Volume 2011, Article ID 754350,25pages doi:10.1155/2011/754350
Research Article
Explicit Bounds to Some New
Gronwall-Bellman-Type Delay Integral Inequalities in Two Independent Variables on Time Scales
Fanwei Meng,
1Qinghua Feng,
1, 2and Bin Zheng
21School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
2School of Science, Shandong University of Technology, Zibo 255049, China
Correspondence should be addressed to Fanwei Meng,[email protected] Received 20 April 2011; Accepted 8 August 2011
Academic Editor: Bernard Geurts
Copyrightq2011 Fanwei Meng 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.
Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which provide a handy tool in the research of qualitative and quantitative properties of solutions of delay dynamic equations on time scales. The established inequalities generalize some of the results in the work of Zhang and Meng 2008, Pachpatte 2002, and Ma 2010.
1. Introduction
During the past decades, with the development of the theory of differential and integral equations, a lot of integral and difference inequalities have been discovered, which play an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. In these established inequalities, Gronwall-Bellman-type inequalities are of particular importance as these inequalities provide explicit bounds for unknown functions, and much effort has been done for developing such inequalitiese.g., see1–13 and the references therein. On the other hand, Hilger 14 initiated the theory of time scales as a theory capable containing both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scalese.g., see 15–17 and the references therein. In these investigations, integral inequalities on time scales have been paid much attention by many authors, and a lot of integral inequalities on time scales have been establishede.g., see18–26, which have been designed to unify continuous and discrete analysis and play an important role in the research of qualitative and quantitative properties of solutions of certain dynamic equations on time scales. But
to our knowledge, Gronwall-Bellman-type delay integral inequalities on time scales have been paid little attention in the literature so far. Recent results in this direction include the works of Li27and Ma and Peˇcari´c28to our best knowledge. Furthermore, nobody has studied Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales.
Our aim in this paper is to establish some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales, which unify some known con- tinuous and discrete analysis. New explicit bounds for unknown functions are obtained due to the presented inequalities. We will also present some applications for our results.
First we will give some preliminaries on time scales and some universal symbols for further use.
Throughout the paper,Rdenotes the set of real numbers and R 0,∞, while Z denotes the set of integers. For two given setsG,H, we denote the set of maps fromGtoH byG, H.
A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper, Tdenotes an arbitrary time scale. OnTwe define the forward and backward jump operators σ∈T,Tandρ∈T,Tbyσt inf{s∈T, s > t}andρt sup{s∈T, s < t}.
Definition 1.1. The graininessμ∈T,Ris defined byμt σt−t.
Remark 1.2. Obviously,μt 0 ifTRwhileμt 1 ifTZ.
Definition 1.3. A point t ∈ T is said to be left-dense ifρt t andt /infT, right-dense if σt tandt /supT, left-scattered ifρt< t, and right-scattered ifσt> t.
Definition 1.4. The setTκ is defined to be T if Tdoes not have a left-scattered maximum, otherwise it isTwithout the left-scattered maximum.
Definition 1.5. A functionf ∈T,Ris called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at left-dense points, while f is called regressive if 1μtft/0.Crddenotes the set of rd-continuous functions, whileRdenotes the set of all regressive and rd-continuous functions, andR {f|f ∈R,1μtft>0,∀t∈T}.
Definition 1.6. For somet∈Tκand a functionf∈T,R, the delta derivative offis denoted by fΔtand satisfies
fσt−fs−fΔtσt−s≤ε|σt−s| for∀ε >0, 1.1
wheres∈U, andUis a neighborhood oftwhich can depend onε.
Similarly, for somey∈Tκand a functionf∈T×T,R, the partial delta derivative off with respect toyis denoted byfx, yΔy orfyΔx, yand satisfies
f x, σ
y
−fx, s−fyΔ x, y
σ y
−s≤εσ y
−s for∀ε >0, 1.2
wheres∈U, andUis a neighborhood ofywhich can depend onε.
Remark 1.7. IfTR, thenfΔtbecomes the usual derivativeft, whilefΔt ft1−ft ifTZ, which represents the forward difference.
Definition 1.8. Fora, b∈Tand a functionf∈T,R, the Cauchy integral offis defined by b
a
ftΔtFb−Fa, 1.3
whereFΔt ft, t∈Tκ.
Similarly, fora, b∈Tand a functionf∈T×T,R, the Cauchy partial integral offwith respect toyis defined by
b
a
f x, y
ΔyFx, b−Fx, a, 1.4
whereFyΔx, y fx, y, y∈Tκ.
Definition 1.9. The cylinder transformationξhis defined by
ξhz
⎧⎪
⎨
⎪⎩
Log1hz
h , if h /0
for z / −1 h
,
z, if h0,
1.5
where Log is the principal logarithm function.
Definition 1.10. Forpx, y∈Rwith respect toy, the exponential function is defined by
ep y, s
exp y
s
ξμτ
px, τ Δτ
, fors, y∈T. 1.6
Remark 1.11. IfTR, then fory∈Rthe following formula holds:
ep
y, s exp
y s
px, τdτ
, for s∈T. 1.7
IfTZ, then, fory∈Z,epy, s y−1
τs1px, τ, fors∈Zands < y.
The following two theorems include some known properties on the exponential function.
Theorem 1.12. Ifpx, y∈Rwith respect toy, then the following conclusions hold:
iepy, y≡1 ande0s, y≡1,
iieps, σy 1μypx, yeps, y,
iiiifp∈Rwith respect toy, theneps, y>0 for alls, y∈T, ivifp∈Rwith respect toy, thenp∈R,
veps, y 1/epy, s epy, s, wherepx, y −px, y/1μypx, y.
Theorem 1.13. Ifpx, y ∈ R with respect toy, y0 ∈ Tis a fixed number, then the exponential functionepy, y0is the unique solution of the following initial value problem:
zΔy x, y
p x, y
z x, y
, z
x, y0
1.
1.8
Theorems1.12-1.13are similar to24, Theorems 5.1-5.2. For more details about the calculus of time scales, we advise to refer to29.
In the rest of this paper, for the convenience of notation, we always assume thatT0 x0,∞
T, T0 y0,∞
T, wherex0, y0∈Tκ, and furthermore assume thatT0⊆Tκ, T0⊆ Tκ.
2. Main Results
We will give some lemmas for further use.
Lemma 2.1. Suppose thatX ∈T0 is a fixed number anduX, y, bX, y ∈ Crd, mX, y ∈R
with respect toy,mX, y≥0; then
u X, y
≤a X, y
b X, y
y y0
mX, tuX, tΔt, y∈T0, 2.1
implies
u X, y
≤a X, y
b X, y
y y0
em y, σt
aX, tmX, tΔt, y∈T0, 2.2
wheremX, y mX, ybX, y, andemy, y0is the unique solution of the following IVP zΔyX,y m
X, y z
X, y , z
X, y0
1. 2.3
The proof of Lemma2.1is similar to that of24, Theorem 5.6, and we omit it here.
Lemma 2.2. Under the conditions of Lemma 2.1and furthermore assuming that ax, y is non- decreasing inyfor every fixedx, bx, y≡1, then one has
u X, y
≤a X, y
em
y, y0
. 2.4
Proof. Sinceax, yis nondecreasing inyfor every fixedx, then from Lemma2.1we have
u X, y
≤a X, y
y
y0
em
y, σt
aX, tmX, tΔt
≤a X, y
1 y
y0
em
y, σt
mX, tΔt
.
2.5
On the other hand, from29, Theorems 2.39 and 2.36iwe have 1y
y0emy, σtmX, tΔt emy, y0.
Then collecting the above information we can obtain the desired inequality.
Lemma 2.3see30. Assume thata≥0, p≥q≥0, andp /0, then for anyK >0
aq/p ≤ q
pKq−p/pa p−q
p Kq/p. 2.6
Theorem 2.4. Suppose thatu, f, g, h, a, b∈CrdT0×T0,Randa, bare nondecreasing.p, q, r, m are constants, andp ≥ q ≥ 0, p ≥ r ≥ 0, p ≥ m ≥ 0, p /0. τ1 ∈ T0,T, τ1x ≤ x, −∞ <
α inf{τ1x, x ∈ T0} ≤ x0.τ2 ∈ T0,T, τ2y ≤ y, −∞ < β inf{τ2y, y ∈ T0} ≤ y0. φ∈Crdα, x0×β, y0
T2,R. If forx, y∈T0×T0,ux, ysatisfies the following inequality:
up x, y
≤a x, y
b x, y
y y0
x
x0
fs, tuqτ1s, τ2t gs, turs, t ΔsΔt b
x, y
y y0
x
x0
t
y0
s
x0
h ξ, η
um τ1ξ, τ2
η
ΔξΔηΔsΔt,
2.7
with the initial condition u
x, y φ
x, y
, ifx∈α, x0
T, or y∈
β, y0 T, φ
τ1x, τ2
y
≤α1/p x, y
, ifτ1x≤x0 orτ2
y
≤y0, ∀ x, y
∈T0×T0,
2.8
then
u x, y
≤
B1
x, y b
x, y
y y0
eB2 y, σt
B2x, tB1x, tΔt 1/p
,
x, y
∈T0×T0, 2.9
where
B1 x, y
a x, y
b x, y
y y0
x
x0
fs, tp−q
p Kq/pgs, tp−r p Kr/p
ΔsΔt b
x, y
y y0
x
x0
t
y0
s
x0
h
ξ, ηp−m
p Km/pΔξΔηΔsΔt, ∀K >0,
2.10
B2
x, y
x
x0
f
s, yq
pKq−p/pg s, yr
pKr−p/p
y
y0
s
x0
h ξ, ηm
pKm−p/pΔξΔη
Δs, ∀K >0,
2.11
B2
x, y b
x, y B2
x, y
. 2.12
Proof. Let the right side of2.7bevx, y. Then
u x, y
≤v1/p x, y
,
x, y
∈T0×T0. 2.13
Ifτ1x≥x0andτ2y≥y0, thenτ1x∈T0, τ2y∈T0, and sincea, bare nondecreasing we have
u
τ1x, τ2
y
≤v1/p
τ1x, τ2
y
≤v1/p x, y
. 2.14
Ifτ1x≤x0orτ2y≤y0, then from2.8we have
u
τ1x, τ2
y φ
τ1x, τ2
y
≤a1/p x, y
≤v1/p x, y
. 2.15
From2.14and2.15we have
u
τ1x, τ2
y
≤v1/p x, y
,
x, y
∈T0×T0. 2.16
FixX∈T0, and letx∈x0, X
T, y∈T0; then
v X, y
a X, y
b X, y
y y0
X
x0
fs, tuqτ1s, τ2t gs, turs, t ΔsΔt b
X, y
y y0
X
x0
t
y0
s
x0
h ξ, η
um τ1ξ, τ2
η
ΔξΔηΔsΔt
≤a X, y
b X, y
y y0
X
x0
fs, tvq/ps, t gs, tvr/ps, t
t
y0
s
x0
h ξ, η
vm/p ξ, η
ΔξΔη
ΔsΔt.
2.17
From Lemma2.3, we have
vq/p x, y
≤ q
pKq−p/pv x, y
p−q p Kq/p, vr/p
x, y
≤ r
pKr−p/pv x, y
p−r p Kr/p, vm/p
x, y
≤ m
pKm−p/pv x, y
p−m
p Km/p, ∀K >0.
2.18
So combining2.17and2.18, it follows that
v X, y
≤a X, y
b X, y
y y0
X
x0
fs, t q
pKq−p/pvs, t p−q p Kq/p
ΔsΔt b
X, y
y y0
X
x0
gs, t r
pKr−p/pvs, t p−r p Kr/p
ΔsΔt b
X, y
y y0
X
x0
t
y0
s
x0
h
ξ, ηm
pKm−p/pv ξ, η
p−m p Km/p
ΔξΔηΔsΔt
≤a X, y
b X, y
y y0
X
x0
fs, tp−q
p Kq/pgs, tp−r p Kr/p
t
y0
s
x0
h
ξ, ηp−m
p Km/pΔξΔη
ΔsΔt b
X, y
y y0
X
x0
fs, tq
pKq−p/pgs, tr
pKr−p/p
t
y0
s
x0
h ξ, ηm
pKm−p/pΔξΔη
Δs
vX, tΔt
B1 X, y
b X, y
y y0
B2X, tvX, tΔt,
2.19 whereB1x, y, B2x, yare defined in2.10and2.11, respectively. ConsideringB2X, y bX, yB2X, y, by application of Lemma2.1, we have
v X, y
≤B1
X, y b
X, y
y y0
eB
2
y, σt
B2X, tB1X, tΔt, y∈T0. 2.20
SinceX ∈T0is arbitrary, then in fact2.20holds for allx∈T0, that is,
v x, y
≤B1
x, y b
x, y
y y0
eB
2
y, σt
B2x, tB1x, tΔt, x, y
∈ T0×T0
.
2.21 Combining2.13and2.21, we obtain
u x, y
≤
B1
x, y b
x, y
y y0
eB
2
y, σt
B2x, tB1x, tΔt 1/p
,
x, y
∈ T0×T0
, 2.22 which is the desired inequality.
If we apply Lemma2.2instead of Lemma2.1at the end of the proof of Theorem2.4, we obtain the following theorem.
Theorem 2.5. Suppose thatu, f, g, h, a, p, q, r, m, τ1, τ2, α, β, φare defined as in Theorem2.4. If that forx, y∈T0×T0, ux, ysatisfies the following inequality:
up x, y
≤a x, y
y
y0
x
x0
fs, tuqτ1s, τ2t gs, turs, t
t
y0
s
x0
h ξ, η
um τ1ξ, τ2
η ΔξΔη
ΔsΔt
2.23
with the initial condition2.8, then
u x, y
≤ B1
x, y eB2
y, y01/p
,
x, y
∈T0×T0, 2.24
where
B1
x, y a
x, y
y
y0
x
x0
fs, tp−q
p Kq/pgs, tp−r p Kr/p
ΔsΔt
y
y0
x
x0
t
y0
s
x0
h
ξ, ηp−m
p Km/pΔξΔηΔsΔt, ∀K >0,
B2
x, y
x
x0
f
s, yq
pKq−p/pg s, yr
pKr−p/p y
y0
s
x0
h ξ, ηm
pKm−p/pΔξΔη
Δs,
∀K >0.
2.25
From Theorems2.4and2.5we can obtain two direct corollaries.
Corollary 2.6. Under the conditions of Theorem2.4, if, for x, y ∈ T0×T0,ux, ysatisfies the following inequality:
u x, y
≤a x, y
y
y0
x
x0
fs, tuτ1s, τ2t gs, tus, t ΔsΔt
y
y0
x
x0
t
y0
s
x0
h ξ, η
u τ1ξ, τ2
η
ΔξΔηΔsΔt,
2.26
with the initial condition2.8(p=1), then
u x, y
≤a x, y
y
y0
eB2
y, σt
B2x, tax, tΔt, x, y
∈T0×T0, 2.27
where
B2
x, y
x
x0
f
s, y g
s, y
y
y0
s
x0
h ξ, η
ΔξΔη
Δs. 2.28
Corollary 2.7. Under the conditions of Theorem2.5, if, for x, y ∈ T0×T0,ux, ysatisfies the following inequality:
u x, y
≤a x, y
y
y0
x
x0
fs, tuτ1s, τ2t gs, tus, t ΔsΔt
y
y0
x
x0
t
y0
s
x0
h ξ, η
u τ1ξ, τ2
η
ΔξΔηΔsΔt,
2.29
with the initial condition2.8(p1), then u
x, y
≤a x, y
eB2 y, y0
,
x, y
∈T0×T0, 2.30
where
B2 x, y
x
x0
f
s, y g
s, y
y
y0
s
x0
h ξ, η
ΔξΔη
Δs. 2.31
Theorem 2.8. Suppose thatu∈CrdT0 × T0,R,f, g, h, τ1, τ2are defined as in Theorem2.4, and τ1x≥x0,τ2y≥y0. If, forx, y∈T0×T0, ux, ysatisfies the following inequality:
u x, y
≤ y
y0
x
x0
fs, tuτ1s, τ2t gs, tus, t t
y0
s
x0
h ξ, η
u τ1ξ, τ2
η ΔξΔη
ΔsΔt,
2.32
thenux, y≡0.
The proof of Theorem2.8is similar to Theorem2.4, and we omit it here.
Based on Theorem 2.4, we will establish a class of Volterra-Fredholm-type integral inequality on time scales.
Theorem 2.9. Suppose that u, fi, gi, hi ∈ CrdT0 ×T0,R, i 1,2. a, p, q, r, m, φ, τ1, τ2, α, β are the same as in Theorem 2.4, and M ∈ T0, N ∈ T0 are two fixed numbers. If, for x, y ∈ x0, M
T×y0, N
T, ux, ysatisfies the following inequality:
up x, y
≤a x, y
y
y0
x
x0
f1s, tuqτ1s, τ2t g1s, turτ1s, τ2t ΔsΔt
y
y0
x
x0
t
y0
s
x0
h1
ξ, η um
τ1ξ, τ2
η
ΔξΔηΔsΔt
N
y0
M
x0
f2s, tuqτ1s, τ2t g2s, turτ1s, τ2t ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2 ξ, η
um
τ1ξ, τ2
η
ΔξΔηΔsΔt,
2.33
with the initial condition2.8, then one has
u x, y
≤
λB6
1−B5
B3
x, y B4
x, y1/p
,
x, y
∈
x0, M T
×
y0, N T ,
2.34
provided thatB5<1, where
λ N
y0
M
x0
f2s, tp−q
p Kq/pg2s, tp−r p Kr/p
t
y0
s
x0
h2
ξ, ηp−m
p Km/pΔξΔη
ΔsΔt,
2.35
B1
x, y a
x, y
y
y0
x
x0
f1s, tp−q
p Kq/pg1s, tp−r p Kr/p
ΔsΔt
y
y0
x
x0
t
y0
s
x0
h1
ξ, ηp−m
p Km/pΔξΔηΔsΔt, ∀K >0,
2.36
B2
x, y
x
x0
f1
s, yq
pKq−p/pg1
s, yr
pKr−p/p
y
y0
s
x0
h1
ξ, ηm
pKm−p/pΔξΔη
Δs, ∀K >0,
2.37
B3
x, y 1
y
y0
eB2
y, σt B2x, tΔt, 2.38
B4
x, y B1
x, y
y
y0
eB2
y, σt B2x, tB1x, tΔt, 2.39
B5 N
y0
M
x0
f2s, tq
pKq−p/pB3s, t g2s, tr
pKr−p/pB3s, t
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2 ξ, ηm
pKm−p/pB3 ξ, η
ΔξΔηΔsΔt,
2.40
B6 N
y0
M
x0
f2s, tq
pKq−p/pB4s, t g2s, tr
pKr−p/pB4s, t
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2
ξ, ηm
pKm−p/pB4
ξ, η
ΔξΔηΔsΔt.
2.41
Proof. Let the right side of2.33bevx, yand
μ N
y0
M
x0
f2s, tuqτ1s, τ2t g2s, turτ1s, τ2t ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2 ξ, η
um τ1ξ, τ2
η
ΔξΔηΔsΔt.
2.42
Then
u x, y
≤v1/p x, y
,
x, y
∈
x0, M T
×
y0, N T
. 2.43
Similar to the process of2.14–2.16one has
u
τ1x, τ2
y
≤v1/p x, y
,
x, y
∈
x0, M T
×
y0, N T
. 2.44
FixX∈x0, M
T, and letx∈x0, X
T, y∈y0, N
T. Then
v X, y
a X, y
μ y
y0
X
x0
f1s, tuqτ1s, τ2t g1s, turτ1s, τ2t ΔsΔt
y
y0
X
x0
t
y0
s
x0
h1
ξ, η um
τ1ξ, τ2
η
ΔξΔηΔsΔt
≤a X, y
μ y
y0
X
x0
f1s, tvq/ps, t g1s, tvr/ps, t ΔsΔt
y
y0
X
x0
t
y0
s
x0
h1
ξ, η vm/p
ξ, η
ΔξΔηΔsΔt.
2.45
Considering the structure of2.45is similar to2.17, then following in a same manner as the process of2.17–2.20we can deduce
v X, y
≤μB1 X, y
y
y0
eB
2
y, σt B2X, t
μB1X, t Δt μ
1
y
y0
eB2
y, σt B2X, tΔt
B1
X, y
y
y0
eB
2
y, σt B2X, tB1X, tΔt, y∈
y0, N T,
2.46
whereB1x, y,B2x, yare defined in2.36and2.37, respectively.
SinceXis selected fromx0, M
Tarbitrarily, then in fact2.46holds for allx∈T0, that is,
v x, y
≤μ
1 y
y0
eB
2
y, σt B2x, tΔt
B1
x, y
y
y0
eB
2
y, σt B2x, tB1x, tΔt
μB3 x, y
B4 x, y
,
x, y
∈
x0, M T
×
y0, N T ,
2.47
whereB3x, y, B4x, yare defined in2.38and2.39, respectively.
On the other hand, from2.18,2.42, and2.44we obtain
μ≤ N
y0
M
x0
f2s, tvq/ps, t g2s, tvr/ps, t t
y0
s
x0
h2 ξ, η
vm/p ξ, η
ΔξΔη
ΔsΔt
≤ N
y0
M
x0
f2s, t
q
pKq−p/pvs, t p−q p Kq/p
g2s, t r
pKr−p/pvs, t p−r p Kr/p
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2
ξ, ηm
pKm−p/pv ξ, η
p−m p Km/p
ΔξΔηΔsΔt λ
N
y0
M
x0
f2s, tq
pKq−p/pvs, t g2s, tr
pKr−p/pvs, t
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2
ξ, ηm
pKm−p/pv ξ, η
ΔξΔηΔsΔt,
2.48
whereλis defined in2.35. Then using2.47in2.48yields
μ≤λ N
y0
M
x0
f2s, tq
pKq−p/p
μB3s, t B4s, t
g2s, tr
pKr−p/p
μB3s, t B4s, t ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2
ξ, ηm
pKm−p/p μB3
ξ, η B4
ξ, η
ΔξΔηΔsΔt λμ
N
y0
M
x0
f2s, tq
pKq−p/pB3s, t g2s, tr
pKr−p/pB3s, t
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2 ξ, ηm
pKm−p/pB3 ξ, η
ΔξΔηΔsΔt
N
y0
M
x0
f2s, tq
pKq−p/pB4s, t g2s, tr
pKr−p/pB4s, t
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2
ξ, ηm
pKm−p/pB4
ξ, η
ΔξΔηΔsΔt λμB5B6,
2.49
which implies
μ≤ λB6
1−B5. 2.50
Combining2.43,2.47, and2.50we can obtain the desired inequality2.34.
In the proof of Theorem2.9, if we let the right side of2.33beax, y vx, yin the first statement, then following in a same process as in Theorem2.9we obtain another bound on the functionux, y, which is shown in the following theorem.
Theorem 2.10. Under the conditions of Theorem2.9, if, forx, y∈x0, MT×y0, NT, ux, ysatisfies2.33with the initial condition2.8, then the following inequality holds:
u x, y
≤
a x, y
μJ1 x, y 1−λ eJ2
y, y0
1/p
,
x, y
∈
x0, M T
×
y0, N T , 2.51
provided thatλ < 1, where
λ N
y0
M
x0
f2s, tq
pKq−p/peJ2
t, y0
g2s, tr
pKr−p/peJ2
t, y0
ΔsΔt
N
y0
M
x0
t
y0
s
x0
h2 ξ, ηm
pKm−p/peJ2 η, y0
ΔξΔηΔsΔt,
μ N
y0
M
x0
f2s, t
q
pKq−p/pas, t p−q p Kq/p
g2s, t r
pKr−p/pas, t p−r p Kr/p
t
y0
s
x0
h2
ξ, ηm
pKm−p/p a
ξ, η v
ξ, η
p−m p Km/p
ΔξΔη
ΔsΔt, J1
x, y
y
y0
x
x0
f1s, t
q
pKq−p/pas, t p−q p Kq/p
g1s, t r
pKr−p/pas, t p−r p Kr/p
ΔsΔt
y
y0
X
x0
t
y0
s
x0
h1
ξ, ηm
pKm−p/pa ξ, η
p−m p Km/p
ΔξΔηΔsΔt, J2
x, y
x
x0
f1
s, yq
pKq−p/pg1
s, yr
pKr−p/p y
y0
s
x0
h1
ξ, ηm
pKm−p/pΔξΔη
Δs.
2.52 Finally, we will establish a more general inequality than that in Theorems2.9-2.10.
Consider the following inequality:
up x, y
≤a x, y
y
y0
x
x0
Ls, t, uτ1s, τ2t t
y0
s
x0
h1 ξ, η
uq
τ1ξ, τ2
η ΔξΔη
ΔsΔt
N
y0
M
x0
Ls, t, uτ1s, τ2t t
y0
s
x0
h2 ξ, η
uq τ1ξ, τ2
η ΔξΔη
ΔsΔt,
2.53 with the initial condition 2.8, where u, a, p, q, φ, α, β, τi, hi, i 1,2 are the same as in Theorem 2.4, M ∈ T0, N ∈ T0 are two fixed numbers. L ∈ T0 × T0 × R,R, and 0≤Ls, t, x−Ls, t, y≤As, t, yx−yforx≥y≥0, whereA∈T0×T0×R,R.
Theorem 2.11. If, for x, y ∈ x0, M
T×y0, N
T, ux, y satisfies2.53, then the following inequality holds:
u x, y
≤
λB6
1−B5
B3
x, y B4
x, y1/p
,
x, y
∈
x0, M T
×
y0, N T , 2.54
provided thatB5<1, where
λ N
y0
M
x0
L
s, t,p−1 p K1/p
t
y0
s
x0
h2
ξ, ηp−q
p Kq/pΔξΔη
ΔsΔt, 2.55 B1
x, y a
x, y
y
y0
x
x0
L
s, t,p−1 p K1/p
t
y0
s
x0
h1
ξ, ηp−q
p Kq/pΔξΔη
ΔsΔt, ∀K >0, 2.56 B2
x, y
x
x0
A
s, y,p−1 p K1/p
1
pK1−p/p
y
y0
s
x0
h1
ξ, ηq
pKq−p/pΔξΔη
Δs, ∀K >0,
2.57
B3 x, y
1 y
y0
eB
2
y, σt B2x, tΔt, 2.58
B4 x, y
B1 x, y
y
y0
eB
2
y, σt B2x, tB1x, tΔt, 2.59
B5 N
y0
M
x0
A
s, t,p−1 p K1/p
1
pK1−p/pB3s, t
t
y0
s
x0
h2
ξ, ηq
pKq−p/pB3
ξ, η ΔξΔη
ΔsΔt,
2.60
B6 N
y0
M
x0
A
s, t,p−1 p K1/p
1
pK1−p/pB4s, t
t
y0
s
x0
h2 ξ, ηq
pKq−p/pB4 ξ, η
ΔξΔη
ΔsΔt.
2.61
Proof. Let the right side of2.53bevx, yand
μ
N
y0
M
x0
Ls, t, uτ1s, τ2t t
y0
s
x0
h2
ξ, η uq
τ1ξ, τ2
η ΔξΔη
ΔsΔt. 2.62
Then
u x, y
≤v1/p x, y
,
x, y
∈
x0, M T
×
y0, N T
. 2.63
Similar to the process of2.14–2.16we have
u
τ1x, τ2
y
≤v1/p x, y
,
x, y
∈
x0, M T
×
y0, N T
. 2.64
FixX∈x0, MT, and letx∈x0, XT, y∈y0, NT. Then
v X, y
a X, y
μ
y
y0
X
x0
Ls, t, uτ1s, τ2t t
y0
s
x0
h1
ξ, η uq
τ1ξ, τ2
η ΔξΔη
ΔsΔt
≤a X, y
μ y
y0
X
x0
L
s, t, v1/ps, t
t
y0
s
x0
h1
ξ, η vq/p
ξ, η ΔξΔη
ΔsΔt.
2.65
From Lemma2.3, we have
vq/p x, y
≤ q
pKq−p/pv x, y
p−q p Kq/p, v1/p
x, y
≤ 1
pK1−p/pv x, y
p−1
p K1/p, ∀K >0.
2.66
Combining2.65and2.66, it follows that
v X, y
≤a X, y
μ y
y0
X
x0
L
s, t,1
pK1−p/pvs, t p−1 p K1/p
ΔsΔt
y
y0
X
x0
t
y0
s
x0
h1
ξ, ηq
pKq−p/pv ξ, η
p−q p Kq/p
ΔξΔηΔsΔt a
X, y μ
y
y0
X
x0
L
s, t,1
pK1−p/pvs, t p−1 p K1/p
−L
s, t,p−1 p K1/p
L
s, t,p−1 p K1/p
ΔsΔt
