Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 157301, 22 pages
doi:10.1155/2012/157301
Research Article
Some Opial Dynamic Inequalities Involving Higher
Order Derivatives on Time Scales
Samir H. Saker
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Correspondence should be addressed to Samir H. Saker, shsaker@mans.edu.eg
Received 25 April 2012; Accepted 16 August 2012
Academic Editor: Eric R. Kaufmann
Copyright q 2012 Samir H. Saker. 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.
We will prove some new Opial dynamic inequalities involving higher order derivatives on time
scales. The results will be proved by making use of Hölder’s inequality, a simple consequence of
Keller’s chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities
will be derived from our results as special cases.
1. Introduction
In the past decade a number of Opial dynamic inequalities have been established by some
authors which are motivated by some applications; we refer to the papers 1–3. The general
idea is to prove a result for a dynamic inequality where the domain of the unknown function
is a so-called time scale T, which may be an arbitrary closed subset of the real numbers R,
to avoid proving results twice, once on a continuous time scale which leads to a differential
inequality and once again on a discrete time scale which leads to a difference inequality. The
three most popular examples of calculus on time scales are differential calculus, difference
calculus, and quantum calculus see 4, that is, when T R, T N and T qN0 {qt :
t ∈ N0 } where q > 1. A cover story article in New Scientist 5 discusses several possible
applications of time scales. In this paper, we will assume that sup T ∞ and define the time
scale interval a, bT by a, bT : a, b ∩ T. Since the continuous and discrete inequalities
involving higher order derivatives are important in the analysis of qualitative properties of
solutions of differential and difference equations 6–8, we also believe that the dynamic
inequalities involving higher order derivatives on time scales will play the same effective
act in the analysis of qualitative properties of solutions of dynamic equations 2, 3, 9. To the
best of the author’s knowledge there are few inequalities involving higher order derivatives
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Discrete Dynamics in Nature and Society
established in the literature 10–13. In the following, we recall some of these results that
serve and motivate the contents of this paper.
In 13 the authors proved that if y : a, bT → R is delta differentiable n times with
yΔi a 0, for i 0, 1, . . . , n − 1, and h is a positive rd-continuous function on a, bT , then
b
b
q
pq
p Δ
q
np
n
ht yt y t Δt ≤
htyΔn t Δt.
b − a
p
q
a
a
1.1
In 10 it is proved that if y : a, bT → R is delta differentiable n times n odd with yΔi a 0, for i 0, 1, . . . , n − 1, then
b
a
⎛ q/p ⎞ b
b t
q
p
ytΔt ≤ ⎝
hn−1 t, σsΔs
Δt⎠ yΔn t Δt,
a
a
1.2
a
where p, q > 1 and satisfy 1/p 1/q 1. Also in 10 it is proved that if y : a, bT → R is
delta differentiable n times with yΔi a 0, for i 0, 1, . . . , n − 1, and |yΔn t| is increasing,
then
b
a
⎛ 1/q
1/p ⎞ b
b t
2q
Δ
p
1/q
Δ
yty n tΔt ≤ b − a ⎝
hn−1 t, σsΔs
Δt⎠
,
y n t Δt
a
a
a
1.3
where p, q > 1 and satisfy 1/p 1/q 1. As a generalization of 1.3 it is proved in 10 that if
y : a, bT → R is delta differentiable n times with yΔmi a 0, for i 0, 1, . . . , n − m − 1, and
|yΔn t| is increasing, then
b
Δm Δn y ty tΔt
a
⎛
≤ b − a1/q ⎝
b t
a
a
1/p
p
hn−m−1 t, σsΔs
⎞
1/q
b Δn 2q
⎠
Δt
,
y t Δt
1.4
a
where p, q > 1 and satisfy 1/p 1/q 1. In 12 the authors proved that if r and s are positive
rd-continuous functions on a, bT such that s is nonincreasing, and y : a, bT → R is delta
differentiable n times with yΔi a 0, for i 0, 1, . . . , n − 1, then
b
p sxyt yΔn tΔt
a
1
≤
b − an−1
p1
b
a
1p/γ b
r 1−γ tΔt
1.5
1p/ν
ν
rtstp/p1 yΔn t Δt
,
a
where p > 0 and 1/γ 1/ν 1. For contributions of different types of dynamic inequalities
on time scales, we refer the reader to the papers 1, 2, 14–17 and the references cited therein.
Discrete Dynamics in Nature and Society
3
Following this trend, to develop the qualitative theory of dynamic inequalities on time
scales, we will prove some new inequalities of Opial’s type involving higher order derivatives
by making use of the Hölder inequality see, 18, Theorem 6.13:
h
ftgtΔt ≤
h
a
1/γ h
ftγ Δt
a
1/ν
gtν Δt
,
1.6
a
where a, h ∈ T and f; g ∈ Crd I, R, γ > 1 and 1/ν 1/γ 1, the formula
Δ
x t γ
γ
1
hxσ 1 − hxγ−1 dhxΔ t,
1.7
0
which is a simple consequence of Keller’s chain rule 18, Theorem 1.90, and the Taylor
monomials on time scales. The results in this paper extend and improve the pervious results
in the sense that our results contain two different weighted functions and do not require
the monotonicity condition on |yΔn t| the results in 10 required that |yΔn t| should be
increasing. Some results on continuous and discrete spaces, which lead to differential and
difference inequalities, will be derived from our results as special cases. This paper is a
continuation of the papers 3, 10–13, 16.
2. Main Results
In this section, we will prove the main results. For completeness, we recall the following
concepts related to the notion of time scales. A time scale T is an arbitrary nonempty closed
subset of the real numbers R. We assume throughout that T has the topology that it inherits
from the standard topology on the real numbers R. The forward jump operator and the
backward jump operator are defined by
σt : inf{s ∈ T : s > t},
ρt : sup{s ∈ T : s < t},
2.1
where sup ∅ inf T. A point t ∈ T is said to be left-dense if ρt t and t > inf T, is rightdense if σt t, is left-scattered if ρt < t and right-scattered if σt > t. The three most
popular examples of calculus on time scales are differential calculus, difference calculus, and
quantum calculus see 4, that is, when T R, T N and T qN0 {qt : t ∈ N0 } where
q > 1. For more details of time scale analysis we refer the reader to the two books by Bohner
and Peterson 18, 19 which summarize and organize much of the time scale calculus.
A function g : T → R is said to be right-dense continuous rd-continuous provided
g is continuous at right-dense points and at left-dense points in T; left-hand limits exist and
are finite. The set of all such rd-continuous functions is denoted by Crd T.
The graininess function μ for a time scale T is defined by μt : σt − t, and for any
function f : T → R the notation f σ t denotes fσt. We will assume that sup T ∞ and
define the time scale interval a, bT by a, bT : a, b ∩ T. Fix t ∈ T and let y : T → R.
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Discrete Dynamics in Nature and Society
Define yΔ t to be the number if it exists with the property that given any > 0 there is a
neighborhood U of t with
yσt − ys − yΔ tσt − s ≤ |σt − s|,
∀s ∈ U.
2.2
In this case, we say yΔ t is the delta derivative of y at t and that y is delta differentiable
at t. We will frequently use the following results which are due to Hilger 20. Assume that
y : T → R and let t ∈ T.
i If y is differentiable at t, then y is continuous at t.
ii If y is continuous at t and t is right-scattered, then y is differentiable at t with
yσt − yt
.
μt
yΔ t 2.3
iii If y is differentiable and t is right-dense, then
yΔ t lim
s→t
yt − ys
.
t−s
2.4
iv If y is differentiable at t, then yσt yt μtyΔ t.
We will make use of the following product and quotient rules for the derivative of
the product fg and the quotient f/g where gg σ /
0, here g σ g ◦ σ of two differentiable
functions f and g:
fg
Δ
Δ
Δ
Δ
Δ
f
f Δ g − fg Δ
.
g
gg σ
Δ σ
f g f g fg f g ,
σ
2.5
In this paper, we will refer to the delta integral which we can define as follows: if GΔ t gt, then the Cauchy delta integral of g is defined by
t
2.6
gsΔs : Gt − Ga.
a
It can be shown see 18 that if g ∈ Crd T, then the Cauchy integral Gt :
t0 ∈ T, and satisfies GΔ t gt, t ∈ T. An infinite integral is defined as
∞
ftΔt lim
b
b→∞
a
ftΔt,
t
t0
gsΔs exists,
2.7
a
and the integration on discrete time scales is defined by
b
a
ftΔt t∈a,b
μtft.
2.8
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5
Now, we define the Taylor monomials or generalized polynomials as defined originally
by Agarwal and Bohner 21. These types of monomials are important because they are
intimately related to Cauchy functions for certain dynamic equations which are important
in variations of constants formulas. The Taylor monomials hk : T × T → R, k ∈ N0 N ∪ {0}
are defined recursively as follows. The function h0 is defined by
h0 t, s 1,
∀s, t ∈ T,
2.9
and given hk for k ∈ N0 , the function hk1 is defined by
hk1 t, s t
hk τ, sΔτ,
∀s, t ∈ T.
2.10
s
If we let hΔ
t, s denote for each fixed s ∈ T, the derivative of ht, s with respect to t, then
k
hΔ
k t, s hk−1 t, s,
k ∈ N, t ∈ T,
2.11
for each fixed s ∈ T. The above definition obviously implies
h1 t, s t − s,
∀s, t ∈ T.
2.12
In the following, we give some formulas of hk t, s as determined in 18. In the case when
T R, then σt t, μt 0, yΔ t y t, and
hk t, s t − sk
,
k!
∀s, t ∈ R.
2.13
In the case when T N, we see that σt t 1, μt 1, yΔ t Δyt yt 1 − yt, and
hk n, s :
n − sk
,
k!
k 0, 1, 2, . . . , t > s,
2.14
where tk tt − 1 · · · t − k 1 is the so-called falling function cf. 22. When T {t : t qn , n ∈ N, q > 1}, we have σt qt, μt q − 1t, yΔ t Δq yt yq t − yt/q − 1t,
and
hk t, s k−1
t − qm s
m j ,
j0 q
m0
∀s, t ∈ T.
2.15
If T hN, h > 0,we see that σt t h, μt h, yΔ t Δh yt yt h − yt/h, and
k−1
hk t, s i0
t − ih − s
,
k!
∀s, t ∈ T, t > s.
2.16
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Discrete Dynamics in Nature and Society
In general for t ≥ s, we have that hk t, s ≥ 0, and
hk t, s ≤
t − sk
,
k!
∀t > s, k ∈ N0 .
2.17
We also consider the Taylor monomials gk : T × T → R, k ∈ N0 N ∪ {0}, which are defined
recursively as follows. The function g0 is defined by
g0 t, s 1,
∀s, t ∈ T,
2.18
and given gk for k ∈ N0 , the function gk1 is defined by
gk1 t, s t
gk στ, sΔτ,
∀s, t ∈ T.
2.19
s
If we let gkΔ t, s denote for each fixed s ∈ T, the derivative of gt, s with respect to t, then
gkΔ t, s gk−1 σt, s,
k ∈ N, t ∈ T,
2.20
for each fixed s ∈ T. By Theorem 1.112 in 18, we see that
hk t, s −1k gk s, t.
2.21
n
We denote by Crd T the space of all functions f ∈ Crd T such that f Δi ∈ Crd T for i 0, 1, 2, . . . , n for n ∈ N. For the function f : T → R, we consider the second derivative f Δ2
provided f Δ is delta differentiable on T with derivative f Δ2 f Δ Δ . Similarly, we define the
nth order derivative f Δn f Δn−1 Δ . Now, we are ready to state the Taylor formula that we
will need to prove the main results in this paper. This formula as proved in 23 states the
n
following. Assuming that f ∈ Crd T and s ∈ T, then
t
n−1
Δk
f shk t, s hn−1 t, στf Δn τΔτ.
ft 2.22
s
k0
As a special case if m < n, then
f
Δm
t n−m−1
f
k0
Δkm
shk t, s t
hn−m−1 t, στf Δn τΔτ.
2.23
s
Now, we are ready to state and prove our main results in this paper. Throughout the
rest of the paper, we will assume that all the integrals that will appear in the inequalities exist
and are finite.
Discrete Dynamics in Nature and Society
7
n
Theorem 2.1. Letting T be a time scale with a, b ∈ T and y ∈ Crd a, b ∩ T. If yΔi a 0, for
i 0, 1, . . . , n − 1, then
b
ytyΔn tΔt ≤
a
1/2 b
b
t
1
Δn 2
2
|hn−1 t, σs| Δs Δt
y t Δt.
2
a
a
a
2.24
Proof. From the Taylor formula 2.22, since yΔi a 0, for i 0, 1, . . . , n − 1, we have
t
hn−1 t, σsyΔn sΔs.
yt :
2.25
a
This implies that
t
Δn y t yt ≤ yΔn t |hn−1 t, σs|yΔn sΔs.
2.26
a
Applying the Hölder inequality 1.6 with γ ν 2, we have
1/2 t
1/2
t
2
Δ
Δn 2
Δ
.
|hn−1 t, σs| Δs
y t yt ≤ y n t
y n s Δs
a
2.27
a
Then
1/2
1/2
b t
b
t Δn Δn Δ n 2
2
Δt. 2.28
|hn−1 t, σs| Δs
y t yt Δt ≤
y t
y s Δs
a
a
a
a
t
2
2
Define zt : a |yΔn s| Δs. This implies that za 0 and |yΔn t| zΔ t. From this and
2.28, we have
1/2
b t
b
1/2
Δn 2
ztzΔ t
Δt.
|hn−1 t, σs| Δs
y t ytΔt ≤
a
a
2.29
a
Applying the Hölder inequality again γ ν 2, we obtain
b t
1/2 b
1/2
b
Δn 2
Δ
ztz tΔt
.
|hn−1 t, σs| Δs Δt
y t ytΔt ≤
a
a
a
2.30
a
From 1.7, we have note that zt > 0 and zΔ t > 0 that
z t
2
Δ
Δ
2 z t
1
0
hzσ 1 − hzdh ≥ 2zΔ tzt.
2.31
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Discrete Dynamics in Nature and Society
Then
b
ztzΔ tΔt ≤
a
1
2
b z2 t
Δ
Δt a
1 2
z b,
2
2.32
where za 0. Substituting 2.32 into 2.30, we have
1/2
b
b
t
1
Δn 2
zb
|hn−1 t, σs| Δs Δt
y t ytΔt ≤
2
a
a
a
1/2 b
b
t
1
Δn 2
2
|hn−1 t, σs| Δs Δt
y t Δt,
2
a
a
a
2.33
which is the desired inequality 2.24. The proof is complete.
n−k
Remark 2.2. Let 0 ≤ k < n, but fixed, and let xt ∈ Crd
0 ≤ i ≤ n − k − 1. Then from 2.24 it follows that
b
a
|xt|xΔn−k tΔt ≤
a, b ∩ T be such that xΔi a 0,
1/2 b
b
t
1
Δn−k 2
2
t Δt.
|hn−k−1 t, σs| Δs Δt
x
2
a
a
a
n−k
Thus for x yΔk , where y ∈ Crd
result.
2.34
a, b, yΔi 0, k ≤ i ≤ n − 1; then we have the following
n
Corollary 2.3. Let T be a time scale with a,b ∈ T and y ∈ Crd a, b ∩ T. If yΔi a 0,
k ≤ i ≤ n − 1, then
1/2 b
b
b
t
1
Δk Δn Δn 2
2
|hn−k−1 t, σs| Δs Δt
y ty tΔt ≤
y t Δt.
2
a
a
a
a
2.35
Note that Theorem 2.1 can be extended to a general inequality with two different
constants p and q that satisfy 1/p 1/q 1, to obtain the following result.
n
Theorem 2.4. Letting T be a time scale with a, b ∈ T and y ∈ Crd a, b ∩ T. If yΔi a 0, for
i 0, 1, . . . , n − 1, then
1/p b
1/q b t
Δn q
p
ytyΔn tΔt ≤ 1
|hn−1 t, σs| Δs Δt
y t Δt.
2
a
a
a
a
b
2.36
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9
Theorem 2.5. Let T be a time scale with a, b ∈ T and let l,m be positive real numbers such that
n
l m > 1, and y ∈ Crd a, b ∩ T. If yΔi a 0, for i 0, 1, . . . , n − 1; then
l/lm b
m/lm b
m
m
Δn lm
lm−1
ytl yΔn t Δt ≤
H
t, sΔt
y t Δt,
lm
a
a
a
2.37
b
where
t
Ht, s :
hn−1 t, σslm/lm−1 Δs.
2.38
a
Proof. From the Taylor formula 2.22 and since yΔi a 0, for i 0, 1, . . . , n − 1, we have
yt ≤
t
hn−1 t, σsyΔn sΔs.
2.39
a
Applying the Hölder inequality with γ l m and ν l m/l m − 1, we have
yt ≤
t
hn−1 t, σs
lm/lm−1
lm−1/lm t
Δs
a
1/lm
Δn lm
y s Δs
.
2.40
a
This implies that
m
ytl yΔn t
t
llm−1/lm t
l/lm
Δn lm
Δn m
lm/lm−1
≤ y t
Δs
.
hn−1 t, σs
y s Δs
a
a
2.41
Then
b l
Δn m y t yt Δt
a
≤
b t
a
llm−1/lm
hn−1 t, σs
lm/lm−1
Δs
l/lm
t Δn m
Δn lm
Δt.
y t
y s Δs
a
a
2.42
Define zt :
t
a
lm
|yΔn s|
Δs. This implies that za 0, and
Δ
y n tm zΔ t m/lm > 0.
2.43
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Discrete Dynamics in Nature and Society
From this and 2.42, we have
b
m
ytl yΔn t Δt
a
≤
b t
a
llm−1/lm
hn−1 t, σslm/lm−1 Δs
zΔ t
m/lm
2.44
ztl/lm Δt.
a
Applying the Hölder inequality again γ l m/l and ν l m/m, we have
b
m
ytl yΔn t Δt
a
⎛
≤⎝
a
×
llm−1/lmlm/l
b t
b
⎞l/lm
Δt⎠
hn−1 t, σslm/lm−1 Δs
a
2.45
m/ml
Δ
z tzt
l/m
Δt
.
a
From 1.7, we have note that zt and zΔ t > 0; see also, page 116 17 that
lm/m
z
t
Δ
lm
≥
m
1
hzσ 1 − hzlm/m−1 zΔ t
0
2.46
lm
≥
ztl/m zΔ t.
m
Then
b
l/m
z
a
m
tz tΔt ≤
lm
Δ
b zlm/m t
a
Δ
Δt m lm/m
z
b,
lm
2.47
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11
where za 0. Substituting 2.47 into 2.45, we have
b
m
ytl yΔn t Δt
a
⎛ llm−1/lmlm/l ⎞l/lm
t
m m/lm ⎝ b
≤
Δt⎠
zb
hn−1 t, σslm/lm−1 Δs
lm
a
a
×
m
lm
m/lm
⎛
⎝
b t
a
lm−1
hn−1 t, σs
lm/lm−1
Δs
⎞l/lm
Δt⎠
a
b Δn lm
y t Δt,
a
2.48
which is the desired inequality 2.37. The proof is complete.
Following Remark 2.2, we can obtain the following result.
Corollary 2.6. Let T be a time scale with a, b ∈ T and let l,m be positive real numbers such that
n
l m > 1 and y ∈ Crd a, b ∩ T. If yΔi a 0, k ≤ i ≤ n − 1; then
l/lm b
b m/lm b
m
Δk l Δn m
Δn lm
lm−1
H
t, sΔt
y t y t Δt ≤
y t Δt,
lm
a
a
a
2.49
where
t
Ht, s :
hn−k−1 t, σslm/lm−1 Δs.
2.50
a
Note that Theorem 2.5 cannot be applied when l m 1. In the following theorem we prove
a new inequality which can be applied in this case.
Theorem 2.7. Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that
n
l m 1 and y ∈ Crd a, b ∩ T. If yΔi a 0, for i 0, 1, . . . , n − 1, then
b
a
b
l b
m
Δn ytl yΔn t Δt ≤ mm
hn−1 t, aΔt
y tΔt.
a
a
2.51
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Proof. Using the fact that |hn t, s| is increasing with respect to its first component for t ≥
σs > a, we have from the Taylor formula 2.22 and yΔi a 0, for i 0, 1, . . . , n − 1, that
yt ≤ hn−1 t, a
t
Δn y sΔs.
2.52
a
This implies that
l
m
m t l Δ
l
Δ
Δ
yt y n t ≤ hn−1 t, a y n t
y n sΔs .
2.53
a
Now applying the Hölder inequality 1.6 with γ 1/l and ν 1/m, we obtain
b
m
ytl yΔn t Δt ≤
a
b
l/m ⎞m
l ⎛ b
t
Δ
Δ
hn−1 t, aΔt ⎝ y n t
Δt⎠ . 2.54
y n sΔs
a
Let zt t
a
a
a
|yΔn s|Δs. Then za 0 and zΔ t |yΔn t| so that
l/m
b
b
t
Δn Δn Δt
zΔ tztl/m Δt,
y t
y sΔs
a
a
2.55
a
and hence
b
m
ytl yΔn t Δt ≤
b
a
hn−1 t, aΔt
l b
a
m
Δ
z tzt
l/m
Δt
.
2.56
a
As in the proof of Theorem 2.5, we have that
b
zΔ tztl/m Δt ≤
a
b
zl/m tzΔ tΔt ≤ m
a
b zlm/m t
Δ
Δt
a
mz1/m b m
b
2.57
1/m
Δn y tΔt
.
a
Substituting into 2.56, we have
b
a
b
l b
m
l Δ
m
Δ
n
n
yt y t Δt ≤ m
hn−1 t, aΔt
y tΔt ,
a
a
which is the desired inequality 2.51. The proof is complete.
Following Remark 2.2, we can obtain the following result.
2.58
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13
Corollary 2.8. Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that
n
l m 1, and y ∈ Crd a, b ∩ T. If yΔi a 0, k ≤ i ≤ n − 1, then
l b
b
b Δk l Δn m
Δn m
|hn−k−1 t, a|Δt
y t y t Δt ≤ m
y tΔt .
a
a
2.59
a
In the following, we will prove some inequalities with two different weighted functions.
Theorem 2.9. Let T be a time scale with a, b ∈ T and let l, m, r be positive real numbers such that
l m > 1 and r > 1. Further, let pt and qt be positive rd-continuous functions defined on
n
a, b ∩ T and y ∈ Crd a, b ∩ T. If yΔi a 0, for i 0, 1, . . . , n − 1, then
b
b
lm/r
m
r
l Δ
Δ
qtyt y n t Δt ≤ Λ1 l, m, r, p, q
psy n s Δs
,
a
2.60
a
where
Λ1 l, m, p, q, r :
m
lm
m/r b
r−m/r
q
r/r−m
tp
−m/r−m
lr−1/r−m
tP t
Δt
,
a
P t :
t
p−1/r−1 shn−1 t, σsr/r−1 Δs.
a
2.61
Proof. From the Taylor formula 2.22, we see that
yt ≤
t
p−1/r shn−1 t, σspl/r syΔn sΔs.
2.62
a
Applying the Hölder inequality on the right hand side with r and r/r − 1, we have
yt ≤
t
p−1/r shn−1 t, σspl/r syΔn sΔs
a
≤
t
r−1/r t
p−1/r−1 shn−1 t, σsr/r−1 Δs
a
2.63
1/r
r
psyΔn s Δs
.
a
This implies that
t
l/r
m
l Δ
Δ n r
Δn m
lr−1/r
n
psy s Δs
.
qt yt y t ≤ qtP
ty t
a
2.64
14
Discrete Dynamics in Nature and Society
Integrating from a to b, we have
b
m
l qtyt yΔn t Δt
a
b
≤
qtP
lr−1/r
2.65
t
l/r
Δ n r
Δn m
psy s Δs
Δt.
ty t
a
a
Let
t
zt :
r
psyΔn s Δs.
2.66
a
r
m
Then za 0, zΔ t pt|yΔn t| and |yΔn t| zΔ tm/r p−m/r t. This implies that
b
m
l qtyt yΔn t Δt
a
≤
b
2.67
m/r
qtP lr−1/r tp−m/r t zΔ t
ztl/r Δt.
a
Applying the Hölder inequality 1.6 with indices r/m and r/r − m, we obtain
b
m/r
qtP lr−1/r tp−m/r t zΔ t
ztl/r Δt
a
≤
b
Δ
z t zt
l/m
Δt
m/r b
r−m/r
q
a
r/r−m
tP
lr−1/r−m
tp
−m/r−m
tΔt
.
a
2.68
Substituting into 2.67, we have
b
m
l qtyt yΔn t Δt
a
≤
b
q
a
r/r−m
tP
lr−1/r−m
tp
−m/r−m
r−m/r b
tΔt
Δ
m/r
l/m
z t zt
Δt
.
a
2.69
Discrete Dynamics in Nature and Society
15
As in the proof of Theorem 2.5, we have
b
m/r
Δ
z tzt
l/m
Δt
a
≤
m
lm
m
lm
b m/r
lm/m
z
Δ
t Δt
2.70
a
m/r
lm/r
zb
m
lm
m/r b
lm/r
r
psyΔn s Δs
.
a
This implies that
b
b
lm/r
m
r
l Δ
Δ
qtyt y n t Δt ≤ Λ1 l, m, p, q, r
psy n s Δs
,
a
2.71
a
which is the desired inequality 2.60 where Λ1 l, m, p, q, r is defined as in 2.61. The proof
is complete.
Following Remark 2.2, we can obtain the following result.
Theorem 2.10. Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that
l m > 1 and r > 1. Further, let pt and qt be positive rd-continuous functions defined on
n
a, b ∩ T and y ∈ Crd a, b ∩ T. If yΔi a 0, k ≤ i ≤ n − 1, then
b
b
lm/r
Δk l Δn m
Δ n r
qty t y t Δt ≤ Λ2 l, m, r, p, q
psy s Δs
,
a
2.72
a
where
Λ2 l, m, p, q, r :
m
lm
m/r b
P t :
r−m/r
q
r/r−m
tp
−m/r−m
lr−1/r−m
tP t
Δt
,
a
t
a
p−1/r−1 shn−k−1 t, σsΔs.
r/r−1
2.73
Note that Theorem 2.10 cannot be applied when r 1 and r < m. In the following
theorem we prove an inequality which can be applied in this case.
16
Discrete Dynamics in Nature and Society
Theorem 2.11. Let T be a time scale with a, b ∈ T and let α, β be positive real numbers such that
n
α β > 1, and let p, q be nonnegative rd-continuous functions on a, bT and y ∈ Crd a, b ∩ T.
If yΔi a 0, for i 0, 1, . . . , n − 1, then
b
β
αβ
α Δ
b
n
qt yt y t Δt ≤ Λ3 a, b, α, β
ptyΔn t Δt,
a
2.74
a
where
Λ3 a, b, α, β
β/αβ
β
αβ
⎛
⎝
b
qαβ/α t
pβ/α t
a
t
a
αβ−1
hn−1 t, σsαβ/αβ−1
Δs
p1/αβ−1 s
⎞α/αβ
Δt⎠
.
2.75
Proof. From the Taylor formula, we see that
yt ≤
t
t
1/αβ Δn hn−1 t, σs hn−1 t, σsyΔn sΔs ps
y sΔs.
1/αβ
a
a ps
2.76
Now, since p is nonnegative on a, bT , it follows from the Hölder inequality 1.6 with
1/αβ Δn gs ps
y s,
hn−1 t, σs
fs 1/αβ ,
ps
αβ
γ
,
αβ−1
2.77
ν αβ
that
yt ≤
t
a
αβ−1/αβ t
hn−1 t, σsαβ/αβ−1
Δs
1/αβ−1
ps
1/αβ
αβ
psyΔn s Δs
.
a
2.78
Then, we get that
ytα ≤
t
a
ααβ−1/αβ t
hn−1 t, σsαβ/αβ−1
Δs
1/αβ−1
ps
α/αβ
αβ
psyΔn s Δs
.
a
2.79
Setting zt :
t
a
αβ
ps|yΔn s|
Δs, we see that za 0, and
αβ
> 0.
zΔ t ptyΔn t
2.80
Discrete Dynamics in Nature and Society
17
This gives us
Δn β
y t zΔ t
pt
β/αβ
2.81
.
Since q is nonnegative on a, bT , we have from 2.79 and 2.81 that
β
α qtyt yΔn t
ααβ−1/αβ
β/αβ
1 β/αβ t hn−1 t, σsαβ/αβ−1
α/αβ Δ
z
≤ qt
.
Δs
zt
t
pt
p1/αβ−1 s
a
2.82
This implies that
b
β
α qtyt yΔn t Δt
a
b
1 β/αβ
qt
≤
pt
a
t
ααβ−1/αβ
β/αβ
hn−1 t, σsαβ/αβ−1
×
Δt.
Δs
ztα/αβ zΔ t
1/αβ−1
p
s
a
2.83
Applying the Hölder inequality 1.6 with indices α β/α and α β/β, we have
b
β
α qtyt yΔn t Δt
a
⎛
≤⎝
b
a
×
sαβ/α t
pβ/α t
b
t
a
αβ−1
hn−1 t, σsαβ/αβ−1
Δs
p1/αβ−1 s
⎞α/αβ
Δt⎠
2.84
β/αβ
α/β
z
Δ
tz tΔt
.
a
From 2.80, the chain rule 1.7 and the fact that zΔ s > 0, we obtain
zα/β tzΔ t ≤
β αβ/β Δ
z
t .
αβ
2.85
18
Discrete Dynamics in Nature and Society
Substituting 2.85 into 2.84 and using the fact that za 0, we have
b
β
α qtyt yΔn t Δt
a
⎛
≤⎝
b
a
×
⎛
⎝
α
αβ
b
a
×
Using zb :
b
a
qαβ/α t
pβ/α t
a
αβ/β
z
s
Δ
αβ−1 ⎞α/αβ
dt⎠
β/αβ
2.86
Δs
a
t
a
hn−1 t, σsαβ/αβ−1
Δs
p1/αβ−1 s
αβ−1
⎞α/αβ
Δt⎠
β/αβ
zb.
αβ
ps|yΔn s|
b
hn−1 t, σsαβ/αβ−1
Δs
p1/αβ−1 s
β/αβ b qαβ/α t
pβ/α t
β
αβ
t
Δs, we have from the last inequality that
β
αβ
α b
qtyt yΔn t Δt ≤ Λ3 a, b, α, β
ptyΔn t Δt,
a
2.87
a
which is the desired inequality 2.74. The proof is complete.
Following Remark 2.2, we can obtain the following result.
Theorem 2.12. Let T be a time scale with a, b ∈ T and let α, β be positive real numbers such that
n
α β > 1, and let p, q be nonnegative rd-continuous functions on a, bT and y ∈ Crd a, b ∩ T.
Δi If y a 0, k ≤ i ≤ n − 1, then
b
αβ
b
Δk α Δn β
qty t y t Δt ≤ Λ4 a, b, α, β
ptyΔn t Δt,
a
2.88
a
where
Λ4 a, b, α, β
β
αβ
β/αβ
⎛
αβ−1 ⎞α/αβ
b αβ/α t
αβ/αβ−1
q
t
σs
h
t,
n−k−1
⎝
Δt⎠
.
Δs
pβ/α t
p1/αβ−1 s
a
a
2.89
Discrete Dynamics in Nature and Society
19
Instead of 2.25, we can use the relation between gn and hn and define
yt : −1n
b
gn−1 σs, tyΔn sΔs.
2.90
t
Proceeding as pervious by using the same arguments and using 2.90 one can obtain some
results when yΔi b 0, for i 0, 1, . . . , n − 1. For example one can get the following results.
n
Theorem 2.13. Letting T be a time scale with a, b ∈ T and y ∈ Crd X, b ∩ T. If yΔi b 0, for
i 0, 1, . . . , n − 1, then
b
ytyΔn tΔt ≤
a
1/2 b
b
b
1
Δn 2
2
gn−1 σs, tΔs Δt
y t Δt.
2
a
t
a
2.91
Theorem 2.14. Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that
n
l m > 1. Let y ∈ Crd a, b ∩ T. If yΔi b 0, for i 0, 1, . . . , n − 1, then
b
m
ytl yΔn t Δt
a
⎛
≤⎝
b b
a
t
lm/lm−1
gn−1
lm−1
σs, tΔs
⎞l/lm
Δt⎠
b Δn lm
y t Δt.
2.92
a
Remark 2.15. Similar results as in Theorems 2.13 and 2.14 can be obtained from the results in
the rest of the paper, but in this case one will use yΔi b 0, for i 0, 1, . . . , n − 1, instead of If
yΔi a 0, for i 0, 1, . . . , n − 1, and gn s, t instead of hn t, s.
Remark 2.16. It is worth mentioning here that the results in this paper can be used to derive
some inequalities on different time scales based on the definition of the corresponding
function hk t, s.
For example if T R, then from Corollary 2.8, Theorems 2.11 and 2.12 and using 2.13,
we get the following inequalities of Opial’s type in R.
Theorem 2.17. Let a, b ∈ R and let l, m be positive real numbers such that l m 1, and y ∈
Cn a, b ∩ R. If yi a 0, k ≤ i ≤ n − 1, then
b
b mm b − aln−k
k l n m
n y t y t dt ≤
y tdt .
n − k!l
a
a
2.93
20
Discrete Dynamics in Nature and Society
Theorem 2.18. Let a, b ∈ R and let α, β be positive real numbers such that α β > 1, and let
p, q be nonnegative continuous functions on a, b and y ∈ Cn a, b ∩ R. If yi a 0, for
i 0, 1, . . . , n − 1, then
b
β
αβ
α b
qtyt yΔn t dt ≤ Λ5 a, b, α, β
ptyΔn t dt,
a
2.94
a
where
Λ5 a, b, α, β
β
αβ
β/αβ
⎛
⎝
b
a
qαβ/α t
pβ/α t
t
a
αβ−1 ⎞α/αβ
t − sn−1αβ/αβ−1
dt⎠
.
ds
n − 1!p1/αβ−1 s
2.95
Theorem 2.19. Let a, b ∈ R and let α, β be positive real numbers such that α β > 1, and let p, q be
nonnegative continuous functions on a, b and y ∈ Cn a, b ∩ R. If yi a 0, k ≤ i ≤ n − 1,
then
b
α β
αβ
b
qtyk t yn t dt ≤ Λ6 a, b, α, β
ptyn t dt,
a
2.96
a
where
Λ6 a, b, α, β β
αβ
β/αβ 1
n − k − 1!
β/αβ
⎛
αβ−1 ⎞α/αβ 2.97
b αβ/α t
n−k−1αβ/αβ−1
q
t
−
s
t
×⎝
dt⎠
.
ds
pβ/α t
p1/αβ−1 s
a
a
When T Z, we have yΔ t Δyt yt1−yt and Δi ΔΔi−1 . Using the fact
that hk t, s ≤ t − sk /k!, we get from Corollary 2.8 and Theorems 2.11 and 2.12 the following
discrete inequalities.
Theorem 2.20. Let a, b ∈ N and let l, m be positive real numbers such that l m 1. If Δi ya 0,
k ≤ i ≤ n − 1, then
b−1 b−1 l m mm b − aln−k k
n
yt
Δ yt Δn yt ≤
Δ
.
l
−
k!
n
ta
ta
2.98
Discrete Dynamics in Nature and Society
21
Theorem 2.21. Let a, b ∈ N and let α, β be positive real numbers such that α β > 1, and let p, q be
nonnegative sequences. If Δi ya 0, for i 0, 1, . . . , n − 1, then
b−1
b−1
β
αβ
α qtyt Δn yt ≤ Λ7 a, b, α, β
ptΔn yt ,
ta
2.99
ta
where
Λ7 a, b, α, β
β
αβ
β/αβ
⎛
αβ−1 ⎞α/αβ
n−1αβ/αβ−1
b−1 αβ/α
t−1
q
t
−
s
t
⎝
⎠
.
β/α t
1/αβ−1 s
sa n − 1!p
ta p
2.100
Theorem 2.22. Let a, b ∈ N and let α, β be positive real numbers such that α β > 1, and let p, q be
nonnegative sequences. If Δi ya 0, 0 ≤ k ≤ i ≤ n − 1, then
b−1
b−1
α β
αβ
qtΔk yt Δn yt dt ≤ Λ8 a, b, α, β
ptΔn yt ,
ta
2.101
ta
where
Λ8 a, b, α, β β
αβ
β/αβ 1
n − k − 1!
β/αβ
⎛
αβ−1 ⎞α/αβ
n−k−1αβ/αβ−1
b−1 αβ/α
t−1
q
t
−
s
t
⎠
×⎝
.
β/α t
p1/αβ−1 s
sa
ta p
2.102
Similar results when T hZ and T qN0 {qt : t ∈ N0 } where q > 1 and different time
scales can be obtained as in Theorems 2.17 and 2.22. The details are left to the reader.
Problem 1. It will be interesting to extend the pervious results and prove some inequalities of
the form
X
a
α β
αp
t
qtyΔk t yΔk1 t Δt ≤ Λ a, b, α, β
ptyΔn t Δt,
2.103
a
where Λ is the constant of the inequality that needs to be determined.
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