondecreasing in each variable. Hence
for any x, y ∈ Δ,
gx x, y α x
βy
βyo γ x
≤ α x
δy
δyo βy
βyo γ x
a αx, t u αx, t dt
b γx, t u γx, t ϕ u γx, t dt
a αx, t h−1 g αx, t dt
δy
δyo b γx, t h−1 g γx, t ϕ h−1 g γx, t dt
βy
≤ α xh g αx, βy
−1
βyo a αx, t dt
2.16
δy γ xh g γx, βy
b γx, t ϕ h−1 g γx, t dt
−1
≤ α xh−1 gx, y
βy
βyo γ xh−1 gx, y
δyo a αx, t dt
δy
δyo b γx, t ϕ h−1 g γx, t dt,
that is,
gx x, y
≤ α x
h−1 gx, y
βy
βyo a αx, t dt γ x
δy
δyo b γx, t ϕ h−1 g γx, t dt.
2.17
Integrating with respect to x over xo , x, we get
x
gx ξ, y
dξ ≤
−1
gξ, y
xo h
x
α ξ
βy
βyo xo
x
γ ξ
δy
αx βy
αxo βyo as, tdt ds
γx δy
γxo δyo Ax, y −1 b γξ, t ϕ h g γξ, t dt dξ
δyo xo
≤
a αξ, t dt dξ
bs, tϕ h−1 gs, t dt ds
γx δy
γxo δyo bs, tϕ h−1 gs, t dt ds.
2.18
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Journal of Inequalities and Applications
Since
x
h−1
gξ ξ, y
gξ, y x
g · gξ ξ, y
gξ,
y
dξ
2 dξ,
−1 gξ, y
h−1 gξ, y xo
xo h
xo
h−1 gξ, y
x
2.19
h is strictly increasing and g > 0 is nondecreasing with respect to each variable. Together with
2.18, we have
x
gx x, y
gx, y x
≥ −1 −1 gx, y
h gx, y xo
xo h
gx, y
gxo , y
− −1 h gxo , y
gx, y
h−1
H h−1 gx, y −
2.20
c
h−1 c
for all x, y ∈ Δ. Hence,
−1
H h
gx, y
c
≤ −1
Ax, y h c
γx δy
γxo δyo bs, tϕ h−1 gs, t dt ds
2.21
for all x, y ∈ Δ.
For any fixed x, y ∈ xo , x1 × yo , y1 , by the fact that A is nondecreasing in each
variable, we have
H h−1 gx, y ≤
γx δy
c
y
bs, tϕ h−1 gs, t dt ds
Ax,
−1
h c
γxo δyo 2.22
for all x, y ∈ xo , x × yo , y. Now by applying Theorem 2.1 to the strictly increasing
function H, we have
−1
h
−1
ΦH ΦH
gx, y ≤ H
−1
c
A x, y Bx, y
h−1 c
2.23
for all x, y ∈ xo , x × yo , y. In particular, this leads to
Φ
ux, y ≤ h−1 g x, y ≤ H −1 Φ−1
H
H
c
A
x,
y
B
x,
y
.
h−1 c
Since x, y ∈ xo , x1 × yo , y1 is arbitrary, this concludes the proof of the theorem.
2.24
Journal of Inequalities and Applications
9
Corollary 2.5. Suppose u ∈ CI, R . Let c ≥ 0 be a constant. If α, γ ∈ C1 I, I, a, b ∈ CI, R ,
and ϕ, h ∈ CR , R are functions satisfying
i α, γ are nondecreasing and α, γ ≤ idI ;
ii ϕ is nondecreasing with ϕr > 0 for r > 0;
iii ht and Ht : ht/t, t > 0, are strictly increasing with Ht → ∞ as t → ∞;
iv for any x ∈ I,
h ux, y ≤ c αx
αxo asusds γx
γxo bsusϕ us ds,
2.25
then we have
ux, y ≤ H
−1
Φ−1
H
ΦH
c
Ax Bx
h−1 c
2.26
for all x ∈ xo , x1 , where
Ax αx
αxo asds,
Bx γx
γxo bsds,
2.27
−1
Φ−1
H is the inverse function of ΦH , and x1 ∈ I is chosen such that ΦH c/h c Ax Bx ∈
−1
DomΦH for all x ∈ xo , x1 .
Remark 2.6. If we choose ht t2 , c k2 , α γ idI , as fs, and bs gs, then
Corollary 2.5 reduces to Theorem 1.2.
Theorem 2.7. Suppose u ∈ CΔ, R . Let c ≥ 0 be a constant. If α, γ ∈ C1 I, I, β, δ ∈ C1 J, J,
a, b ∈ CΔ, R , and ϕ, h ∈ CR , R are functions satisfying
i α, β, γ, δ are nondecreasing and α, γ ≤ idI , β, δ ≤ idJ ;
ii ϕ is nondecreasing with ϕr > 0 for r > 0;
: h ◦ Q−1 /t, t > 0, are strictly increasing
iii h, Q, Ht : ht/t, Ht : h ◦ Q−1 and H
with Qr > 0 when r > 0 and Ht
→ ∞ as t → ∞;
iv for any x, y ∈ Δ,
h ux, y ≤ c αx βy
αxo βyo γx δy
γxo δyo as, tQ us, t dt ds
bs, tQ us, t ϕ us, t dt ds,
2.28
10
Journal of Inequalities and Applications
then we have
−1 Ψ−1 Ψ ux, y ≤ Q−1 ◦ H
H
H
c
−1
H c
Ax, y Bx, y
2.29
for all x, y ∈ xo , x1 ×yo , y1 , where Ax, y, Bx, y are the same as in Theorem 2.4, ψ : ϕ◦Q−1 ,
−1
−1
and x1 , y1 ∈ Δ is chosen such that ΨH
c/H c Ax, y Bx, y ∈ DomΨ for all
H
x, y ∈ xo , x1 × yo , y1 .
Proof. It suffices to consider only the case c > 0. For any r > 0, define ψr : ϕQ−1 r. Then
clearly ψ satisfies condition ii of Theorem 2.4. Let v Qu, from 2.18, we have
h Q−1 vx, y ≤ c αx βy
αxo βyo γx δy
γxo δyo as, tvs, tdt ds
bs, tvs, tϕ Q−1 us, t dt ds
2.30
or
H vx, y ≤ c αx βy
αxo βyo γx δy
γxo as, tvs, tdt ds
δyo bs, tvs, tψ us, t dt ds.
2.31
From Theorem 2.4, we have
−1 Ψ−1 Ψ vx, y ≤ H
H
H
c
−1
H c
Ax, y Bx, y ,
2.32
that is,
−1 Ψ−1 Ψ ux, y ≤ Q−1 ◦ H
H
H
c
−1
H c
Ax, y Bx, y
for all x, y ∈ xo , x1 × yo , y1 .
Remark 2.8. Theorem 2.7 is a generalization of Theorem 1.5.
2.33
Journal of Inequalities and Applications
11
3. Application to Boundary Value Problems
In this section, we use the results obtained in Section 2 to study certain properties of positive
solutions of the following boundary value problem BVP:
h ux, y xy F x, y, u αx, γy ,
ux, yo fx,
uxo , y gy,
fxo gyo 0,
3.1
where h is defined as in Theorem 2.1, F ∈ CΔ × R, R, f ∈ CI, R, and g ∈ CI, R are given.
Our first result deals with the boundedness of solutions.
Theorem 3.1. Consider BVP 3.1. If
i |Fx, y, u| ≤ bx, yϕ|u| for some b ∈ CΔ, R ;
ii |hfx hgy| ≤ K for some K ≥ 0,
then all positive solutions to BVP 3.1 satisfy
ux, y ≤ h−1 Φ−1
h ΦK Bx, y ,
x, y ∈ Δ,
3.2
are defined as in Theorem 2.1, and
where ϕ, Φh , and Φ−1
h
Bx, y x y
bs, tdt ds.
xo
3.3
yo
In particular, if Bx, y is bounded on Δ, then every solution to BVP 3.1 is bounded on Δ.
Proof. It is easily seen that u ux, y solves BVP 3.1 if and only if it satisfies the integral
equation
h ux, y h fx h gy x y
xo
F s, t, u αx, γy dt ds.
3.4
yo
Hence by i and ii,
h ux, y ≤ K x y
xo
bs, tϕ u αs, γt dt ds.
3.5
yo
Changing variables by letting σ αs, τ γt, we get
h ux, y ≤ K αx γy
αxo γyo b α−1 σ, γ −1 τ ϕ uσ, τ α−1 σ γ −1 τdτ dσ. 3.6
12
Journal of Inequalities and Applications
From Theorem 2.1, we have
αx γy
−1
−1 −1 −1
ux, y ≤ h−1 Φ−1
K
b
α
σ,
γ
τ
α
σ
γ
τdτ
dσ
, 3.7
Φ
h
h
αxo γyo that is,
ux, y ≤ h−1 Φ−1
h Φh K Bx, y .
3.8
The next result is about the quantitative property of solutions.
Theorem 3.2. Consider BVP 3.1. If
F x, y, u1 − F x, y, u2 ≤ bx, yh u1 − h u2 3.9
for some b ∈ CΔ, R , then (BVP) 3.1 has at most one solution on Δ.
Proof. Assume that u1 x, y and u2 x, y are two solutions to BVP 3.1. By 3.4, we have
h u1 x, y − h u2 x, y ≤
x y
xo
≤
yo
x y
xo
F1 s, t, u αx, γy − F2 s, t, u αx, γy dt ds
bs, th u1 αs, γt − h u2 αs, γt dt ds.
yo
Changing variables by letting σ αs, τ γt, we get
h u1 x, y − h u2 x, y ≤
αx γy
αxo γyo b α−1 σ, γ −1 τ h u1 σ, τ − h u2 σ, τ α−1 σ γ −1 τdτ dσ.
3.10
From Corollary 2.2, we have
h u1 x, y − h u2 x, y ≤ 0,
3.11
that is,
u1 x, y u2 x, y,
∀x, y ∈ Δ.
3.12
Finally, we investigate the continuous dependence of the solutions of BVP 3.1 on the
functional F and the boundary data f and g. For this, we consider the following variation of
Journal of Inequalities and Applications
13
BVP 3.1:
h ux, y xy F x, y, u αx, γy ,
ux, yo fx,
uxo , y gy,
fxo gyo 0,
3.13
where h is defined as in Theorem 2.1, F ∈ CΔ × R, R, f ∈ CI, R, and g ∈ CI, R are given.
Theorem 3.3. Consider BVP 3.1 and BVP 3.13. If
i |Fx, y, u − Fx, y, u| ≤ bx, y|hu − hu| for some b ∈ CΔ, R ;
ii |hfx − hfx hgx − hgx| ≤ /2;
iii for all solutions ux, y of BVP 3.13,
x y
xo
F s, t, u αx, γy − F s, t, u αx, γy dt ds ≤ ,
2
yo
3.14
then
h ux, y − h ux, y ≤ eBx,y ,
3.15
where Bx, y is as defined in Theorem 3.1. Hence hux, y depends continuously on F, f, and g.
Proof. Let ux, y and ux, y be solutions to BVP 3.1 and BVP 3.13, respectively. Then
h ux, y h fx h gy x y
xo
h ux, y h fx h gy F s, t, u αx, γy dt ds,
yo
x y
3.16
F s, t, u αx, γy dt ds.
xo
yo
Hence from assumption ii, we have
h ux, y − h ux, y ≤ h fx − h fx h gy − h gy x y
F s, t, u αx, γy − F s, t, u αx, γy dt ds
xo
≤
yo
x y
F s, t, u αx, γy − F s, t, u αx, γy dt ds
2
xo yo
x y
F s, t, u αx, γy − F s, t, u αx, γy dt ds
xo
≤
yo
x y
xo
yo
bx, yh ux, y − h ux, y dt ds.
3.17
14
Journal of Inequalities and Applications
From Corollary 2.2, we have
|h ux, y − h ux, y ≤ eBx,y .
3.18
If we restrict to any compact subset A of Δ, then Bx, y is bounded, and hence
h ux, y − h ux, y ≤ K
3.19
for some K > 0 for all x, y ∈ A. Therefore, hux, y depends continuously on F, f, and
g.
Remark 3.4. The uniqueness of solution is often a direct consequence of the continuous
dependence on parameters. In fact, let BVP 3.13 be coincide with 3.1, then according to
Theorem 3.3,
h ux, y − h ux, y ≤ eBx,y
∀ > 0,
3.20
thus ux, y ux, y, and so Theorem 3.2 can be viewed as a corollary of Theorem 3.3.
Acknowledgments
The authors express their gratitude to the referees for their careful reading and many useful
comments and suggestions. Research is supported in part by the Research Grants Council of
the Hong Kong SAR, China Project no. HKU7016/07P.
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