Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 464976, 13 pages
doi:10.1155/2010/464976
Research Article
On Some Integral Inequalities on Time Scales and
Their Applications
Run Xu,1 Fanwei Meng,1 and Cuihua Song2
1
2
School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, China
School of Chemistry and Chemical Engineering, Qufu Normal University, Qufu, 273165 Shandong, China
Correspondence should be addressed to Run Xu, xurun 2005@163.com
Received 15 January 2010; Revised 3 March 2010; Accepted 18 March 2010
Academic Editor: Martin Bohner
Copyright q 2010 Run Xu 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.
The purpose of this paper is to investigate some new dynamic inequalities on time scales.
We establish some new dynamic inequalities; the results unify and extend some continuous
inequalities and their corresponding discrete analogues. The inequalities given here can be used as
tools in the qualitative theory of certain dynamic equations. Some examples are given in the end
of this paper.
1. Introduction
The theory of time scales was introduced by Hilger 1 in 1988 in order to contain both
difference and differential calculus in a consistent way. Recently, many authors have extended
some fundamental integral inequalities used in the theory of differential and integral
equations on time scales. For example, we refer the reader to the papers 2–12 and the
references cited there in.
In this paper, we investigate some nonlinear integral inequalities on time scales,
which extend some inequalities established by Li and Sheng 8 and Li 9. The obtained
inequalities can be used as important tools in the study of dynamic equations on time scales.
Throughout this paper, let us assume that we have already acquired the knowledge
of time scales and time scales notation; for an excellent introduction to the calculus on time
scales, we refer the reader to Bohner and Peterson 4 for general overview.
2. Some Preliminaries on Time Scales
In what follows, R denotes the set of real numbers, Z denotes the set of integers, N0 denotes
the set of nonnegative integers, C denotes the set of complex numbers, and CM, S denotes
2
Journal of Inequalities and Applications
the class of all continuous functions defined on set M with range in the set S. T is an arbitrary
time scale. If T has a right-scattered maximum m, then the set Tk T − {m}; otherwise,
Tk T. Crd denotes the set of rd-continuous functions; R denotes the set of all regressive and
rd-continuous functions. We define the set of all positively regressive functions by R {p ∈
R : 1 μtpt > 0, t ∈ T}. Obviously, if p ∈ Crd and pt ≥ 0 for t ∈ T, then p ∈ R .
For f : T → R and t ∈ Tk , we define f Δ t as follows provided it exists:
f σ t − fs
;
s→t
σt − s
f Δ t : lim
2.1
we call f Δ t the delta derivative of f at t.
The following lemmas are very useful in our main results.
Lemma 2.1 see 4. If p ∈ R and fix t0 ∈ T, then the exponential function ep ·, t0 is for the unique
solution of the initial value problem
xΔ ptx,
2.2
xt0 1 on T.
Lemma 2.2 see 4. Let t0 ∈ Tk and w : T × Tk → R be continuous at t, t, where 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,
2.3
where wΔ denotes the derivative of w with respect to the first variable, then
t
gt :
2.4
wt, τΔτ
t0
implies
Δ
g t :
t
wΔ t, τΔτ wσt, t.
2.5
t0
The following theorem is a foundational result in dynamic inequalities.
Lemma 2.3 Comparison Theorem 4. Suppose u, b ∈ Crd , a ∈ R ; then
uΔ t ≤ atut bt,
t ≥ t0 , t ∈ Tk ,
2.6
implies
ut ≤ ut0 ea t, t0 t
t0
bτea t, στΔτ,
t ≥ t0 , t ∈ Tk .
2.7
Journal of Inequalities and Applications
3
The following lemma is useful in our main results.
Lemma 2.4 see 7. Let a ≥ 0, p ≥ q > 0, then
aq/p ≤
q q−p/p
p − q q/p
K
K ,
a
p
p
2.8
K > 0.
3. Main Results
In this section, we study some integral inequalities on time scales. We always assume that
p, q, r, m are constants, p ≥ q > 0, p ≥ m > 0, p ≥ r > 0, and t ≥ t0 , t ∈ Tk .
Theorem 3.1. Assume that u, a, b, f, g, h ∈ Crd ; ut, at, bt, ft, gt, and ht are nonnegative; then
u t ≤ at bt
p
t fsu s gsu s q
r
t0
s
hτu τΔτ Δs,
m
t ∈ Tk ,
3.1
t0
implies
ut ≤
at bt
1/p
t
BτeAτ t, στΔτ
,
K > 0, t ∈ Tk ,
3.2
t0
where
t
q q−p/p
r
m
K
ft K r−p/p gt bt K m−p/p
bτhτΔτ,
p
p
p
t0
p − q q/p
p − r p/r
q q−p/p
r r−p/p
K
K
at at gt K
Bt ft K
p
p
p
p
t p − m m/p
m m−p/p
K
K
aτ hτΔτ, t ∈ Tk .
p
t0 p
At 3.3
Proof. Define zt by
s
t q
r
m
fsu s gsu s hτu τΔτ Δs,
zt t0
3.4
t0
then zt0 0, and 3.1 can be restated as
up t ≤ at btzt.
3.5
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Journal of Inequalities and Applications
Using Lemma 2.1, for any k > 0, we obtain
uq t ≤ at btztq/p ≤
q q−p/p
p − q q/p
K
K ,
at btzt p
p
ur t ≤ at btztr/p ≤
p − r r/p
r r−p/p
K
K ,
at btzt p
p
um t ≤ at btztm/p ≤
3.6
p − m m/p
m m−p/p
K
K .
at btzt p
p
It follows from 3.4 and 3.6 that
zΔ t ftuq t gtur t t
hτum τΔτ
t0
p − q q/p
q q−p/p
K
≤ ft K
at btzt p
p
p − r r/p
r
K
gt K r−p/p at btzt p
p
t
p − m m/p
m m−p/p
K
K
hτ
Δτ
aτ bτzτ p
p
t0
≤ Bt Atzt,
3.7
t ∈ Tk ,
where At, and Bt are defined as in 3.3 and At is regressive obviously.
From Lemma 2.3 and 3.7, noting zt0 0, we obtain
zt ≤
t
3.8
BτeAτ t, στΔτ.
t0
Therefore, the desired inequality 3.2 follows from 3.5 and 3.8.
Remark 3.2. Theorem 3.1 extends some known inequalities on time scales. If q 1, r 0, ht 0, then Theorem 3.1 reduces to 7, Theorem 3.1. If q p, ht 0, then Theorem 3.1 reduces
to 8, Theorem 3.2.
Remark 3.3. The result of Theorem 3.1 holds for an arbitrary time scale. If T R, then
Theorem 3.1 becomes the Theorem 1 established by Yuan et al. 13. If T Z, we can have the
following Corollary.
Corollary 3.4. Let T Z and assume that ut, at, bt, ft, gt, and ht are nonnegative
functions defined for t ∈ N0 . Then the inequality
u t ≤ at bt
p
t−1
s0
s−1
m
fsu s gsu s hτu τ ,
q
r
τ0
t ∈ N0 ,
3.9
Journal of Inequalities and Applications
5
implies
t−1
t−1
1 Aτ
at bt Bs
ut ≤
s0
1/p
K > 0, t ∈ N0 ,
,
3.10
τs1
where
At t−1
q q−p/p
r
m
K
ft K r−p/p gt bt K m−p/p bτhτ,
p
p
p
τ0
q q−p/p
p − q q/p
p − r p/r
r r−p/p
K
K
Bt ft K
at at gt K
p
p
p
p
t−1 m
τ0
p
K m−p/p aτ p − m m/p
K
hτ,
p
3.11
t ∈ N0 .
Corollary 3.5. Let T lZ ∩ 0, ∞, where lZ {lk : k ∈ Z, l > 0}. We assume that
ut, at, bt, ft, gt,and ht are nonnegative functions defined for t ∈ T. Then the inequality
u t ≤ at bt
p
t/l−1
flsu ls glsu ls q
r
s0
s/l−1
hlτu lτ ,
m
t∈T
3.12
τ0
implies
ut ≤
at bt
1/p
t/l−1
t/l−1
s0
τs/l1
Bls
1 Aτ
,
K > 0, t ∈ T,
3.13
where
At t/l−1
q q−p/p
r
m
K
ft K r−p/p gt bt K m−p/p
blτhlτ,
p
p
p
τ0
q
p − q q/p
p − r p/r
r
K
K
Bt ft K q−p/p at gt K r−p/p at p
p
p
p
t/l−1
τ0
p − m m/p
m m−p/p
K
K
alτ hlτ,
p
p
t ∈ T.
3.14
6
Journal of Inequalities and Applications
Theorem 3.6. Assume that u, a, b, f, h are defined as in Theorem 3.1, Lt, y, Mt, y : Tk × R →
R are continuous functions, and Lt, y is nondecreasing about the second variable and satisfies
0 ≤ Lt, x − L t, y ≤ M t, y x − y
3.15
for t ∈ Tk and x ≥ y ≥ 0; then
t s
q
r
m
u t ≤ at bt
fsu s Ls, u s hτu τΔτ Δs,
p
t0
t ∈ Tk
3.16
t0
implies
ut ≤
at bt
t
1/p
B1 τeA1 t, στΔτ
,
K > 0, t ∈ Tk ,
3.17
t0
where
A1 t q q−p/p
m
K
ftbt K m−p/p
p
p
t
hτbτΔτ
t0
p − r r/p
r r−p/p
r r−p/p
K
M t, K
at bt,
K
p
p
p
B1 t ft
p − q q/p
q q−p/p
K
K
at p
p
t
hτ
t0
3.18
p − m m/p
m m−p/p
K
K
aτ Δτ
p
p
p − r r/p
r r−p/p
K
at ,
L t, K
p
p
t ∈ Tk .
Proof. Define zt by
t s
q
r
m
zt fsu s Ls, u s hτu τΔτ Δs,
t0
then zt0 0, and 3.16 can be written as 3.5.
t0
3.19
Journal of Inequalities and Applications
7
Therefore, from 3.6 and 3.19, we have
zΔ t ftuq t Lt, ur t t
hτum τΔτ
t0
q
p − q q/p
K
≤ ft K q−p/p at btzt p
p
p − r r/p
r
K
L t, K r−p/p at btzt p
p
p − r r/p
r
K
− L t, K r−p/p at p
p
t
t0
p − m m/p
m m−p/p
K
K
Δτ
aτ bτzτ p
p
hτ
p − r r/p
r
L t, K r−p/p at K
p
p
≤ ft
t
p − q q/p
p − m m/p
q q−p/p
m m−p/p
K
K
K
K
at hτ
aτ Δτ
p
p
p
p
t0
q q−p/p
m
K
ftbt K m−p/p
p
p
t
hτbτΔτ zt
t0
p − r r/p r r−p/p
r
M t, K r−p/p at K
K
btzt
p
p
p
p − r r/p
r
L t, K r−p/p at K
A1 tzt B1 t,
p
p
t ∈ Tk ,
3.20
where A1 t, and B1 t are defined as in 3.18 and A1 t is regressive obviously.
From Lemma 2.3 and 3.20, noting zt0 0, we obtain
zt ≤
t
B1 τeA1 τ t, στΔτ.
t0
Therefore, the desired inequality 3.17 follows from 3.5 and 3.21.
3.21
8
Journal of Inequalities and Applications
Remark 3.7. If T R, then Theorem 3.6 becomes 13, Theorem 3. If T Z, we can have the
following Corollary.
Corollary 3.8. Let T Z and assume that ut, at, bt, ft, gt, and ht are nonnegative
functions defined for t ∈ N0 . L, M ∈ CR2 , R satisfy
0 ≤ Lt, x − L t, y ≤ M t, y x − y
3.22
for x ≥ y ≥ 0 and Lt, y is nondecreasing about the second variable. Then the inequality
u t ≤ at bt
p
t−1
s−1
m
fsu s Ls, u s hτu τ ,
q
t ∈ N0
3.23
K > 0, t ∈ N0 ,
3.24
r
s0
τ0
implies
ut ≤
t−1
t−1
at bt B1 τ
1 A1 τ
s0
1/p
,
τs1
where
A1 t t−1
q q−p/p
m
K
ftbt K m−p/p hτbτ
p
p
τ0
p − r r/p
r r−p/p
r
K
K
M t, K r−p/p at bt,
p
p
p
q − p q/p
q q−p/p
K
K
B1 t ft
at p
p
t−1
p − m m/p
m m−p/p
K
K
aτ hτ
p
p
τ0
p − r r/p
r r−p/p
K
at ,
L t, K
p
p
t ∈ N0 .
3.25
Theorem 3.9. Assume that ut, at, bt, ft, gt, and ht are defined as in Theorem 3.1, wt, s
is defined as in Lemma 2.2 such that wσt, t ≥ 0, wΔ t, s ≥ 0 for t, s ∈ T with s ≤ t; then
u t ≤ at bt
p
t
t0
wt, s fsu s gsu s q
r
s
hτu τΔτ Δs,
m
t ∈ Tk ,
t0
3.26
Journal of Inequalities and Applications
9
implies
ut ≤
at bt
t
1/p
B2 τeA2 t, στΔτ
,
K > 0, t ∈ Tk ,
3.27
t0
where
A2 t wσt, t
t
q q−p/p
r r−p/p
m m−p/p t
K
ft K
gt bt K
bτhτΔτ
p
p
p
t0
wΔ t, s
t0
q q−p/p
r
K
fs K r−p/p gs bs
p
p
m
K m−p/p
p
s
bτhτΔτ Δs,
t0
p − q q/p
p − r p/r
q q−p/p
r r−p/p
K
K
K
K
B2 t wσt, t ft
at at gt
p
p
p
p
p − m m/p
m m−p/p
K
K
hτ
aτ Δτ
p
p
t0
t
t
p − q q/p
q q−p/p
K
K
wΔ t, s fs
as p
p
t0
p − r p/r
r r−p/p
K
K
gs
as p
p
p − m m/p
m m−p/p
K
K
hτ
aτ Δτ Δs,
p
p
t0
s
t ∈ Tk .
3.28
Proof. Define zt by
zt t
wt, s fsu s gsu s q
t0
then zt0 0, and 3.26 can be written as 3.5.
r
s
t0
hτu τΔτ Δs,
m
3.29
10
Journal of Inequalities and Applications
Therefore, from 3.6 and 3.29 we have
Δ
z t wσt, t ftu t gtu t q
r
t
hτu τΔτ
m
t0
t
Δ
w t, s fsu s gsu s q
r
t0
s
hτu τΔτ Δs
m
t0
p − q q/p
q q−p/p
K
K
≤ wσt, t ft
at btzt p
p
p − r r/p
r r−p/p
K
K
gt
at btzt p
p
t
p − m m/p
m m−p/p
K
K
hτ
Δτ
aτ bτzτ p
p
t0
3.30
t
q q−p/p
p − q q/p
K
K
wΔ t, s fs
as bszs p
p
t0
p − r r/p
r r−p/p
gs
K
K
as bszs p
p
s
p − m m/p
m m−p/p
K
K
hτ
Δτ Δs
aτ bτzτ p
p
t0
≤ B2 t A2 tzt,
t ∈ Tk ,
where A2 t, and B2 t are defined by 3.28 and A2 t is regressive obviously.
From Lemma 2.3 and 3.30, noting zt0 0, we obtain
zt ≤
t
3.31
B2 τeA2 τ t, στΔτ.
t0
Therefore, the desired inequality 3.27 follows from 3.5 and 3.31.
Remark 3.10. If q p, ht 0, then Theorem 3.9 reduces to 8, Theorem 3.8.
Using our results, we can also obtain many dynamic inequalities for some peculiar
time scales; here, we omit them.
4. Some Applications
In this section, we present some applications of Theorem 3.9 to investigate certain properties
of solution ut of the following dynamic equation:
Δ
u t F t, Ut, ut,
p
t
Hs, usΔs ,
t0
up t0 C,
t ∈ Tk ,
4.1
Journal of Inequalities and Applications
11
where C is a constant, F : Tk × R × R → R is a continuous function, and U : Tk × R → R, H :
Tk × R → R are also continuous functions.
Example 4.1. Assume that
|Ft, U, V | ≤ |U| |V |,
|Ut, u| ≤ ft|u|q gt|u|r ,
|Ht, u| ≤ ht|u|m ,
4.2
t ∈ Tk ,
where p, q, r, and m are constants, p ≥ q > 0, and p ≥ m > 0, p ≥ r > 0. f, g, h ∈ Crd , ft, gt
and ht are nonnegative. Then every solution ut of 4.1 satisfies
|ut| ≤
|C| t
1/p
BτeAτ t, στΔτ
,
K > 0, t ∈ Tk ,
4.3
t0
where A, B are defined as in 3.3 with at |C|, bt 1.
Indeed, the solution ut of 4.1 satisfies the following equivalent equation
u t C p
t
τ
F τ, Uτ, uτ,
t0
Hs, usΔs Δτ,
t ∈ Tk .
4.4
t0
It follows from 4.2 and 4.4 that
τ
t Hs, usΔs Δτ
|u t| ≤ |C| F τ, Uτ, uτ,
t0 t0
p
t τ
q
r
m
≤ |C| fτ|uτ| gτ|uτ| hs|us| Δs Δτ.
t0
4.5
t0
Using Theorem 3.1, the inequality 4.3 is obtained from 4.5.
Example 4.2. Assume that
|Ft, U1 , V1 − Ft, U2 , V2 | ≤ |U1 − U2 | |V1 − V2 |,
p
p
|Ut, u1 − Ut, u2 | ≤ ftu1 − u2 ,
p
p
|Ht, u1 − Ht, u2 | ≤ htu1 − u2 , t ∈ Tk ,
4.6
p, f, and h are defined as in Example 4.1. If p m/n m, n ∈ N and m is odd, then 4.1 has
at most one solution; otherwise, the two solutions u1 t, and u2 t of 4.1 have the relation
p
p
u1 t u2 t.
12
Journal of Inequalities and Applications
Proof. Let u1 t, and u2 t be two solutions of 4.1. Then we have
p
u1 t
−
p
u2 t
t τ
F τ, Uτ, u1 τ,
Hs, u1 sΔs
t0
t0
−F τ, Uτ, u2 τ,
τ
4.7
Hs, u2 sΔs
Δτ,
t ∈ Tk .
t0
It follows from 4.6 and 4.7 that
t
τ
p
p
p
p
p
p
fτu1 τ − u2 τ hsu1 s − u2 sΔs Δτ,
u1 t − u2 t ≤
t0
t ∈ Tk .
4.8
t0
p
p
By Theorem 3.1, we have u1 t − u2 t ≡ 0, t ∈ Tk . The results are obtained.
Example 4.3. Consider the equation
u at bt
p
t
s
F t, s, Us, u,
t0
Hτ, uΔτ Δs,
t ∈ Tk .
4.9
t0
If
|Ft, s, U, V | ≤ wt, s|U| |V |,
|Ut, u| ≤ ft|u|q gt|u|r ,
|Ht, u| ≤ ht|u|m ,
4.10
t ∈ Tk ,
where p, q, r, m are constants, p ≥ q > 0, p ≥ m > 0, p ≥ r > 0. a, b, f, g, h ∈
Crd , at, bt, ft, gt and ht are nonnegative, wt, s is defined as in Lemma 2.2 such that
wσt, t ≥ 0, wΔ t, s ≥ 0 for t, s ∈ T with s ≤ t.
Then we have the estimate of the solution ut of 4.9 that
|ut| ≤
at bt
t
1/p
B2 τeA2 t, στΔτ
,
K > 0, t ∈ Tk ,
4.11
t0
where A2 , B2 are defined as in 3.28.
Proof. From 4.10 and 4.9, we have
p
|ut| ≤ at bt
t
q
r
wt, s fs|us| gs|us| t0
s
m
hτ|uτ| Δτ Δs,
t ∈ Tk .
t0
4.12
By Theorem 3.9 and 4.12, we have that 4.11 holds.
Journal of Inequalities and Applications
13
Acknowledgments
The authors thank the referees very much for their careful comments and valuable
suggestions on this paper. This research is supported by the National Natural Science
Foundation of China 10771118, the Natural Science Foundation of Shandong Province
ZR2009AM011, and the Science Foundation of the Education Department of Shandong
Province, China J07yh05.
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