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 inequa
Trang 1Volume 2010, Article ID 464976, 13 pages
doi:10.1155/2010/464976
Research Article
On Some Integral Inequalities on Time Scales and Their Applications
1 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, China
2 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
Copyrightq 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 Sheng8 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 Peterson4 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, N0denotes the set of nonnegative integers,C denotes the set of complex numbers, and CM, S denotes
Trang 2the 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 T k T − {m}; otherwise,
Tk T Crddenotes 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 byR {p ∈
R : 1 μtpt > 0, t ∈ T} Obviously, if p ∈ Crdand pt ≥ 0 for t ∈ T, then p ∈ R
For f : T → R and t ∈ T k , we define fΔt as follows provided it exists:
fΔt : lim
s → t
f σ t − fs
σ t − s ; 2.1
we call fΔt 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, xt0 1 on T. 2.2
Lemma 2.2 see 4 Let t0 ∈ Tk and w : T × Tk → R be continuous at t, t, where t ≥ t0 Assume that wΔt, · is rd-continuous on t0, σ t If for any ε > 0, there exists a neighborhood U of
t, independent of τ ∈ t0, σ t, such that
wσt, τ − ws, τ − wΔt, τσt − s ≤ ε|σt − s|, s ∈ U, 2.3
where wΔdenotes the derivative of w with respect to the first variable, then
g t :
t
t0
w t, τΔτ 2.4
implies
gΔt :
t
t0
wΔt, τΔτ wσt, t. 2.5
The following theorem is a foundational result in dynamic inequalities
Lemma 2.3 Comparison Theorem 4 Suppose u, b ∈ Crd, a∈ R; then
uΔt ≤ atut bt, t ≥ t0, t∈ Tk , 2.6
implies
u t ≤ ut0eat, t0
t
t
b τeat, στΔτ, t ≥ t0, t∈ Tk 2.7
Trang 3The following lemma is useful in our main results.
Lemma 2.4 see 7 Let a ≥ 0, p ≥ q > 0, then
a q/p≤ q
p K
q−p/p ap − q
p K
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
nonnega-tive; then
u p t ≤ at bt
t
t0
f su q s gsu r s
s
t0
h τu m τΔτ
Δs, t ∈ T k , 3.1
implies
u t ≤
a t bt
t
t0
B τeA τ t, στΔτ
1/p
, K > 0, t∈ Tk , 3.2
where
A t
q
p K
q−p/p f t r
p K
r−p/p g t
b t m
p K
m−p/pt
t0
b τhτΔτ,
B t ft
q
p K
q−p/p a t p − q
p K
q/p
gt
r
p K
r−p/p a t p − r
p K
p/r
t
t0
m
p K
m−p/p a τ p − m
m/p
h τΔτ, t ∈ T k
3.3
Proof Define z t by
z t
t
t0
f su q s gsu r s
s
t0
h τu m τΔτ
Δs, 3.4
then zt0 0, and 3.1 can be restated as
u p t ≤ at btzt. 3.5
Trang 4UsingLemma 2.1, for any k > 0, we obtain
u q t ≤ at btzt q/p ≤ q
p K
q−p/p at btzt p − q
p K
q/p ,
u r t ≤ at btzt r/p≤ r
p K
r−p/p at btzt p − r
p K
r/p ,
u m t ≤ at btzt m/p≤ m
p K
m−p/p at btzt p − m
m/p
3.6
It follows from3.4 and 3.6 that
zΔt ftu q t gtu r t
t
t0
h τu m τΔτ
≤ ft
q
p K
q−p/p at btzt p − q
p K
q/p
gt
r
p K
r−p/p at btzt p − r
p K
r/p
t
t0
h τ
m
p K
m−p/p aτ bτzτ p − m
m/p
Δτ
≤ Bt Atzt, t ∈ T k ,
3.7
where At, and Bt are defined as in 3.3 and At is regressive obviously.
z t ≤
t
t0
B τeA τ t, στΔτ. 3.8
Therefore, the desired inequality3.2 follows from 3.5 and 3.8
Remark 3.2. Theorem 3.1extends some known inequalities on time scales If q 1, r 0, ht
0, thenTheorem 3.1reduces to7, Theorem 3.1 If q p, ht 0, thenTheorem 3.1reduces
to8, Theorem 3.2.
Remark 3.3 The result of Theorem 3.1 holds for an arbitrary time scale If T R, then
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 p t ≤ at bt t−1
s0
f su q s gsu r s s−1
τ0
h τu m τ
, t∈ N0, 3.9
Trang 5u t ≤
a t bt t−1
s0
B s t−1
τ s1
1 Aτ
1/p , K > 0, t∈ N0, 3.10
where
A t
q
p K
q−p/p f t r
p K
r−p/p g t
b t m
p K
m−p/p t−1
τ0
b τhτ,
B t ft
q
p K
q−p/p a t p − q
p K
q/p
gt
r
p K
r−p/p a t p − r
p K
p/r
t−1
τ0
m
p K
m−p/p a τ p − m
m/p
h τ, t ∈ N0.
3.11
Corollary 3.5 Let T lZ ∩ 0, ∞, where lZ {lk : k ∈ Z, l > 0} We assume that
u t, at, bt, ft, gt,and ht are nonnegative functions defined for t ∈ T Then the inequality
u p t ≤ at bt t/l−1
s0
f lsu q ls glsu r ls s/l−1
τ0
h lτu m lτ
, t∈ T 3.12
implies
u t ≤
a t bt t/l−1
s0
B ls t/l −1
τ s/l1
1 Aτ
1/p , K > 0, t ∈ T, 3.13
where
A t
q
p K
q−p/p f t r
p K
r−p/p g t
b t m
p K
m−p/p t/l−1
τ0
b lτhlτ,
B t ft
q
p K
q−p/p a t p − q
p K
q/p
gt
r
p K
r−p/p a t p − r
p K
p/r
t/l−1
τ0
m
p K
m−p/p a lτ p − m
m/p
h lτ, t ∈ T.
3.14
Trang 6Theorem 3.6 Assume that u, a, b, f, h are defined as in Theorem 3.1 , L t, y, Mt, y : T k× R →
Rare continuous functions, and L t, 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
u p t ≤ at bt
t
t0
f su q s Ls, u r s
s
t0
h τu m τΔτ
Δs, t ∈ T k 3.16
implies
u t ≤
a t bt
t
t0
B1τeA1t, στΔτ
1/p
, K > 0, t∈ Tk , 3.17
where
A1t q
p K
q−p/p f tbt m
p K
m−p/pt
t0
h τbτΔτ
r
p K
r−p/p M
t, r
p K
r−p/p a t p − r
p K
r/p
b t,
B1t ft
q
p K
q−p/p a t p − q
p K
q/p
t
t0
h τ
m
p K
m−p/p a τ p − m
m/p
Δτ
L
t, r
p K
r−p/p a t p − r
p K
r/p
, t∈ Tk
3.18
Proof Define z t by
z t
t
t0
f su q s Ls, u r s
s
t0
h τu m τΔτ
Δs, 3.19
then zt0 0, and 3.16 can be written as 3.5
Trang 7Therefore, from3.6 and 3.19, we have
zΔt ftu q t Lt, u r t
t
t0
h τu m τΔτ
≤ ftq
p K
q−p/p at btzt p − q
p K
q/p
L
t, r
p K
r−p/p at btzt p − r
p K
r/p
− L
t, r
p K
r−p/p a t p − r
p K
r/p
t
t0
h τ
m
p K
m−p/p aτ bτzτ p − m
m/p
Δτ
L
t, r
p K
r−p/p a t p − r
p K
r/p
≤ ftq
p K
q−p/p a t p − q
p K
q/p
t
t0
h τ
m
p K
m−p/p a τ p − m
m/p
Δτ
q
p K
q−p/p f tbt m
p K
m−p/pt
t0
h τbτΔτ
z t
M
t, r
p K
r−p/p a t p − r
p K
r/p
r
p K
r−p/p b tzt
L
t, r
p K
r−p/p a t p − r
p K
r/p
A1tzt B1t, t ∈ T k ,
3.20
where A1t, and B1t are defined as in 3.18 and A1t is regressive obviously.
z t ≤
t
t0
B1τeA1τ t, στΔτ. 3.21
Therefore, the desired inequality3.17 follows from 3.5 and 3.21
Trang 8Remark 3.7 IfT R, thenTheorem 3.6becomes13, 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 p t ≤ at bt t−1
s0
f su q s Ls, u r s s−1
τ0
h τu m τ
, t∈ N0 3.23
implies
u t ≤
a t bt t−1
s0
B1τ t−1
τ s1
1 A1τ
1/p , K > 0, t∈ N0, 3.24
where
A1t q
p K
q−p/p f tbt m
p K
m−p/p t−1
τ0
h τbτ
r
p K
r−p/p M
t, r
p K
r−p/p a t p − r
p K
r/p
b t,
B1t ft
q
p K
q−p/p a t q − p
p K
q/p
t−1
τ0
h τ
m
p K
m−p/p a τ p − m
m/p
L
t, r
p K
r−p/p a t p − r
p K
r/p
, t∈ N0.
3.25
Theorem 3.9 Assume that ut, at, bt, ft, gt, and ht are defined as in Theorem 3.1 , w t, s
is defined as in Lemma 2.2 such that w σt, t ≥ 0, wΔt, s ≥ 0 for t, s ∈ T with s ≤ t; then
u p t ≤ at bt
t
t0
w t, s
f su q s gsu r s
s
t0
h τu m τΔτ
Δs, t ∈ T k ,
3.26
Trang 9u t ≤
a t bt
t
t0
B2τeA2t, στΔτ
1/p
, K > 0, t∈ Tk , 3.27
where
A2t wσt, t
q
p K
q−p/p f t r
p K
r−p/p g t
b t m
p K
m−p/pt
t0
b τhτΔτ
t
t0
wΔt, s
q
p K
q−p/p f s r
p K
r−p/p g s
b s
m
p K
m−p/ps
t0
b τhτΔτ
Δs,
B2t wσt, t
f t
q
p K
q−p/p a t p − q
p K
q/p
gt
r
p K
r−p/p a t p − r
p K
p/r
t
t0
h τ
m
p K
m−p/p a τ p − m
m/p
Δτ
t
t0
wΔt, s
f s
q
p K
q−p/p a s p − q
p K
q/p
gs
r
p K
r−p/p a s p − r
p K
p/r
s
t0
h τ
m
p K
m−p/p a τ p − m
m/p
Δτ
Δs, t ∈ T k
3.28
Proof Define z t by
z t
t
t0
w t, s
f su q s gsu r s
s
t0
h τu m τΔτ
Δs, 3.29
then zt0 0, and 3.26 can be written as 3.5
Trang 10Therefore, from3.6 and 3.29 we have
zΔt wσt, t
f tu q t gtu r t
t
t0
h τu m τΔτ
t
t0
wΔt, s
f su q s gsu r s
s
t0
h τu m τΔτ
Δs
≤ wσt, t
f t
q
p K
q−p/p at btzt p − q
p K
q/p
gt
r
p K
r−p/p at btzt p − r
p K
r/p
t
t0
h τ
m
p K
m−p/p aτ bτzτ p − m
m/p
Δτ
t
t0
wΔt, s
f s
q
p K
q−p/p as bszs p − q
p K
q/p
gs
r
p K
r−p/p as bszs p − r
p K
r/p
s
t0
h τ
m
p K
m−p/p aτ bτzτ p − m
m/p
Δτ
Δs
≤ B2t A2tzt, t∈ Tk ,
3.30
where A2t, and B2t are defined by 3.28 and A2t is regressive obviously.
z t ≤
t
t0
B2τeA2τ t, στΔτ. 3.31 Therefore, the desired inequality3.27 follows from 3.5 and 3.31
Remark 3.10 If q p, ht 0, thenTheorem 3.9reduces to8, 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 ofTheorem 3.9to investigate certain properties
of solution ut of the following dynamic equation:
u p tΔ F
t, U t, ut,
t
t
H s, usΔs
, u p t0 C, t ∈ T k , 4.1
Trang 11where C is a constant, F :Tk × R × R → R is a continuous function, and U : T k × 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 , t∈ Tk ,
4.2
where p, q, r, and m are constants, p ≥ q > 0, and p ≥ m > 0, p ≥ r > 0 f, g, h ∈ Crd, f t, gt and ht are nonnegative Then every solution ut of 4.1 satisfies
|ut| ≤
|C|
t
t0
B τeA τ t, στΔτ
1/p
, K > 0, t∈ Tk , 4.3
where A, B are defined as in3.3 with at |C|, bt 1.
Indeed, the solution ut of 4.1 satisfies the following equivalent equation
u p t C
t
t0
F
τ, U τ, uτ,
τ
t0
H s, usΔs
Δτ, t ∈ T k 4.4
It follows from4.2 and 4.4 that
|u p t| ≤ |C|
t
t0
F
τ, U τ, uτ,
τ
t0
H s, usΔs
Δτ
≤ |C|
t
t0
f τ|uτ| q gτ|uτ| r
τ
t0
h s|us| m Δs
Δτ.
4.5
UsingTheorem 3.1, the inequality4.3 is obtained from 4.5
Example 4.2 Assume that
|Ft, U1, V1 − Ft, U2, V2| ≤ |U1− U2| |V1− V2|,
|Ut, u1 − Ut, u2| ≤ ftu p
1− u p
2,
|Ht, u1 − Ht, u2| ≤ htu p
1− u p
2, t ∈ T k ,
4.6
p, f, and h are defined as inExample 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 u1t, and u2t of 4.1 have the relation
u p1t u p
2t.
Trang 12Proof Let u1t, and u2t be two solutions of 4.1 Then we have
u p1t − u p
2t
t
t0
F
τ, U τ, u1τ,
τ
t0
H s, u1sΔs
−F
τ, U τ, u2τ,
τ
t0
H s, u2sΔs
Δτ, t ∈ T k
4.7
It follows from4.6 and 4.7 that
u p1t − u p2t ≤t
t0
f τu p
1τ − u p2τ τ
t0
h su p
1s − u p2sΔsΔτ, t ∈ T k 4.8
ByTheorem 3.1, we have u p1t − u p
2t ≡ 0, t ∈ T k The results are obtained.
Example 4.3 Consider the equation
u p at bt
t
t0
F
t, s, U s, u,
s
t0
H τ, uΔτ
Δs, t ∈ T k 4.9
If
|Ft, s, U, V | ≤ wt, s|U| |V |,
|Ut, u| ≤ ft|u| q gt|u| r ,
|Ht, u| ≤ ht|u| m , t∈ Tk ,
4.10
where p, q, r, m are constants, p ≥ q > 0, p ≥ m > 0, p ≥ r > 0 a, b, f, g, h ∈
Crd, a t, bt, ft, gt and ht are nonnegative, wt, s is defined as inLemma 2.2such that
w σt, t ≥ 0, wΔt, s ≥ 0 for t, s ∈ T with s ≤ t.
Then we have the estimate of the solution ut of 4.9 that
|ut| ≤
a t bt
t
t0
B2τeA2t, στΔτ
1/p
, K > 0, t∈ Tk , 4.11
where A2, B2are defined as in3.28
Proof From4.10 and 4.9, we have
|ut| p ≤ at bt
t
t0
w t, s
f s|us| q gs|us| r
s
t0
h τ|uτ| m Δτ
Δs, t ∈ T k
4.12
... dynamic inequalities for some peculiar time scales; here, we omit them4 Some Applications
In this section, we present some applications ofTheorem 3.9to investigate certain...
where p, q, r, and m are constants, p ≥ q > 0, and p ≥ m > 0, p ≥ r > f, g, h ∈ Crd, f t, gt and ht are nonnegative Then every solution ut of 4.1... f, and h are defined as inExample 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 u1t, and