Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 484185, 16 pages
doi:10.1155/2009/484185
Research Article
Bounds for Certain New Integral Inequalities on
Time Scales
Wei Nian Li1, 2
1
2
Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China
Department of Mathematics, Binzhou University, Shandong 256603, China
Correspondence should be addressed to Wei Nian Li, wnli@263.net
Received 31 March 2009; Accepted 10 June 2009
Recommended by Victoria Otero-Espinar
Our aim in this paper is to investigate some new integral inequalities on time scales, which provide
explicit bounds on unknown functions. Our results unify and extend some integral inequalities
and their corresponding discrete analogues. The inequalities given here can be used as handy
tools to study the properties of certain dynamic equations on time scales.
Copyright q 2009 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.
1. Introduction
The study of dynamic equations on time scales, which goes back to its founder Hilger 1,
is an area of mathematics that has recently received a lot of attention. For example, we
refer the reader to literatures 2–8 and the references cited therein. At the same time, some
fundamental integral inequalities used in analysis on time scales have been extended by
many authors 9–14. In this paper, we investigate some new nonlinear integral inequalities
on time scales, which unify and extend some continuous inequalities and their corresponding
discrete analogues. Our results can be used as handy tools to study the properties of certain
dynamic equations on time scales.
Throughout this paper, a knowledge and understanding of time scales and time scale
notation is assumed. For an excellent introduction to the calculus on time scales, we refer the
reader to monographes 2, 3.
2. Main Results
In what follows, R denotes the set of real numbers, R 0, ∞, Z denotes the set of integers,
N0 {0, 1, 2, . . .} denotes the set of nonnegative integers, CM, S denotes the class of all
continuous functions defined on set M with range in the set S, T is an arbitrary time scale,
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Advances in Difference Equations
Crd denotes the set of rd-continuous functions, R denotes the set of all regressive and rdcontinuous functions, and R {p ∈ R : 1μtpt > 0, ∀t ∈ T}. We use the usual conventions
that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper,
we always assume that p ≥ 1, 0 < q ≤ p, p, and q are real constants, and t ≥ t0 , t0 ∈ Tκ .
We firstly introduce the following lemmas, which are useful in our main results.
Lemma 2.1 15 Bernoulli’s inequality. Let 0 < α ≤ 1 and x > −1. Then 1 xα ≤ 1 αx.
Lemma 2.2 2. Let t0 ∈ Tκ and w : T × Tκ → R be continuous at t, t, where t ≥ t0 , t ∈ Tκ with
t > t0 . Assume that wΔ t, · is rd-continuous on t0 , σt. If for any ε > 0, there exists a neighborhood
U of t, independent of τ ∈ t0 , σt, such that
wσt, τ − ws, τ − wΔ t, τσt − s ≤ ε|σt − s|,
∀s ∈ U,
2.1
where wΔ denotes the derivative of w with respect to the first variable, then
t
wt, τΔτ
2.2
wΔ t, τΔτ wσt, t.
2.3
gt :
t0
implies
g Δ t t
t0
Lemma 2.3 2 Comparison Theorem. Suppose u, b ∈ Crd , a ∈ R . Then
uΔ t ≤ atut bt,
t ≥ t0 , t ∈ Tκ ,
2.4
implies
ut ≤ ut0 ea t, t0 t
ea t, στbτΔτ,
t ≥ t0 , t ∈ Tκ .
2.5
t0
Lemma 2.4 see 13. Let u, f, g ∈ Crd , ut, ft, and gt be nonnegative. If ft is
nondecreasing, then
ut ≤ ft t
gτuτΔτ,
t ∈ Tκ ,
2.6
t0
implies
ut ≤ fteg t, t0 ,
t ∈ Tκ ,
2.7
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3
Next, we establish our main results.
Theorem 2.5. Assume that u, a, b, g, h ∈ Crd , at > 0, ut, bt, gt and ht are nonnegative.
Then
u t ≤ at bt
p
t
gτuq τ hτ Δτ,
t ∈ Tκ ,
2.8
t0
implies
ut ≤ a
1/p
1
t a1/p−1 tbt
p
t
eB t, στFτΔτ,
t ∈ Tκ ,
2.9
t0
where
Ft gtaq/p t ht,
Bt q q/p−1
a
tbtgt,
p
2.10
t ∈ Tκ .
2.11
Proof. Define a function zt by
zt t
gτuq τ hτ Δτ.
2.12
t0
Then 2.8 can be restated as
btzt
up t ≤ at btzt at 1 .
at
2.13
Using Lemma 2.1, from the above inequality, we easily obtain
ut ≤ a1/p t 1 1/p−1
a
tbtzt,
p
2.14
uq t ≤ aq/p t q q/p−1
a
tbtzt.
p
2.15
It follows from 2.12 and 2.15 that
zΔ t gtuq t ht
q
≤ gt aq/p t aq/p−1 tbtzt ht
p
Ft Btzt,
t ∈ Tκ ,
2.16
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Advances in Difference Equations
where Ft and Bt are defined as in 2.10 and 2.11, respectively. Using Lemma 2.3 and
noting zt0 0, from 2.16 we have
t
zt ≤
eB t, στFτΔτ,
2.17
t ∈ Tκ .
t0
Therefore, the desired inequality 2.9 follows from 2.14 and 2.17. This completes
the proof of Theorem 2.5.
Corollary 2.6. Assume that u, g ∈ Crd , ut, and gt are nonnegative. If α > 0 is a constant, then
u t ≤ α p
t
gτuq τΔτ,
2.18
t ∈ Tκ ,
t0
implies
ut ≤ α
1/p
1 1
1 − eB t, t0 ,
q q
t ∈ Tκ ,
2.19
where
q αq/p−1 gt,
Bt
p
t ∈ Tκ .
2.20
Proof. Letting at α, bt 1, and ht 0 in Theorem 2.5, we obtain
Ft αq/p gt,
Bt q q/p−1
α
gt : Bt,
p
t ∈ Tκ .
Therefore,
ut ≤ α
1/p
1
α1/p−1
p
α1/p 1 1/p−1
α
p
t
t0
t
t0
eB t, στFτΔτ
eB t, σταq/p gτΔτ
2.21
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1
α1/p α1/p
q
1
α1/p α1/p
q
t
q
eB t, στ αq/p−1 gτΔτ
p
t0
t
t0
eB t, στBτΔτ
1
α1/p α1/p eB t, t0 − eB t, t
q
1
1
α1/p α1/p eB t, t0 − α1/p
q
q
1 1
1/p
α
1 − eB t, t0 , t ∈ Tκ .
q q
2.22
The proof of Corollary 2.6 is complete.
Remark 2.7. The result of Theorem 2.5 holds for an arbitrary time scale. Therefore, using
Theorem 2.5, we can obtain many results for some peculiar time scales. For example, letting
T R and T Z, respectively, we have the following two results.
Corollary 2.8. Let T R and assume that ut, at, bt, gt, ht ∈ CR , R , and at > 0.
Then the inequality
t
up t ≤ at bt
gsuq s hs ds,
2.23
t ∈ R ,
0
implies
ut ≤ a
1/p
1
t a1/p−1 tbt
p
t
t
Fθ exp
0
Bsds dθ,
t ∈ R ,
2.24
θ
where Ft and Bt are defined as in Theorem 2.5.
Corollary 2.9. Let T Z and assume that at > 0, ut, bt, gt, and ht are nonnegative
functions defined for t ∈ N0 . Then the inequality
up t ≤ at bt
t−1
gsuq s hs ,
t ∈ N0 ,
2.25
s0
implies
ut ≤ a1/p t t−1
t−1
1 1/p−1
a
tbt Fθ
1 Bs,
p
θ0
sθ1
where Ft and Bt are defined as in Theorem 2.5.
t ∈ N0 ,
2.26
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Advances in Difference Equations
Investigating the proof procedure of Theorem 2.5 carefully, we easily obtain the
following more general result.
Theorem 2.10. Assume that u, a, b, gi , hi ∈ Crd , at > 0, ut, bt, gi t, and hi t are
nonnegative, i 1, 2, . . . , n. If there exists a series of positive real numbers q1 , q2 , . . . , qn such that
p ≥ qi > 0, i 1, 2, . . . , n, then
u t ≤ at bt
p
n t
i1
gi τuqi τ hi τ Δτ,
t ∈ Tκ ,
2.27
t0
implies
ut ≤ a
1/p
1
t a1/p−1 tbt
p
t
eB∗ t, στF ∗ τΔτ,
t ∈ Tκ ,
2.28
t0
where
F ∗ t n gi taqi /p t hi t ,
n
qi qi /p−1
a
B∗ t bt
tgi t.
p
i1
i1
2.29
Theorem 2.11. Assume that u, a, b, f, g, m ∈ Crd , at > 0, ut, bt, ft, gt and mt are
nonnegative. If wt, s is defined as in Lemma 2.2 such that wt, s ≥ 0 and wΔ t, s ≥ 0 for t, s ∈ T
with s ≤ t, then
t
u t ≤ at bt wt, τ fτup τ gτuq τ mτ Δτ,
t ∈ Tκ ,
p
2.30
t0
implies
ut ≤ a1/p t 1 1/p−1
a
tbt
p
t
eA t, στGτΔτ,
t ∈ Tκ ,
2.31
t0
where
q
At wσt, tbt ft aq/p−1 tgt
p
t
q
wΔ t, τbτ fτ aq/p−1 τgτ Δτ,
p
t0
Gt wσt, t atft gtaq/p t mt
t
t0
wΔ t, τ aτfτ gτaq/p τ mτ Δτ.
2.32
2.33
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Proof. Define a function zt by
zt t
wt, τ fτup τ gτuq τ mτ Δτ,
t ∈ Tκ .
2.34
t0
Then zt0 0. As in the proof of Theorem 2.5, we easily obtain 2.14 and 2.15. Using
Lemma 2.2 and combining 2.34 and 2.15, we have
zΔ t wσt, t ftup t gtuq t mt
t
wΔ t, τ fτup τ gτuq τ mτ Δτ
t0
q
≤ wσt, t atft gtaq/p t mt bt ft aq/p−1 tgt zt
p
t
wΔ t, τ aτfτ gτaq/p τ mτ
t0
q
bτ fτ aq/p−1 tgτ zτ Δτ
p
q
≤ wσt, tbt ft aq/p−1 tgt
p
t
q q/p−1
Δ
w t, τbτ fτ a
tgτ Δτ zt
p
t0
wσt, t atft gtaq/p t mt
t
2.35
wΔ t, τ aτfτ gτaq/p τ mτ Δτ
t0
Atzt Gt,
t ∈ Tκ ,
where At and Gt are defined as in 2.32 and 2.33, respectively. Therefore, in the above
inequality, using Lemma 2.3 and noting zt0 0, we get
zt ≤
t
eA t, στGτΔτ,
t ∈ Tκ .
2.36
t0
It is easy to see that the desired inequality 2.31 follows from 2.14 and 2.36. This
completes the proof of Theorem 2.11.
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Corollary 2.12. Let T R and assume that ut, at, bt, ft, gt, mt ∈ CR , R , at > 0.
If wt, s and its partial derivative ∂wt, s/∂t are real–valued nonnegative continuous functions for
t, s ∈ R with s ≤ t, then the inequality
t
u t ≤ at bt wt, s fsup s gsuq s ms ds,
t ∈ R ,
p
2.37
0
implies
ut ≤ a
1/p
t
t
1 1/p−1
Aτdτ ds,
t a
tbt Gs exp
p
0
s
t ∈ R ,
2.38
where
q
At wt, tbt ft aq/p−1 tgt
p
t
q q/p−1
∂wt, s
bs fs a
sgs ds,
∂t
p
0
Gt wt, t atft gtaq/p t mt
2.39
t
∂wt, s asfs gsaq/p s ms ds.
∂t
0
Corollary 2.13. Let T Z and assume that at > 0, ut, bt, ft, gt and mt are nonnegative
functions defined for t ∈ N0 . If wt, s and Δ1 wt, s are real-valued nonnegative functions for t, s ∈
N0 with s ≤ t, then the inequality
up t ≤ at bt
t−1
wt, s fsup s gsuq s ms ,
t ∈ N0 ,
2.40
s0
implies
ut ≤ a1/p t t−1 t−1
1 1/p−1
,
a
1 Aτ
tbt Gs
p
s0
τs1
t ∈ N0 ,
2.41
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where Δ1 wt, s wt 1, s − wt, s for t, s ∈ N0 with s ≤ t,
wt 1, tbt ft q aq/p−1 tgt
At
p
t−1
q
Δ1 wt, sbs fs aq/p−1 sgs ,
p
s0
wt 1, t atft gtaq/p t mt
Gt
2.42
t−1
Δ1 wt, s asfs gsaq/p s ms .
s0
Corollary 2.14. Suppose that ut, at, and wt, s are defined as in Theorem 2.11. If at is
nondecreasing for t ∈ Tκ , then
t
wt, τuq τΔτ,
t ∈ Tκ ,
2.43
1 1
t 1 − eA t, t0 ,
q q
t ∈ Tκ ,
2.44
u t ≤ at p
t0
implies
ut ≤ a
1/p
where
q
At p
1/p−1
wσt, ta
t t
Δ
w t, τa
1/p−1
τΔτ .
2.45
t0
Proof. Letting bt 1, ft 0, gt 1, and mt 0 in Theorem 2.11, we obtain
q
At p
1/p−1
wσt, ta
t t
Δ
w t, τa
τΔτ
: At,
t0
Gt wσt, taq/p t t
wΔ t, τaq/p τΔτ
t0
≤ at wσt, ta1/p−1 t t
t0
1/p−1
p
atAt,
q
t ∈ Tκ ,
wΔ t, τa1/p−1 τΔτ
2.46
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where the inequality holds because at is nondecreasing for t ∈ Tκ . Therefore, using
Theorem 2.11 and noting 2.46, we easily have
t
1 1/p−1
a
t eA t, στGτΔτ
p
t0
t
p
1
≤ a1/p t a1/p−1 t e t, στ aτAτΔτ
A
p
q
t0
t
1 1/p
1/p
≤ a t a t e t, στAτΔτ
A
q
t0
1
e t, t0 − e t, t
a1/p t 1 A
q A
1 1
a1/p t 1 − e t, t0 , t ∈ Tκ .
q q A
ut ≤ a1/p t 2.47
The proof of Corollary 2.14 is complete.
By Theorem 2.11, we can establish the following more general result.
Theorem 2.15. Assume that u, a, b, f, gi , m ∈ Crd , at > 0, ut, bt, ft, gi t, and mt are
nonnegative, i 1, 2, . . . , n, and there exists a series of positive real numbers q1 , q2 , . . . , qn such that
p ≥ qi > 0, i 1, 2, . . . , n. If wt, s is defined as in Lemma 2.2 such that wt, s ≥ 0 and wΔ t, s ≥ 0
for t, s ∈ T with s ≤ t, then
u t ≤ at bt
p
t
wt, τ fτu τ p
n
t0
gi τu τ mτ Δτ,
qi
t ∈ Tκ ,
2.48
i1
implies
ut ≤ a
1/p
1
t a1/p−1 tbt
p
t
eA∗ t, στG∗ τΔτ,
t ∈ Tκ ,
2.49
t0
where
∗
A t wσt, tbt ft t
n
qi
i1
p
Δ
w t, τbτ fτ t0
qi /p−1
a
n
qi
i1
p
tgi t
qi /p−1
a
τgi τ Δτ,
n
G∗ t wσt, t atft gi taqi /p t mt
t
t0
Δ
i1
w t, τ aτfτ n
i1
2.50
gi τa
qi /p
τ mτ Δτ.
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Theorem 2.16. Let u, a, r ∈ Crd , ut and rt be nonnegative, at > 0, and at be nondecreasing.
Assume that there exists a series of positive real numbers q1 , q2 , . . . , qn such that p ≥ qi > 0, i 1, 2, . . . , n. If Si : Tκ × R → R is a continuous function such that
0 ≤ Si t, xi − Si t, yi ≤ Hi t, yi xi − yi ,
2.51
for t ∈ Tκ and xi ≥ yi ≥ 0, i 1, 2, . . . , n, where Hi : Tκ × R → R is a nonnegative continuous
function, i 1, 2, . . . , n, then
t
up t ≤ at rτup τΔτ t0
n t
i1
t ∈ Tκ ,
Si τ, uqi τΔτ,
2.52
t0
implies
1
ut ≤ R1/p t a1/p t a1/p−1 tLteJ t, t0 ,
p
t ∈ Tκ ,
2.53
where
Rt er t, t0 ,
n t
Si τ, Rqi /p τaqi /p τ Δτ,
Lt i1
Jt 2.55
t0
n
qi
i1
2.54
p
Hi t, Rqi /p taqi /p t Rqi /p taqi /p−1 t.
2.56
Proof. Let
vt n t
i1
Si τ, uqi τΔτ,
zt at vt,
t ∈ Tκ .
2.57
t0
Then 2.52 can be restated as
u t ≤ zt p
t
rτup τΔτ,
t ∈ Tκ .
2.58
t0
It is easy to see that zt ∈ Crd , zt > 0, and zt is nondecreasing. Using Lemma 2.4, from
2.58, we have
up t ≤ Rtzt,
t ∈ Tκ ,
2.59
where Rt is defined as in 2.54. It follows from 2.57 and 2.59 that
up t ≤ Rtat vt.
2.60
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Using Lemma 2.1 to the above inequality, we obtain
ut ≤ R1/p tat vt1/p
1 1/p−1
1/p
1/p
≤ R t a t a
tvt ,
p
2.61
uqi t ≤ Rqi /p tat vtqi /p
qi
≤ Rqi /p t aqi /p t aqi /p−1 tvt ,
p
2.62
t ∈ Tκ .
Noting the hypotheses on Si , from 2.62, we get
vt ≤
n t
qi
Si τ, Rqi /p τ aqi /p τ aqi /p−1 τvτ Δτ
p
t0
i1
n t qi qi /p−1
qi /p
qi /p
Si τ, R τ a τ a
τvτ
p
i1 t0
−Si τ, R
qi /p
τa
qi /p
n t
Si τ, Rqi /p τaqi /p τ Δτ
τ Δτ i1
≤ Lt t0
n t
i1
qi
Hi τ, Rqi /p τaqi /p τ Rqi /p τ aqi /p−1 τvτΔτ,
p
t0
t ∈ Tκ ,
2.63
where Lt is defined by 2.55. Clearly, Lt ≥ 0 and Lt are nondecreasing. Therefore, for
any ε > 0, from 2.63, we obtain
n
vt
≤1
Lt ε
i1
t
qi
vτ
Δτ,
Hi τ, Rqi /p τaqi /p τ Rqi /p τ aqi /p−1 τ
p
Lτ ε
t0
t ∈ Tκ . 2.64
Let
ψt vt
,
Lt ε
t ∈ Tκ ,
2.65
and define kt by the right hand of 2.64. Then kt > 0, kt0 1, ψt ≤ kt, and
kΔ t ≤
qi
Hi t, Rqi /p taqi /p t Rqi /p t aqi /p−1 tψt
p
i1
n
qi
Hi t, Rqi /p taqi /p t Rqi /p t aqi /p−1 tkt
p
i1
n
ktJt,
t ∈ Tκ ,
2.66
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where Jt is defined by 2.56. Using Lemma 2.3 and noting kt0 1, from 2.66, we have
kt ≤ eJ t, t0 ,
2.67
t ∈ Tκ .
Therefore,
vt ≤ Lt εeJ t, t0 ,
2.68
t ∈ Tκ .
It follows from 2.61 and 2.68 that
1
ut ≤ R1/p t a1/p t a1/p−1 tLt εeJ t, t0 ,
p
t ∈ Tκ .
2.69
Letting ε → 0 in 2.69, we immediately obtain the desired inequality 2.53. This completes
the proof of Theorem 2.16.
Corollary 2.17. Let T R, u, a, r ∈ CR , R , at > 0, and at be nondecreasing. Assume that
there exists a series of positive real numbers q1 , q2 , . . . , qn such that p ≥ qi > 0, i 1, 2, . . . , n. If
Si : R × R → R is a continuous function such that
0 ≤ Si t, xi − Si t, yi ≤ Hi t, yi xi − yi ,
2.70
for t ∈ R and xi ≥ yi ≥ 0, i 1, 2, . . . , n, where Hi : R × R → R is a continuous function,
i 1, 2, . . . , n, then
up t ≤ at t
rτup τdτ 0
n t
i1
t ∈ R ,
Si τ, uqi τdτ,
2.71
0
implies
1/p
ut ≤ R
t a
1/p
1
t a1/p−1 tLt exp
p
t
Jsds
,
t ∈ R ,
2.72
0
where Jt is defined as in 2.56,
t
Rt exp
rsds ,
0
Lt n t
i1
Jt 0
n
qi
i1
qi /p
Si τ, R τaqi /p τ dτ,
p
qi /p
qi /p
Hi t, R taqi /p t R taqi /p−1 t.
2.73
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Corollary 2.18. Let T Z, at > 0, at be nondecreasing, ut and rt be nonnegative functions
defined for t ∈ N0 . Assume that there exists a series of positive real numbers q1 , q2 , . . . , qn such that
p ≥ qi > 0, i 1, 2, . . . , n. If Si : N0 × R → R such that
0 ≤ Si t, xi − Si t, yi ≤ Hi t, yi xi − yi ,
2.74
for t ∈ N0 and xi ≥ yi ≥ 0, i 1, 2, . . . , n, where Hi : N0 × R → R , i 1, 2, . . . , n, then
up t ≤ at t−1
rτup τ τ0
t−1
n Si τ, uqi τ,
t ∈ N0 ,
2.75
i1 τ0
implies
1/p
ut ≤ R
t a
1/p
t−1 1 1/p−1 ,
1 Js
t a
tLt
p
s0
t ∈ N0 ,
2.76
where Jt is defined as in 2.56,
Rt
t−1
1 rs,
s0
Lt
t−1 n qi /p τaqi /p τ ,
Si τ, R
2.77
i1 τ0
Jt
n
qi
i1
p
qi /p taqi /p t R
qi /p taqi /p−1 t.
Hi t, R
Remark 2.19. Using our main results, we can obtain many dynamic inequalities for some
peculiar time scales. Due to limited space, their statements are omitted here.
3. Some Applications
In this section, we present two applications of our main results.
Example 3.1. Consider the inequality as in 2.25 with at αt 1, bt αt2 1, gt t, ht 0, p 2, q 1, α 10−6 , and we compute the values of ut from 2.25 and also
we find the values of ut by using the result 2.26. In our computations we use 2.25 and
2.26 as equations and as we see in Table 1 the computation values as in 2.25 are less than
the values of the result 2.26.
From Table 1, we easily find that the numerical solution agrees with the analytical
solution for some discrete inequalities. The program is written in the programming Matlab
7.0.
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Table 1
t
1
2
5
7
10
12
14
17
20
25
2.25
1.414213562373095e − 003
2.661293464584210e − 003
5.486250637546570e − 002
2.670738191264154e − 001
1.527219045903506e 000
3.720520602864323e 000
7.856747926470754e 000
1.997586843703775e 001
4.331228422296512e 001
1.241251179017371e 002
2.26
1.414213562373095e − 003
2.910562109546456e − 003
1.103460932829943e − 001
5.410171853718061e − 001
3.137697944498020e 000
8.436559692675310e 000
2.187361362745254e 001
9.900992670086097e 001
5.854191578762491e 002
2.937887676184530e 004
Example 3.2. Consider the following initial value problem on time scales:
up tΔ Mt, ut,
ut0 β,
t ∈ Tκ ,
3.1
where p ≥ 1 and β / 0 are constants, and M : Tκ × R → R is a continuous function.
Assume that
3.2
|Mt, ut| ≤ gt|uq t|,
where gt is defined as in Corollary 2.6, 0 < q ≤ p is a constant. If ut is a solution of IVP
3.1, then
1 1
|ut| ≤ β 1 − eV t, t0 ,
q q
t ∈ Tκ ,
3.3
where
V t q q−p
β
gt,
p
t ∈ Tκ .
3.4
In fact, the solution ut of IVP 3.1 satisfies the following equation:
up t βp t
Mτ, uτΔτ,
t ∈ Tκ .
3.5
t0
Using the assumption 3.2, from 3.5, we have
p
|ut|p ≤ β t
gτ|uτ|q Δτ,
t ∈ Tκ .
t0
Now a suitable application of Corollary 2.6 to 3.6 yields 3.2.
3.6
16
Advances in Difference Equations
Acknowledgments
This work is supported by the Natural Science Foundation of Shandong Province
Y2007A08, the National Natural Science Foundation of China 60674026, 10671127, China
Postdoctoral Science Foundation Funded Project 20080440633, 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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