Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 345701, 12 pages
doi:10.1155/2010/345701
Research Article
Some Nonlinear Weakly Singular Integral
Inequalities with Two Variables and Applications
Hong Wang and Kelong Zheng
School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China
Correspondence should be addressed to Kelong Zheng, zhengkelong@swust.edu.cn
Received 23 October 2010; Accepted 22 December 2010
Academic Editor: Radu Precup
Copyright q 2010 H. Wang and K. Zheng. 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.
Some nonlinear weakly singular integral inequalities in two variables which generalize some
known results are discussed. The results can be used as powerful tools in the analysis of certain
classes of differential equations, integral equations, and evolution equations. An example is
presented to show boundedness of solution of a differential equation here.
1. Introduction
Various singular integral inequalities play an important role in the development of the
theory of differential equations, functional differential equations, and integral equations. For
example, Henry 1 proposed a linear integral inequality with singular kernel to investigate
some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu
2 gave a modified version of Henry type inequality. However, such results are expressed by
a complicated power series which are sometimes inconvenient for their applications. To avoid
the shortcoming of these results, Medveď 3 presented a new method to discuss nonlinear
singular integral inequalities of Henry type and their Bihari version as follows:
ut ≤ at t
t − sβ−1 sγ −1 Fsusds,
0
ut ≤ at t
1.1
t − sβ−1 Fswusds,
0
and the estimates of solutions are given, respectively. In 4, Medveď also generalized his
results to an analogue of the Wendroff inequalities for functions in two variables. From then
2
Journal of Inequalities and Applications
on, more attention has been paid to such inequalities with singular kernel see 5–9. In
particular, Ma and Yang 8 used a modification of Medveď method to obtain pointwise
explicit bounds on solutions of more general weakly singular integral inequalities of the
Volterra type, and later Ma and Pečarić 9 used this method to study nonlinear inequalities
of Henry type. Recently, Cheung et al. 10 applied the modified Medveď method to
investigate some new weakly singular integral inequalities of Wendroff type and applications
to fractional differential and integral equations.
In this paper, motivated mainly by the work of Ma et al. 8, 9 and Cheung et al. 10,
we discuss more general form of nonlinear weakly singular integral inequality of Wendroff
type for functions in two variables
u x, y ≤ a x, y x y
0
β−1 γ −1 t f x, y, s, t wus, tds dt.
xα − sα β−1 sγ −1 yα − tα
1.2
0
Our results can generalize some known results and be used more effectively to study the
qualitative properties of the solutions of certain partial differential and integral equations.
Moreover, an example is presented to show the usefulness of our results.
2. Main Result
In what follows, R denotes the set of real numbers, and R 0, ∞. CX, Y denotes the
collection of continuous functions from the set X to the set Y . D1 zx, y and D2 zx, y denote
the first-order partial derivatives of zx, y with respect to x and y, respectively.
Before giving our result, we cite the following definition and lemmas.
Definition 2.1 see 8. Let x, y, z be an ordered parameter group of nonnegative real
numbers. The group is said to belong to the first-class distribution and is denoted by
x, y, z ∈ I if conditions x ∈ 0, 1, y ∈ 1/2, 1, and z ≥ 3/2 − y are satisfied; it is said to
belong to the second-class distribution and is denoted by x, y, z ∈ II if conditions x ∈ 0, 1,
y ∈ 0, 1/2 and z > 1 − 2y2 /1 − y2 are satisfied.
Lemma 2.2 see 8. Let α, β, γ , and p be positive constants. Then,
t
α pβ−1 pγ −1
t − s α
s
0
p γ −1 1 tθ
,p β − 1 1 ,
ds B
α
α
t ∈ R ,
2.1
1
where Bξ, η 0 sξ−1 1 − sη−1 ds (Re ξ > 0, Re η > 0) is well-known B-function and
θ pαβ − 1 γ − 1 1.
Lemma 2.3 see 8. Suppose that the positive constants α, β, γ , p1 , and p2 satisfy the following
conditions:
1 if α, β, γ ∈ I, p1 1/β;
2 if α, β, γ ∈ II, p2 1 4β/1 3β.
Journal of Inequalities and Applications
3
Then, for i 1, 2,
pi γ − 1 1 , pi β − 1 1 ∈ 0, ∞,
B
α
θi pi α β − 1 γ − 1 1 ≥ 0
2.2
are valid.
Assume that
A1 ax, y ∈ CR2 , R and fx, y, s, t ∈ CR4 , R ;
A2 wu ∈ CR , R is nondecreasing and w0 0.
y, s, t max 0≤τ≤x,0≤η≤y fτ, η, s, t.
Let a
x, y max 0≤τ≤x,0≤η≤y aτ, η and fx,
Theorem 2.4. Under assumptions (A1 ) and (A2 ), if um, n ∈ CR2 , R satisfies 1.2, then
(1) for α, β, γ ∈ I,
1−β
1/1−β
x y −1
ds dt
f x, y, s, t
u x, y ≤ W1 W1 A1 x, y B1 x, y
0
2.3
0
for 0 ≤ x ≤ X and 0 ≤ y ≤ Y , where
1 β γ − 1 2β − 1
,
,
M1 B
α
αβ
β
1/1−β
A1 x, y 2β/1−β a
x, y
,
2.4
1/βαβ−1γ −11 β/1−β
B1 x, y 2β/1−β M12 xy
,
W1−1 is the inverse of W1 ,
W1 u
u0
dξ
w1/1−β
ξ1−β
,
u ≥ u0 > 0,
2.5
and X, Y ∈ R are chosen such that
W1 A1 x, y B1 x, y
x y
0
0
1/1−β
ds dt ∈ Dom W1−1 ,
f
x, y, s, t
2.6
(2) for α, β, γ ∈ II,
β/14β
14β/β
x y −1
ds dt
f x, y, s, t
u x, y ≤ W2 W2 A2 x, y B2 x, y
0
0
2.7
4
Journal of Inequalities and Applications
for 0 ≤ x ≤ X and 0 ≤ y ≤ Y , where
γ 1 4β − β 4β2
1
,
M2 B
,
α
1 3β
α 1 3β
14β/β
A2 x, y 213β/β a
x, y
,
2.8
14β/13βαβ−1γ −11 13β/β
B2 x, y 213β/β M22 xy
,
W2−1 is the inverse of W2 ,
W2 u
u0
dξ
β/14β ,
ξ
u ≥ u0 > 0,
2.9
14β/β
f
x, y, s, t
ds dt ∈ Dom W2−1 .
2.10
w14β/β
and X, Y ∈ R are chosen such that
W2 A2 x, y B2 x, y
x y
0
0
y, s, t, clearly, a
x, y and f
x, y, s, t are
Proof. With the definition of a
x, y and fx,
y, s, t ≥
nonnegative and nondecreasing in x and y. Furthermore, a
x, y ≥ ax, y and fx,
fx, y, s, t. From 1.2, we have
u x, y ≤ a
x, y x y
0
β−1 γ −1 t f
x, y, s, t wus, tds dt.
xα − sα β−1 sγ −1 yα − tα
2.11
0
Next, for convenience, we introduce indices pi , qi . Denote that if α, β, γ ∈ I, then let p1 1/β
and q1 1/1 − β; if α, β, γ ∈ II, then let p2 1 4β/1 3β and q2 1 4β/β.
Then 1/pi 1/qi 1 holds for i 1, 2.
Using the Hölder inequality with indices pi , qi to 2.11, we get
u x, y ≤ a
x, y x y
0
p β−1 pi γ −1
t
ds dt
xα − sα pi β−1 spi γ −1 yα − tα i
1/pi
0
x y 1/qi
qi
f
x, y, s, t
×
.
wus, tqi ds dt
0
2.12
0
By
A1 A2 · · · An r ≤ nr−1 Ar1 Ar2 · · · Arn ,
Ai ≥ 0, r ≥ 1,
2.13
Journal of Inequalities and Applications
5
from 2.12 and Lemma 2.2, we have
uqi x, y
q
≤ 2qi −1 a
x, y i x y
0
p β−1 pi γ −1
t
ds dt
xα − sα pi β−1 spi γ −1 yα − tα i
qi /pi
0
x y qi
q
i
f
x, y, s, t
×
wus, t ds dt
0
2.14
0
q
θ qi /pi
2qi −1 a
x, y i 2qi −1 Mi2 xy i
x y 0
qi
f
x, y, s, t
wus, tqi ds dt ,
0
where
pi γ − 1 1 1
Mi B
, pi β − 1 1
α
α
2.15
and θi is given in Lemma 2.3 for i 1, 2.
Since qi ≥ 0 and θi ≥ 0 i 1, 2, then a
x, yqi and xyθi qi /pi are also nondecreasing
in x and y. Taking any arbitrary x
and y
with x
≤ X, y
≤ Y , we obtain
θ qi /pi
q
uqi x, y ≤ 2qi −1 a
x,
y
i 2qi −1 Mi2 x
y
i
x y 0
qi
f
x,
y,
s, t
wus, tqi ds dt
0
2.16
for 0 ≤ x ≤ x,
0 ≤ y ≤ y.
Denote
q
y
i ,
Ai x,
y
2qi −1 a
x,
2.17
and let
θ qi /pi
zi x, y Ai x,
y
2qi −1 Mi2 x
y
i
x y 0
qi
qi
f x,
y,
s, t
wus, t ds dt .
0
2.18
1/q
y,
and zi x, y
Then, uqi x, y ≤ zi x, y or ux, y ≤ zi i x, y. Meanwhile, zi 0, y Ai x,
is nondecreasing in x and y. Considering
θ qi /pi
D1 zi x, y 2qi −1 Mi2 x
y
i
2
≤ 2qi −1 Mi x
y
θi
y 0
q /p y i
qi
f
x,
y,
x, t
wux, tqi ds dt
i
0
qi qi 1/qi
dt ,
f x,
y,
x, t
wzi x, t
2.19
6
Journal of Inequalities and Applications
we have
y θi qi /pi
qi
D1 zi x, y
qi −1
2
1/q dt ,
Mi x
y
f x,
y,
x, t
≤ 2
0
wqi zi i x, y
2.20
1/q
where we apply the fact that wqi zi i x, y is nondecreasing in y. Integrating both sides of
the above inequality from 0 to x, we obtain
θ qi /pi
Wi zi x, y ≤ Wi zi 0, y 2qi −1 Mi2 x
y
i
x y 0
θ qi /pi
Wi Ai x,
y
2qi −1 Mi2 x
y
i
qi
f
x,
y,
s, t
ds dt
0
x y 0
qi
f x,
y,
s, t
ds dt
2.21
0
for 0 ≤ x ≤ x,
0 ≤ y ≤ y,
where
Wi u u
u0
dξ
,
wqi ξ1/qi
2.22
u ≥ u0 > 0.
From assumption A2 , Wi is strictly increasing so its inverse Wi−1 is continuous and increasing in its corresponding domain. Replacing x and y by x
and y,
we have
θ qi /pi
Wi zi x,
y
≤ Wi Ai x,
y
2qi −1 Mi2 x
y
i
x
y
0
qi
ds dt .
f x,
y,
s, t
0
2.23
Since x
and y
are arbitrary, we replace x
and y
by x and y, respectively, and get
θ qi /pi
Wi zi x, y ≤ Wi Ai x, y 2qi −1 Mi2 xy i
x y 0
qi
f x, y, s, t
ds dt .
2.24
0
for 0 ≤ x ≤ X and 0 ≤ y ≤ Y . The above inequality can be rewritten as
x y qi
θ qi /pi
f
x, y, s, t
zi x, y ≤ Wi−1 Wi Ai x, y 2qi −1 Mi2 xy i
ds dt
.
0
0
2.25
Therefore, we have
1/q u x, y ≤ zi i x, y
x y 1/qi
θ qi /pi
qi
ds dt
≤ Wi−1 Wi Ai x, y 2qi −1 Mi2 xy i
f
x, y, s, t
0
0
2.26
Journal of Inequalities and Applications
7
for 0 ≤ x ≤ X and 0 ≤ y ≤ Y .
Finally, considering two situations for i 1, 2 and using parameters α, β, γ to denote
pi , qi , Mi , and θi in the above inequality, we can obtain the estimations, respectively. we omit
the details here.
Remark 2.5. Medveď 4, Theorem 2.2 investigated the special case α γ 1, fx, y, s, t Fs, t
of inequality 1.2 under the assumption that “ wu satisfies the condition q.” However, in our
result, the q condition is eliminated. If we take α 1 and wu u, then we can obtain the result of
linear case 4, Theorem 2.4.
Remark 2.6. Let up x, y vx, y, then ux, y v1/p x, y or uq x, y vq/p x, y. Therefore,
if we take wv vq/p , the formula 2.6 in 10 is the special case of inequality 1.2, and we can
obtain more concise results than 2.7 and 2.9 in 10. Moreover, here the condition p ≥ q also can
be eliminated.
Remark 2.7. When α, β, γ does not belong to I or II, there are some technical problems which we do
not discuss here.
3. Some Corollaries
Corollary 3.1. Let functions ux, y, ax, y, fx, y, s, t be defined as in Theorem 2.4, and let k be
a constant with 0 < k ≤ 1. Suppose that
u x, y ≤ a x, y x y
0
β−1 γ −1 t f x, y, s, t us, tk ds dt.
xα − sα β−1 sγ −1 yα − tα
3.1
0
Then,
(1) for α, β, γ ∈ I,
if k 1,
x y 1/1−β
β f x, y, s, t
ds dt ,
u x, y ≤ 2 a
x, y exp 1 − β B1 x, y
0
3.2
0
if 0 < k < 1,
u x, y ≤
1−k/1−β
2 a
x, y
1 − kB1 x, y
β
x y 1−β/1−k
1/1−β
ds dt
f x, y, s, t
0
0
3.3
for x ≥ 0, y ≥ 0, where a
x, y, f
x, y, s, t, B1 x, y are defined as in Theorem 2.4,
(2) for α, β, γ ∈ II,
if k 1,
u x, y ≤ 213β/14β a
x, y exp
β
B2 x, y
1 4β
x y 0
0
14β/β
f
x, y, s, t
ds dt ,
3.4
8
Journal of Inequalities and Applications
if 0 < k < 1,
u x, y
≤
14β 1−k/β
213β a
x, y
1 − kB2 x, y
x y 0
14β/β
f
x, y, s, t
ds dt
β/14β1−k
0
,
3.5
for x ≥ 0, y ≥ 0, where a
x, y, f
x, y, s, t, B2 x, y are defined as in Theorem 2.4.
Proof. Clearly, inequality 3.1 is the special case of 1.2. Taking wu uk , we can get 3.1.
i If k 1,
u
Wi u u0
u
dξ
ln ,
ξ
u0
3.6
Wi−1 u u0 eu ,
u ≥ u0 > 0,
Dom Wi−1 0, ∞,
i 1, 2.
ii If 0 < k < 1,
Wi u u
u0
Wi−1 u u1−k
1 − ku
0
dξ
1 1−k
1−k
u
,
−
u
0
1−k
ξk
1/1−k
Dom Wi−1 0, ∞,
,
3.7
i 1, 2.
Therefore, the positive numbers X and Y in 2.6 and 2.10 can be taken as ∞, and the results
can be obtained by simple computation. We omit the details.
Corollary 3.2. Let functions ux, y, ax, y, fx, y, s, t be defined as in Theorem 2.4. Suppose that
gx, y, s, t ∈ CR4 , R and ux, y satisfies
u x, y ≤ a x, y x y
x y
0
0
α β−1 γ −1 α
x − s α
0
μ−1 τ−1 t g x, y, s, t us, tds dt
xσ − sσ μ−1 sτ−1 yσ − tσ
s
y −t
α β−1 γ −1
t
f x, y, s, t wus, tds dt.
3.8
0
Then,
i if α, β, γ , σ, μ, τ ∈ I,
1/1−β −1
u x, y ≤ W1 W1 A1 x, y Ω1 x, y
1/1−β Ω1 x, y
B1 x, y
x y
0
0
1/1−β
f
x, y, s, t
ds dt
1−β
3.9
Journal of Inequalities and Applications
9
for 0 ≤ x ≤ X1 and 0 ≤ y ≤ Y1 , where
x y 1/1−μ
g
x, y, s, t
ds dt ,
Ω1 x, y 2μ exp 1 − μ B1 x, y
0
3.10
0
W1 , W1−1 , A1 x, y, B1 x, y are defined as in Theorem 2.4, and X1 , Y1 ∈ R are chosen such that
1/1−β W1 A1 x, y Ω1 x, y
1/1−β Ω1 x, y
B1 x, y
x y
0
0
3.11
1/1−β
f
x, y, s, t
ds dt ∈ Dom W1−1 ,
ii if α, β, γ , σ, μ, τ ∈ II,
14β/β u x, y ≤ W2−1 W2 A2 x, y Ω2 x, y
14β/β Ω2 x, y
B2 x, y
x y
0
14β/β
f
x, y, s, t
ds dt
β/14β
3.12
0
for 0 ≤ x ≤ X2 and 0 ≤ y ≤ Y2 , where
Ω2 x, y 213μ/14μ exp
μ
B2 x, y
1 4μ
x y
0
14μ/μ
g
x, y, s, t
ds dt ,
3.13
0
W2 , W2−1 , A2 x, y, B2 x, y are defined as in Theorem 2.4, and X2 , Y2 ∈ R are chosen such that
14β/β W2 A2 x, y Ω2 x, y
14β/β Ω2 x, y
B2 x, y
x y
0
14β/β
f
x, y, s, t
ds dt
0
3.14
∈ Dom W2−1 .
Proof. By the two mentioned lemmas, it follows from 3.8 that
u x, y ≤ Pi x, y x y
0
μ−1 τ−1 t g
x, y, s, t us, tds dt,
xσ − sσ μ−1 sτ−1 yσ − tσ
3.15
0
where g
x, y, s, t max0≤τ≤x,0≤η≤y gτ, η, s, t and
θ 1/pi
Pi x, y a
x, y Mi2 xy i
i For α, β, γ , σ, μ, τ ∈ I,
x y
0
0
f
qi x, y, s, t wus, tqi ds dt
1/qi
.
3.16
10
Journal of Inequalities and Applications
applying Corollary 3.1 to 3.15, we have
1/1−μ
x y g
x, y, s, t
ds dt .
u x, y ≤ 2μ P1 x, y exp 1 − μ B1 x, y
0
3.17
0
Letting
1/1−μ
x y Ω1 x, y 2μ exp 1 − μ B1 x, y
g
x, y, s, t
ds dt ,
0
3.18
0
we get
u x, y ≤ P1 x, y Ω1 x, y
a
x, y Ω1 x, y
θ 1/p1
Ω1 x, y M12 xy 1
x y
0
f
q1 x, y, s, t wus, tq1 ds dt
3.19
1/q1
.
0
Since inequality 3.19 is similar to 2.12, we can repeat the procedure of proof in Theorem 2.4
and get 3.9.
ii As for the case that α, β, γ , σ, μ, τ ∈ II, the proof is similar to the argument in the
proof of case i with suitable modification. We omit the details.
Remark 3.3. When α, β, γ ∈ I, σ, μ, τ ∈ II or α, β, γ ∈ II, σ, μ, τ ∈ I, we can get the results
which are similar to that in Corollary 3.2 and omit them here.
4. Application
In this section, we will apply our result to discuss the boundedness of certain partial integral
equation with weakly singular kernel.
Suppose that ux, y ∈ CR2 , R satisfies the inequality as follow:
1
u x, y ≤ 2
x y
0
−1/3 −1/6 −s−2t t
e
us, tds dt
x − s−1/3 s−1/6 y − t
4.1
0
for x ≥ 0, y ≥ 0. Then, 4.1 is the special case of inequality 1.2 that is,
1
a x, y ,
2
α 1,
f x, y, s, t e−s−2t ,
2
5
,
γ ,
3
6
wu us, t.
β
4.2
Journal of Inequalities and Applications
11
Obviously, α, β, γ 1, 2/3, 5/6 ∈ I. Letting p1 3/2, q1 3, we have
1
f
x, y, s, t e−s−2t ,
a
x, y ,
2
3
1
1
3 1
A1 x, y 22
M1 B , ,
,
2
2
4 2
2
3 1 2 1/4
3 1 4
B1 x, y 22
xy
B ,
4 B ,
xy,
4 2
4 2
u
√
√ dξ
W1 u 2 u − u0 ,
ξ
u0
√
u 2
u0 , Dom W1−1 0, ∞.
W1−1 u 2
4.3
Applying 2.3 in Theorem 2.4, we get for x ≥ 0, y ≥ 0
1/3
x y −s−2t 3
u x, y ≤ W1−1 W1 A1 x, y B1 x, y
e
ds dt
0
W1−1
0
1/3
x y
3 1 4
1
−3 −6t
4 B ,
xy
e e ds dt
W1
2
4 2
0 0
√
−1
W1
1/3
√
3 1 4 2
−3x
−6y
B ,
2 − 2 u0 xy 1 − e
1−e
9
4 2
4.4
2/3
3 1 4 2 1
−3x
−6y
1−e
xy 1 − e
B ,
2
9
4 2
√
which implies that ux, y in 4.1 is bounded.
Acknowledgments
This work is supported by Scientific Research Fund of Sichuan Provincial Education Department no. 09ZC109. The authors are very grateful to the referees for their helpful comments
and valuable suggestions.
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