Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 897087, 14 pages
doi:10.1155/2009/897087
Research Article
Bounds for Certain Nonlinear Dynamic
Inequalities on Time Scales
Wei Nian Li1, 2 and Maoan Han2
1
2
Department of Mathematics, Binzhou University, Shandong 256603, China
Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China
Correspondence should be addressed to Wei Nian Li, wnli@263.net
Received 29 July 2009; Accepted 19 November 2009
Recommended by Guang Zhang
We investigate some new nonlinear dynamic inequalities on time scales. Our results unify and
extend some integral inequalities and their corresponding discrete analogues. The inequalities
given here can be used to investigate the properties of certain dynamic equations on time scales.
Copyright q 2009 W. N. Li and M. Han. 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
To unify the theory of continuous and discrete dynamic systems, in 1988, Hilger 1 first
introduced the calculus on time scales. Motivated by the paper 1, many authors have
expounded on various aspects of the theory of dynamic equations on time scales. For
example, we refer the reader to the literatures 2–7 and the references cited therein. At
the same time, a few papers 8–13 have studied the theory of dynamic inequalities on time
scales.
The main purpose of this paper is to investigate some nonlinear dynamic inequalities
on time scales, which unify and extend some integral inequalities and their corresponding
discrete analogues. Our work extends some known results of dynamic inequalities 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 6, 7.
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
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continuous functions defined on set M with range in the set S, T is an arbitrary time scale,
Crd denotes the set of rd-continuous functions, R denotes the set of all regressive and rdcontinuous functions, and R {p ∈ R : 1 μtpt > 0 for all 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 ≥ q > 0, p and q are real constants, and t ≥ t0 , t0 ∈ Tκ .
Firstly, we introduce the following lemmas, which are useful in our main results.
Lemma 2.1. Let a ≥ 0. Then
q/p
a
≤
q q−p/p
p − q q/p
K
K
a
p
p
for any K > 0.
2.1
Proof. If a 0, then we easily see that the inequality 2.1 holds. Thus we only prove that the
inequality 2.1 holds in the case of a > 0.
Letting
fK p − q q/p
q q−p/p
K
K ,
a
p
p
2.2
K > 0,
we have
q p − q q−2p/p
f K K
K − a.
p2
2.3
It is easy to see that
f K ≥ 0,
K > a,
f K 0,
K a,
f K ≤ 0,
2.4
0 < K < a.
Therefore,
2.5
fK ≥ fa aq/p .
The proof of Lemma 2.1 is complete.
Lemma 2.2 see 6. Let t0 ∈ Tκ and w : T × Tκ → R be continuous at t, t, t ∈ Tκ with t > t0 .
Assume that w1Δ 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 − w1Δ t, τσt − s ≤ ε|σt − s|
for all s ∈ U,
2.6
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3
where w1Δ denotes the derivative of w with respect to the first variable, then
t
wt, τΔτ
2.7
w1Δ t, τΔτ wσt, t.
2.8
υt :
t0
implies
Δ
υ t t
t0
Lemma 2.3 Comparison theorem 6. Suppose u, b ∈ Crd , a ∈ R . Then
uΔ t ≤ atut bt
for all t ∈ Tκ
2.9
implies
ut ≤ ut0 ea t, t0 t
ea t, στbτΔτ
for all t ∈ Tκ .
2.10
t0
Next, we establish our main results.
Theorem 2.4. Assume that u, a, b, g, h ∈ Crd , and ut, at, bt, gt and ht are nonnegative.
Then
t
u t ≤ at bt
gτuq τ hτ Δτ
p
for all t ∈ Tκ
E1
t0
implies
ut ≤
at bt
t
1/p
eB t, στFτΔτ
for any K > 0, t ∈ Tκ ,
2.11
t0
where
Ft gt
p − q q/p
qat
K p
pK p−q/p
ht,
2.12
and also
Bt qbtgt
pK p−q/p
for all t ∈ Tκ .
2.13
Proof. Obviously, if t t0 , then the inequality 2.11 holds. Therefore, in the next proof, we
always assume that t > t0 , t ∈ Tκ .
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Define a function zt by
zt t
gτuq τ hτ Δτ.
2.14
t0
Then E1 can be restated as
up t ≤ at btzt.
2.15
Using Lemma 2.1, from 2.15, for any K > 0, we easily obtain
uq t ≤ at btztq/p
≤
2.16
p − q q/p
qat
qbtzt
K .
p−q/p
p
pK
pK p−q/p
It follows from 2.14 and 2.16 that
Δ
z t ≤ gt
p − q q/p
qat
qbtzt
K p−q/p
p
pK
pK p−q/p
ht
2.17
Ft Btzt,
where Ft and Bt are defined as in 2.12 and 2.13, respectively. Using Lemma 2.3 and
noting zt0 0, from 2.17 we have
zt ≤
t
eB t, στFτΔτ,
for any K > 0, t ∈ Tκ .
2.18
t0
Therefore, the desired inequality 2.11 follows from 2.15 and 2.18. This completes
the proof of Theorem 2.4.
Remark 2.5. By letting p q 1 in Theorem 2.4, it is easy to observe that the bound obtained
in 2.11 reduces to the bound obtained in 9, Theorem 3.1.
As a particular case of Theorem 2.4, we immediately obtain the following result.
Corollary 2.6. Assume that u, g ∈ Crd , and ut and gt are nonnegative. If α > 0 is a constant,
then
up t ≤ α t
t0
gτuq τΔτ
for all t ∈ Tκ
E 1
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5
implies
ut ≤
α
t
t0
1/p
eB t, στFτΔτ
for any K > 0, t ∈ Tκ ,
2.19
where
gt
Ft
Bt
p − q q/p
qα
K p−q/p
p
pK
qgt
pK p−q/p
,
for all t ∈ Tκ .
2.20
2.21
Remark 2.7. The result of Theorem 2.4 holds for an arbitrary time scale. Therefore, using
Theorem 2.4, we immediately 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 . Then the
inequality
t
up t ≤ at bt
gsuq s hs ds,
t ∈ R
2.22
0
implies
ut ≤ at bt
t
t
1/p
Fθ exp
Bsds dθ
0
for any K > 0, t ∈ R ,
2.23
θ
where Ft and Bt are defined as in Theorem 2.4.
Corollary 2.9. Let T Z and assume that ut, at, 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
2.24
s0
implies
1/p
t−1
t−1
ut ≤ at bt Fθ
1 Bs
θ0
sθ1
where Ft and Bt are defined as in Theorem 2.4.
for any K > 0, t ∈ N0 ,
2.25
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Discrete Dynamics in Nature and Society
Investigating the proof procedure of Theorem 2.4 carefully, we can obtain the
following result.
Theorem 2.10. Assume that u, a, b, gi , h ∈ Crd , and ut, at, bt, gi t, and ht 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
t n
t0
gi τu τ hτ Δτ
for all t ∈ Tκ
qi
E 1
i1
implies
ut ≤
at bt
1/p
t
∗
for any K > 0, t ∈ Tκ ,
e t, στF τΔτ
B∗
2.26
t0
where
n
gi t
F t ∗
i1
B∗ t p − qi qi /p
qi at
K
p
pK p−qi /p
n
qi btgi t
i1
pK p−qi /p
ht,
2.27
for all t ∈ Tκ .
2.28
Theorem 2.11. Assume that u, a, b, f, g, m ∈ Crd , ut, at, bt, ft, gt, and mt are
nonnegative, and wt, s is defined as in Lemma 2.2 such that wt, s ≥ 0 and w1Δ t, s ≥ 0 for
t, s ∈ T with s ≤ t. If, for any ε > 0, there exists a neighborhood U of t, independent of τ ∈ t0 , σt,
such that for all s ∈ U,
wσt, τ − ws, τ − w1Δ t, τσt − s fτup τ gτuq τ mτ ≤ ε|σt − s|,
2.29
then
u t ≤ at bt
p
t
wt, τ fτup τ gτuq τ mτ Δτ,
t ∈ Tκ
E2
t0
implies
ut ≤
at bt
t
t0
1/p
eA t, στGτΔτ
for any K > 0, t ∈ Tκ ,
2.30
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7
where
At wσt, tbt ft qgt
pK p−q/p
t
t0
w1Δ t, τbτ
fτ qgτ
pK p−q/p
Δτ,
2.31
and also
p − q K q/p
qat
mt
p
pK p−q/p
Gt wσt, t atft gt
t
t0
w1Δ t, τ aτfτ gτ
p − q K q/p
qaτ
mτ Δτ.
p
pK p−q/p
2.32
Proof. Define a function zt by
zt t
kt, τΔτ
for all t ∈ Tκ ,
2.33
t0
where
kt, τ wt, τ fτup τ gτuq τ mτ .
2.34
Then zt0 0. As in the proof of Theorem 2.4, we easily obtain 2.15 and 2.16.
It follows from 2.34 that
kσt, t wσt, t fτup τ gτuq τ mτ ,
2.35
k1Δ t, τ w1Δ t, τ fτup τ gτuq τ mτ .
2.36
and also
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Discrete Dynamics in Nature and Society
Therefore, noting the condition 2.29, using Lemma 2.2 and combining 2.33–2.36, 2.15,
and 2.16, we have
zΔ t kσt, t t
t0
k1Δ t, τΔτ
wσt, t ftup t gtuq t mt
t
w1Δ t, τ fτup τ gτuq τ mτ Δτ
t0
p − q K q/p
qat
≤ wσt, t atft gt
mt
p
pK p−q/p
qgt
bt ft zt
pK p−q/p
t
p − q K q/p
qaτ
Δ
mτ
w1 t, τ aτfτ gτ
p
pK p−q/p
t0
qgτ
bτ fτ zτ Δτ
pK p−q/p
t
qgt
qgτ
Δ
≤ wσt, tbt ft w1 t, τbτ fτ Δτ zt
pK p−q/p
pK p−q/p
t0
p − q K q/p
qat
wσt, t atft gt
mt
p
pK p−q/p
t
p − q K q/p
qaτ
Δ
w1 t, τ aτfτ gτ
mτ Δτ
p
pK p−q/p
t0
Atzt Gt
for all t ∈ Tκ ,
2.37
where At and Gt are defined as in 2.31 and 2.32, respectively. Therefore, using
Lemma 2.3 and noting zt0 0, we get
zt ≤
t
eA t, στGτΔτ
for all t ∈ Tκ .
2.38
t0
It is easy to see that the desired inequality 2.30 follows from 2.15 and 2.38. This
completes the proof of Theorem 2.11.
Remark 2.12. Letting p q 1, ft 0 in Theorem 2.11, we easily obtain 9, Theorem 3.10.
The following two corollaries are easily established by using Theorem 2.11.
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Corollary 2.13. Let T R and assume that ut, at, bt, ft, gt, mt ∈ CR , R . If wt, s
and its partial derivative ∂/∂twt, s are real-valued nonnegative continuous functions for t, s ∈ R
with s ≤ t, then the inequality
up t ≤ at bt
t
wt, s fsup s gsuq s ms ds,
t ∈ R
2.39
for any K > 0, t ∈ R ,
2.40
0
implies
ut ≤
at bt
t
t
Gs exp
0
1/p
Aτdτ ds
s
where
At wt, tbt ft qgt
pK p−q/p
qgs
∂
ds,
wt, sbs fs pK p−q/p
0 ∂t
t
2.41
and also
p − q K q/p
qat
mt
Gt wt, t atft gt
p
pK p−q/p
t
∂
wt, s asfs gs
∂t
0
p−q K
p
q/p
qas
pK p−q/p
2.42
ms ds.
Remark 2.14. Letting p q 1, ft 0 in Corollary 2.13, we easily obtain 14, Theorem 1.4.3.
Corollary 2.15. Let T Z and assume that ut, at, 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 ,
2.43
s0
implies
ut ≤
t−1 t−1
at bt Gs
1 Aτ
s0
τs1
1/p
for any K > 0, t ∈ N0 ,
2.44
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Discrete Dynamics in Nature and Society
where Δ1 wt, s wt 1, s − wt, s for t, s ∈ N0 with s ≤ t,
At wt 1, tbt ft qgt
pK p−q/p
t−1
Δ1 wt, sbs fs s0
qgs
pK p−q/p
,
2.45
p − q K q/p
qat
mt
Gt wt 1, t atft gt
p
pK p−q/p
t−1
p − q K q/p
qas
Δ1 wt, s asfs gs
ms .
p
pK p−q/p
s0
2.46
Remark 2.16. By letting p q 1, ft 0 in Corollary 2.15, it is very easy to obtain 15,
Theorem 1.3.4.
Corollary 2.17. Suppose that ut, at, and wt, s are defined as in Theorem 2.11, and let at be
nondecreasing for all t ∈ Tκ . If, for any ε > 0, there exists a neighborhood Uof t, independent of
τ ∈ t0 , σt, such that for all s ∈ U,
q
u τ wσt, τ − ws, τ − w1Δ t, τσt − s ≤ ε|σt − s|,
2.47
then
u t ≤ at p
t
for all t ∈ Tκ
wt, τuq τΔτ
E 2
t0
implies
ut ≤
1/p
1 K p − q qat e t, t0 − K p − q
A
q
for any K > 0, t ∈ Tκ ,
2.48
where
At q
pK p−q/p
wσt, t t
t0
w1Δ t, τΔτ
.
2.49
Proof. Letting bt 1, ft 0, gt 1, and mt 0 in Theorem 2.11, we obtain
At q
pK p−q/p
wσt, t t
t0
w1Δ t, τΔτ
: At,
2.50
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11
and also
Gt 1
pK p−q/p
wσt, t K p − q qat t
t0
w1Δ t, τ
K p − q qaτ Δτ
t
K p − q qat
Δ
≤
wσt, t w1 t, τΔτ
pK p−q/p
t0
1 K p − q qat At
q
2.51
for any K > 0, t ∈ Tκ ,
where the inequality holds because at is nondecreasing for all t ∈ Tκ . Therefore, using
Theorem 2.11 and noting 2.50 and 2.51, we easily have
ut ≤
at t
1/p
eA t, στGτΔτ
t0
≤
1
at q
≤
t
t0
e t, στ K p − q qaτ AτΔτ
1/p
A
1 at K p − q qat
q
t
t0
e t, στAτΔτ
1/p
2.52
A
1/p
1 at K p − q qat , e t, t0 − e t, t
A
A
q
1/p
1 for any K > 0, t ∈ Tκ .
K p − q qat e t, t0 − K p − q
A
q
The proof of Corollary 2.17 is complete.
Remark 2.18. In Corollary 2.17, letting wt, s ws, p q 1, we immediately obtain 12,
Theorem 3.1.
From the proof procedure of Theorem 2.11, we can obtain the following result.
Theorem 2.19. Assume that u, a, b, f, gi , m ∈ Crd , ut, at, 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. Let wt, s be defined as in Lemma 2.2 such that wt, s ≥ 0 and
w1Δ t, s ≥ 0 for t, s ∈ T with s ≤ t. If, for any ε > 0, there exists a neighborhood U of t, independent
of τ ∈ t0 , σt, such that for all s ∈ U,
n
wσt, τ − ws, τ − w1Δ t, τσt − s fτup τ gi τuqi τ mτ ≤ ε|σt − s|,
i1
2.53
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then
u t ≤ at bt
p
n
qi
wt, τ fτu τ gi τu τ mτ Δτ,
t
p
t0
t ∈ Tκ
E 2
i1
implies
ut ≤
at bt
t
1/p
∗
for any K > 0, t ∈ Tκ ,
eA∗ t, στG τΔτ
2.54
t0
where
∗
A t wσt, tbt ft i1
t
t0
pK p−qi /p
w1Δ t, τbτ fτ n
qi gi τ
i1
n
G t wσt, t atft gi t
i1
t0
w1Δ t, τ aτfτ pK p−qi /p
2.55
Δτ,
∗
t
n
qi gi t
n
p − qi Kqi /p
qi at
mt
p
pK p−qi /p
gi τ
i1
p − qi K qi /p
qi aτ
mτ Δτ.
p
pK p−qi /p
2.56
Remark 2.20. 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. An Application
In this section, we present an application of Corollary 2.6 to obtain the explicit estimates on
the solutions of a dynamic equation on time scales.
Example 3.1. Consider the dynamic equation
up tΔ Ht, ut,
ut0 C, t ∈ Tκ ,
3.1
where p and C are constants, p > 0, and H : Tκ × R → R is a continuous function.
Assume that
|Ht, ut| ≤ gt|uq t|,
3.2
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13
where gt ∈ Crd , gt is nonnegative, and 0 < q ≤ p is a constant. If ut is a solution of 3.1,
then
|ut| ≤
p
|C| t
t0
1/p
for any K > 0, t ∈ Tκ ,
eB t, στJτΔτ
3.3
is defined as in 2.21, and
where Bt
Jt gt
p − q q/p
q|C|p
K p
pK p−q/p
for all t ∈ Tκ .
3.4
In fact, the solution ut of 3.1 satisfies the following equivalent equation:
u t C p
p
t
Hτ, uτΔτ,
t ∈ Tκ .
3.5
t0
Using the assumption 3.2, we have
|ut|p ≤ |C|p t
gτ|uτ|q Δτ,
t ∈ Tκ .
3.6
t0
Now a suitable application of Corollary 2.6 to 3.6 yields 3.3.
Acknowledgments
This work is supported by the National Natural Science Foundation of China 10971018,
10971139, the Natural Science Foundation of Shandong Province Y2009A05, 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.
References
1 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
2 M. Bohner, L. Erbe, and A. Peterson, “Oscillation for nonlinear second order dynamic equations on a
time scale,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 491–507, 2005.
3 Y. Xing, M. Han, and G. Zheng, “Initial value problem for first-order integro-differential equation of
Volterra type on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 3, pp.
429–442, 2005.
4 R. P. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
5 R. P. Agarwal, D. O’Regan, and S. H. Saker, “Oscillation criteria for second-order nonlinear neutral
delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 203–
217, 2004.
14
Discrete Dynamics in Nature and Society
6 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
7 M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston,
Mass, USA, 2003.
8 R. P. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical
Inequalities & Applications, vol. 4, no. 4, pp. 535–557, 2001.
9 E. Akin–Bohner, M. Bohner, and F. Akin, “Pachpatte inequalities on time scales,” Journal of Inequalities
in Pure and Applied Mathematics, vol. 6, no. 1, article 6, 23 pages, 2005.
10 W. N. Li, “Some new dynamic inequalities on time scales,” Journal of Mathematical Analysis and
Applications, vol. 319, no. 2, pp. 802–814, 2006.
11 W. N. Li and W. Sheng, “Some nonlinear dynamic inequalities on time scales,” Proceedings of the Indian
Academy of Sciences. Mathematical Sciences, vol. 117, no. 4, pp. 545–554, 2007.
12 D. B. Pachpatte, “Explicit estimates on integral inequalities with time scale,” Journal of Inequalities in
Pure and Applied Mathematics, vol. 7, no. 4, article 143, 8 pages, 2006.
13 W. N. Li, “Some Pachpatte type inequalities on time scales,” Computers & Mathematics with
Applications, vol. 57, no. 2, pp. 275–282, 2009.
14 B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197 of Mathematics in Science and
Engineering, Academic Press, San Diego, Calif, USA, 1998.
15 B. G. Pachpatte, Inequalities for Finite Difference Equations, vol. 247 of Monographs and Textbooks in Pure
and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2002.