Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 362526, 23 pages
doi:10.1155/2012/362526
Research Article
New Inequalities of Opial’s Type on
Time Scales and Some of Their Applications
Samir H. Saker
College of Science Research Centre, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Correspondence should be addressed to Samir H. Saker, ssaker@ksu.edu.sa
Received 31 December 2011; Accepted 14 March 2012
Academic Editor: Hassan A. El-Morshedy
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 dynamic inequalities of Opial’s type on time scales. The results not only
extend some results in the literature but also improve some of them. Some continuous and discrete
inequalities are derived from the main results as special cases. The results will be applied on
second-order half-linear dynamic equations on time scales to prove several results related to the
spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or
its derivative. The results also yield conditions for disfocality of these equations.
1. Introduction
In 1960 Opial 1 proved that if y is an absolutely continuous function on a, b with ya yb 0, then
b
yty tdt ≤ b − a
4
a
b
2
y t dt.
1.1
a
In further simplifying the proof of the Opial inequality which had already been simplified by
Olech 2, Beesack 3, Levinson 4, Mallows 5, and Pederson 6, it is proved that if y is
real absolutely continuous on 0, b and with y0 0, then
b
yty tdt ≤ b
2
0
b
2
y t dt.
1.2
0
Since the discovery of Opial’s inequality much work has been done, and many papers which
deal with new proofs, various generalizations, extensions, and their discrete analogues have
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appeared in the literature. The discrete analogy of the 1.1 has been proved in 7 and the
discrete analogy of 1.2 has been proved in 8, Theorem 5.2.2. It is worth to mention here
that many results concerning differential inequalities carry over quite easily to corresponding
results for difference inequalities, while other results seem to be completely different from
their continuous counterparts.
In recent years, there has been much research activity concerning the qualitative theory
of dynamic equations on time scales. It has been created in 9 in order to unify the study of
differential and difference equations, and it also extends these classical cases to cases “in
between,” for example, to the so-called q-difference equations. The general idea is to prove a
result for a dynamic equation or 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. A cover story article in New Scientist 10 discusses several possible applications
of time scales. The three most popular examples of calculus on time scales are differential
calculus, difference calculus, and quantum calculus see Kac and Cheung 11, that is, when
T R, T N and T qN0 {qt : t ∈ N0 } where q > 1.
One of the main subjects of the qualitative analysis on time scales is to prove some
new dynamic inequalities. These on the one hand generalize and on the other hand furnish
a handy tool for the study of qualitative as well as quantitative properties of solutions of
dynamic equations on time scales. In the following, we recall some results obtained for
dynamic inequalities on time scales that serve and motivate the contents of this paper.
Bohner and Kaymakçalan in 12 established the time scale analogy of 1.2 and
proved that if y : 0, b ∩ T → R is delta differentiable with y0 0, then
h
h 2
yt yσ tyΔ tΔt ≤ h yΔ t Δt.
0
1.3
0
Also in 12 the authors proved that if r and q are positive rd-continuous functions on
b
0, bT , 0 Δt/rt < ∞, q nonincreasing and y : 0, b ∩ T → R is delta differentiable with
y0 0, then
b
0
b
2
Δt b
qσ t yt yσ t yΔ tΔt ≤
rtqtyΔ t Δt.
rt
0
0
1.4
Since the discovery of the inequalities 1.3 and 1.4 some papers which deal with new
proofs, various generalizations, and extensions of 1.3 and 1.4 have appeared in the
literature, we refer to the results in 13–15 and the references cited therein. Karpuz et al.
13 proved an inequality similar to the inequality 1.4 of the form
b
b 2
qt yt yσ t yΔ tΔt ≤ Kq a, b yΔ t Δt,
a
1.5
a
where q is a positive rd-continuous function on a, b, and y : a, b ∩ T → R is delta
differentiable with ya 0, and
1/2
b
2
.
Kq a, b 2 q uσu − aΔu
a
1.6
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Wong et al. 14 and Sirvastava et al. 15 proved that if r is a positive rd-continuous function
on a, b, we have
b
q
p rtyt yΔ t Δt ≤
a
q
b − ap
pq
b
pq
rtyΔ t Δt,
1.7
a
where y : a, b ∩ T → R is delta differentiable with ya 0.
In 16 the author proved that if y : a, X∩T → R is delta differentiable with ya 0,
then
X
q
pq
p X
sxyx yΔ x Δx ≤ K1 a, X, p, q
rxyΔ x Δx,
a
1.8
a
where p, q are positive real numbers such that p q > 1, and let r, s be nonnegative rdX
continuous functions on a, XT such that a r −1/pq−1 tΔt < ∞ and
q/pq
q
K1 a, X, p, q pq
X
x
pq−1 p/pq
pq/p
−q/p
−1/pq−1
×
Δt
Δx
.
sx
rx
rt
a
a
1.9
If ya is replaced by yb, then
b
pq
p Δ q
b
sx yx y x Δx ≤ K2 X, b, p, q
rxyΔ x Δx,
X
1.10
X
where
K2 X, b, p, q q/pq
q
pq
⎛
b
pq−1 ⎞p/pq
b
× ⎝ sxpq/p rx−q/p
Δx⎠
.
rt−1/pq−1 Δt
X
x
1.11
For contributions of different types of inequalities on time scales, we also refer the reader to
the papers 16–24 and the references cited therein.
The paper is organized as follows: in Section 2, we will prove some new dynamic
inequalities of Opial’s type on time scales of the form
X
a
pq
p Δ q
X
σ
sx yx y x y x Δx ≤ K a, X, p, q
rxyΔ x Δx,
a
1.12
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where p, q are positive real numbers such that p ≤ 1, p q > 1 and Ka, X, p, q is
the coefficient of the inequality. As special cases, we derive some differential and discrete
inequalities on continuous and discrete time scales. In Section 3, we will apply the obtained
inequalities in Section 2 on the second-order half-linear dynamic equation
γ Δ
γ
qt yσ t 0,
rt yΔ t
1.13
on a, bT ,
where T is an arbitrary time scale, 0 < γ ≤ 1 is a quotient of odd positive integers, r and q are
real rd-continuous functions defined on T with rt > 0. In particular, we will prove several
results related to the problems:
i obtain lower bounds for the spacing β−α where y is a solution of 1.13 and satisfies
yα yΔ β 0, or yΔ α yβ 0,
ii obtain lower bounds for the spacing between consecutive zeros of solutions of
1.13.
Our motivation comes from that fact that the inequalities obtained in the literature
cannot be applied on the half-linear dynamic equation 1.13 to prove results related to the
problems i-ii.
2. Main Results
The main inequalities will be proved in this section by making use of the Hölder inequality
see 25, Theorem 6.13
h
a
ftgtΔt ≤
h
1/γ h
ftγ Δt
a
1/ν
gtν Δt
,
2.1
a
where a, h ∈ T and f; g ∈ Crd I, R, γ > 1 and 1/ν 1/γ 1, and the inequality see 8,
page 51
2r−1 ar br ≤ a br ≤ ar br ,
where a, b ≥ 0, 0 ≤ r ≤ 1.
2.2
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},
2.3
where sup ∅ inf T. A point t ∈ T is said to be left dense if ρt t and t > inf T, is right dense
if σt t, is left scattered if ρt < t and right scattered if σt > t.
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.
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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 x : T → R. Define xΔ t to be the number if it exists with the
property that given any > 0 there is a neighborhood U of t with
xσt − xs − xΔ tσt − s ≤ |σt − s|,
∀s ∈ U.
2.4
In this case, we say xΔ t is the delta derivative of x at t and that x is delta differentiable
at t.
We will frequently use the results in the following theorem which is due to Hilger 9.
Assume that g : T → R and let t ∈ T.
i If g is differentiable at t, then g is continuous at t.
ii If g is continuous at t and t is right scattered, then g is differentiable at t with
g Δ t gσt − gt
.
μt
2.5
iii If g is differentiable and t is right-dense, then
g Δ t lim
s→t
gt − gs
.
t−s
2.6
iv If g is differentiable at t, then gσt gt μtg Δ t.
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
gsΔs : Gt − Ga.
2.7
a
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
2.8
We say that a function p : T → R is regressive provided 1μtpt /
0, t ∈ T. The integration
by parts formula is given by
b
a
b
ftg tΔt ftgt a −
Δ
b
a
f Δ tg σ tΔt.
2.9
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t
It can be shown see 25 that if g ∈ Crd T, then the Cauchy integral Gt : t0 gsΔs exists,
t0 ∈ T, and satisfies GΔ t gt, t ∈ T. The integration on discrete time scales is defined by
b
ftΔt a
2.10
μtft.
t∈a,b
To prove the main results, we need the formula
xγ tΔ γ
1
hxσ 1 − hxγ−1 dhxΔ t,
2.11
γ > 0,
0
which is a simple consequence of Keller’s chain rule 25, Theorem 1.90. The books on the
subject of time scales by Bohner and Peterson 25, 26 summarize and organize much of
time scale calculus and contain some results for dynamic equations on time scales. To admit
functions such that y and yΔ may change sign on a, bT , we note that if
yx x
yΔ xΔx,
then vx a
x
Δ y xΔx ≥ yx,
2.12
a
with equality if and only if yΔ does not change sign. The same result holds if y b
b
− x yΔ xΔx and vx x |yΔ x|Δx. Now, we are ready to state and prove the main
results.
Theorem 2.1. Let T be a time scale with a, X ∈ T and p, q be positive real numbers such that
p ≤ 1, p q > 1 and let r, s be nonnegative rd-continuous functions on a, XT such that
X −1/pq−1
r
tΔt < ∞. If y : a, X ∩ T → R is delta differentiable with ya 0, then one
a
has
X
pq
p Δ q
X
σ
sx yx y x y x Δx ≤ K1 a, X, p, q
rxyΔ x Δx,
a
2.13
a
where
K1 a, X, p, q sup
μp x
a≤x≤X
×
X
a
sx
rx
2p
sxpq/p
rxq/p
q
pq
x
r
q/pq
−1/pq−1
pq−1
tΔt
2.14
p/pq
Δx
.
a
Proof. Since
yx ≤
x
Δ y tΔt,
a
for x ∈ a, XT ,
2.15
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and r is nonnegative on a, XT , then it follows from the Hölder inequality 2.1 with
ft gt rt1/pq yΔ t,
1
rt
,
1/pq
γ
pq
,
pq−1
ν p q,
2.16
that
x
x
Δ y tΔt ≤
a
a
pq−1/pq x
1
rt
Δt
1/pq−1
pq 1/pq
rtyΔ t Δt
.
2.17
a
Then, for a ≤ x ≤ X, we get noting that ya 0 that
yxp ≤
x
a
1
ppq−1/pq x
Δt
rt1/pq−1
pq p/pq
rtyΔ t Δt
.
2.18
a
Since yσ y μyΔ , we have
yx yσ x 2yx μyΔ x.
2.19
Applying the inequality 2.2, we get where p ≤ 1 that
p
p
y yσ p 2yx μyΔ x ≤ 2p yp μp yΔ .
2.20
Setting
x
pq
rtyΔ t Δt,
2.21
pq
> 0.
zΔ x rxyΔ x
2.22
zx :
a
we see that za 0, and
From this, we get that
Δ
Δ pq z x
,
y x
rx
Δ q
y x zΔ x
rx
q/pq
.
2.23
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Thus, since s is nonnegative on a, XT , we have from 2.20 and 2.23 that
pq
q
q
p p sxyx yσ x yΔ x ≤ 2p sxyx yΔ x μp xsxyΔ q/pq x
ppq−1/pq
1
1
Δt
≤ 2p sx
×
1/pq−1 t
rx
a r
q/pq
zΔ x
p/pq
Δ
p
.
z x
× zx
μ xsx
rx
2.24
This implies that
X
a
q
p sxyx yσ x yΔ x Δx
ppq−1/pq
q/pq x
1
1
≤2
Δt
sx
×
1/pq−1 t
rx
a
a r
X
q/pq
sx Δ
z xΔx
× zxp/pq zΔ x
Δx μp
rx
a
ppq−1/pq
X
q/pq x
1
1
p
sx
×
Δt
≤2
1/pq−1
rx
a
a rt
X
q/pq
sx
× zxp/pq zΔ x
Δx sup μp
zΔ xΔx.
rx
a
a≤x≤X
p
X
2.25
Supposing that the integrals in 2.25 exist and again applying the Hölder inequality 2.1
with indices p q/p and p q/q on the first integral on the right-hand side of 2.25, we have
X
a
q
p sxyx yσ x yΔ x Δx
⎛
pq−1 ⎞p/pq
q/p x
1
1
Δt
Δx⎠
≤ 2p ⎝ sxpq/p
1/pq−1
rx
a
a rt
X
q/pq
X
sx
p/q Δ
×
sup μp
zΔ xΔx.
zx z xΔx
rx
a
a
a≤x≤X
X
2.26
From 2.22, and the chain rule 2.11, we obtain
zp/q xzΔ x ≤
q pq/q Δ
z
x .
pq
2.27
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Substituting 2.27 into 2.26 and using the fact that za 0, we have that
X
q
p sxyx yσ x yΔ x Δx
a
⎛
≤ 2p ⎝
X
1
rx
q/p x
pq−1
1
⎞p/pq
Δt
Δx⎠
rt 1/pq−1
q/pq X
q/pq X Δ
p
sx
pq/q
×
Δt
μp
zΔ xΔx
zt
pq
rx
a
a
⎛
pq−1 ⎞p/pq
X
q/p x
1
1
pq/p
Δt
Δx⎠
⎝ sx
1/pq−1
rx
a
a rt
q/pq
q
sx
zX.
× 2p
zX sup μp
pq
rx
a≤x≤X
sxpq/p
a
a
2.28
Using 2.21, we have from the last inequality that
X
q
pq
p X
sxyx yσ x yΔ x Δx ≤ K1 a, X, p, q
rxyΔ x Δx,
a
2.29
a
which is the desired inequality 2.13. The proof is complete.
Here, we only state the following theorem, since its proof is the same as that of
b
Theorem 2.1, with a, X replaced by b, X and |yx| x |yΔ t|Δt.
Theorem 2.2. Let T be a time scale with X, b ∈ T and let p, q be positive real numbers such
that p ≤ 1, p q > 1 and let r, s be nonnegative rd-continuous functions on X, bT such that
b −1/pq−1
r
tΔt < ∞. If y : X, b ∩ T → R is delta differentiable with yb 0, then one has
X
b
pq
p Δ q
b
σ
sx yx y x y x Δx ≤ K2 X, b, p, q
rxyΔ x Δx,
X
2.30
X
where
q/pq
q
sx
2p
μp x
rx
pq
X≤x≤b
⎛
pq−1 ⎞p/pq
b
b
pq/p
sx
−1/pq−1
×⎝
Δt
Δx⎠
.
rt
rxq/p
X
x
K2 X, b, p, q sup
2.31
Note that when T R, we have yσ y and μx 0. Then from Theorems 2.1 and 2.2
we have the following differential inequalities.
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Discrete Dynamics in Nature and Society
Corollary 2.3. Assume that p, q are positive real numbers such that p ≤ 1, p q > 1 and let r, s be
X
nonnegative continuous functions on a, XR such that a rt−1/pq−1 dt < ∞. If y : a, X∩R →
R is differentiable with ya 0, then one has
X
p q
sxyx y x dx ≤ C1 a, X, p, q
X
a
pq
rxy x dx,
2.32
a
where
C1 a, X, p, q 2
p
q
pq
q/pq
×
X
a
sxpq/p
rxq/p
x
rt
−1/pq−1
p/pq
pq−1
dt
dx
.
a
2.33
Corollary 2.4. Assume that p, q are positive real numbers such that p ≤ 1, p q > 1 and let r, s be
b
nonnegative continuous functions onX, bR such that X rt−1/pq−1 dt < ∞. If y : X, b ∩ R →
R is delta differentiable with yb 0, then one has
b
p q
sxyx y x dx ≤ C2 X, b, p, q
X
b
pq
rxy x dx,
2.34
X
where
C2 X, b, p, q 2p
q
pq
q/pq
⎛
×⎝
b
X
pq/p
sx
rx
q/p
b
pq−1
rt−1/pq−1 dt
⎞p/pq
dx⎠
.
x
2.35
In the following, we assume that there exists h ∈ a, b which is the unique solution of
the equation
K p, q K1 a, h, p, q K2 h, b, p, q < ∞,
2.36
where K1 a, h, p, q and K2 h, b, p, q are defined as in Theorems 2.1 and 2.2.
Theorem 2.5. Let T be a time scale with a, b ∈ T and let p, q be positive real numbers such
that p ≤ 1, p q > 1 and let r, s be nonnegative rd-continuous functions on a, bT such that
b
rt−1/pq−1 Δt < ∞. If y : a, b ∩ T → R is delta differentiable with ya 0 yb, then
a
one has
b
a
pq
p Δ q
b
σ
sx yx y x y x Δx ≤ K p, q
rxyΔ x Δx.
a
2.37
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Proof. Since
b
X
q
p Δ q
p σ
sx yx y x y x Δx sxyx yσ x yΔ x Δx
a
a
b
q
p sxyx yσ x yΔ x Δx,
2.38
X
then the rest of the proof will be a combination of Theorems 2.1 and 2.2 and hence is omitted.
The proof is complete.
As a special case if r s in Theorem 2.1, then we obtain the following result.
Corollary 2.6. Let T be a time scale with a, X ∈ T and let p, q be positive real numbers such
that p ≤ 1, p q > 1 and let r be a nonnegative rd-continuous function on a, XT such that
X
rt−1/pq−1 Δt < ∞. If y : a, X ∩ T → R is delta differentiable with ya 0, then one has
a
X
a
q
pq
p X
rxyx yσ x yΔ x Δx ≤ K1∗ a, X, p, q
rxyΔ x Δx,
2.39
a
where
K1∗ a, X, p, q sup μp x 2p
a≤x≤X
×
X
x
rx
r
a
q
pq
q/pq
−1/pq−1
tΔt
pq−1
2.40
p/pq
Δx
.
a
From Theorems 2.2 and 2.5 one can derive similar results by setting r s. The details are left
to the reader.
On a time scale T, we note from the chain rule 2.11 that
t − a
pq Δ
pq
≥ pq
1
hσt − a 1 − ht − apq−1 dh
0
1
ht − a 1 − ht − apq−1 dh
2.41
0
p q t − apq−1 .
This implies that
X
a
x − apq−1 Δx ≤
X
a
Δ
1 X − apq
x − apq Δx .
pq
pq
2.42
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From this and 2.40 by putting rt 1, we get that
K1∗
a, X, p, q 2
p
≤2
p
q/pq
q
pq
×
q/pq q
pq
X
p/pq
x − a
a
X − apq
pq
pq−1
Δx
p/pq
max μp x
2.43
a≤x≤X
qq/pq
max μp x 2p
X − ap .
a≤x≤X
pq
So setting r 1 in 2.39 and using 2.43, we have the following result.
Corollary 2.7. Let T be a time scale with a, X ∈ T and let p, q be positive real numbers such that
p ≤ 1 and p q > 1. If y : a, X ∩ T → R is delta differentiable with ya 0, then one has
X
q
X Δ pq
yx yσ xp yΔ x Δx ≤ L a, X, p, q
y x Δx,
a
2.44
a
where
L a, X, p, q :
pq
2
q/pq
pq
p
× X − a sup μ x .
p
2.45
a≤x≤X
Remark 2.8. Note that when T R, we have yσ y, μx 0 and then the inequality 2.44
becomes
X
a
q/pq
yxp y xq dx ≤ q
× X − ap
pq
X
pq
y x dx.
2.46
a
Note also that when p 1 and q 1, then the inequality 2.46 becomes
X
a
yxy xdx ≤ X − a
2
X
2
y x dx,
2.47
a
which is the Opial inequality 1.2.
When T N, we have form 2.44 the following discrete Opial’s type inequality.
Corollary 2.9. Assume that p, q are positive real numbers such that p ≤ 1, p q > 1 and {ri }0≤i≤N
are a nonnegative real sequence. If {yi }0≤i≤N is a sequence of positive real numbers with y0 0,
then
N−1
p q
rnyn yn 1 Δyn ≤
n1
pq
2
N−1
pq
N − ap
rnΔyn .
1
pq
n0
q/pq
2.48
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The inequality 2.44 has an immediate application to the case where ya yb 0.
Choose X ab/2 and apply 2.40 to a, X and X, b and adding we obtain the following
inequality.
Corollary 2.10. Let T be a time scale with a, b ∈ T and let p, q be positive real numbers such that
p ≤ 1 and p q > 1. If y : a, b ∩ T → R is delta differentiable with ya 0 yb, then one has
b
q
b Δ pq
yx yσ xp yΔ x Δx ≤ F a, b, p, q
y x Δx,
a
2.49
a
where
qq/pq
F a, b, p, q :
b − ap sup μp x .
pq
a≤x≤b
2.50
From this inequality, we have the following discrete Opial type inequality.
Corollary 2.11. Assume that p, q are positive real numbers such that p ≤ 1 and p q > 1. If
{yi }0≤i≤N is a sequence of real numbers with y0 0 yN, then
N−1
p q
rnyn yn 1 Δyn ≤
n1
N−1
pq
qq/pq
p
rnΔyn .
N − a 1
pq
n0
2.51
By setting p q 1 in 2.49 we have the following Opial type inequality on a time
scale.
Corollary 2.12. Let T be a time scale with a, b ∈ T. If y : a, X ∩ T → R is delta differentiable
with ya 0 yb, then one has
b
yx yσ xyΔ xΔx ≤
a
b
b − a
Δ 2
sup μx
y x Δx.
2
a
a≤x≤b
2.52
As special cases from 2.52 on the continuous and discrete spaces, that is, when T R and
T N, one has the following inequalities.
Corollary 2.13. If y : a, b ∩ T → R is differentiable with ya 0 yb, then one has the Opial
inequality
b
yxy xdx ≤ b − a
4
a
b
2
y x dx.
2.53
a
Corollary 2.14. If {yi }0≤i≤N is a sequence of real numbers with y0 0 yN, then
N−1
yn yn 1Δyn ≤
n1
N−1
N
Δyn2 .
1
2
n0
2.54
14
Discrete Dynamics in Nature and Society
3. Applications
Our aim in this section, is to apply the dynamic inequalities of Opial’s type proved in
Section 2 to prove several results related to the problems i-ii for the second-order halflinear dynamic equation
γ Δ
γ
qt yσ t 0,
rt yΔ t
on a, bT ,
3.1
on an arbitrary time scale T, where 0 < γ ≤ 1 is a quotient of odd positive integers, r and q are
real rd-continuous functions defined on T with rt > 0. The terminology half-linear arises
because of the fact that the space of all solutions of 3.1 is homogeneous, but not generally
additive. Thus, it has just “half” of the properties of a linear space. It is easily seen that if yt
is a solution of 3.1, then so also is cyt.
By a solution of 3.1 on an interval I, we mean a nontrivial real-valued function y ∈
1
I and satisfies 3.1 on I. We say that a
Crd I, which has the property that rtyΔ t ∈ Crd
solution y of 3.1 has a generalized zero at t if yt 0 and has a generalized zero in t, σt
in case ytyσ t < 0 and μt > 0. Equation 3.1 is disconjugate on the interval t0 , bT , if
there is no nontrivial solution of 3.1 with two or more generalized zeros in t0 , bT .
Equation 3.1 is said to be nonoscillatory on t0 , ∞T if there exists c ∈ t0 , ∞T such
that this equation is disconjugate on c, dT for every d > c. In the opposite case 3.1 is
said to be oscillatory on t0 , ∞T . The oscillation of solutions of 3.1 may equivalently be
defined as follows: a nontrivial solution yt of 3.1 is called oscillatory if it has infinitely
many isolated generalized zeros in t0 , ∞T ; otherwise it is called nonoscillatory. So that
the solution yt of 3.1 is said to be oscillatory if it is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory. This means that the property of oscillation
or nonoscillation is the behavior in the neighborhood of the infinite points.
We say that 3.1 is right disfocal left disfocal on α, βT if the solutions of 3.1 such
that yΔ α 0yΔ β 0 have no generalized zeros in α, βT .
Note that 3.1 in its general form involves some different types of differential and
difference equations depending on the choice of the time scale T. For example when T R,
3.1 becomes a second-order half-linear differential equation and σt t. When T Z, 3.1
becomes a second-order half-linear difference equation and σt t 1. When T hN, 3.1
becomes a generalized difference equation and σt t h. When T {t : t k , k ∈ N0 , >
1}, 3.1 becomes a quantum difference equation see 11 and σt t. Note also that the
√
results in this paper can be applied on the time scales T N2 {t2 : t ∈ N}, T2 { n : n ∈
√
N0 }, T3 { 3 n : n ∈ N0 }, and when T Tn {tn : n ∈ N0 } where {tn } is the set of harmonic
√
numbers. In these cases we see that when T N20 {t2 : t ∈ N0 }, we have σt t 12
and when T Tn {tn : n ∈ N} where tn } is the harmonic numbers that are defined by
n
√
T2 { n : n ∈ N0 }, we have
t0 0 and
√ tn k1 1/k, n ∈ N√0 , we have σtn tn1 . When
√
3
σt t2 1, and when T3 { 3 n : n ∈ N0 }, we have σt t3 1.
Perhaps the best known existence results of types i-ii for a special case of 3.1
when γ 1 and rt 1 is due to Bohner et al. 27, where they extended the Lyapunov
inequality obtained for differential equations in 28. In particular the authors in 27
considered the dynamic equation
yΔΔ t qtyσ t 0,
3.2
Discrete Dynamics in Nature and Society
15
where qt is a positive rd-continuous function defined on T and proved that if yt is a
solution of 3.2 with ya yb 0 a < b, then
b
ptΔt >
a
4
.
b−a
3.3
Karpuz et al. 13 proved that if q ∈ Crd a, bT , R and yt is a solution of 3.2 with ya yΔσ b 0, then
σb
1/2
2
2
Q uσu − aΔu
≥ 1,
where Qu σb
a
qtΔt.
3.4
u
Saker 22 considered the second-order half-linear dynamic equation
Δ
rtϕ xΔ
qtϕxσ t 0,
3.5
on an arbitrary time scale T, where ϕu |u|γ−1 u, γ ≥ 1 is a positive constant, r and q are real
rd-continuous positive functions defined on T and proved that if xt is a positive solution of
3.1 which satisfies xa xb 0, xt /
0 for t ∈ a, b and xt has a maximum at a point
c ∈ a, b, then
b
a
Δt
r γ t
γ b
3.6
qtΔt ≥ 2γ1 .
a
Of particular interest in this paper is when q is oscillatory which is different from the
conditions imposed on q in 12, 22, 27. The results also yield conditions for disfocality for
3.1 on time scales. As special cases, the results include some results obtained for differential
equations and give new results for difference equations on discrete time scales.
Now, we are ready to state and prove the main results in this section. To simplify the
presentation of the results, we define
|Qt|
,
M β : sup μγ t
rt
α≤t≤β
where Qt |Qt|
,
Mα : sup μ t
rt
α≤t≤β
where Qt γ
β
qsΔs,
t
3.7
t
qsΔs.
α
Note that when T R, we have Mα 0 Mβ, and when T Z, we have
M β sup
α≤t≤β
β−1
st qs
rt
,
Mα sup
α≤t≤β
t−1
sα qs
rt
.
3.8
16
Discrete Dynamics in Nature and Society
Theorem 3.1. Suppose that y is a nontrivial solution of 3.1. If yα yΔ β 0, then
2
1/γ1 ×
γ 1
β
where Qt t
t
α
α
|Qx|γ1/γ
r 1/γ x
x
α
Δt
1/γ
r t
γ/γ1
γ
Δx
21−γ M β ≥ 1,
3.9
qsΔs. If yΔ α yβ 0, then
2
1/γ1
γ 1
where Qt β
β
α
|Qx|γ1/γ
r 1/γ x
β
x
Δt
1/γ
r t
γ
γ/γ1
Δx
21−γ Mα ≥ 1,
3.10
qsΔs.
Proof. We prove 3.9. Without loss of generality we may assume that yt > 0 in α, βT .
Multiplying 3.1 by yσ and integrating by parts, we have
β β β
γ1
γ Δ
γ
rt yΔ t
yσ tΔt rt yΔ t yt −
rt yΔ t
Δt
α
−
β
α
qt y t
σ
γ1
α
3.11
Δt.
α
Using the assumptions that yα yΔ β 0 and Qt β
β
t
qsΔs, we have
β
β
γ1
γ1
γ1
rt yΔ t
Δt qt yσ t
Δt − QΔ t yσ t
Δt.
α
α
3.12
α
Integrating by parts the right-hand side see 2.9, we see that
β
β
γ1
Δ
γ1 β
rt yΔ t
Δt −Qt yt
Qt yγ1 t Δt.
α
α
3.13
α
Again using the facts that yα 0 Qβ, we obtain
β
α
β
γ1
Δ
Δ
rt y t
dt Qt yγ1 t dt.
α
3.14
Discrete Dynamics in Nature and Society
17
Applying the chain rule formula 2.11 and the inequality 2.2, we see that
γ1 Δ 1 σ
y t ≤ γ 1
hy t 1 − hytγ dhyΔ t
0
1 γ
Δ 1 σ γ
≤ γ 1 y t hy t dh γ 1 yΔ t 1 − hyt dh
0
0
γ
γ yΔ tyσ t yΔ tyt
γ ≤ 21−γ yσ t yt yΔ t.
3.15
This and 3.14 imply that
β
β
γ1
γ rtyΔ t Δt ≤ 21−γ |Qt|yt yσ t yΔ tΔt.
α
3.16
α
Applying the inequality 2.13 with st |Qt|, p γ and q 1, we have
β
γ1
γ1
β
rtyΔ t Δt ≤ 21−γ K1 α, β, γ, 1
rtyΔ t Δt,
α
3.17
α
where
K1 α, β, γ, 1 M β 2γ
×
β
1
γ 1
1/γ1
γ1/γ −1/γ
|Qx|
r
x
x
r
α
2
M β tΔt
3.18
γ/γ1
γ
Δx
.
α
Then, we have from 3.17 after cancelling the term
1−γ
−1/γ
2
1/γ1 ×
γ 1
β
α
|Qx|γ1/γ
r 1/γ x
β
α
γ1
rt|yΔ t|
x
α
Δt
r 1/γ t
Δt, that
γ/γ1
γ
Δx
≥ 1,
3.19
which is the desired inequality 3.9. The proof of 3.10 is similar to 3.9 by using the
integration by parts and 2.30 of Theorem 2.2 and 2.31 instead of 2.14. The proof is
complete.
As a special case of Theorem 3.1, when rt 1, we have the following result.
Corollary 3.2. Suppose that y is a nontrivial solution of
yΔ t
γ Δ
γ
qt yσ t 0,
t ∈ α, β T .
3.20
18
Discrete Dynamics in Nature and Society
If yα yΔ β 0, then
2
1/γ1 ×
γ 1
where Qt β
t
t
α
γ/γ1
|Qt|
1γ/γ
γ
t − α Δt
21−γ sup μγ t|Qt| ≥ 1,
α
3.21
α≤t≤β
qsΔs. If yΔ α yβ 0, then
2
1/γ1
γ 1
where Qt β
β
|Qt|
1γ/γ γ
γ/γ1
β − t Δt
21−γ sup μγ t|Qt| ≥ 1,
α
3.22
α≤t≤β
qsΔs.
As a special case of Theorem 3.1, when γ 1, we have the following result.
Corollary 3.3. Suppose that y is a nontrivial solution of
rtyΔ t
Δ
qtyσ t 0,
t ∈ α, β T .
3.23
If yα yΔ β 0, then
√
β
2
α
where Qt β
t
β
2
α
t
α
α
1/2
Δt
Δx
M β ≥ 1,
rt
3.24
qsΔs. If yΔ α yβ 0, then
√
where Qt x
|Qx|2
rx
|Qx|2
rx
β
x
Δt
rt
1/2
Δx
Mα ≥ 1,
3.25
qsΔs.
As a special case when T R, we have Mα Mβ 0 and then the results in
Corollary 3.3 reduce to the following results obtained by Brown and Hinton 29 for the
second-order differential equation
y qtyt 0,
α ≤ t ≤ β.
3.26
Corollary 3.4 see 29. If y is a solution of 3.26 such that yα y β 0, then
β
2
α
Q2 ss − αds > 1,
3.27
Discrete Dynamics in Nature and Society
where Qt β
t
19
qsds. If instead y α yβ 0, then
β
2
Q2 s β − s ds > 1,
3.28
α
where Qt t
α
qsds.
Remark 3.5. Note that if T N, then μt 1 and 3.1 when rt 1 becomes
Δ2 yn qnyn 1 0,
3.29
and as a special case of Corollary 3.3, we have the following result.
Corollary 3.6. If y is a solution of 3.29 such that yα Δyβ 0, then
√
⎞1/2
β−1
2
2⎝ Qn n − α⎠ sup |Qn| > 1,
⎛
where Qn β−1
sn
qs. If instead Δyα yβ 0, then
⎞1/2
⎛
β−1
√
2
2⎝ Qn β − n ⎠ sup |Qn| > 1,
n−1
sα
3.31
α≤n≤β
nα
where Qn 3.30
α≤n≤β
nα
qs.
Remark 3.7. By using the maximum of |Q| on α, βT and
X
x − a
pq−1
a
Δx ≤
X
a
Δ
1 X − apq
x − apq Δx ,
pq
pq
3.32
in 3.21 and 3.22 with p γ and q 1, we have the following results.
Corollary 3.8. Suppose that y is a nontrivial solution of 3.20, where γ ≤ 1 is a quotient of odd
positive integers. If yα yΔ β 0, then
γ
β
β
2 β−α
1−γ
γ
max qsΔs 2 sup μ t qsΔs ≥ 1,
γ 1 α≤t≤β t
t
α≤t≤β
3.33
and if yΔ α yβ 0, then
γ
t
t
2 β−α
1−γ
γ
max qsΔs 2 sup μ t qsΔs ≥ 1.
γ 1 α≤t≤β α
α
α≤t≤β
3.34
20
Discrete Dynamics in Nature and Society
As a special when T R, we have Mα Mβ 0 and then the results in
Corollary 3.8 reduce to the following results for the second-order half-linear differential
equation:
y t
γ γ
qt yt 0,
α ≤ t ≤ β,
3.35
where γ ≤ 1 is a quotient of odd positive integers.
Corollary 3.9. Assume that γ ≤ 1 is a quotient of odd positive integers. Suppose that y is a nontrivial
solution of 3.35. If yα y β 0, then
β
γ
2 β − α sup qsds ≥ 1.
γ 1
α≤t≤β t
3.36
If instead y α yβ 0, then
t
γ
2 β − α sup qsds ≥ 1.
γ 1
α≤t≤β α
3.37
As a special case of Corollary 3.9 when γ 1, we have the following results that has
been established by Harris and Kong 30.
Corollary 3.10 see 30. If y is a solution of the equation
y qtyt 0,
a ≤ t ≤ b,
3.38
with no zeros in a, b and such that ya y b 0, then
b
b − a sup qsds > 1.
a≤t≤b t
3.39
t
b − a sup qsds > 1.
a≤t≤b a
3.40
If instead y a yb 0, then
As a special when T Z, we see that Mα and Mβ are defined as in 3.8 and then
the results in Corollary 3.8 reduce to the following results for the second-order half-linear
difference equation
γ γ
Δ Δyn
qn yn 1 0,
where γ ≤ 1 is a quotient of odd positive integers.
α ≤ n ≤ β,
3.41
Discrete Dynamics in Nature and Society
21
Corollary 3.11. Suppose that y is a nontrivial solution of 3.41, where γ ≤ 1 is a quotient of odd
positive integers. If yα Δyβ 0, then
⎞
⎛
γ
β−1
β−1
2 β−α
max qs 21−γ sup ⎝ qs⎠ ≥ 1,
γ 1 α≤n≤β sn
sn
α≤n≤β
3.42
and if Δyα yβ 0, then
γ
n−1
n−1
2 β−α
1−γ
max qs 2 sup qs ≥ 1.
sα
γ 1 α≤n≤β sα
α≤n≤β
3.43
Remark 3.12. The above results yield sufficient conditions for disfocality of 3.1, that is,
sufficient conditions so that there does not exist a nontrivial solution y satisfying either
yα yΔ β 0 or yΔ α yβ 0.
In the following, we employ Theorem 2.5, to determine the lower bound for the
distance between consecutive zeros of solutions of 3.1. Note that the applications of the
above results allow the use of arbitrary antiderivative Q in the above arguments. In the
following, we assume that QΔ t qt and there exists h ∈ α, β which is the unique solution
of the equation
K1 α, β K1 α, β, h K1 α, h, β < ∞,
3.44
where
h
γ γ/γ1
β
Δt
2γ
|Qx|γ1/γ
Δx
,
1/γ1 ×
1/γ t
r 1/γ x
α
α r
γ 1
β
β
γ γ/γ1
Δt
2γ
|Qx|γ1/γ
K1 α, h, β Δx
.
1/γ1
1/γ t
r 1/γ x
α
h r
γ 1
K1 α, β, h 3.45
Theorem 3.13. Assume that QΔ t qt and y is a nontrivial solution of 3.1. If yα yβ 0,
then
K1 α, β ≥ 1,
3.46
where K1 α, β is defined as in 3.44.
Proof. Multiplying 3.1 by yσ t, proceed as in Theorem 3.1 and use yα yβ 0, to get
β
α
β
β
γ1
γ1
γ1
Δ
rt y t
Δt qt yt
Δt QΔ t yσ t
Δt.
α
α
3.47
22
Discrete Dynamics in Nature and Society
Integrating by parts the right hand side see 2.9, we see that
β
β
γ1
Δ
γ1 β
rt yΔ t
Δt Qt yt
−Qt yγ1 t Δt.
α
α
3.48
α
Again using the facts that yα 0 yβ, we obtain
β
β
γ Δ γ1
rty t Δt ≤ |Qt|yt yσ t yΔ tΔt.
α
3.49
α
Applying the inequality 2.37 with st |Qt|, p γ and q 1, we have
β
γ1
β
Δ γ1
1−γ
rty t dt ≤ 2 K1 α, β
rtyΔ t Δt.
α
From this inequality, after cancelling
completes the proof.
3.50
α
β
α
|yΔ t|
γ1
Δt, we get the desired inequality 3.46. This
Problem 1. It would be interesting to extend the above results to cover the delay equation with
oscillatory coefficients
Δ
γ−1
rtxΔ t xΔ t
qt|xτt|γ−1 xτt 0,
3.51
where the delay function τt satisfies τt < t and limt → ∞ τt ∞.
Acknowledgment
This project was supported by the Research Centre, College of Science, King Saud University.
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