Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 754350, 25 pages
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,1 Qinghua Feng,1, 2 and Bin Zheng2
1
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
School of Science, Shandong University of Technology, Zibo 255049, China
Correspondence should be addressed to Fanwei Meng, fengqinghua1978@126.com
Received 20 April 2011; Accepted 8 August 2011
Academic Editor: Bernard Geurts
Copyright q 2011 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 inequalities e.g., see 1–13 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 scales e.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 established e.g., see 18–26, 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
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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 Li 27 and Ma and Pečarić 28 to 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 continuous 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, 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 by σt inf{s ∈ T, s > t} and 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 is denoted by
f Δ t and satisfies
fσt − fs − f Δ tσt − s ≤ ε|σt − s|
for ∀ε > 0,
1.1
where s ∈ U, and U is a neighborhood of t which can depend on ε.
Similarly, for some y ∈ Tκ and a function f ∈ T × T, R, the partial delta derivative of f
Δ
with 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
for ∀ε > 0,
1.2
where s ∈ U, and U is a neighborhood of y which can depend on ε.
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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Definition 1.8. For a, b ∈ T and a function f ∈ T, R, the Cauchy integral of f is defined by
b
ftΔt Fb − Fa,
1.3
a
where F Δ t ft, t ∈ Tκ .
Similarly, for a, b ∈ T and a function f ∈ T × T, R, the Cauchy partial integral of f with
respect to y is defined by
b
f x, y Δy Fx, b − Fx, a,
1.4
a
where FyΔ x, y fx, y, y ∈ Tκ .
Definition 1.9. The cylinder transformation ξh is defined by
⎧
1
⎪
⎨ Log1 hz , if h 0
for
z
−
,
/
/
h
h
ξh z ⎪
⎩z,
if h 0,
1.5
where Log is the principal logarithm function.
Definition 1.10. For px, y ∈ R with respect to y, the exponential function is defined by
y
ep y, s exp
ξμτ px, τ Δτ ,
for s, y ∈ T.
1.6
s
Remark 1.11. If T R, then for y ∈ R the following formula holds:
ep y, s exp
y
px, τdτ ,
for s ∈ T.
1.7
s
If T Z, then, for y ∈ Z, ep y, s y−1
τs 1
px, τ, for s ∈ Z and s < y.
The following two theorems include some known properties on the exponential
function.
Theorem 1.12. If px, y ∈ R with respect to y, then the following conclusions hold:
i ep y, y ≡ 1 and e0 s, y ≡ 1,
ii ep s, σy 1 μypx, yep s, y,
iii if p ∈ R with respect to y, then ep s, y > 0 for all s, y ∈ T,
iv if p ∈ R with respect to y, then p ∈ R ,
v ep s, y 1/ep y, s ep y, s, where px, y −px, y/1 μypx, y.
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Theorem 1.13. If px, y ∈ R with respect to y, y0 ∈ T is a fixed number, then the exponential
function ep y, y0 is the unique solution of the following initial value problem:
zΔ
y x, y p x, y z x, y ,
z x, y0 1.
1.8
Theorems 1.12-1.13 are similar to 24, Theorems 5.1-5.2. For more details about the
calculus of time scales, we advise to refer to 29.
In the rest of this paper, for the convenience of notation, we always assume that T0 0 y0 , ∞ T, where x0 , y0 ∈ Tκ , and furthermore assume that T0 ⊆ Tκ , T
0 ⊆
x0 , ∞ T, T
κ
T .
2. Main Results
We will give some lemmas for further use.
Lemma 2.1. Suppose that X ∈ T0 is a fixed number and uX, 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
y0
2.1
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 IVP
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 that of 24, Theorem 5.6, and we omit it here.
Lemma 2.2. Under the conditions of Lemma 2.1 and furthermore assuming that 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
em y, σt aX, tmX, tΔt
y0
≤ a X, y 1 y
em y, σt mX, tΔt .
y0
2.5
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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 30. Assume that a ≥ 0, p ≥ q ≥ 0, and p /
0, then for any K > 0
q q−p/p
p − q q/p
K
K .
a
p
p
aq/p ≤
2.6
0 , R and a, b are nondecreasing. p, q, r, m
Theorem 2.4. Suppose that u, f, g, h, a, b ∈ Crd T0 × T
are constants, and p ≥ q ≥ 0, p ≥ r ≥ 0, p ≥ m ≥ 0, p /
0. τ1 ∈ T0 , T, τ1 x ≤ x, −∞ <
0 , T, τ2 y ≤ y, −∞ < β inf{τ2 y, y ∈ T
0 } ≤ y0 .
α inf{τ1 x, x ∈ T0 } ≤ x0 . τ2 ∈ T
2
φ ∈ Crd α, x0 ×β, y0 T , R . If for x, y ∈ T0 × T0 , ux, y satisfies the following inequality:
up x, y ≤ a x, y b x, y
y x
y0
b x, y
x0
y x t s
y0
x0
y0
fs, tuq τ1 s, τ2 t gs, tur s, t ΔsΔt
2.7
h ξ, η um τ1 ξ, τ2 η ΔξΔηΔsΔt,
x0
with the initial condition
u x, y φ x, y ,
if x ∈ α, x0 φ τ1 x, τ2 y ≤ α1/p x, y ,
T,
T, or y ∈ β, y0
2.8
0,
if τ1 x ≤ x0 or τ2 y ≤ y0 , ∀ x, y ∈ T0 × T
then
1/p
y
eB2 y, σt B2 x, tB1 x, tΔt
,
u x, y ≤ B1 x, y b x, y
0,
x, y ∈ T0 × T
y0
2.9
where
B1 x, y a x, y b x, y
b x, y
x y x p − r r/p
p − q q/p
K gs, t
K
fs, t
ΔsΔt
p
p
y0 x0
y x t s
y0
x0
y0
p − m m/p
K ΔξΔηΔsΔt,
h ξ, η
p
x0
r
q
f s, y K q−p/p g s, y K r−p/p
p
p
x0
y s
m m−p/p
K
h ξ, η
ΔξΔη Δs,
p
y0 x0
B2 x, y b x, y B2 x, y .
B2 x, y 2.10
∀K > 0,
2.11
∀K > 0,
2.12
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Proof. Let the right side of 2.7 be vx, y. Then
u x, y ≤ v1/p x, y ,
0.
x, y ∈ T0 × T
2.13
0 , and since a, b are nondecreasing we
If τ1 x ≥ x0 and τ2 y ≥ y0 , then τ1 x ∈ T0 , τ2 y ∈ T
have
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 2.8 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 have
u τ1 x, τ2 y ≤ v1/p x, y ,
Fix X ∈ T0 , and let x ∈ x0 , X
v X, y a X, y b X, y
y X
x0
≤ a X, y b X, y
2.16
y0
fs, tuq τ1 s, τ2 t gs, tur s, t ΔsΔt
x0
y X t s
y0
0.
x, y ∈ T0 × T
0 ; then
T, y ∈ T
y0
b X, y
h ξ, η um τ1 ξ, τ2 η ΔξΔηΔsΔt
x0
y X
fs, tvq/p s, t gs, tvr/p s, t
y0
2.17
x0
t s
y0
m/p h ξ, η v
ξ, η ΔξΔη ΔsΔt.
x0
From Lemma 2.3, we have
p − q q/p
q
K ,
vq/p x, y ≤ K q−p/p v x, y p
p
p − r r/p
r
vr/p x, y ≤ K r−p/p v x, y K ,
p
p
p − m m/p
m
vm/p x, y ≤ K m−p/p v x, y K ,
p
p
∀K > 0.
2.18
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7
So combining 2.17 and 2.18, it follows that
v X, y ≤ a X, y b X, y
y X
y0
b X, y
x0
y X
gs, t
y0
fs, t
x0
q q−p/p
p − q q/p
K
K
vs, t ΔsΔt
p
p
p − r r/p
r r−p/p
K
K
vs, t ΔsΔt
p
p
y X t s
m m−p/p p − m m/p
K
K
h ξ, η
v ξ, η ΔξΔηΔsΔt
p
p
y 0 x 0 y 0 x0
y X
p − q q/p
p − r r/p
fs, t
≤ a X, y b X, y
K gs, t
K
p
p
y0 x0
t s
p − m m/p
K ΔξΔη ΔsΔt
h ξ, η
p
y0 x0
X y
q
r
fs, t K q−p/p gs, t K r−p/p
b X, y
p
p
y0
x0
t s
m m−p/p
K
h ξ, η
ΔξΔη Δs vX, tΔt
p
y0 x0
y
B1 X, y b X, y
B2 X, tvX, tΔt,
b X, y
y0
2.19
where B1 x, y, B2 x, y are defined in 2.10 and 2.11, respectively. Considering B2 X, y bX, yB2 X, y, by application of Lemma 2.1, we have
v X, y ≤ B1 X, y b X, y
y
y0
eB2 y, σt B2 X, tB1 X, tΔt,
0.
y∈T
2.20
Since X ∈ T0 is arbitrary, then in fact 2.20 holds for all x ∈ T0 , that is,
v x, y ≤ B1 x, y b x, y
y
y0
eB2 y, σt B2 x, tB1 x, tΔt,
0 .
x, y ∈ T0 × T
2.21
Combining 2.13 and 2.21, we obtain
1/p
y
u x, y ≤ B1 x, y b x, y
eB2 y, σt B2 x, tB1 x, tΔt
,
0 ,
x, y ∈ T0 × T
y0
2.22
which is the desired inequality.
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Journal of Applied Mathematics
If we apply Lemma 2.2 instead of Lemma 2.1 at the end of the proof of Theorem 2.4,
we obtain the following theorem.
Theorem 2.5. Suppose that u, f, g, h, a, p, q, r, m, τ1 , τ2 , α, β, φ are defined as in Theorem 2.4. If that
0 , ux, y satisfies the following inequality:
for x, y ∈ T0 × T
y x fs, tuq τ1 s, τ2 t gs, tur s, t
u x, y ≤ a x, y p
y0
x0
t s
y0
m
h ξ, η u τ1 ξ, τ2 η ΔξΔη ΔsΔt
2.23
x0
with the initial condition 2.8, then
1/p
u x, y ≤ B1 x, y eB2 y, y0
,
0,
x, y ∈ T0 × T
2.24
where
B1 x, y a x, y y x fs, t
y0
y x t s
y0
B2 x, y x0
p − r r/p
p − q q/p
K gs, t
K
ΔsΔt
p
p
x x0
x0
y0
p − m m/p
K ΔξΔηΔsΔt,
h ξ, η
p
x0
q
r
f s, y K q−p/p g s, y K r−p/p p
p
∀K > 0,
m m−p/p
K
h ξ, η
ΔξΔη Δs,
p
x0
y s
y0
∀K > 0.
2.25
From Theorems 2.4 and 2.5 we can obtain two direct corollaries.
0 , ux, y satisfies the
Corollary 2.6. Under the conditions of Theorem 2.4, if, for x, y ∈ T0 × T
following inequality:
u x, y ≤ a x, y y x
y0
x0
y0
fs, tuτ1 s, τ2 t gs, tus, t ΔsΔt
x0
y x t s
y0
h ξ, η u τ1 ξ, τ2 η ΔξΔηΔsΔt,
2.26
x0
with the initial condition 2.8 (p=1), then
u x, y ≤ a x, y y
y0
0,
eB2 y, σt B2 x, tax, tΔt, x, y ∈ T0 × T
2.27
Journal of Applied Mathematics
9
where
x B2 x, y f s, y g s, y
x0
y s
y0
h ξ, η ΔξΔη Δs.
2.28
x0
0 , ux, y satisfies the
Corollary 2.7. Under the conditions of Theorem 2.5, if, for x, y ∈ T0 × T
following inequality:
u x, y ≤ a x, y y x
y0
x0
y0
fs, tuτ1 s, τ2 t gs, tus, t ΔsΔt
x0
y x t s
y0
h ξ, η u τ1 ξ, τ2 η ΔξΔηΔsΔt,
2.29
x0
with the initial condition 2.8 (p 1), then
u x, y ≤ a x, y eB2 y, y0 ,
0,
x, y ∈ T0 × T
2.30
where
y s
x B2 x, y f s, y g s, y h ξ, η ΔξΔη Δs.
x0
y0
2.31
x0
0 , R , f, g, h, τ1 , τ2 are defined as in Theorem 2.4, and
Theorem 2.8. Suppose that u ∈ Crd T0 × T
0 , ux, y satisfies the following inequality:
τ1 x ≥ x0 , τ2 y ≥ y0 . If, for x, y ∈ T0 × T
u x, y
y x t s
≤
fs, tuτ1 s, τ2 t gs, tus, t h ξ, η u τ1 ξ, τ2 η ΔξΔη ΔsΔt,
y0
x0
y0
x0
2.32
then ux, y ≡ 0.
The proof of Theorem 2.8 is similar to Theorem 2.4, and we omit it here.
Based on Theorem 2.4, we will establish a class of Volterra-Fredholm-type integral
inequality on time scales.
0 , R , i 1, 2. a, p, q, r, m, φ, τ1 , τ2 , α, β
Theorem 2.9. Suppose that u, fi , gi , hi ∈ Crd T0 × T
0 are two fixed numbers. If, for x, y ∈
are the same as in Theorem 2.4, and M ∈ T0 , N ∈ T
x0 , M T × y0 , N T, ux, y satisfies the following inequality:
up x, y ≤ a x, y y x
y0
x0
y0
f1 s, tuq τ1 s, τ2 t g1 s, tur τ1 s, τ2 t ΔsΔt
x0
y x t s
y0
x0
h1 ξ, η um τ1 ξ, τ2 η ΔξΔηΔsΔt
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Journal of Applied Mathematics
N M
y0
f2 s, tuq τ1 s, τ2 t g2 s, tur τ1 s, τ2 t ΔsΔt
x0
N M t s
y0
x0
y0
h2 ξ, η um τ1 ξ, τ2 η ΔξΔηΔsΔt,
x0
2.33
with the initial condition 2.8, then one has
u x, y
≤
1/p
λ B6 ,
B3 x, y B4 x, y
1 − B5
2.34
T ,
x, y ∈ x0 , M T × y0 , N
provided that B5 < 1, where
N M
p − q q/p
p − r r/p
K g2 s, t
K
p
p
y0 x0
t s
p − m m/p
K ΔξΔη ΔsΔt,
h2 ξ, η
p
y0 x0
y x p − q q/p
p − r r/p
K g1 s, t
K
f1 s, t
ΔsΔt
B1 x, y a x, y p
p
y 0 x0
λ
y x t s
y0
f2 s, t
x0
x p − m m/p
K ΔξΔηΔsΔt,
h1 ξ, η
p
x0
y0
q
r
f1 s, y K q−p/p g1 s, y K r−p/p
p
p
x0
y s
m m−p/p
K
h1 ξ, η
ΔξΔη Δs,
p
y 0 x0
y
eB2 y, σt B2 x, tΔt,
B3 x, y 1 B2 x, y y0
B4 x, y B1 x, y y
y0
B5 N M
y0
x0
y0
2.37
∀K > 0,
q
r
f2 s, t K q−p/p B3 s, t g2 s, t K r−p/p B3 s, t ΔsΔt
p
p
x0
y0
2.36
∀K > 0,
eB2 y, σt B2 x, tB1 x, tΔt,
N M t s
2.35
m m−p/p K
h2 ξ, η
B3 ξ, η ΔξΔηΔsΔt,
p
x0
2.38
2.39
2.40
Journal of Applied Mathematics
B6 N M
y0
x0
11
q
r
f2 s, t K q−p/p B4 s, t g2 s, t K r−p/p B4 s, t ΔsΔt
p
p
N M t s
y0
x0
m m−p/p K
h2 ξ, η
B4 ξ, η ΔξΔηΔsΔt.
p
x0
y0
2.41
Proof. Let the right side of 2.33 be vx, y and
μ
N M
y0
f2 s, tuq τ1 s, τ2 t g2 s, tur τ1 s, τ2 t ΔsΔt
x0
N M t s
h2 ξ, η u
y0
x0
y0
m
τ1 ξ, τ2 η ΔξΔηΔsΔt.
2.42
x0
Then
u x, y ≤ v1/p x, y ,
x, y ∈ x0 , M T × y0 , N
T .
2.43
Similar to the process of 2.14–2.16 one has
u τ1 x, τ2 y ≤ v1/p x, y ,
Fix X ∈ x0 , M
T, and let x ∈ x0 , X
v X, y a X, y μ y X
y0
y X t s
y0
x0
y0
≤ a X, y μ x0
y0
T, y ∈ y0 , N
T. Then
f1 s, tuq τ1 s, τ2 t g1 s, tur τ1 s, τ2 t ΔsΔt
x0
x0
y X
f1 s, tvq/p s, t g1 s, tvr/p s, t ΔsΔt
y X t s
y0
2.44
h1 ξ, η um τ1 ξ, τ2 η ΔξΔηΔsΔt
y0
T .
x, y ∈ x0 , M T × y0 , N
x0
x0
h1 ξ, η vm/p ξ, η ΔξΔηΔsΔt.
2.45
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Considering the structure of 2.45 is similar to 2.17, then following in a same manner as
the process of 2.17–2.20 we can deduce
y
v X, y ≤ μ B1 X, y y0
μ 1
y
y0
y
y0
eB2
eB2 y, σt B2 X, t μ B1 X, t Δt
y, σt B2 X, tΔt B1 X, y
eB2 y, σt B2 X, tB1 X, tΔt,
2.46
T,
y ∈ y0 , N
where B1 x, y, B2 x, y are defined in 2.36 and 2.37, respectively.
Since X is selected from x0 , M T arbitrarily, then in fact 2.46 holds for all x ∈ T0 ,
that is,
y
y
eB2 y, σt B2 x, tΔt B1 x, y eB2 y, σt B2 x, tB1 x, tΔt
v x, y ≤ μ 1 y0
y0
μB3 x, y B4 x, y ,
T ,
x, y ∈ x0 , M T × y0 , N
2.47
where B3 x, y, B4 x, y are defined in 2.38 and 2.39, respectively.
On the other hand, from 2.18, 2.42, and 2.44 we obtain
N M
t s
m/p q/p
r/p
f2 s, tv s, t g2 s, tv s, t μ≤
h2 ξ, η v
ξ, η ΔξΔη ΔsΔt
y0
≤
x0
y0
x0
N M
p − q q/p
q q−p/p
K
K
vs, t f2 s, t
p
p
y0 x0
p − r r/p
r r−p/p
K
K
vs, t g2 s, t
p
p
ΔsΔt
N M t s
y0
x0
y0
m m−p/p p − m m/p
K
K
ΔξΔηΔsΔt
h2 ξ, η
v ξ, η p
p
x0
N M
q
r
λ
f2 s, t K q−p/p vs, t g2 s, t K r−p/p vs, t ΔsΔt
p
p
y0 x0
N M t s
y0
x0
y0
m m−p/p K
h2 ξ, η
v ξ, η ΔξΔηΔsΔt,
p
x0
2.48
Journal of Applied Mathematics
13
where λ is defined in 2.35. Then using 2.47 in 2.48 yields
μ≤λ
N M
y0
x0
q
f2 s, t K q−p/p μB3 s, t B4 s, t
p
r
g2 s, t K r−p/p μB3 s, t B4 s, t ΔsΔt
p
N M t s
y0
x0
y0
m m−p/p μB3 ξ, η B4 ξ, η ΔξΔηΔsΔt
K
h2 ξ, η
p
x0
N M q
r
λμ
f2 s, t K q−p/p B3 s, t g2 s, t K r−p/p B3 s, t ΔsΔt
p
p
y 0 x0
y0
N M t s
x0
y0
2.49
m m−p/p K
h2 ξ, η
B3 ξ, η ΔξΔηΔsΔt
p
x0
N M
q
r
f2 s, t K q−p/p B4 s, t g2 s, t K r−p/p B4 s, t ΔsΔt
p
p
y0 x0
N M t s
y0
x0
y0
m m−p/p K
h2 ξ, η
B4 ξ, η ΔξΔηΔsΔt
p
x0
λ μB5 B6 ,
which implies
μ≤
λ B6
.
1 − B5
2.50
Combining 2.43, 2.47, and 2.50 we can obtain the desired inequality 2.34.
In the proof of Theorem 2.9, if we let the right side of 2.33 be ax, y vx, y in the
first statement, then following in a same process as in Theorem 2.9 we obtain another bound
on the function ux, y, which is shown in the following theorem.
Theorem 2.10. Under the conditions of Theorem 2.9, if, for x, y ∈ x0 , M T × y0 , N T,
ux, y satisfies 2.33 with the initial condition 2.8, then the following inequality holds:
u x, y ≤
1/p
J1 x, y
μ
a x, y ,
eJ2 y, y0
1 − λ
T ,
x, y ∈ x0 , M T × y0 , N
2.51
14
Journal of Applied Mathematics
provided that λ < 1, where
λ N M
y0
x0
q
r
f2 s, t K q−p/p eJ2 t, y0 g2 s, t K r−p/p eJ2 t, y0 ΔsΔt
p
p
N M t s
y0
x0
y0
N M
m m−p/p K
h2 ξ, η
eJ2 η, y0 ΔξΔηΔsΔt,
p
x0
p − q q/p
q q−p/p
K
K
as, t f2 s, t
μ
p
p
y0 x0
p − r r/p
r r−p/p
K
K
as, t g2 s, t
p
p
t s
m m−p/p K
h2 ξ, η
a ξ, η v ξ, η
p
y0 x0
p − m m/p
K
ΔξΔη ΔsΔt,
p
y x p − q q/p
q q−p/p
K
K
as, t J1 x, y f1 s, t
p
p
y0 x0
p − r r/p
r r−p/p
K
K
as, t ΔsΔt
g1 s, t
p
p
y X t s
m m−p/p p − m m/p
K
K
h1 ξ, η
a ξ, η ΔξΔηΔsΔt,
p
p
y0 x 0 y 0 x0
y s
x q q−p/p
r r−p/p
m m−p/p
f1 s, y K
J2 x, y K
g1 s, y K
h1 ξ, η
ΔξΔη Δs.
p
p
p
x0
y0 x0
2.52
Finally, we will establish a more general inequality than that in Theorems 2.9-2.10.
Consider the following inequality:
up x, y ≤ a x, y
y x y0
Ls, t, uτ1 s, τ2 t t s
x0
y0
N M
Ls, t, uτ1 s, τ2 t y0
x0
q
x0
t s
y0
h1 ξ, η u τ1 ξ, τ2 η ΔξΔη ΔsΔt
h2 ξ, η u τ1 ξ, τ2 η ΔξΔη ΔsΔt,
q
x0
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
0 are two fixed numbers. L ∈ T0 × T
0 × R , R , and
Theorem 2.4, M ∈ T0 , N ∈ T
0 ≤ Ls, t, x − Ls, t, y ≤ As, t, yx − y for x ≥ y ≥ 0, where A ∈ T0 × T0 × R , R .
Journal of Applied Mathematics
Theorem 2.11. If, for x, y ∈ x0 , M
following inequality holds:
u x, y ≤
15
T × y0 , N
1/p
λ B6 ,
B3 x, y B4 x, y
1 − B5
T, ux, y satisfies 2.53, then the
T ,
x, y ∈ x0 , M T × y0 , N
2.54
provided that B5 < 1, where
λ N M t s
p − q q/p
p − 1 1/p
L s, t,
K
K ΔξΔη ΔsΔt,
h2 ξ, η
p
p
y0 x0
y 0 x0
2.55
B1 x, y a x, y
y x t s
p − q q/p
p − 1 1/p
L s, t,
K
K ΔξΔη ΔsΔt,
h1 ξ, η
p
p
y 0 x0
y0 x0
∀K > 0,
2.56
B2 x, y x p − 1 1/p 1 1−p/p
K
K
A s, y,
p
p
x0
y0
B3 x, y 1 q q−p/p
h1 ξ, η K
ΔξΔη Δs,
p
x0
y s
y
y0
y
y0
2.58
eB2 y, σt B2 x, tB1 x, tΔt,
N M p − 1 1/p 1 1−p/p A s, t,
K
K
B3 s, t
p
p
y0 x0
q q−p/p h2 ξ, η K
B3 ξ, η ΔξΔη ΔsΔt,
p
x0
t s
y0
B6 ∀K > 0,
eB2 y, σt B2 x, tΔt,
B4 x, y B1 x, y B5 2.57
N M p − 1 1/p 1 1−p/p K
K
A s, t,
B4 s, t
p
p
y0 x0
q q−p/p h2 ξ, η K
B4 ξ, η ΔξΔη ΔsΔt.
p
x0
t s
y0
2.59
2.60
2.61
Proof. Let the right side of 2.53 be vx, y and
N M
t s
q
Ls, t, uτ1 s, τ2 t μ
h2 ξ, η u τ1 ξ, τ2 η ΔξΔη ΔsΔt.
y0
x0
y0
x0
2.62
16
Journal of Applied Mathematics
Then
u x, y ≤ v1/p x, y ,
x, y ∈ x0 , M T × y0 , N
T .
2.63
Similar to the process of 2.14–2.16 we have
u τ1 x, τ2 y ≤ v1/p x, y ,
Fix X ∈ x0 , M
T .
x, y ∈ x0 , M T × y0 , N
T, and let x ∈ x0 , X
T, y ∈ y0 , N
v X, y a X, y μ
y X
Ls, t, uτ1 s, τ2 t y0
x0
t s
y0
2.64
T. Then
h1 ξ, η u τ1 ξ, τ2 η ΔξΔη ΔsΔt
q
x0
y X t s 1/p
q/p
ξ, η ΔξΔη ΔsΔt.
L s, t, v s, t h1 ξ, η v
≤ a X, y μ
y0
x0
y0
x0
2.65
From Lemma 2.3, we have
p − q q/p
q
vq/p x, y ≤ K q−p/p v x, y K ,
p
p
v
1/p
p − 1 1/p
1
x, y ≤ K 1−p/p v x, y K ,
p
p
∀K > 0.
Combining 2.65 and 2.66, it follows that
v X, y ≤ a X, y μ
y X p − 1 1/p
1
K
L s, t, K 1−p/p vs, t ΔsΔt
p
p
y 0 x0
y X t s
y0
x0
y0
q q−p/p p − q q/p
K
K
h1 ξ, η
v ξ, η ΔξΔηΔsΔt
p
p
x0
a X, y μ
y X p − 1 1/p
p − 1 1/p
1
K
K
L s, t, K 1−p/p vs, t − L s, t,
p
p
p
y 0 x0
p − 1 1/p
K
L s, t,
ΔsΔt
p
2.66
Journal of Applied Mathematics
17
y X t s
y0
x0
q q−p/p p − q q/p
K
K
h1 ξ, η
v ξ, η ΔξΔηΔsΔt
p
p
x0
y0
≤ a X, y μ
y X p − 1 1/p
p − 1 1/p 1 1−p/p
K
K
K
vs, t L s, t,
A s, t,
ΔsΔt
p
p
p
y 0 x0
y X t s
y0
x0
q q−p/p
h1 ξ, η K
ΔξΔη vX, tΔsΔt
p
x0
y0
y X t s
y0
x0
p − q q/p
K ΔξΔηΔsΔt
h1 ξ, η
p
x0
y0
≤ a X, y μ
y X
y0
p − 1 1/p 1 1−p/p
K
K
A s, t,
Δs vX, tΔt
p
p
x0
y X p − 1 1/p
K
L s, t,
ΔsΔt
p
y 0 x0
y X t s
y0
x0
q q−p/p
h1 ξ, η K
ΔξΔηΔs vX, tΔt
p
x0
y0
y X t s
y0
x0
y0
p − q q/p
K ΔξΔηΔsΔt
h1 ξ, η
p
x0
μ
B1 X, y y
B2 X, tvX, tΔt,
y0
2.67
where B1 x, y, B2 x, y are defined in 2.56 and 2.57, respectively.
We notice the structure of 2.67 is similar to 2.19, so following in a same manner as
in 2.19–2.21 we obtain
v x, y
≤μ
1
y
y0
eB2 y, σt B2 x, tΔt B1 x, y μ
B3 x, y B4 x, y ,
y
y0
eB2 y, σt B2 x, tB1 x, tΔt
T ,
x, y ∈ x0 , M T × y0 , N
where B3 x, y, B4 x, y are defined in 2.58 and 2.59, respectively.
2.68
18
Journal of Applied Mathematics
On the other hand, from 2.62, 2.64, and 2.66 we have
N M t s 1/p
q/p
L s, t, v
μ
≤
h2 ξ, η v
ξ, η ΔξΔη ΔsΔt
y0
x0
y0
x0
N M p − 1 1/p
1
L s, t, K 1−p/p vs, t K
≤
p
p
y0 x0
q q−p/p p − q q/p
K
K
h2 ξ, η
v ξ, η ΔξΔη ΔsΔt
p
p
x0
t s
y0
N M p − 1 1/p
p − 1 1/p
1
K
K
L s, t, K 1−p/p vs, t − L s, t,
p
p
p
y0 x0
p − 1 1/p
K
L s, t,
ΔsΔt
p
N M t s
q q−p/p p − q q/p
K
K
h2 ξ, η
v ξ, η ΔξΔηΔsΔt
p
p
y0 x0 y 0 x 0
≤
N M p − 1 1/p
p − 1 1/p 1 1−p/p
K
K
K
vs, t L s, t,
A s, t,
ΔsΔt
p
p
p
y0 x0
N M t s
y0
x0
q q−p/p p − q q/p
K
K
h2 ξ, η
v ξ, η ΔξΔηΔsΔt
p
p
x0
y0
N M p − 1 1/p 1 1−p/p
A s, t,
K
K
λ vs, t
p
p
y0 x0
q q−p/p h2 ξ, η K
v ξ, η ΔξΔη ΔsΔt,
p
x0
t s
y0
2.69
where λ is defined in 2.55. Then using 2.68 in 2.69 yields
μ
≤ λ N M
y0
x0
p − 1 1/p 1 1−p/p K
K
μ
B3 s, t B4 s, t ΔsΔt
A s, t,
p
p
N M t s
y0
λ μ
x0
y0
N M
y0
x0
t s
y0
q
B3 ξ, η B4 ξ, η ΔξΔηΔsΔt
h2 ξ, η K q−p/p μ
p
x0
p − 1 1/p 1 1−p/p K
K
A s, t,
B3 s, t
p
p
q
h2 ξ, η K q−p/p B3 ξ, η ΔξΔηΔsΔt
p
x0
Journal of Applied Mathematics
N M y0
x0
19
p − 1 1/p 1 1−p/p B4 s, t
A s, t,
K
K
p
p
q q−p/p h2 ξ, η K
B4 ξ, η ΔξΔη ΔsΔt
p
x0
t s
y0
λ μ
B5 B6 ,
2.70
which implies
μ
≤
λ B6
.
1 − B5
2.71
Combining 2.63, 2.68, and 2.71, we obtain the desired inequality 2.54.
Remark 2.12. The established above inequalities generalize many known results including
both integral inequalities for continuous functions and discrete inequalities. For example, if
we take T R, p q 1, gx, y hx, y ≡ 0, then Theorem 2.4 reduces to 1, Theorem 2.2,
which is one case of integral inequality for continuous function. If we take T R, hx, y ≡
0, ax, y ≡ C, then Corollary 2.7 reduces to 2, Theorem 3 c1, which is another case of
inequality for continuous function. If we take T R, p q 1, g1 x, y g2 x, y f2 x, y h1 x, y h2 x, y ≡ 0, then Theorem 2.10 reduces to 1, Theorem 2.2 with slight difference.
If we take T Z, g1 x, y h1 x, y f2 x, y h2 x, y ≡ 0, τ1 x x, τ2 y y, then
Theorem 2.10 reduces to 3, Theorem 2.1, which is a discrete inequality.
Remark 2.13. Since T is an arbitrary time scale, then if we take T for some peculiar cases, such
as T R or T Z, we can deduce a series of corollaries according to Theorem 2.4–2.11. Due
to the limited space, we omit them here.
3. Some Applications
In this section, we will present some applications for the results we have established
previously. New explicit bounds for solutions of certain dynamic equations are derived in
the first two examples, while the quantitative property of solutions is focused on in the final
example.
Example 3.1. Consider the following delay dynamic differential equation:
ΔΔ
up x, y yx
F s, t, uτ1 s, τ2 t,
t s
y0
x0
!
0,
W ξ, η, u τ1 ξ, τ2 η ΔξΔη , x, y ∈ T0 × T
3.1
20
Journal of Applied Mathematics
with the initial condition
Δ
up x0 , y y bΔ y ,
up x, y0 ax,
u x, y φ x, y ,
φ τ1 x, τ2 y ≤ k x, y 1/p ,
if x ∈ α, x0 T, or y ∈ β, y0
T,
3.2
0,
if τ1 x ≤ x0 , or τ2 y ≤ y0 , ∀ x, y ∈ T0 × T
0 , R, a ∈ Crd T0 , R, b ∈ Crd T
0 , R, b is delta differential, and by0 where u ∈ Crd T0 × T
0 , R , p > 0 is a constant, φ ∈ Crd α, x0 × β, y0 T2 , R, F ∈ T0 × T
0 ×
0, k ∈ Crd T0 × T
2
R , R, W ∈ T0 × T0 × R, R. α, β, τ1 , τ2 are the same as in Theorem 2.4.
Theorem 3.2. Suppose that ux, y is a solution of 3.1-3.2, |ax by| ≤ kx, y, and
|Fs, t, x, y| ≤ fs, t|x|q |y|, |Wξ, η, x| ≤ hξ, η|x|m , where f, h, q, m are defined as in
Theorem 2.4; then
1/p
y
u x, y ≤ B1 x, y eB2 y, σt B2 x, tB1 x, tΔt
,
0,
x, y ∈ T0 × T
3.3
y0
where
B1 x, y k x, y y x y0
x0
p − q q/p
fs, t
K p
p − m m/p
K ΔξΔη ΔsΔt,
h ξ, η
p
x0
t s
y0
∀K > 0,
3.4
and B2 x, y is defined as in Theorem 2.4 (with gx, y ≡ 0).
Proof. The equivalent integral equation of 3.1 can be denoted by
up x, y ax b y
y x
y0
x0
F s, t, uτ1 s, τ2 t,
t s
y0
!
W ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt.
x0
3.5
Then
p
u x, y !
y x t s
W ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
≤ k x, y F s, t, uτ1 s, τ2 t,
y0 x0 y 0 x0
Journal of Applied Mathematics
≤ k x, y
y x y0
x0
21
t s
fs, t|uτ1 s, τ2 t| W ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
y0 x0
q
y x ≤ k x, y fs, t|uτ1 s, τ2 t|q
y0
x0
t s
y0
m
h ξ, η u τ1 ξ, τ2 η
ΔξΔη ΔsΔt,
x0
3.6
and a suitable application of Theorem 2.4 to 3.6 yields the desired inequality 3.3.
Theorem 3.3. Under the conditions of Theorem 3.2, one has
u x, y ≤ B1 x, y eB y, y0 1/p ,
2
0,
x, y ∈ T0 × T
3.7
where B1 , B2 are defined as in Theorem 3.2.
Proof. The desired inequality can be obtained by an application of Theorem 2.5 to 3.6.
Example 3.4. Consider the following delay dynamic integral equation:
u x, y C p
y x
y0
N M
y0
F1 s, t, uτ1 s, τ2 t,
x0
F2 s, t, uτ1 s, τ2 t,
x0
t s
y0
x0
t s
y0
!
W1 ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
!
W2 ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt,
x0
T ,
x, y ∈ x0 , M T × y0 , N
3.8
with the initial condition
u x, y φ x, y , if x ∈ α, x0 T, or y ∈ β, y0
T,
φ τ1 x, τ2 y ≤ |C|1/p , if τ1 x ≤ x0 , or τ2 y ≤ y0 , ∀ x, y ∈ T0 × T
0,
3.9
0 , R, p > 0 is a constant, C up x0 , y0 , M ∈ T0 , N ∈ T
0 are two fixed
where u ∈ Crd T0 × T
2
2
0 × R, R, i numbers. φ ∈ Crd α, x0 × β, y0 T , R, Fi ∈ T0 × T0 × R , R, Wi ∈ T0 × T
1, 2. α, β, τ1 , τ2 are the same as in Theorem 2.4.
22
Journal of Applied Mathematics
Theorem 3.5. Suppose ux, y is a solution of 3.8-3.9 and |Fi s, t, x, y| ≤ Ls, t, |x| |y|, |Wi ξ, η, x| ≤ hi ξ, η|x|q , i 1, 2, where L, hi , i 1, 2, q are defined the same as in
Theorem 2.11; then the following inequality holds:
u x, y ≤
1/p
λ B6 ,
B3 x, y B4 x, y
1 − B5
T ,
x, y ∈ x0 , M T × y0 , N
3.10
B2 x, y, B3 x, y, B4 x, y, B5 , B6 are defined the same as in
provided that B5 < 1, where λ,
Theorem 2.11, and
B1 x, y |C| y x t s
p − q q/p
p − 1 1/p
L s, t,
K
K ΔξΔη ΔsΔt,
h1 ξ, η
p
p
y0 x0
y0 x0
∀K > 0.
3.11
Proof. From 3.8 we have
p
u x, y ≤ |C|
!
y x t s
W1 ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
F1 s, t, uτ1 s, τ2 t,
y0 x0
y0 x0
!
N M t s
W2 ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
F2 s, t, uτ1 s, τ2 t,
y0 x0
y 0 x0
≤ |C|
y x t s
Ls, t, |uτ1 s, τ2 t| W1 ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
y0 x0
y0 x0
N M
t s
Ls, t, |uτ1 s, τ2 t| W2 ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt
y0 x0
y0 x0
≤ |C|
y x t s
q
Ls, t, |uτ1 s, τ2 t| h1 ξ, η u τ1 ξ, τ2 η
ΔξΔη ΔsΔt
y0
x0
N M
y0
x0
y0
Ls, t, |uτ1 s, τ2 t| x0
t s
y0
q
h2 ξ, η u τ1 ξ, τ2 η
ΔξΔη ΔsΔt.
x0
3.12
Journal of Applied Mathematics
23
So by use of Theorem 2.11 we obtain the desired inequality 3.10.
Example 3.6. Consider the following delay dynamic integral equation:
u x, y
C
y x
y0
F s, t, uτ1 s, τ2 t,
x0
t s
y0
!
W ξ, η, u τ1 ξ, τ2 η ΔξΔη ΔsΔt,
3.13
x0
0 , R, C up x0 , y0 , F ∈ T0 × T
0 × R2 , R, W ∈ T0 × T
0 × R, R. τ1 , τ2
where u ∈ Crd T0 × T
are the same as in Theorem 2.4.
Theorem 3.7. Assume that |Fs, t, u1 , v1 −Fs, t, u2 , v2 | ≤ fs, t|u1 −u2 ||v1 −v2 |, |Ws, t, u1 −
Ws, t, u2 | ≤ hs, t|u1 − u2 |, where f, h are defined as in Theorem 2.4, and; furthermore, assume
that τ1 x ≥ x0 , τ2 y ≥ y0 , then 3.13 has at most one solution.
Proof. Suppose that u1 x, y, u2 x, y are two solutions of 3.13; then we have
u1 x, y − u2 x, y !
t s
y x
F s, t, u1 τ1 s, τ2 t
W ξ, η, u1 τ1 ξ, τ2 η ΔξΔη
≤
y0 x0
y0 x0
−F s, t, u2 τ1 s, τ2 t
t s
y0
W ξ, η, u2 τ1 ξ, τ2 η ΔξΔη
x0
ΔsΔt
!
!
y x t s
≤
W ξ, η, u1 τ1 ξ, τ2 η ΔξΔη
F s, t, u1 τ1 s, τ2 t
y0 x0 y 0 x0
−F s, t, u2 τ1 s, τ2 t
y0
≤
y x
y0
!
W ξ, η, u2 τ1 ξ, τ2 η ΔξΔη ΔsΔt
x0
t s
fs, t|u1 τ1 s, τ2 t − u2 τ1 s, τ2 t|ΔsΔt
x0
y x t s
y0
x0
y0
x0
W ξ, η, u1 τ1 ξ, τ2 η
− W ξ, η, u2 τ1 ξ, τ2 η ΔξΔηΔsΔt
24
Journal of Applied Mathematics
≤
y x
y0
fs, t|u1 τ1 s, τ2 t − u2 τ1 s, τ2 t|ΔsΔt
x0
y x t s
y0
x0
y0
h ξ, η u1 τ1 ξ, τ2 η − u2 τ1 ξ, τ2 η ΔξΔηΔsΔt, .
x0
3.14
Then a suitable application of Theorem 2.8 yields |u1 x, y − u2 x, y| ≤ 0, that is, u1 x, y ≡
u2 x, y, and the proof is complete.
4. Conclusions
In this paper, we established some new Gronwall-Bellman type integral inequalities on time
scales. As one can see, the presented results provide a handy tool in the quantitative as well
as qualitative analysis of solutions of certain delay dynamic equations on time scales. The
established inequalities unify some known continuous and discrete inequalities.
Acknowledgments
This work is supported by Natural Science Foundation of Shandong Province
ZR2009AM011 China and Specialized Research Fund for the Doctoral Program of
Higher Education 20103705110003 China. The authors thank the referees very much for
their careful comments and valuable suggestions on this paper.
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