Model 3G
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Computers and Mathematics with Applications
PR
OO
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journal homepage: www.elsevier.com/locate/camwa
New comparison and oscillation theorems for ∧second-order half-linear
dynamic equations on time scales
Jia Baoguo a,b , Lynn Erbe a , Allan Peterson a,∗
a
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
b
School of Mathematics and Computer Science, Zhongshan University, Guangzhou, 510275, China
article
info
a b s t r a c t
Article history:
Received 28 March 2008
Accepted 27 May 2008
ED
Let T be a time scale (i.e., a closed nonempty subset of R) with sup T = +∞. Consider the
second-order half-linear dynamic equation
∧
(r (t )(x∆ (t ))α )∆ + p(t )xα (σ (t )) = 0,
Keywords:
Half-linear dynamic equation
Condition (A)
Condition (Â)
Condition (B)
R∞
1
RE
CT
where r (t ) > 0, p(t ) are continuous, t (r (t ))− α ∆t = ∞, α is a quotient of odd positive
0
integers. In particular, no explicit sign assumptions are made with respect to the coefficient
p(t ). We give conditions under which every positive solution of the equations is strictly
increasing. For α = 1, T = R, the result improves the original theorem [see: [Lynn Erbe,
Oscillation theorems for second-order linear differential equation, Pacific J. Math. 35 (2)
∧
(1970) 337–343]]. As applications,
we get two comparison theorems and an oscillation
theorem for half-linear dynamic equations which improve and extend earlier results. Some
examples are given to illustrate our theorems.
© 2008 Published by Elsevier Ltd
1. Introduction
∆
CO
R
Let T be a time scale (i.e., a closed nonempty subset of R) with sup T = ∞. Consider the second-order half-linear dynamic
∧
equation
α ∆
α
(r (t )(x (t )) ) + p(t )x (σ (t )) = 0,
(1.1)
R∞
1
where r (t ) > 0, p(t ) are continuous, t (r (t ))− α ∆t = ∞, α is a quotient of odd positive integers. We emphasize that no
0
explicit sign assumptions are made with respect to the coefficient p(t ).
For completeness, we recall some basic results for dynamic equations and the calculus on time scales. The forward jump
operator is defined by
UN
σ (t ) = inf{s ∈ T : s > t },
and the backward jump operator is defined by
ρ(t ) = sup{s ∈ T : s < t },
where inf ∅ = sup T, where ∅ denotes the empty set. If σ (t ) > t, we say t is right-scattered, while if ρ(t ) < t we say t is
left-scattered. If σ (t ) = t we say t is right-dense, while if ρ(t ) = t and t 6= inf T we say t is left-dense. Given a time scale
∗
Corresponding author.
E-mail addresses: [email protected] (B. Jia), [email protected] (L. Erbe), [email protected] (A. Peterson).
0898-1221/$ – see front matter © 2008 Published by Elsevier Ltd
doi:10.1016/j.camwa.2008.05.014
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Definition 1. We say that a function g : T → R satisfies condition (A) if the following condition holds:
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
t
Z
t →∞
We wish to extend this notion to a triple of functions (α, p, r ), so we introduce the following definition:
27
CT
26
Definition 2. We say that the triple (α, p, r ) satisfies condition (Â), if there exists a continuously differentiable function
h : T → R, such that either h∆ (t ) is of one sign for all t ∈ T or h∆ (t ) ≡ 0 and is such that p(t )hα+1 (σ (t )) − r (t )(h∆ (t ))α+1
satisfies condition (A).
Notice that if h(t ) = 1, α = 1, then this means that p(t ) satisfies condition (A).
A continuous version of the following definition appeared in [7], Page 814.
28
29
32
33
34
35
36
37
38
39
40
41
42
DefinitionR 3. We say that a function p : T → R satisfies condition (B) in case there exists a sequence {τn } ⊂ T, τn → ∞,
t
such that τ p(s)∆s ≥ 0, for t ≥ τn .
n
It is obvious that condition (A) implies both condition (Â) and condition (B) (see [6]), but the converse is not true (see
Examples 1.1 and 1.2). In Section 2, we prove that if p(t ) satisfies condition (B) and the triple (α, p, r ) satisfies condition
(Â), then positive solutions of (1.1) are strictly increasing. This improves and extends a result of [6].
In Sections 3 and 4, we prove two comparison theorems that improve two main results of [8] and give two examples to
illustrate that our theorems are new.
In Section 5, we obtain an oscillation theorem that extends the results of [4,9,10] and give several examples to illustrate
our theorem.
The following examples show that the class of functions which satisfy condition (Â) and condition (B) but do not satisfy
condition (A) is nonempty.
CO
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31
UN
30
Example 1.1. Let q > 1. Consider the time scale T = qN0 := {qk : k ∈ N0 }. In this case, σ (t ) = qt, µ(t ) = (q − 1)t for all
t ∈ T. (Recall that any dynamic equation on the time scale qN0 is called a q-difference equation.) Let
p(t ) =
43
44
47
λ
β(−1)n
+
,
t (σ (t ))b
t (σ (t ))b
n :=
ln t
ln q
where λ > 0, 0 < b < 3. Let α = 3. Consider the q-difference equation
((x∆ (t ))3 )∆ + p(t )x3 (σ (t )) = 0.
45
46
6≡ 0,
for all large T .
24
25
and
T
RE
23
g (s)∆s ≥ 0
lim inf
22
ED
2
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20
interval [c , d] := {t ∈ T : c ≤ t ≤ d} in T the notation [c , d]κ denotes the interval [c , d] in case ρ(d) = d and denotes
the interval [c , d) in case ρ(d) < d. The graininess function µ for a time scale T is defined by µ(t ) = σ (t ) − t, and for any
function f : T → R the notation f σ (t ) denotes f (σ (t )).
The theory of time scales was introduced by Stefan Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and
discrete analysis (see [1]). Not only does this unify the theories of differential equations and difference equations, but it also
extends these classical situations to cases ‘‘in between’’— e.g., to the so-called q-difference equations which are important
in the theory of orthogonal polynomials. Moreover, the theory can be applied to numerous other time scales. We refer to
the two books on the subject of time scales by Bohner and Peterson [2,3] which summarize and organize much of time scale
calculus and applications to dynamic equations.
A function f : T → R is said to be rd-continuous provided it is continuous at right-dense points in T and its left-sided
limits exist (finite) at left-dense points in T. The set of rd-continuous functions f : T → R will be denoted by Crd . The set of
functions f : T → R that are delta differentiable on [c , d]κ and whose delta derivative is rd-continuous on [c , d]κ is denoted
∧
1
by Crd
.
We recall that a solution of Eq. (1.1) is said to be oscillatory on [a, ∞) in case it is neither eventually positive nor eventually
negative. Otherwise, the solution is said to be nonoscillatory. Eq. (1.1) is said to be oscillatory in case all of its solutions are
oscillatory. The study of the oscillatory and nonoscillatory properties of Eq. (1.1) and its many generalizations and extensions
is voluminous and we refer to [4,5] and the references therein.
The following condition (A) was introduced in [6] for the continuous case in order to obtain some new oscillation and
comparison results for the linear homogeneous differential equation in the case when the function p(t ) can take on both
positive and negative values for large t.
1
Let
m :=
qb − 1
qb + 1
,
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and assume further that 0 < λ < mβ . Then we have, for tn = qn ,
∞
X
1
p(s)∆s =
qb
tn
k=n
1
qk(1+b)
=
(q − 1)
=
(q − 1)
qnb
qnb
[λ + β(−1)k ](q − 1)qk
λ
β(−1)n
+ b
b
q −1
q +1
×
1
qb
1
−1
2
3
(λ + mβ(−1)n ).
PR
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F
∞
Z
4
Notice that this last expression may be negative, for large n, since 0 < λ < mβ . Hence, p(t ) does not satisfy condition (A).
b
4
Take h(t ) = t , r (t ) = 1. Then we have, for t = q
t
Z
n
{p(s)h4 (σ (s)) − r (s)[h∆ (s)]4 }∆s =

Z t
λ
1
1
β(−1)
+
s
s
ln s
ln q
"
−
b
q4 − 1
#4
q−1
6


1
∆s → ∞.
s4−b 
7
So the triple (3, p, 1) satisfies condition (Â).
Rt
Let τn = q2n . It is easy to see that τ p(s)∆s ≥ 0, for t ≥ τn and so p(t ) satisfies condition (B).
n
8
9
(ii) Let h(t ) = t
, r (t ) = 1. Then
t
Z
3
{g (s)h2 (s) − r (s)[h0 (s)]2 }ds = t 4
T
3
3
+ t− 4
t
3
s 4 sin sds
T
3
lim sup t − 4
t →∞
Take =
−
10
9
t
3
1
.
9
3
= 1 − ≤ t− 4
Z
t
3
4
3
3
+ t− 4
Z
t
1 −5
4 3
1 −5
t 4 − T4 −
T 4.
3
80
80
12
(1.2)
t
Z
s 4 sin s ds ≤ 1 + =
3
s 4 sin s ds
T
≥
1
9
∞
10
9
,
Similarly, we also can get
∞
{g (s)h4 (s) − r (s)[h0 (s)]4 }ds = ∞.
UN
Z
15
16
17
18
19
3
t4.
{g (s)h2 (s) − r (s)[h0 (s)]2 }ds = ∞.
T
14
3
s 4 sin sds = −1.
Hence by (1.2), we obtain
Z
13
T
CO
R
3
t →∞
We have
for large t.
Therefore, for large t, we have
t4
lim inf t − 4
T
T
3
s 4 sin sds = 1,
+
11
CT
Integrating by parts twice, it is easy to see that
Z
10
RE
∧
4
Z
ED
Example 1.2. Let T be the real interval [1, ∞),Rg (t ) = 1 + t sin t. Then we have R
∞
∞
(i) g (t ) does not satisfy condition (A), since T g (t )dt does not converge and T g (t )dt 6= ∞.
− 18
5
20
21
22
23
24
T
Therefore the triple (1, g , 1) and (3, g , 1) satisfy the condition (Â).
(iii) In the following, we show that g (t ) satisfies condition (B).
Assume that 0 < t1 < t2 < · · · < t2k < t2k+1 < t2k+2 < · · · are the positive zero points of g (t ). It suffices to prove that
Z
t2k+2
g (s)ds ≥ 0,
25
26
27
28
t2k
i.e.,
29
Z
t2k+1
t2k
g (s)ds ≥ −
Z
t2k+2
g (s)ds,
t2k+1
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for large k. That is,
t2k+1 − t2k+1 cos t2k+1 + sin t2k+1 − (t2k − t2k cos t2k + sin t2k )
≥ −(t2k+2 − t2k+2 cos t2k+2 + sin t2k+2 ) + (t2k+1 − t2k+1 cos t2k+1 + sin t2k+1 ).
3
4
5
6
7
Using the fact that sin tj =
t2k+2 −
1
−
t2k+2
q
−1
tj
2
t2k
+2 − 1 ≥ t2k −
1
t2k
−
q
2
t2k
− 1.
If we set
1
f (x) = x −
x
−
(1.3)
and rearranging, we see that (1.3) is equivalent to
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B. Jia et al. / Computers and Mathematics with Applications xx (xxxx) xxx–xxx
p
x2 − 1,
(1.4)
8
then it is easy to see that f 0 (x) > 0 for large x and therefore it follows that (1.4) holds for large k. This completes the proof.
9
2. Lemma
10
Lemma 2.1. Let x(t ) be a nonoscillatory solution of (1.1) and assume that p(t ) satisfies condition (B), the triple (α, p, r ) satisfies
11
condition (Â), and
15
16
17
18
n
Let us assume, for the sake of contradiction, that x∆ (t ) is not strictly positive for all large t. First consider the case when
x∆ (t ) < 0 for all large t. Then without loss of generality, we can assume that x∆ (t ) < 0 for t ≥ τk ≥ t0 , where k is large and
fixed. An integration of Eq. (1.1) for t > τk gives
p(s)xα (σ (s))∆s = r (τk )(x∆ (τk ))α .
∧
t
Z
τk
21
τk
Now integrating by parts, we have
19
20
t
Z
r (t )(x∆ (t ))α +
p(s)xα (σ (s))∆s = xα (t )
t
Z
τk
(xα (t ))∆ =
1
Z
Z
t
τk
30
31
32
33
34
α
∆
(x (t ))
Z
s
(2.2)
τk
p(u)∆u ∆s ≤ 0,
p(s)xα (σ (s))∆s ≥ xα (t )
Z
t
τk
p(s)∆s ≥ 0.
Consequently, from (2.1), we have
UN
29
τk
p(u)∆u ∆s.
and so from (2.2), we have
25
28
s
CO
R
t
τk
27
Z
since (x(t ) + hµ(t )x (t ))α−1 ≥ 0 and x∆ (t ) < 0. Hence, it follows that
23
26
τk
(xα (s))∆s
α(x(t ) + hµ(t )x∆ (t ))α−1 dh x∆ (t ) ≤ 0,
∆
Z
24
t
Z
(2.1)
By the Pötzsche Chain Rule, ([2] Theorem 1.90) we have
0
22
p(s)∆s −
ED
14
Proof. Suppose that x is a nonoscillatory solution of (1.1) and withoutRloss of generality, assume x(t ) > 0 for t ≥ t0 . Since
t
p(t ) satisfies condition (B), let τn be the corresponding sequence with τ p(s)∆s ≥ 0, for t ≥ τn .
CT
13
t0
RE
12
1
(r (t ))− α ∆t = ∞. Then there exists T ≥ t0 such that x(t )x∆ (t ) > 0, for t ≥ T .
R∞
r (t )(x∆ (t ))α ≤ r (τk )(x∆ (τk ))α ,
t ≥ τk .
Hence
x(t ) ≤ x(τk ) +
1
(r (τk )) α x∆ (τk )
Z t
1
τk
r (s)
α1
∆s → −∞,
as t → ∞, which is a contradiction.
So x∆ (t ) is not negative for all large t and since we are assuming x∆ (t ) is not positive for all large t, it follows that x∆ (t )
must change sign infinitely often.
Make the substitution
ω(t ) = r (t )
x∆ (t )
x(t )
α
hα+1 (t ),
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for t ≥ T1 . We may suppose that T1 is sufficiently large so that
t
Z
lim inf
t →∞
1
{p(s)hα+1 (σ (s)) − r (s)[h∆ (s)]α+1 }∆s ≥ 0,
(2.3)
holds and is such that ω(T1 ) ≤ 0, (i.e., x∆ (T1 ) ≤ 0).
ω∆ (t ) = r (t )
x∆ (t )
α ∆
x(t )
3
hα+1 (σ (t )) + r (t )
x∆ (t )
α
x(t )
(hα+1 (t ))∆
4
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OO
F
= −p(t )hα+1 (σ (t )) + r (t )(h∆ (t ))α+1
∆ α
x (t )
(x∆ (t ))α (xα (t ))∆ α+1
∆
α+1
α+1
∆
− r (t ) (h (t ))
−
(h (t )) + α
h
(σ (t )) .
x(t )
x (t )xα (σ (t ))
If we define (omitting arguments)
∆ α
x
(x∆ )α (xα )∆
(hα+1 )∆ + α σ α (hσ )α+1 ,
F (t ) := r (h∆ )α+1 −
x
x (x )
then we have
ω∆ (t ) = −p(t )hα+1 (σ (t )) + r (t )[h∆ (t )]α+1 − F (t ).
5
6
7
8
9
(2.4)
(i) Suppose that t ∈ T is right-dense. Then (hα+1 (t ))∆ = (α + 1)hα (t )h∆ (t ), so we have (again omitting arguments)
h ∆ i α+α 1
∆ α
( x x h )α

 [h∆ ]α+1
x
h
∆
.

F (t ) = (α + 1)r 
−h
+
α+
1

α+1
x
α
ED
We use Young’s inequality [11], which says that
p
− uv +
|v|
q
≥ 0,
p > 1, q > 1,
1
p
p
with equality if and only if v = uα , α := q .
So if we let
v=
∆
x (t )h(t )
x(t )
then we have that F (t ) ≥ 0 and
iff
x (t )h(t )
,
1
q=
(h
∆
(t )) =
F (t ) =
15
16
α+1
,
α
17
= h∆ (t ).
x(t )
hα+1 (σ (t ))−hα+1 (t )
.
µ(t )
r (t )hα+1 (t )aα+1
µα+1 (t )
14
18
19
(ii) Suppose next that t ∈ T is right-scattered. Then x∆ (t ) =
α+1
13
= 1,
q
p = α + 1,
CO
R
F (t ) = 0
∆
α
+
RE
u = h∆ (t ),
Let us put a :=
h(σ (t ))
,
h(t )
b :=
x(σ (t ))
.
x(t )
x(σ (t ))−x(t )
,
µ(t )
(xα (t ))∆ =
xα (σ (t ))−xα (t )
,
µ(t )
h∆ (t ) =
h(σ (t ))−h(t )
,
µ(t )
Then after substituting and rearranging we have
f (a, b)
UN
24
25
∂f
∂f
It follows that if a > b, then ∂ a (a, b) > 0, and so f (a, b) > 0. Likewise, if a < b, then ∂ a (a, b) < 0, and so f (a, b) > 0.
In other words, f (a, b) ≥ 0 and
h(σ (t ))
h(t )
=
x(σ (t ))
x(t )
∆
⇔
x (t )
x(t )
∆
=
h (t )
h(t )
.
From (i) and (ii), we obtain that F (t ) ≥ 0 and
∧
∆
iff
h (t )
h(t )
∆
=
x (t )
x(t )
21
23
∂f
(α + 1)a−2
(a, b) =
[(a − 1)α − (b − 1)α ].
∂a
aα
f (a, b) = 0 ⇔ a = b ⇔
20
22
where f (a, b) := (1 − a−1 )α+1 − (b − 1)α (1 − a−(α+1) ) + (b − 1)α (1 − b−α ).
Notice that f (a, a) = 0 and
F (t ) = 0
11
12
CT
|u|
q
10


p
2
T1
.
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Integrating both sides of (2.4) from T1 to t, we have
ω(t ) − ω(T1 ) = −
2
Z
t
{p(s)hα+1 (σ (s)) − r (s)[h∆ (s)]α+1 }∆s −
Z
In the following, we will consider two cases:
Case (I)
We then have
h∆ (s)
7
8
9
10
11
12
13
14
15
h(s)
18
≡
x∆ (s)
x(s)
.
So x(s) = Ch(s). Without loss of generality we assume that h(s) > 0, for s ≥ T1 , since the other case is similar. Therefore,
we have C > 0.
(i) If h∆ (s) > 0, for s ∈ [T1 , ∞), we have x∆ (t ) > 0, which is a contradiction to the assumption that x∆ (t ) changes sign
infinitely often.
(ii) If h∆ (s) ≡ 0, we will have p(s) ≡ 0, which contradicts the definition of condition (Â).
(iii) If h∆ (s) < 0, for s ∈ [T1 , ∞), we have x∆ (t ) < 0. which is also a contradiction to the assumption that x∆ (t ) changes
sign infinitely often.
Case (II)
F (s) 6≡ 0,
16
17
s ≥ T1 .
PR
OO
F
F (s) ≡ 0,
5
6
for s ≥ T1 .
In this case we can choose > 0 and T2 > T1 such that for t ≥ T2 ,
Z
t
F (s)∆s > .
19
CT
T1
By (2.3), there exists T3 > T2 such that for t ≥ T3 ,
20
Z
t
21
{p(s)hα+1 (σ (s)) − r (s)[h∆ (s)]α+1 }∆s ≥ − .
2
RE
T1
22
(2.5)
ED
4
F (s)∆s.
T1
T1
3
t
So by (2.5), when t > T3 , we have
ω(t ) ≤ ω(T1 ) +
23
2
− < 0,
26
3. Comparison theorems
CO
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25
which implies that x∆ (t ) < 0 for all large t > T3 , which is again a contradiction to the assumption that x∆ (t ) changes sign
infinitely often. This completes the proof of Lemma 2.1. We are now in a position to obtain some comparison results. Consider the second-order half-linear dynamic equations
27
29
∧
(r (t )(x∆ (t ))α )∆ + p(t )xα (σ (t )) = 0,
28
and
(3.1)
UN
24
(R(t )(x∆ (t ))α )∆ + a(t )P (t )xα (σ (t )) = 0,
30
(3.2)
34
where r (t ) > 0, R(t ) > 0, p(t ), P (t ) are continuous, a(t ) is continuously differentiable, and α is a quotient of odd positive
integers.
The following two lemmas from [8] are very useful in establishing oscillation, nonoscillation, and comparison results for
second-order linear and half-linear dynamic equations on time scales.
35
Lemma 3.1 (Riccati Technique). Eq. (3.1) is nonoscillatory if and only if there exists T ∈ [t0 , ∞) and a continuously differentiable
36
function ω : [T , ∞) → R such that r α (t ) + µ(t )ω α (t ) > 0 holds and
31
32
33
∧
1
37
ω∆ (t ) + p(t ) + S [ω, r ](t ) ≤ 0,
1
for t ∈ [T , ∞),
(3.3)
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where
1
at right-dense t,
2
at right-scattered t.
PR
OO
F
)
(
α+1
αω α



(t )

1

rα
!)
S [ω, r ](t ) = (


r
 ω 1−

(t )

1
1
µ
[µω α + r α ]α
If in Lemma 2.1, we let h(t ) ≡ 1 then it is easy to obtain the expression for S [ω, r ](t ) from the expression for F (t ).
Lemma 3.2 (Sturm–Picone Comparison Theorem). Consider the equation
α ∆
∆
α
[r̃ (t )(x (t )) ] + p̃(t )x (σ (t )) = 0,
3
4
(3.4)
where r̃ and p̃ satisfy the same assumptions as r and p. Suppose that 0 < r̃ (t ) ≤ r (t ) and p(t ) ≤ p̃(t ) on [T , ∞) for all large T .
Then (3.4) is nonoscillatory on [t0 , ∞) implies that (3.1) is nonoscillatory on [t0 , ∞).
The proofs of the following two theorems may be found in [8]:
(i) the function p(t ) satisfies condition (A),
1
R∞
(r (t ))− α ∆t = ∞,
t0
(iii) 0 < a(t ) ≤ 1, a∆ (t ) ≤ 0.
6
7
8
1
, 0 < r (t ) ≤ R(t ), P (t ) ≤ p(t ) for t ∈ [t0 , ∞) and
Theorem A. Assume a ∈ Ccd
9
10
11
12
ED
(ii)
5
Then (3.1) is nonoscillatory on [t0 , ∞) implies that (3.2) is nonoscillatory on [t0 , ∞).
13
1
, 0 < R(t ) ≤ r (t ), p(t ) ≤ P (t ) for t ∈ [t0 , ∞) and
Theorem B. Assume a ∈ Ccd
14
(i) the function aP satisfies condition (A),
1
R∞
15
16
CT
(R(t ))− α ∆t = ∞,
t0
(iii) a(t ) ≥ 1, a∆ (t ) ≥ 0, t ∈ [t0 , ∞).
(ii)
17
Then (3.1) is oscillatory on [t0 , ∞) implies (3.2) is oscillatory on [t0 , ∞).
18
RE
Our goal in this section is to show that condition (A) (i.e., condition (i)) in Theorems A and B can be weakened to the
assumptions that condition (B) and condition (Â) hold for the triple (α, p, r ).
1
, r (t ) ≤ R(t ), P (t ) ≤ p(t ) and
Theorem 3.3. Assume a ∈ Ccd
1
R∞
CO
R
(r (t ))− α ∆t = ∞,
t0
(iii) 0 < a(t ) ≤ 1, a∆ (t ) ≤ 0.
22
23
24
Then (3.1) is nonoscillatory on [t0 , ∞) implies (3.2) is nonoscillatory on [t0 , ∞).
25
∆
Proof. The assumptions of the theorem imply that there exists a solution x of (3.1) and T ∈ T such that x(t ) > 0 and x (t ) >
x∆ ( t )
1
UN
0 ≥ ω a + pa + S [aω, ar ](t )
27
28
29
30
∆
≥ ω a + Pa + S [aω, ar ](t )
≥ ω∆ a + ωa∆ + Pa + S [aω, ar ](t )
= (ωa)∆ + Pa + S [aω, ar ](t )
31
32
33
for t ∈ [T , ∞). Hence the function ϕ = ωa satisfies the generalized Riccati inequality,
34
∆
ϕ + P (t )a(t ) + S [ϕ, ar ](t ) ≤ 0
1
26
1
0 on [T , ∞) by Lemma 2.1. Therefore, the function ω(t ) = r (t )( x(t ) )α > 0 satisfies (3.3) with r α (t ) + µ(t )ω α (t ) > 0. We
have aS [ω, r ] = S [aω, ar ] (see Lemma 3.1).
Now, multiplying (3.3) by a(t ), we get
∆
20
21
(i) p(t ) satisfies condition (B), the triple (α, p, r ) satisfies condition (Â),
(ii)
19
35
1
with (ar ) α (t ) + µ(t )ϕ α (t ) > 0, for t ∈ [T , ∞). Therefore the equation
∆
α ∆
36
α
(a(t )r (t )(x (t )) ) + a(t )P (t )x (σ (t )) = 0,
is nonoscillatory by Lemma 3.1 and so Eq. (3.2) is nonoscillatory by Lemma 3.2 since a(t )r (t ) ≤ r (t ) ≤ R(t ).
37
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2
3
4
The corresponding ‘‘oscillation’’ result is
1
Theorem 3.4. Assume a ∈ Ccd
, R(t ) ≤ r (t ), p(t ) ≤ P (t ) and
(i) aP satisfies condition (B), the triple (α, aP , r ) satisfies condition (Â),
(ii)
R∞
t0
1
(R(t ))− α ∆t = ∞,
5
(iii) a(t ) ≥ 1, a∆ (t ) ≥ 0.
6
Then (3.1) is oscillatory on [t0 , ∞) implies (3.2) is oscillatory on [t0 , ∞).
7
8
9
10
11
Proof. The proof of Theorem 3.4 follows from Theorem 3.3. If we let b =
is nonoscillatory, then from Theorem 3.3, it follows that
(R(t )(x∆ (t ))α )∆ + b(t )a(t )P (t )xα (σ (t )) = 0,
is also nonoscillatory. That is,
(R(t )(x∆ (t ))α )∆ + P (t )xα (σ (t )) = 0,
PR
OO
F
1
B. Jia et al. / Computers and Mathematics with Applications xx (xxxx) xxx–xxx
1
,
a
then b(t ) ≤ 1 and b∆ (t ) ≤ 0. Therefore if (3.2)
is nonoscillatory. But then since P (t ) ≥ p(t ) and R(t ) ≤ r (t ), Lemma 3.2 (the Sturm–Picone comparison theorem) implies
that Eq. (3.1) is also nonoscillatory. That is a contradiction and completes the proof. 14
4. Examples
18
19
20
21
22
23
24
25
26
27
CT
17
In this section, we will give several examples to illustrate Theorems 3.3 and 3.4. Since Example 4.1 is somewhat involved,
we give the basic idea of its construction. We would also like to point out that in [12] the linear case (α = 1) for the case
T = R as well as several other illustrative time scales was extensively investigated and a wide class of functions of the form
t
p(t ) = ta2 + b sin
was determined which are such that the triple (1, p, 1) satisfies condition (Â). This was shown to lead to
t
a number of very useful comparison and oscillation results for the linear case. The following examples deal in an analogous
way with the case α 6= 1.
t
,
Example 4.1. Let α = 3, and let T be the real interval. Let us consider a function p(t ) of the form p(t ) = ta4 + b sin
t3
a > 0, b > 0. It is easy to observe that if a > 3b then p(t ) satisfies condition (A). So we seek to find conditions on a and b
such that p(t ) satisfies the conditions of Theorem 3.3 but does not satisfy condition (A). For simplicity, We consider the case
r ≡ 1 so that (1.1) becomes
((x0 )3 )0 (t ) + p(t )x3 (t ) = 0.
Let h(t ) = t γ , γ <
I (T ) = lim inf
= T 4γ −3
28
31
32
35
36
[p(t )h4 (t ) − (h0 (t ))4 ]dt
a − γ4
3 − 4γ
∞
p(t )dt = T
+ bT 3−4γ
Z
∞
T
sin t
dt .
t 3−4γ
−3
a
3
+ bT
3
Z
T
∞
sin t
(4.2)
t3
dt ,
it follows that p(t ) does not satisfy condition (A), if
(iii) Also we have
lim sup t 3
lim inf t 3
t →∞
∞
Z
t →∞
37
t
T
T
34
(4.1)
The basic idea of constructing Example 4.1 is based on the following steps (i)–(iv).
(i) By Theorem 5.2, when I (T ) = +∞, (4.1) is oscillatory. Therefore, in order that (4.1) be nonoscillatory, we choose
γ < 34 .
(ii) Since
Z
33
Denote
UN
30
3
.
4
Z
t →∞
29
RE
16
CO
R
15
ED
13
12
p(s)ds =
t
Z
t
∞
p(s)ds =
a
3
a
3
a
3
< b.
+ b,
− b.
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9
By Hille’s Theorem [11], if
−
2α + 1
α+1
1
α
α
α+1
∞
Z
< lim inf t 3
t →∞
p(s)ds
2
t
≤ lim sup t 3
∞
Z
t →∞
p(s)ds <
t
1
α+1
α
α+1
α
,
3
then Eq. (4.1) is nonoscillatory. Therefore, if we choose
7
3
4
43
− ×
<
a
3
−b<
a
3
+b<
3
3
44
4
,
then Eq. (4.1) is nonoscillatory. Note that a > 0, b > 0, a + b <
holds if we choose a > 0, b > 0 and a + b <
33
.
44
33
44
PR
OO
F
3
implies
a
3
− b > − 74 ×
Therefore, from (i)–(iv), if we choose 0 <
Therefore, Hille’s condition
That is, Eq. (4.1) is nonoscillatory.
a−γ 4
b.
3−4γ
a−γ 4
3
with 3−4γ
b, then it
4
1
, it follows that p t
16
(iv) From (4.2), we see that the triple (α, p, r ) satisfies the condition (Aˆ ), if we take
< b with
a
3
33
.
43
5
a
3
+b <
33
, and
44
1
1
,b
,
4
64
γ <
7
>
>
8
follows that the
triple (α, p, r ) satisfies condition (Â). In particular, if we take a =
=
γ =
()=
+
is
such that Eq. (4.1) is nonoscillatory.
Now if we set a(t ) := ct −d (log t )β , c > 0, d > 0, β ∈ R, then we have 0 < a(t ) ≤ 1, a0 (t ) ≤ 0, for large t. So by
Theorem 3.3, the equation
c (log t )β
4t 4+d
+
c (log t )β sin t
64t 3+d
x3 (t ) = 0
is nonoscillatory on (2, ∞) for all c > 0, d > 0, β ∈ R.
P (t ) =
a
t 1+b+c
sin t
+
t b+ c
,
a(t ) = t c ,
b
.
where a > 0, 0 < b < 3, c = 3−
2
We have
∞
a(s)P (s)ds = T −b
T
a
b
+ Tb
Z
∞
sin t
tb
T
b
dt
.
RE
Z
With h(t ) = t 4 , r (t ) = 1. Then we have
t
Z
4
t
(Â).
Take h1 (t ) = t
t
13
15
16
17
18
19
20
22
< 1, 0 < b < 3, a(t )P (t ) does not satisfy condition (A), but the triple (3, aP , 1) does satisfy condition
24
b+c
4
4
×
b−3
sb−3 |tT → ∞,
25
, r (t ) = 1. Then we have
{P (s)[h1 (s)]4 − r (s)[h01 (s)]4 }ds → ∞,
UN
Z
12
23
sin sds +
T
a
b
1
11
(t → ∞).
= a ln s |tT +
So when 0 <
b
10
21
{a(s)P (s)h4 (s) − r (s)[h0 (s)]4 }ds
T
CO
R
Z
9
14
CT
Example 4.2. Let α = 3, T = [1, ∞). Let
sin t
64t 3
ED
((x0 )3 )0 (t ) +
1
4t 4
6
26
(t → ∞),
27
T
b
where a > 0, 0 < b < 3, c = 3−
. By Theorem 5.2, ((x0 )3 )0 (t ) + P (t )x3 (t ) = 0 is oscillatory. Since a(t ) = t c ≥ 1, for t ≥ 1
2
0
and a (t ) ≥ 0, it follows by Theorem 3.4 that ((x0 )3 )0 (t ) + a(t )P (t )x3 (t ) = 0 is oscillatory.
γ
λ(−1)t
Example 4.3. Let T = Z, α = 3, p(t ) = t (σ (t ))3 + (σ (t ))3 , r (t ) = 1, γ > 0, λ > 0. We have
Z
∞
p(s)∆s =
t
∞ X
k=n
γ
(−1)k
+
λ
.
k(k + 1)3
(k + 1)3
k=n
29
30
Note that
∞ X
28
31
32
γ
γ
∼ 3,
3
k(k + 1)
3n
for large n.
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Also, we have
2
(2m + 1)3
(
= (2m)
→ 1−
4
1
(−1)k
→ −1 +
3
(k + 1)
2
k=2m+1
∞
X
(2m + 1)3
γ
So, in this case, if
Furthermore,
lim sup t
9
3
p(s)∆s =
p(s)∆s =
is nonoscillatory.
Let h(t ) = t β , β <
I (n) = lim inf
=
16
∞ X
k=n
∞ X
k=n
∞
Z
1
2
,
(n → ∞).
1
1
dx = − .
(1 + x)
2
2
0
−
,
2
λ
2
.
γ
3
+
λ
2
33
,
44
<
n
∼
β
lim n
1
(3 − 4β)n3−4β
−1
(2m + 5)3
3
2m+4
+ o
1
2m+4
5 3
#)
+ ···
(2m + )
#)
+ ···
then equation
.
CO
R
3−4β
γ
λ(−1)k
β
β 4
+ 3−4β − [(k + 1) − k ] .
k(k + 1)3−4β
k
k1−β
[(k + 1)β − kβ ]4 =
So
+
3
1
2m+4
, for large k, we have
β4
k4−4β
+
β4
k4−4β
o(1).
∞
∞ X
X
β4
β
β 4
3−4β
[(k + 1) − k ] = lim n
(4−4β)
n→∞
k=n
=
k
k=n
β4
.
3 − 4β
Therefore
∞
X
[(k + 1)β − kβ ]4 ∼
β4
.
(3 − 4β)n3−4β
γ −β 4
So the triple (3, p, 1) will satisfy condition (Â) if we take 3−4β > λ2 .
<
28
requirements. In particular, if we take γ =
1
,
3
29
is nonoscillatory.
Therefore, choosing 0 < β <
3
4
, 0<
γ
3
27
1
2m+2
3 3
1+
p(s)h4 (σ (s)) − (h∆ (s))4 ∆s
k(k + 1)3−4β
k=n
26
dx =
Denote
UN
25
t
1
n→∞
24
3
.
4
Z
Since (k + 1)β − kβ ∼
20
23
λ
+
Note that
18
22
3
t
t →∞
21
+ o
+
(2m + )
((x∆ )3 )∆ (t ) + p(t )x3 (σ (t )) = 0
15
19
3
By Hille’s Theorem [11], if we choose
12
17
γ
γ
∞
Z
t →∞
14
∞
Z
t
lim inf t 3
10
13
3
2m+2
< λ2 , p(t ) does not satisfy condition (A).
3
t →∞
11
1
0
(2m + 3)3
−
(1 + x)2
−1
CT
8
2
Similarly, we have
6
7
∞
Z
1
3
1
2m+2
RE
5
(2m + 1)3
1+
−
"
1
3
3
"
1
PR
OO
F
(
∞
X
(−1)k
= (2m)3
(2m)
3
(
k
+
1
)
k=2m
3
ED
1
B. Jia et al. / Computers and Mathematics with Applications xx (xxxx) xxx–xxx
λ
2
with
λ=
3
,
4
γ
3
+
β=
λ
2
<
7
.
10
4
33 γ −β
,
44 3−4β
γ
λ(−1)t
satisfies all the
t (t +1)3
(t +1)3
3(−1)t
1
is such that Eq. (3.1)
3t (t +1)3
4(t +1)3
> λ2 , then p(t ) =
it follows that p(t ) =
+
+
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5. Oscillation theorem
1
In this section, by means of Lemma 2.1, we obtain an oscillation theorem which extends some earlier results. The
following theorem may be found in [9], Theorem 5.81. (See also [10,4].)
∞
Z
[p(t )u2 (t ) − r (t )(u0 (t ))2 ]dt = +∞.
t0
t0
r −1 (t )dt = ∞ and there exists
Theorem 5.2. Assume that p(t ) satisfies condition (B) and assume
t0
t0
then all solutions of (1.1) are oscillatory.
x (t ) > 0 for t ≥ T . Make the substitution ω(t ) = r (t )
ED
∧
α
x∆ (t )
,
x( t )
where F (t ) ≥ 0. So
α+1
∆
ω (t ) ≤ −p(t )h
∆
(σ (t )) + r (t )[h (t )]
α+1
Integrating from T to t gives
t
21
CO
R
[(x ) ] (t ) + (1 + t sin t )x (t ) = 0,
3
λ
24
27
n
+
28
29
((x∆ (t ))3 )∆ + p(t )x3 (σ (t )) = 0.
UN
(5.2)
b
4
In Section 1, we have proved that p(t ) satisfies condition (B) and for h(t ) = t , r (t ) = 1, t = qn
t
23
26
β(−1)
t σ (t )b
where λ > 0, 0 < b < 3. Let α = 3. Consider the q-difference equation
Z
22
25
are oscillatory.
Let q > 1. Consider the time scale T = qN0 := {qk : k ∈ N0 }. Let
t σ (t )b
15
18
For T = R, we proved in Section 1 that p(t ) = 1 + t sin t satisfies condition (B) and the triple (3, p(t ), 1) satisfies
condition (Â). So by Theorem 5.2 all solutions of
p(t ) =
14
20
∧
0 3 0
13
19
But now the left-hand side is unbounded and the right-hand side is bounded. this contradiction proves the theorem.
∧
11
17
{p(t )hα+1 − r (t )[h∆ (t )]α+1 }∆t ≤ (ωhα+1 )(T ) − (ωhα+1 )(t ) ≤ (ωhα+1 )(T ).
T
9
10
16
.
RE
Z
for t ≥ T . By the proof of Lemma 2.1, we have
CT
ω∆ (t ) = −p(t )hα+1 (σ (t )) + r (t )[h∆ (t )]α+1 − F (t )
8
12
Proof. Let us suppose that (1.1) is nonoscillatory and x is a solution of (1.1). To be specific, suppose that x(t ) > 0 for all
large t, since the other case is similar.
By (5.1), we obtain that the triple (α, p, r ) satisfies condition
h
i (Â). In view of Lemma 2.1, we may then suppose also that
∆
7
1
(r (t ))− α ∆t = ∞. If there exists a continuously
differentiable function h : T → R, such that either h (t ) is of one sign for all t ∈ T or h∆ (t ) ≡ 0, and is such that
Z ∞
[p(t )hα+1 (σ (t )) − r (t )(h∆ (t ))α+1 ]∆t = +∞,
(5.1)
∆
4
6
Analogous to the above theorem, we may obtain a corresponding version for half-linear dynamic equations on time scales
which we state as follows:
R∞
3
5
PR
OO
F
Theorem 5.1. The equation (r (t )x0 (t ))0 + p(t )x = 0 is oscillatory on the interval [t0 , ∞), if
a continuously differentiable function u(t ) > 0 such that
R∞
2
{p(s)h4 (σ (s)) − r (s)[h∆ (s)]4 }∆s → ∞.
30
31
32
1
So by Theorem 5.1, all solutions of (5.2) are oscillatory.
Example 5.1. Let T = R, α = 3, r (t ) = 1 and p(t ) =
b
4
a
t 1+b
33
+
c sin t
,
tb
where 0 < b < 3, a > 0, c ∈ R. It is easy to see that p(t )
satisfies condition (B). Take h(t ) = t . We have
Z
34
35
∞
{p(s)[h(s)]4 − r (s)[h0 (s)]4 }ds = ∞.
T
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So by Theorem 5.2, all solutions of the second-order half-linear differential equations
((x ) ) (t ) +
0 3 0
2
a
t 1+b
+
∧
x3 (t ) = 0,
c sin t
tb
3
are oscillatory for all 0 < b < 3, a > 0, c ∈ R.
4
Note that
∞
Z
p(s)ds = t −b
5
b
t
6
a
+ ct b
∞
Z
sin s
sb
t
ds .
So for 0 < b < 3, a > 0,
lim inf t
7
3
Z
t →∞
∞
p(s)ds =
t


+∞
if

−∞
if
a
b
a
b
> |c |
< |c |.
By Hille’s Theorem [4], if
8
lim inf t 3
9
Z
t →∞
∞
p(s)ds >
t
1
4
3
3
×
4
,
that is :
a
b
> |c |,
PR
OO
F
1
B. Jia et al. / Computers and Mathematics with Applications xx (xxxx) xxx–xxx
then Eq. (4.1) is oscillatory.
Therefore the oscillation conditions of Eq. (5.3) that we get improve the oscillation conditions of Hille’s theorem.
12
Example 5.2. Consider the generalized Euler–Cauchy dynamic equation
((x∆ )α )∆ +
13
ED
11
10
β
xα (σ (t )) = 0,
(σ (t ))α+1
α
for t ∈ T. Take h(t ) = t α+1 . Then
∞
Z
15
{hα+1 (σ (t ))p(t ) − [h∆ (t )]α+1 r (t )}∆t
T
(
=
16
T
)
∆ α+1
α
β
∆t .
− t α+1
σ (t )
α
α
t
and in this case (t α+1 )∆ = α+
1
∞
(
19
T
22
23
24
25
26
1
− α+
1
. Therefore (5.5) can be rewritten as
α+1 #
Z ∞ "
∆ α+1
α
1
α
β
− t α+1
∆t =
β−
∆t = ∞
σ (t )
t
α+1
T
α α+1
α α+1
provided that β > ( α+
) . Hence every solution of (5.3) oscillates if β > ( α+
) , which agrees with the well-known
1
1
∧
oscillatory behavior of (5.3).
If T = Z, then (5.3) is the half-linear Euler–Cauchy difference equation
((x∆ )α )∆ +
β
x(t + 1) = 0,
(t + 1)α+1
α ∆
α
α
and we have t α+1
= (t + 1) α+1 − t α+1 . Therefore (5.5) can be rewritten as
Z ∞
Z ∞
h α
iα+1 i
α
α
β
1 h
− (t α+1 )∆
∆t =
β − (t + 1)[(t + 1) α+1 − t α+1 ]α+1 ∆t .
σ (t )
t +1
T
T
Note that
h
α
α
lim (t + 1) (t + 1) α+1 − t α+1
27
iα+1
=
t →∞
28
β
)
UN
21
(5.5)
CO
R
Z
20
(5.4)
If T = R, then the dynamic equation (5.3) is the half-linear Euler–Cauchy differential equation ((x∆ )α )∆ + t α+1 xα (t ) = 0
17
18
RE
∞
Z
CT
14
(5.3)
α
α+1
α+1
.
So
Z
29
T
∞
1
t +1
h
β − (t + 1)[(t + 1)
α
α+1
−t
α
α+1 α+1
]
i
∆t = ∞
Please cite this article in press as: B. Jia, et al., New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales,
Computers and Mathematics with Applications (2008), doi:10.1016/j.camwa.2008.05.014 ∧
CAMWA: 4402
ARTICLE IN PRESS
B. Jia et al. / Computers and Mathematics with Applications xx (xxxx) xxx–xxx
13
α α+1
α α+1
provided that β > ( α+
) . Hence every solution of (5.3) oscillates if β > ( α+
) , which agrees with the well-known
1
1
∧
oscillatory behavior of (5.3).
N
2
If T = q0 = {1, q, q , . . .}, q > 1. Then the dynamic equation (5.3) is the q-difference equation
((x∆ )α )∆ +
1
2
3
β
x(qt ) = 0,
(qt )α+1
4
and in this case, (5.5) can be rewritten as
∞
T
Z ∞
∆ α+1
α
β
∆t =
− t α+1
σ (t )
T
)
provided that β > q
α
q α+1 −1
q−1
α
q α+1 −1
q−1
α+1
α+1
5

#α+1 
" α


1
q α+1 − 1
β −q
∆t = ∞

qt 
q−1
PR
OO
F
Z
(
. Hence every solution of (5.3) oscillates if β > q
α
q α+1 −1
q− 1
6
α+1
.
7
α α+1
is different from ( α+
)
which is the well-known critical constant from the continuous and
1
8
the discrete cases.
The interested reader may give additional examples. We remark that the results in the example above may not be
obtained by any existing criteria, as far as the authors are aware.
10
References
12
Note that q
11
13
CT
ED
S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56.
M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
W.A. Coppel, Disconjugacy, in: Lecture Notes in Mathematics, vol. 220, Springer-Verlag, 1971.
L. Erbe, Oscillation criteria for second order linear equations on a time scale, Canad. Appl. Math. Q. 9 (2001) 1–31.
Lynn Erbe, Oscillation theorems for second order linear differential equation, Pacific J. Math. 35 (2) (1970) 337–343.
Lynn Erbe, Oscillation theorems for second order nonlinear differential equations, Proc. Amer. Math. Soc. 24 (1970) 811–814.
P. Řehák, Half-linear dynamic equations on time scales, Habilitation Thesis, 2005.
Miloš Ráb, Kriterien für die Oszillation der Lösungen der Differentialgleichung [p(x)y0 ]0 + q(x)y = 0, Časopis Pěst. 84 (1959) 335–370; 85 (1960) 91.
Aurel Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949) 115–117.
O. Došlý, P. Řehák, Half-linear Differential Equation, in: North-Holland Mathematics Studies, vol. 202, Elsevier, Amsterdam, 2005.
J. Baoguo, L. Erbe, A. Peterson, Some new comparison results for second order linear dynamic equations (submitted for publication).
UN
CO
R
RE
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
9
Please cite this article in press as: B. Jia, et al., New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales,
Computers and Mathematics with Applications (2008), doi:10.1016/j.camwa.2008.05.014 ∧
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Q1
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