doi:10.1155/2010/984141
Research Article
Some Nonlinear Integral Inequalities in Two Independent Variables
Wei Nian Li
1, 21Department of Mathematics, Binzhou University, Shandong 256603, China
2School of Mathematics Science, Qufu Normal University, Shandong 273165, China
Correspondence should be addressed to Wei Nian Li,[email protected] Received 14 January 2010; Accepted 29 May 2010
Academic Editor: Binggen Zhang
Copyrightq2010 Wei Nian Li. 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 investigate some new nonlinear integral inequalities in two independent variables. The inequalities given here can be used as tools in the qualitative theory of certain nonlinear partial differential equations.
1. Introduction
It is well known that the integral inequalities involving functions of one and more than one independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of differential equations. In the past few years, a number of integral inequalities had been established by many scholars, which are motivated by certain applications. For details, we refer to literatures 1–10 and the references therein. In this paper we investigate some new nonlinear integral inequalities in two independent variables, which can be used as tools in the qualitative theory of certain partial differential equations.
2. Main Results
In what follows,Rdenotes the set of real numbers andR 0,∞is the given subset ofR.
The first-order partial derivatives of azx, ydefined forx, y∈Rwith respect toxandyare denoted byzxx, y, andzyx, yrespectively. Throughout this paper, all the functions which appear in the inequalities are assumed to be real-valued and all the integrals involved exist on the respective domains of their definitions,CM, Sdenotes the class of all continuous
functions defined on set M with range in the set S, p and q are constants, and p ≥ 1, 0< q≤p.
We firstly introduce two lemmas, which are useful in our main results.
Lemma 2.1Bernoulli’s inequality11. Let 0< α≤1 andx >−1. Then1xα≤1αx.
Lemma 2.2see7. Letut, at,andbtbe nonnegative and continuous functions defined for t∈R.
iAssume thatatis nondecreasing fort∈R. If
ut≤at
t
0
bsusds, 2.1
fort∈R, then
ut≤atexp t
0
bsds
, 2.2
fort∈R.
iiAssume thatatis nonincreasing fort∈R. If
ut≤at
∞
t
bsusds, 2.3
fort∈R, then
ut≤atexp ∞
t
bsds
, 2.4
fort∈R.
Next, we establish our main results.
Theorem 2.3. Letux, y, ax, y, bx, y, fx, y, gx, y∈CR2,Randax, y>0.
iIf
up x, y
≤a x, y
b x, y
x 0
∞
y
fs, tups, t gs, tuqs, t
dtds, x, y∈R, E1
then
u x, y
≤a1/p x, y
1
pa1/p−1 x, y
b x, y
h x, y
exp x
0
∞
y
Fs, tdtds
, x, y∈R, 2.5
where
h x, y
x
0
∞
y fs, tas, t gs, taq/ps, t
dtds, 2.6
F x, y
b x, y
f x, y
q
paq/p−1 x, y
g x, y
. 2.7
iiIf
up x, y
≤a x, y
b x, y
∞ x
∞
y
fs, tups, t gs, tuqs, t
dtds, x, y∈R, E1
then
u x, y
≤a1/p x, y
1
pa1/p−1 x, y
b x, y
h x, y
exp ∞
x
∞
y
Fs, tdtds
, x, y∈R, 2.5
where
h x, y
∞
x
∞
y
fs, tas, t gs, taq/ps, t
dtds, 2.6
andFx, yis defined by2.7.
Proof. We only give the proof ofi. The proof ofiican be completed by following the proof ofi.
iDefine a functionzx, yby
z x, y
x
0
∞
y
fs, tups, t gs, tuqs, t
dtds, x, y∈R. 2.8
ThenE1can be restated as
up x, y
≤a x, y
b x, y
z x, y
a x, y
1b x, y
z x, y a
x, y
. 2.9
UsingLemma 2.1, from2.9, we easily obtain
u x, y
≤a1/p x, y
1
pa1/p−1 x, y
b x, y
z x, y
, 2.10
uq x, y
≤aq/p x, y
q
paq/p−1 x, y
b x, y
z x, y
. 2.11
Combining2.8,2.9, and2.11, we have
z x, y
≤ x
0
∞
y
fs, tas, t bs, tzs, t gs, t
aq/ps, t q
paq/p−1s, tbs, tzs, t
dtds
h x, y
x
0
∞
y
Fs, tzs, tdtds, x, y∈R,
2.12
wherehx, yand Fx, yare defined by 2.6and 2.7, respectively. Obviously, hx, yis nonnegative, continuous, nondecreasing inx, and nonincreasing inyforx, y∈R.
Firstly, we assume thathx, y>0 forx, y∈R. From2.12, we easily observe that
z x, y h
x, y ≤1 x
0
∞
y
Fs, tzs, t
hs, t dtds. 2.13
Letting
v x, y
1 x
0
∞
y
Fs, tzs, t
hs, t dtds, 2.14
we easily see thatvx, yis nonincreasing iny, y∈R, and
z x, y
≤h x, y
v x, y
, x, y∈R. 2.15
Therefore,
vx x, y
∞
y
Fx, tzx, t
hx, t dt
≤ ∞
y
Fx, tvx, tdt
≤v x, y
∞ y
Fx, tdt.
2.16
Treatingy,y∈R, fixed in2.16, dividing both sides of2.16byvx, y, settingxs, and integrating the resulting inequality from 0 tox, x∈R, we have
v x, y
≤exp x
0
∞
y
Fs, tdtds
. 2.17
It follows from2.15and2.17that
z x, y
≤h x, y
exp x
0
∞
y
Fs, tdtds
. 2.18
Therefore, the desired inequality2.5follows from2.10and2.18.
Ifhx, yis nonnegative, we carry out the above procedure with hx, y ε instead ofhx, y, whereε > 0 is an arbitrary small constant, and subsequently pass to the limit as ε → 0 to obtain2.5. This completes the proof.
Theorem 2.4. Assume that ux, y, ax, y, bx, y, fx, y, gx, y ∈ CR2,R, and L ∈ CR3,R. Letax, y>0, and
0≤Ls, t, u−Ls, t, v≤Ks, t, vu−v, 2.19
foru≥v≥0, whereK∈CR3,R. iIf
up x, y
≤a x, y
b x, y
x 0
∞
y
fs, tuqs, t Ls, t, us, t
dtds, x, y∈R, E2
then
u x, y
≤a1/p x, y
1
pa1/p−1 x, y
b x, y
G x, y
exp x
0
∞
y
Hs, tdtds
, x, y∈R, 2.20
where
G x, y
x
0
∞
y
fs, taq/ps, t L
s, t, a1/ps, t
dtds, 2.21
H x, y
q
pf x, y
aq/p−1 x, y
K
x, y, a1/p x, y1
pa1/p−1 x, y
b x, y
. 2.22
iiIf
up x, y
≤a x, y
b x, y
∞ x
∞
y
fs, tuqs, t Ls, t, us, t
dtds, x, y∈R, E2
then
u x, y
≤a1/p x, y
1
pa1/p−1 x, y
b x, y
G x, y
×exp ∞
x
∞
y
Hs, tdtds
, x, y∈R,
2.20
where
G x, y
∞
x
∞
y
fs, taq/ps, t L
s, t, a1/ps, t
dtds, 2.21
andHx, yis defined by2.22.
Proof. We only prove the parti. The proof ofiican be completed by following the proof of i.
iDefine a functionzx, yby
z x, y
x
0
∞
y
fs, tuqs, t Ls, t, us, t
dtds. 2.23
Then, as in the proof ofTheorem 2.3, we obtain2.9–2.11. Therefore, we have
x
0
∞
y
fs, tuqs, tdtds≤ x
0
∞
y
fs, t
aq/ps, t q
paq/p−1s, tbs, tzs, t
dtds, 2.24 x
0
∞
y
Ls, t, us, tdtds≤ x
0
∞
y
L
s, t, a1/ps, t 1
pa1/p−1s, tbs, tzs, t
− L
s, t, a1/ps, t L
s, t, a1/ps, t dtds
≤ x
0
∞
y
L
s, t, a1/ps, t dtds
x
0
∞
y
K
s, t, a1/ps, t1
pa1/p−1s, tbs, tzs, tdtds.
2.25
It follows from2.23–2.25that
z x, y
≤ x
0
∞
y
fs, taq/ps, t L
s, t, a1/ps, t dtds
x
0
∞
y
q
pfs, taq/p−1s, tbs, t K
s, t, a1/ps, t1
pa1/p−1s, tbs, t
zs, tdtds
G x, y
x
0
∞
y
Hs, tzs, tdtds,
2.26
whereGx, yandHx, yare defined by2.21and2.22, respectively.
It is obvious that Gx, y is nonnegative, continuous, nondecreasing in x, and nonincreasing in y for x, y ∈ R. By following the proof of Theorem 2.3, from 2.26, we have
z x, y
≤G x, y
exp x
0
∞
y
Hs, tdtds
. 2.27
Combining2.10and2.27, we obtain the desired inequality2.20. The proof is complete.
Theorem 2.5. Let ax, y, ux, y, Ls, t, u, and Ks, t, u be the same as in Theorem 2.4, and rx, y∈CR2,R.
iAssume thatax, yis nondecreasing inx, x∈R, and the condition2.19holds. If
up x, y
≤a x, y
x
0
r s, y
up s, y
ds x
0
∞
y
Ls, t, uqs, tdtds, x, y∈R, E3
then
u x, y
≤B1/p x, y
×
a1/p x, y
1
pa1/p−1 x, y
J x, y
exp x
0
∞
y
Ms, tdtds
, x, y∈R, 2.28
where
B x, y
exp x
0
r s, y
ds
, 2.29
J x, y
x
0
∞
y
L
s, t, Bq/ps, taq/ps, t
dtds, 2.30
M
x, y K
x, y, Bq/p x, y
aq/p x, y
Bq/p x, yq
paq/p−1 x, y
. 2.31
iiAssume thatax, yis nonincreasing inx, x∈R, and the condition2.19holds. If
up x, y
≤a x, y
∞
x
r s, y
up s, y
ds ∞
x
∞
y
Ls, t, uqs, tdtds, x, y∈R, E3
then u
x, y
≤B1/p x, y
×
a1/p x, y
1
pa1/p−1 x, y J
x, y exp
∞
x
∞
y
Ms, t dtds
, x, y∈R, 2.28
where
B x, y
exp ∞
x
r s, y
ds
, 2.29
J x, y
∞
x
∞
y
L
s, t,Bq/ps, taq/ps, t
dtds, 2.30
M x, y
K
x, y,Bq/p x, y
aq/p
x, y Bq/p x, yq
paq/p−1 x, y
. 2.31
Proof. iDefine a functionzx, yby z
x, y a
x, y v
x, y
, 2.32
where
v x, y
x
0
∞
y
Ls, t, uqs, tdtds. 2.33
ThenE3can be restated as
up x, y
≤z x, y
x
0
r s, y
up s, y
ds. 2.34
Noting the assumption thatax, yis nondecreasing inx, x ∈R, we easily see that zx, yis a nonnegative and nondecreasing function inx, x ∈R.Therefore, treatingy, y ∈ R, fixed in??and using partiofLemma 2.2to??, we get
up x, y
≤B x, y
z x, y
. 2.35
that is,
up x, y
≤B x, y
a x, y
v x, y
B x, y
a x, y
1v x, y a
x, y
, 2.36
whereBx, yis defined by2.29. UsingLemma 2.1, from2.35we have
u x, y
≤B1/p x, y
a1/p x, y
1
pa1/p−1 x, y
v x, y
, 2.37
uq x, y
≤Bq/p x, y
aq/p x, y
q
paq/p−1 x, y
v x, y
. 2.38
Combining2.33and2.37, and noting the hypotheses2.19, we obtain
v x, y
≤ x
0
∞
y
L
s, t, Bq/ps, t
aq/ps, t q
paq/p−1s, tvs, t
−L
s, t, Bq/ps, taq/ps, t L
s, t, Bq/ps, taq/ps, t dtds
≤J x, y
x
0
∞
y
Ms, tvs, tdtds,
2.39
whereJx, yandMx, yare defined by2.30and2.31, respectively.
It is obvious that Jx, y is nonnegative, continuous, nondecreasing in x and nonincreasing in y for x, y ∈ R. By following the proof of Theorem 2.3, from 2.38, we obtain
v x, y
≤J x, y
exp x
0
∞
y
Ms, tdtds
, x, y∈R. 2.40
Obviously, the desired inequality2.28follows from2.36and2.39.
ii Noting the assumption thatax, yis nonincreasing inx, x ∈ R,and using the partiiofLemma 2.2, we can complete the proof by following the proof ofiwith suitable changes. Therefore, the details are omitted here.
By using the ideas of the proofs of Theorems2.5and2.3, we easily prove the following theorem.
Theorem 2.6. Letax, y, ux, y, rx, y, fx, y, gx, y∈CR2,R, andax, y>0.
iAssume thatax, yis nondecreasing inx, x∈R. If
up x, y
≤a x, y
x
0
r s, y
up s, y
ds
x
0
∞
y
fs, tups, t gs, tuqs, t
dtds, x, y∈R,
E4
then
u x, y
≤B1/p x, y
×
a1/p x, y
1
pa1/p−1 x, y
H x, y
exp x
0
∞
y
Ps, tdtds
, x, y∈R, 2.41
where
H x, y
x
0
∞
y
fs, tBs, tas, t gs, tBq/ps, taq/ps, t dtds,
P x, y
f x, y
B x, y
g x, y
Bq/p x, yq
paq/p−1 x, y
,
2.42
andBx, yis defined by2.29.
iiAssume thatax, yis nonincreasing inx, x∈R. If
up x, y
≤a x, y
∞
x
r s, y
up s, y
ds
∞
x
∞
y
fs, tups, t gs, tuqs, t
dtds, x, y∈R,
E4
then
u x, y
≤B1/p x, y
×
a1/p x, y
1
pa1/p−1
x, yH x, y
exp ∞
x
∞
y
Ps, tdt ds
, x, y∈R, 2.40
where
H x, y
∞
x
∞
y
fs, tBs, tas, t gs, tBq/ps, taq/ps, t dtds, P
x, y f
x, y B x, y
g
x, y Bq/p x, yq
paq/p−1 x, y
,
2.41
andBx, y is defined by25.
Remark 2.7. Noting thatp andqare constants, andp ≥ 1,0 < q ≤ p, we can obtain many special integral inequalities by using our main results. For example, letp 1, q 1/4, and pq2, respectively; fromTheorem 2.3, we obtain the following corollaries.
Corollary 2.8. Letux, y, ax, y, bx, y, fx, y, gx, y∈CR2,Randax, y>0.
iIf
u x, y
≤a x, y
b x, y
x 0
∞
y
fs, tus, t gs, tu1/4s, t
dtds, x, y∈R, E5
then
u x, y
≤a x, y
b x, yh
x, y exp
x 0
∞
y
Fs, tdt ds
, x, y∈R, 2.43
where
h x, y
x
0
∞
y
fs, tas, t gs, ta1/4s, t
dtds, 2.44
F x, y
b x, y
f x, y
1 4a−3/4
x, y g
x, y
. 2.45
iiIf
u x, y
≤a x, y
b x, y
∞ x
∞
y
fs, tus, t gs, tu1/4s, t
dtds, x, y∈R, E5
then
u x, y
≤a x, y
b x, yh
x, y exp
x 0
∞
y
Fs, tdt ds
, x, y∈R, 2.42
where
h x, y
∞
x
∞
y
fs, tas, t gs, ta1/4s, t
dtds, 2.43
andFx, y is defined by2.44.
Corollary 2.9. Letux, y, ax, y, bx, y, gx, y∈CR2,Randax, y>0.
iIf
u2 x, y
≤a x, y
b x, y
x 0
∞
y
gs, tu2s, tdtds, x, y∈R, E6
then
u x, y
≤a1/2 x, y
1 2a−1/2
x, y b
x, y m
x, y
×exp x
0
∞
y
bs, tgs, tdtds
, x, y∈R,
2.46
where
m x, y
x
0
∞
y
gs, tas, tdtds. 2.47
iiIf
u2 x, y
≤a x, y
b x, y
∞ x
∞
y
gs, tu2s, tdtds, x, y∈R, E6
then
u x, y
≤a1/2 x, y
1 2a−1/2
x, y b
x, y m
x, y
×exp x
0
∞
y
bs, tgs, tdtds
, x, y∈R,
2.45
where
m
x, y
∞
x
∞
y
gs, tas, tdtds. 2.46
Remark 2.10. If we addax, y>0 to the assumptions of7, Theorems 2.2–2.4, then we easily see that 7, Theorems 2.2–2.4 are special cases of Theorems 2.3,2.5, and2.6, respectively.
Therefore, our paper gives some extensions of the results of7in a sense.
3. An Application
In this section, using Theorem 2.3, we obtain the bound on the solution of a nonlinear differential equation.
Example 3.1. Consider the partial differential equation:
pup−1 x, y
uxy
x, y p
p−1 up−2
x, y ux
x, y uy
x, y h
x, y, u x, y
r x, y
,
ux,∞ σx, u
∞, y τ
y
, u∞,∞ d,
3.1 whereh∈CR2×R,R, r ∈CR2,R, σ, τ ∈CR,R, anddis a real constant, andp≥1 is a constant.
Suppose that h
x, y, u
x, y≤f
x, yu
x, ypg
x, yu x, yq,
σx τ y
−d ∞
x
∞
y
rs, tdtds ≤a
x, y ,
3.2
whereax, y, fx, y, gx, y ∈ CR2,Randax, y > 0 forx, y ∈ R, and 0< q ≤ pis a constant. Letux, ybe a solution of3.1forx, y∈R; then
u
x, y≤a1/p x, y
1
pa1/p−1 x, y
E x, y
exp ∞
x
∞
y
Ws, tdtds
, x, y∈R, 3.3
where
E x, y
∞
x
∞
y
fs, tas, t gs, taq/ps, t dtds,
W x, y
f x, y
q
paq/p−1 x, y
g x, y
.
3.4
In fact, ifux, yis a solution of3.1, then it can be written assee1, page 80
u x, yp
σx τ y
−d ∞
x
∞
y
hs, t, us, t rs, tdtds, 3.5
forx, y∈R.
It follows from3.2and3.5that u
x, yp≤a x, y
∞
x
∞
y
fs, t|us, t|pgs, t|us, t|q
dtds. 3.6
Now, a suitable application of part ii of Theorem 2.3 to 3.6 yields the required estimate in3.3.
Acknowledgment
This work is supported by the National Natural Science Foundation of China10971018, the Natural Science Foundation of Shandong ProvinceZR2009AM005, China Postdoctoral Science Foundation Funded Project20080440633, Shanghai Postdoctoral Scientific Program 09R21415200, the Project of Science and Technology of the Education Department of Shandong Province J08LI52, and the Doctoral Foundation of Binzhou University 2006Y01.
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