Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 659563, 18 pages
doi:10.1155/2011/659563
Research Article
Some New Delay Integral Inequalities in
Two Independent Variables on Time Scales
Bin Zheng, Yaoming Zhang, and Qinghua Feng
School of Science, Shandong University of Technology, Zibo, Shandong, 255049, China
Correspondence should be addressed to Bin Zheng, zhengbin2601@126.com
Received 9 August 2011; Accepted 10 October 2011
Academic Editor: C. Conca
Copyright q 2011 Bin Zheng 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 can be used as a handy tool in the research of boundedness
of solutions of delay dynamic equations on time scales. Some of the established results are 2D
extensions of several known results in the literature, while some results unify existing continuous
and discrete analysis.
1. Introduction
In the research of solutions of certain differential and difference equations, if the solutions
are unknown, then it is necessary to study their qualitative and quantitative properties such
as boundedness, uniqueness, and continuous dependence on initial data. The GronwallBellman inequality 1, 2 and its various generalizations, which provide explicit bounds, play
a fundamental role in the research of this domain. During the past decades, much effort has
been done for developing such inequalities e.g., see 3–15 and the references therein. On
the other hand, Hilger 16 initiated the theory of time scales as a theory capable to contain
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 scales e.g., see
17–19 and the references therein. In these investigations, integral inequalities on time
scales have been paid much attention by many authors, which play a fundamental role in
the research of quantitative as well as qualitative properties of solutions of certain dynamic
equations on time scales. A lot of integral inequalities on time scales have been established
e.g., see 20–26, which have been designed to unify continuous and discrete analysis.
But to our best knowledge, the 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 work of Li 27 and that of Ma and Pečarić 28. Furthermore, nobody has studied
2
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the Gronwall-Bellman-type delay integral inequalities in two independent variables on time
scales.
The aim of this paper is to establish some new Gronwall-Bellman-type delay integral
inequalities in two independent variables on time scales, which provide new bounds for
the unknown functions concerned. Some of our results are 2D extensions of many known
inequalities in the literature, while some results unify existing continuous and discrete
analysis. For illustrating the validity of the established results, we will present some
applications of them.
First we will give some preliminaries on time scales and some universal symbols for
further use.
Throughout this paper, R denotes the set of real numbers and R 0, ∞, while Z
denotes the set of integers. For two given sets G, H, we denote the set of maps from G to H
by G, H.
A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper,
T denotes an arbitrary time scale. On T we define the forward and backward jump operators
σ ∈ T, T and ρ ∈ T, T such that σt inf{s ∈ T, s > t}, ρt sup{s ∈ T, s < t}.
Definition 1.1. The graininess μ ∈ T, R is defined by μt σt − t.
Remark 1.2. Obviously, μt 0 if T R while μt 1 if T Z.
Definition 1.3. A point t ∈ T is said to be left-dense if ρt t and t / inf T, right-dense if
σt t and t /
sup T, left-scattered if ρt < t, and right-scattered if σt > t.
Definition 1.4. The set Tκ is defined to be T if T does not have a left-scattered maximum;
otherwise it is T without the left-scattered maximum.
Definition 1.5. A function f ∈ T, R is 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
1 μtft /
0. Crd denotes the set of rd-continuous functions, while R denotes the set of all
regressive and rd-continuous functions, and R {f | f ∈ R, 1 μtft > 0, ∀t ∈ T}.
Definition 1.6. For some t ∈ Tκ and a function f ∈ T, R, the delta derivative of f at t is
denoted by f Δ t provided it exists with the property such that for every ε > 0 there exists
a neighborhood U of t satisfying
fσt − fs − f Δ tσt − s ≤ ε|σt − s|
∀s ∈ U.
1.1
Similarly, for some y ∈ Tκ and a function f ∈ T × T, R, the partial delta of f with
Δ
respect to y is denoted by fx, yΔ
y or fy x, y and satisfies
f x, σ y − fx, s − fyΔ x, y σ y − s ≤ εσ y − s
∀ε > 0,
1.2
where s ∈ U and U is a neighborhood of y.
Remark 1.7. If T R, then f Δ t becomes the usual derivative f t, while f Δ t ft1−ft
if T Z, which represents the forward difference.
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3
For more details about the calculus of time scales, see 29. In the rest of this paper,
0 y0 , ∞ T,
for the convenience of notation, we always assume that T0 x0 , ∞ T, T
0 ⊆ Tκ .
where x0 , y0 ∈ Tκ and furthermore assume T0 ⊆ Tκ , T
2. Main Results
We will give some lemmas for further use.
Lemma 2.1. Suppose X ∈ T0 is a fixed number, and uX, y, aX, y, bX, y ∈ Crd , mX, y ∈
R with respect to y, mX, y ≥ 0, then
u X, y ≤ a X, y b X, y
y
mX, tuX, tΔt,
0,
y∈T
2.1
y0
implies
u X, y ≤ a X, y b X, y
y
em y, σt aX, tmX, tΔt,
0,
y∈T
2.2
y0
where mX, y mX, ybX, y: and em y, y0 is the unique solution of the following equation
zΔ
y X, y m X, y z X, y , z X, y0 1.
2.3
The proof of Lemma 2.1 is similar to 26, Theorem 5.6.
Lemma 2.2. Under the conditions of Lemma 2.1, and furthermore assuming ax, y is nondecreasing
in y for every fixed x, bx, y ≡ 1, then one has
u X, y ≤ a X, y em y, y0 .
2.4
Proof. Since ax, y is nondecreasing in y for every fixed x, then from Lemma 2.1 we have
u X, y ≤ a X, y y
y
em y, σt aX, tmX, tΔt ≤ a X, y 1 em y, σt mX, tΔt .
y0
y0
2.5
y
On the other hand, from 29, Theorems 2.39 and 2.36 i we have 1 y0 em y, σtmX, tΔt em y, y0 . Then collecting the above information, we can obtain the desired inequality.
Lemma 2.3 see 11. Assume that a ≥ 0, p ≥ q ≥ 0, and p / 0; then, for any K > 0
aq/p ≤
q q−p/p
p − q q/p
K
K .
a
p
p
2.6
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Lemma 2.4. Let h : T × R → R be continuous and nondecreasing in the second variable, and assume
X is a fixed number in T. Suppose vX, y and wX, y satisfy the dynamics inequalities:
vyΔ ≤ h y, v ,
wyΔ ≥ h y, w .
2.7
Then vX, y0 ≤ wX, y0 for some y0 ∈ T implies vX, y ≤ wX, y for all y ∈ T.
The proof of Lemma 2.4 is similar to 26, Theorem 5.7.
0 , R , and ax, y, bx, y are nondecreasing. p
Theorem 2.5. Suppose u, a, b, f ∈ Crd T0 × T
is a constant, and p ≥ 1. τ1 ∈ T0 , T, τ1 x ≤ x, −∞ < α inf{τ1 x, x ∈ T0 } ≤ x0 . τ2 ∈
0 } ≤ y0 . φ ∈ Crd α, x0 × β, y0 T2 , R . If for
0 , T, τ2 y ≤ y, −∞ < β inf{τ2 y, y ∈ T
T
0 , ux, y satisfies the following inequality:
x, y ∈ T0 × T
u x, y ≤ a x, y b x, y
p
y x
y0
fs, tuτ1 s, τ2 t Δs Δt,
x0
2.8
with the initial condition
u x, y φ x, y ,
if x ∈ α, x0 φ τ1 x, τ2 y ≤ a1/p x, y ,
T or y ∈ β, y0
T,
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
∀ x, y ∈ T0 × T
2.9
then
1/p
y
u x, y ≤ H1 x, y b x, y
eH 2 y, σt H2 x, tH1 x, tΔt
,
0,
x, y ∈ T0 × T
y0
2.10
where
H1 x, y a x, y b x, y
y x
fs, t
y0
x
x0
p − 1 1/p
K Δs Δt,
p
∀K > 0,
1
f s, y K 1−p/p Δs,
p
x0
H 2 x, y b x, y H2 x, y .
H2 x, y Proof. Fix X ∈ T0 , and x ∈ x0 , X
2.11
0 . Let
T, y ∈ T
v x, y a x, y b x, y
y x
y0
x0
fs, tuτ1 s, τ2 t Δs Δt.
2.12
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5
Then
u x, y ≤ v1/p x, y ≤ v1/p X, y ,
If τ1 x ≥ x0 and τ2 y ≥ y0 , then τ1 x ∈ x0 , X
∀x ∈ x0 , X
0.
T, y ∈ T
2.13
0 , and
T, τ2 y ∈ T
u τ1 x, τ2 y ≤ v1/p τ1 x, τ2 y ≤ v1/p x, y .
2.14
If τ1 x ≤ x0 or τ2 y ≤ y0 , then from 2.9 we have
u τ1 x, τ2 y φ τ1 x, τ2 y ≤ a1/p x, y ≤ v1/p x, y .
2.15
From 2.14 and 2.15 we always have
u τ1 x, τ2 y ≤ v1/p x, y ,
x ∈ x0 , X
0.
T, y ∈ T
2.16
Moreover
v X, y a X, y b X, y
y X
y0
≤ a X, y b X, y
fs, tuτ1 s, τ2 t Δs Δt
x0
2.17
y X
fs, tv
y0
1/p
s, tΔs Δt.
x0
From Lemma 2.3, we have
v1/p s, t ≤
p − 1 1/p
1 1−p/p
K
K ,
vs, t p
p
∀K > 0.
2.18
So
v X, y ≤ a X, y b X, y
y X
y0
≤ a X, y b X, y
y X
fs, t
x0
y X
fs, t
y0
x0
p − 1 1/p
1 1−p/p
K
K
Δs Δt
vs, t p
p
p − 1 1/p
K Δs Δt
p
1
b X, y
fs, t K 1−p/p Δs vX, tΔt
p
y0
x0
y
H2 X, tvX, tΔt.
H1 X, y b X, y
y0
2.19
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Then applying Lemma 2.1 to 2.19, we obtain
v X, y ≤ H1 X, y b X, y
y
y0
eH 2 y, σt H2 X, tH1 X, tΔt.
2.20
So
1/p
y
1/p
u x, y ≤ v
eH 2 y, σt H2 X, tH1 X, tΔt
,
X, y ≤ H1 X, y b X, y
y0
x ∈ x0 , X
T,
2.21
0.
y∈T
Setting x X in 2.21, it follows that
u X, y ≤ H1 X, y b X, y
y
y0
eH 2
1/p
y, σt H2 X, tH1 X, tΔt
.
2.22
Replacing X with x in 2.22, we obtain the desired inequality.
Remark 2.6. Theorem 2.5 is the 2D extension of 27, Theorem 1. For its special case T R,
the established bound for ux, y in 2.10 is a new bound compared with the result in 12,
Theorem 2.2.
Remark 2.7. Assume bx, y ≡ 1 in Theorem 2.5. If we apply Lemma 2.2 instead of Lemma 2.1
to 2.19 in the proof of Theorem 2.5, then we obtain another bound for ux, y as follows:
1/p
,
u x, y ≤ H1 x, y eH2 y, y0
0.
x, y ∈ T0 × T
2.23
Now we will establish a more general inequality than that in Theorem 2.5.
Theorem 2.8. Suppose u, a, b, f, φ, τ1 , τ2 , α, β are the same as in Theorem 2.5, and g ∈ Crd T0 ×
0 , ux, y satisfies
0 , R . p, q, r are constants, and p ≥ q ≥ 0, p ≥ r ≥ 0, ρ T
/ 0. If for x, y ∈ T0 × T
the following inequality:
up x, y ≤ a x, y b x, y
y x
y0
fs, tuq τ1 s, τ2 t gs, tur τ1 s, τ2 t Δs Δt,
x0
2.24
with the initial condition 2.9, then
1/p
y
u x, y ≤ H1 x, y b x, y
e y, σt H2 x, tH1 x, tΔt
,
y0
H2
0,
x, y ∈ T0 × T
2.25
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7
where
y x p − q q/p
p − r r/p
K gs, t
K
Δs Δt,
p
p
y0 x0
x r
q
H2 x, y f s, y K q−p/p g s, y K r−p/p Δs,
p
p
x0
1 x, y a x, y b x, y
H
fs, t
2.26
2 x, y b x, y H
2 x, y .
H
Proof. Fix X ∈ T0 , and x ∈ x0 , X
v x, y a x, y b x, y
0 . Let
T, y ∈ T
y x
y0
fs, tuq τ1 s, τ2 t gs, tur τ1 s, τ2 t Δs Δt.
x0
2.27
Then
u x, y ≤ v1/p x, y ≤ v1/p X, y ,
∀x ∈ x0 , X
0.
T, y ∈ T
2.28
Similar to 2.14–2.16, we obtain
u τ1 x, τ2 y ≤ v1/p x, y ,
x ∈ x0 , X
0.
T, y ∈ T
2.29
So
v X, y a X, y b X, y
y X
y0
≤ a X, y b X, y
fs, tuq τ1 s, τ2 t gs, tur τ1 s, τ2 t Δs Δt
x0
y X
fs, tvq/p s, t gs, tvr/p s, t Δs Δt.
y0
x0
2.30
From Lemma 2.3, we have
p − q q/p
q q−p/p
K
K ,
vs, t p
p
∀K > 0,
p − r r/p
r
K ,
s, t ≤ K r−p/p vs, t p
p
∀K > 0.
vq/p s, t ≤
v
r/p
2.31
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Combining 2.30 and 2.31 we get that
y X
p − q q/p
q q−p/p
K
K
vs, t p
p
y0 x0
p − r r/p
r r−p/p
gs, t
K
K
vs, t Δs Δt
p
p
y X
p − r r/p
p − q q/p
K gs, t
K
≤ a X, y b X, y
fs, t
Δs Δt
p
p
y0 x0
X y
q q−p/p
r r−p/p
b X, y
gs, t K
fs, t K
Δs vX, tΔt
p
p
y0
x0
y
1 X, y b X, y
2 X, tvX, tΔt.
H
H
v X, y ≤ a X, y b X, y
fs, t
2.32
y0
Applying Lemma 2.1 to 2.32 yields
1 X, y b X, y
v X, y ≤ H
y
y0
1 X, tΔt.
2 X, tH
e y, σt H
H2
2.33
Then
1/p
y
1/p
1 X, tΔt
2 X, tH
1 X, y b X, y
u x, y ≤ v
e y, σt H
,
X, y ≤ H
y0
H2
x ∈ x0 , X
T,
2.34
0.
y∈T
Setting x X in 2.34 yields
1 X, y b X, y
u X, y ≤ H
y
y0
e
H2
1/p
1 X, tΔt
2 X, tH
y, σt H
.
2.35
Considering X ∈ T0 is arbitrary and replacing X with x in 2.35, we obtain the desired
inequality.
Remark 2.9. Assume bx, y ≡ 1 in Theorem 2.8. If we apply Lemma 2.2 instead of Lemma 2.1
to 2.32 in the proof of Theorem 2.8, then we obtain another bound for ux, y as follows:
1/p
1 x, y e y, y0
u x, y ≤ H
,
H2
0.
x, y ∈ T0 × T
Remark 2.10. Theorem 2.8 is the 2D extension of 27, Theorem 3.
2.36
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9
Theorem 2.11. Suppose u, f, α, β, φ, τ1 , τ2 are the same as in Theorem 2.5, and C > 0 is a constant.
0 , ux, y satisfies the following inequality:
If for x, y ∈ T0 × T
u2 x, y ≤ C y x
y0
fs, tuτ1 s, τ2 t uτ1 s, στ2 tΔs Δt
x0
2.37
with the initial condition
u x, y φ x, y ,
φ τ1 x, τ2 y ≤ C1/2 ,
if x ∈ α, x0 T,
T or y ∈ β, y0
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
∀ x, y ∈ T0 × T
2.38
then
√
u x, y ≤ C y x
y0
fs, tΔs Δt,
0 .
x, y ∈ T0 × T
2.39
x0
Proof. Let the right side of 2.37 be v2 x, y. Then
u x, y ≤ v x, y ,
0 .
∀ x, y ∈ T0 × T
2.40
0 , if τ1 x ≥ x0 and τ2 y ≥ y0 , then τ1 x ∈ T0 and τ2 y ∈ T
0 , and from
For x, y ∈ T0 × T
2.40 we have
u τ1 x, τ2 y ≤ v τ1 x, τ2 y ≤ v x, y .
2.41
If τ1 x ≤ x0 or τ2 y ≤ y0 , from 2.38 we have
u τ1 x, τ2 y φ τ1 x, τ2 y ≤ a1/2 τ1 x, τ2 y ≤ a1/2 x, y ≤ v x, y .
2.42
So from 2.41 and 2.42, we always have
u τ1 x, τ2 y ≤ v x, y ,
0 .
∀ x, y ∈ T0 × T
2.43
0 , and from
Similarly, when τ1 x ≥ x0 and στ2 y ≥ y0 , then τ1 x ∈ T0 and στ2 y ∈ T
2.40 we have
u τ1 x, σ τ2 y
≤ v τ1 x, σ τ2 y
≤ v x, σ y .
2.44
When τ1 x ≤ x0 or στ2 y ≤ y0 , considering στ2 y ≥ τ2 y ≥ β, from 2.38 it follows
that
u τ1 x, σ τ2 y
φ τ1 x, σ τ2 y
≤ C1/2 ≤ v x, y ≤ v x, σ y .
2.45
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Journal of Applied Mathematics
Combining 2.44 and 2.45, we always have
0 .
∀ x, y ∈ T0 × T
u τ1 x, σ τ2 y
≤ v x, σ y ,
2.46
By 2.43 and 2.46, we obtain
v2 x, y ≤ C y x
y0
0.
x ∈ T0 , y ∈ T
fs, tvs, t vs, σtΔs Δt,
x0
2.47
Let the right side of 2.47 be z2 x, y. Then
0 ,
v x, y ≤ z x, y , ∀ x, y ∈ T0 × T
x
Δ
z2 x, y
f s, y v s, y v s, σ y Δs
y
x0
≤
x
f s, y Δs
x0
≤
x
f s, y Δs
v x, y v x, σ y
2.48
2.49
z x, y z x, σ y .
x0
Considering zx, y zx, σy ≥ zx0 , y0 C > 0, and z2 x, yΔ
y zx, y zx, σyzx, yΔ
,
from
2.49
it
follows
that
y
Δ
z x, y y ≤
x
f s, y Δs.
2.50
x0
integration of 2.50 with respect to y from y0 to y yields zx, y − zx, y0 ≤
An
y x
fs, tΔs Δt.
y0 x0
√
Considering zx, y0 C, it follows that
√
z x, y ≤ C y x
y0
fs, tΔs Δt.
x0
2.51
Then combining 2.40, 2.48, and 2.51, we obtain
√
u x, y ≤ v x, y ≤ z x, y ≤ C y x
y0
fs, tΔs Δt,
x0
2.52
and the proof is complete.
Remark 2.12. If we take T R, then Theorem 2.11 becomes the extension of the known OuIang’s inequality 13 to the 2D case.
The following theorem provides a more general result than Theorem 2.11.
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11
Theorem 2.13. Suppose p is a positive integer, and p ≥ 2. Under the conditions of Theorem 2.11, if
ux, y satisfies
u x, y ≤ C
p
y x
fs, t
y0
x0
p−1 ul τ1 s, τ2 tup−1−l τ1 s, στ2 t Δs Δt,
l0
0 ,
x, y ∈ T0 × T
with the initial condition
u x, y φ x, y , if x ∈ α, x0 T or y ∈ β, y0
T,
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
φ τ1 x, τ2 y ≤ C1/p , ∀ x, y ∈ T0 × T
2.53
2.54
then
u x, y ≤ C1/p y x
y0
0.
x ∈ T0 , y ∈ T
fs, tΔs Δt,
2.55
x0
The proof of Theorem 2.13 is similar to Theorem 2.11. As long as we notice a delta d
ifferentiable function zx, y, the following formula 26, Equation 6.2 holds:
p−1 Δ p
Δ zl x, y zp−1−l x, σ y .
z x, y y z x, y y
2.56
l0
Then following a similar manner as in Theorem 2.11, we can deduce the desired result.
Theorem 2.14. Suppose u, f, τ1 , τ2 , φ, α, β are the same as in Theorem 2.5, ω ∈ CR , R is
nondecreasing, and p, C are constants with p ≥ 1, C > 0. Furthermore, define a bijective function
Δ
1/p
0 , ux, y satisfies the
x, y. If for x, y ∈ T0 × T
G such that Gzx, yΔ
y zx, yy /ωz
following inequality:
u x, y ≤ C p
y x
y0
fs, tωuτ1 s, τ2 t Δs Δt,
x0
2.57
with the initial condition
u x, y φ x, y , if x ∈ α, x0 T or y ∈ β, y0
T,
1/p
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
φ τ1 x, τ2 y ≤ C , ∀ x, y ∈ T0 × T
then
u x, y ≤
−1
G
GC y
y0
where η1 x, y x
x0
fs, yΔs.
1/p
η1 x, tΔt
,
0 ,
x, y ∈ T0 × T
2.58
2.59
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Journal of Applied Mathematics
Proof. Fix X ∈ T0 , and x ∈ x0 , X
v x, y C 0 . Let
T, y ∈ T
y x
y0
fs, t Δs Δt,
x ∈ x0 , X
0.
T, y ∈ T
2.60
0.
T, y ∈ T
2.61
x0
Then
u x, y ≤ v1/p x, y ≤ v1/p X, y ,
∀x ∈ x0 , X
Similar to 2.14–2.16, we obtain
u τ1 x, τ2 y ≤ v1/p x, y ,
x ∈ x0 , X
0.
T, y ∈ T
2.62
Moreover,
vyΔ
X, y X
f s, y ω u τ1 s, τ2 y
Δs
x0
X
Δs
f s, y ω v1/p s, y
≤
x0
≤
X
2.63
f s, y Δs ω v1/p X, y η1 X, y ω v1/p X, y .
x0
Let vX, y be the solution of the following problem:
1/p vΔ
X, y ,
y X, y η1 X, y ω v
v X, y0 C.
2.64
Considering vX, y0 C and ω is nondecreasing and continuous, then from 2.63, 2.64,
and Lemma 2.4, we have
v X, y ≤ v X, y ,
0.
y∈T
2.65
Δ
On the other hand, from the definition of G we have GvX, yΔ
y v y X, y/ωvX, y η1 X, y. Then an integration with respect to y from y0 to y yields
G v X, y − G v X, y0 y
η1 X, tΔt,
y0
2.66
that is,
−1
v X, y ≤ G
GC y
y0
η1 X, tΔt ,
0.
y∈T
2.67
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13
Combining 2.61, 2.65, and 2.67, we have
−1
u x, y ≤
G
GC 1/p
y
η1 X, tΔt
,
x ∈ x0 , X
0.
T, y ∈ T
2.68
y0
Setting x X in 2.68, we get the desired result.
Remark 2.15. If we take T R, then Theorem 2.14 reduces to 14, Theorem 2.1, while
Theorem 2.14 reduces to 15, Theorem 2.1 if we take T Z.
Theorem 2.16. Suppose u, f, τ1 , τ2 , φ, α, β are the same as in Theorem 2.5, and furthermore, u is delta
0 , R . ω ∈ CR , R is nondecreasing, and ω is
0 with respect to y, g ∈ Crd T0 × T
differential on T
is a bijective
submultiplicative, that is, ωαβ ≤ ωαωβ, for all α, β ∈ R . C > 0 is a constant. G
Δ
Δ
0 , ux, y satisfies
function such that Gzx,
yy zx, yy /ωzx, y. If for x, y ∈ T0 × T
the following inequality:
u x, y ≤ C y x
y0
fs, tωuτ1 s, τ2 t gs, tuτ1 s, τ2 t Δs Δt,
x0
2.69
with the initial condition
T,
if x ∈ α, x0 T or y ∈ β, y0
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
∀ x, y ∈ T0 × T
u x, y φ x, y ,
φ τ1 x, τ2 y ≤ C,
2.70
then
−1
u x, y ≤ G
GC
y
η2 x, tΔt eB1 y, y0 ,
0 ,
x, y ∈ T0 × T
2.71
y0
x
x
where B1 x, y x0 gs, yΔs, η2 x, y ωeB1 y, y0 x0 fs, yΔs and eB1 y, y0 is the unique
solution of the following equation:
zΔ
y x, y B1 x, y z x, y ,
Proof. Fix X ∈ T0 , and x ∈ x0 , X
v x, y C y x
y0
z x, y0 1.
2.72
0 . Let
T, y ∈ T
fs, tωuτ1 s, τ2 t gs, tuτ1 s, τ2 t Δs Δt.
x0
2.73
Then
u x, y ≤ v x, y ≤ v X, y ,
∀x ∈ x0 , X
0.
T, y ∈ T
2.74
14
Journal of Applied Mathematics
Similar to 2.14–2.16, we can obtain
u τ1 x, τ2 y ≤ v x, y ,
x ∈ x0 , X
0.
T, y ∈ T
2.75
Furthermore we have
v X, y C y X
y0
≤C
x0
y X
y0
≤C
fs, tωuτ1 s, τ2 t gs, tuτ1 s, τ2 t Δs Δt
fs, tωvs, t gs, tvs, t Δs Δt
x0
y X
y0
Let B2 X, y C fs, tωvs, tΔs Δt x0
y X
y0 x0
gs, tΔs vX, tΔt,
y0
y X
2.76
0.
y∈T
x0
fs, tωvs, tΔs Δt. Then from 2.76 it follows that
v X, y ≤ B2 X, y y
B1 X, tvX, tΔt,
0.
y∈T
y0
2.77
Considering B2 X, y is nondecreasing in y, by applying Lemma 2.2 to 2.77, we obtain
v X, y ≤ B2 X, y eB1 y, y0 ,
0.
y∈T
2.78
On the other hand,
B2 X, y
Δ
y
X
f s, y ω v s, y Δs ≤
X
x0
≤
X
f s, y Δs ω v X, y
x0
f s, y Δs ω B2 X, y eB1 y, y0
x0
≤
X
2.79
f s, y Δs ω B2 X, y ω eB1 y, y0
x0
ω B2 X, y η2 X, y .
Let vX, y be the solution of the following equation:
vΔ
y X, y η2 X, y ω v X, y ,
v X, y0 C.
2.80
Considering B2 X, y0 C and ω is nondecreasing and continuous, then from 2.79, 2.80,
and Lemma 2.4, we have
B2 X, y ≤ v X, y ,
0.
y∈T
2.81
Journal of Applied Mathematics
15
Δ
and 2.80, we have GvX,
yΔ
From the definition of G
y v y X, y/ωvX, y η2 X, y.
Then similar to 2.66 and 2.67, we obtain
−1
GC
B2 X, y ≤ v X, y ≤ G
y
η2 X, tΔt ,
0.
y∈T
2.82
y0
Combining 2.74, 2.78, and 2.82, we have
y
−1 u x, y ≤ G GC η2 X, tΔt eB1 y, y0 ,
x ∈ x0 , X
0.
T, y ∈ T
2.83
y0
Setting x X in 2.83, we obtain
−1
u X, y ≤ G
GC
y
η2 X, tΔt eB1 y, y0 ,
0.
y∈T
2.84
y0
Replacing X with x in 2.84 yields the desired inequality 2.71.
Theorem 2.17. Under the conditions of Theorem 2.16, if p, C are constants with p > 0, C > 0, and
0 , ux, y satisfies the following inequality:
for x, y ∈ T0 × T
up x, y ≤ C y x
y0
fs, tωuτ1 s, τ2 t gs, tup τ1 s, τ2 t Δs Δt,
x0
2.85
with the initial condition 2.58, then
u x, y ≤
−1
G
GC y
η3 x, tΔt eJ1 y, y0
1/p
,
0,
x, y ∈ T0 × T
2.86
y0
x
gs, yΔs, η3 x, y
defined as in Theorem 2.14, J1 x, y
x0
x
1/p
ωeJ1 y, y0 x0 fs, yΔs, and eJ1 y, y0 is the unique solution of the following equation:
where
G
is
zΔ
y x, y J1 x, y z x, y ,
z x, y0 1.
2.87
The proof of Theorem 2.17 is similar to that of Theorem 2.16, and we omit it here.
3. Some Simple Applications
In this section, we will present some examples to illustrate the validity of our results in
deriving explicit bounds of solutions of certain delay dynamic equations on time scales.
16
Journal of Applied Mathematics
Example 3.1. Consider the following delay dynamic integral equation:
u x, y C p
y x
y0
Ms, t, uτ1 s, τ2 tΔs Δt,
0,
x, y ∈ T0 × T
x0
with the initial condition
T,
u x, y φ x, y , if x ∈ α, x0 T or y ∈ β, y0
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
φ τ1 x, τ2 y ≤ |C|1/p , ∀ x, y ∈ T0 , T
3.1
3.2
0 , R, φ, α, β, τ1 , τ2 are the same as in Theorem 2.8, and M ∈ Crd T0 × T
0 ×
where u ∈ Crd T0 × T
q
r
0 , R ,
R, R. Furthermore, assume |Ms, t, u| ≤ fs, t|u| gs, t|u| , where f, g ∈ Crd T0 × T
and p, q, r are the same as in Theorem 2.8.
From 3.1 we have
p
u x, y ≤ |C| y x
y0
≤ |C| |Ms, t, uτ1 s, τ2 t|Δs Δt
x0
y x
y0
fs, t|uτ1 s, τ2 t|q gs, t|uτ1 s, τ2 t|r Δs Δt.
3.3
x0
Then according to Theorem 2.8, we can obtain the following estimate:
1/p
y
u x, y ≤ H
1 x, y eH
,
2 y, σt H2 x, tH1 x, tΔt
0,
x, y ∈ T0 × T
3.4
y0
where
y x p − r r/p
p − q q/p
K gs, t
K
fs, t
Δs Δt,
p
p
y0 x0
x r
q
2 x, y H
f s, y K q−p/p g s, y K r−p/p Δs, ∀K > 0.
p
p
x0
1 x, y |C| H
∀K > 0,
3.5
Example 3.2. Considering the following delay dynamic integral equation:
u x, y C 3
y x
y0
Ns, t, uτ1 s, τ2 t, uτ1 s, στ2 tΔs Δt,
0,
x, y ∈ T0 × T
x0
3.6
with the initial condition
T;
u x, y φ x, y , if x ∈ α, x0 T or y ∈ β, y0
1/3
0 , if τ1 x ≤ x0 or τ2 y ≤ y0 ,
φ τ1 x, τ2 y ≤ |C| , ∀x ∈ T0 , y ∈ T
0 , R, and N ∈ Crd T0 × T
0 × R2 , R.
where u ∈ Crd T0 × T
3.7
Journal of Applied Mathematics
17
0 , R , then from 3.6
Assume |Nx, y, u, v| ≤ fx, y|u|2 |v|2 , where f ∈ Crd T0 × T
we have
y x
3
|Ns, t, uτ1 s, τ2 t, uτ1 s, στ2 t|Δs Δt
u x, y ≤ |C| y0
≤ |C| y0
≤ |C| x0
y x
x0
y x
y0
fs, t |uτ1 s, τ2 t|2 |uτ1 s, στ2 t|2 Δs Δt
fs, t |uτ1 s, τ2 t|2 |uτ1 s, στ2 t|2
3.8
x0
|uτ1 s, τ2 t||uτ1 s, στ2 t|Δs Δt.
According to Theorem 2.13 p 3, we can reach the following estimate:
u x, y ≤ |C|1/3 y x
y0
x0
fs, tΔs Δt,
0.
x, y ∈ T0 × T
3.9
4. Conclusions
In this paper, we established some new Gronwall-Bellman-type delay integral inequalities
in two independent variables on time scales. As one can see, the presented results provide
a handy tool for deriving bounds for solutions of certain delay dynamic equations on time
scales. Furthermore, the process of constructing Theorems 2.5, 2.8, 2.14, 2.16 and 2.17 can be
applied to the situation with n independent variables.
Acknowledgments
This work is supported by the Natural Science Foundation of Shandong Province
ZR2010AZ003 China. The authors thank the referees very much for their careful
comments and valuable suggestions on this paper.
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