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ETNA
Electronic Transactions on Numerical Analysis.
Volume 27, pp. 51-70, 2007.
Copyright 2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
LYNN ERBE AND ALLAN PETERSON
Abstract. We discuss a number of recent results for second order linear and nonlinear dynamic equations on
time scales.
Key words. measure chains, Riccati equation, oscillation, nonoscillation, time scales
AMS subject classification. 34B10, 39A10
1. Introduction and Preliminary Results. In this paper we will study certain linear
and nonlinear dynamic equations. In sections 2 and 3 we study the second order nonlinear
dynamic equation
(1.1)
"!#
#
where and
are real–valued, right–dense continuous
functions on a time scale $%'& with
,/.
'0
()+*
In sections 2 and 3 we also assume
is continuously differentiable
$
&213&
and satisfies
8
9
54
76
(1.2)
!
for
:
<"
;
!.
Although in section 2 we shall assume is a positive function we do not make any explicit
sign assumptions on in contrast to most know results on nonlinear oscillations. In sections
4 and 5 we consider (1.1) under slightly different hypotheses. In section 6 we consider an
example with damping and in section 7 we study comparison theorems for linear dynamic
equations. Finally, in section 8 we are concerned with the oscillation of an Euler–Cauchy
dynamic equation.
For completeness, we recall the following concepts related to the notion of time scales;
see [4], [5] for more details. A time scale $ is an arbitrary nonempty closed subset of the real
numbers & and, since boundedness and>
oscillation
of solutions is our primary concern, we
,/.
make the blanket assumption that (=)* $
We assume throughout
that $ has the topology
.
that it inherits from the standard topology on the real numbers & The forward and backward
jump operators are defined by:
?
@TACUV0W
70/@BACEDGFIH
LKM#ON70
:
UXY@TAC
#
(=)*
U
DGFJH
#PFRQSLKM#
$
ZH
#
$
$
and (=)*
where denotes the
empty set. A point
N9[\]#
RQ
I\]#
()+*
?
:
$
$
isN9said
to
be
left–dense
if
right–dense
if
and
left–
^Q_
].
0
:
$\1a&
scattered if
and right–scattered if ?
A function `
is said to be
right–dense continuous# (rd–continuous) provided ` is continuous at right–dense points and at
left–dense points in $ left hand
limits exist and are finite. The set of all such rd–continuous
].
functions
is
denoted
by
b7cLd
$
The graininess
function e for a time
scale $ 8 is defined
by
70
gfh
i0
.
?
?
e
, and for any function
$<1j&
the
notation
denotes
The assumption (1.2) allows to be of superlinear growth, say
where
@TAC
(1.3)
#
(=)*
0JF
$
$
8
/
kPlnmon#aph6"qr.
s
Received May 1, 2003. Accepted for publication November 25, 2003. Recommended by A. Ruffing. This work
was supported by NSF Grant 0072505.
Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, NE 68588-0323 (lerbe,
apeterso@math.unl.edu).
51
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52
L. ERBE AND A. PETERSON
In sections 4 and 5 we assume the nonlinearity has the property
8
9
(1.4)
!
:
and
8
6Su
t
t
t
v
;
!+#
for
t
for some
u
!.
:
This essentially says that the equation is, in some sense, not too far from being linear.
We shall see that one may relate oscillation and boundedness of solutions of the nonlinear
equation (1.1) to the linear equation
yx w (1.5)
z
Sz+
/!#
!
where :
, for which many oscillation criteria are known (see e.g. [1],[3], [4], [6], [8],
[11], [18], and [20]). In particular, we will obtain the time scale analogues of the results
& . We shall restrict attention to solutions of
due to Erbe [10] for the continuous case $
#],.
H
$
where |5}
may depend on the
(1.1) which exist on some interval of the form { |~}
particular solution.
#
On an arbitrary time scale $ the usual chain rule from calculus is no longer valid (see
Bohner and Peterson [4], pp 31). One form of the extended chain rule, due to S. Keller [26]
and generalized to measure chains by C. Pötzsche [31], is as follows. (See also Bohner and
Peterson [4], pp 32.)
0
$Y1&
is delta
differentiable
on $ . Assume further that
L EMMA 1.1. Assume `
v0
0
&13&
`
$1O&
is continuously differentiable. Then
is delta differentiable and
satisfies
o
(1.6)
g
`
4 S
`
e
`
].
`
We shall also need the following integration by parts formula (cf. [4]), which is a simple
consequence of the product
rule and which we
formulate
as follows:
#P[H
#
H
$
`
b cPd . Then
L EMMA 1.2. Let
and assume
(1.7)
S
8
?
R
`
8
{
`
S
f
9I
].
`
2. A Nonlinear Dynamic Equation. Before stating
our next results, we recall that a
#L,S
solution of equation (1.1) is said to be oscillatory on {
in case it is neither eventually
positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory.
Equation
!
:
(1.1) is said to be oscillatory in case all of its solutions are oscillatory. Since
we
shall consider both cases
q
S
(2.1)
g,
and
q
(2.2)
Q/,v.
We also introduce the following condition:
(2.3)
@T@BAC
FGF6!
¡
and
¢
;
!
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RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
for all large | . It can be shown that (2.3) implies either £
FGF¤>R,
53
or that
S
FGFI
@T
¡
FGF
¡
F¥=FJ6'!
exists and satisfies £ ¡
for all large | .
We have the following lemma which describes the behavior of a nonoscillatory solution
of (1.1) for the case when (2.1) and (2.3) hold. The following results in this section appear in
[17].
L EMMA 2.1. Let be a nonoscillatory
solution of (1.1) and assume conditions (2.1)
6
|
and (2.3) hold. Then there exists | o
such that
!
6
for
:
o
|
.
The first result is a boundedness
result for (1.1).
z
!
:
T HEOREM 2.2.
Let
and
assume that Equation (1.5)
is oscillatory.
Assume
that.
!
6
:
(1.2) holds and let be a nonoscillatory solution of (1.1) with
for all
|
Then
8
@T
(2.4)
0Wv¦¨§z.
z
!
C OROLLARY 2.3. Let :
and assume that Equation (1.5) is oscillatory and (2.1)
and (2.3) hold. Suppose that is a nonoscillatory solution of the generalized Emden–Fowler
equation
(2.5)
w x k=lrmo
"!.
Then
@B
t
v¦i§zª«© ¬
t
.
z
T HEOREM 2.4. Assume that Equation (1.5) is oscillatory for all
(1.2), (2.1), and (2.3) hold. Then all solutions of (1.1) oscillate.
The next theorem deals with the case when (2.2) holds.
z
T HEOREM 2.5. Assume that Equation (1.5) is oscillatory for all
(1.2), (2.2), and (2.3) hold. In addition, assume that
!
and suppose that
:
!
:
and suppose that
q
(2.6)
S­
FG
¡
®®F¤,/.
¡
Then every solution of (1.1) is either oscillatory or converges to zero on
#],.
{
3. Examples. Clearly, equation (1.5) is oscillatory iff equation
¯
(3.1)
q
°
"!
z
is oscillatory. It was shown in Erbe [11, Corollary 7] (see also Bohner and Peterson [4]) that
(3.2)
hS
"!
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54
L. ERBE AND A. PETERSON
is oscillatory if there exists a sequence
that
D¥P±MK
¯+³
@T
±
(3.3)
³
R0
@T^±
with
%/$
±²,
and e
±n
!
:
such
±r
±n8f
(=)*
>,/#
°
± e
FGFM.
where
We can therefore conclude that all solutions of (1.1) oscillate in
£ ´
case (1.4), (2.1), and (2.3) hold along with
¯+³
@T
±
(3.4)
z
±
°
±
z
,/#
e
!.
±nf
(=)*
We note that there is no assumption on the boundedness of and e . If (1.2),
for all :
(2.2), and (2.3) hold along with (3.4), then every solution on (1.1) oscillates or converges
to zero. One may also apply averaging techniques or the telescoping principle to give some
more sophistcated results (see Erbe, Kong, and Kong [13] and Erbe [10]).
]±H
$
As
a second
example, suppose that $ #Pu is such that there
exists a sequence
of points
±
,
P±n§
±n¶6u
1
µ
µ
e
with
and positive numbers
such
that
and
.
Then
if
±r±r^¸,
(1.2) and (2.3) hold and · o e
it follows from results
of
Erbe,
Kong,
and
z
!
Kong [13, Corollary 4.1] that all solutions of (1.5) are oscillatory for :
. Consequently,
all solutions of (1.1) are oscillatory.
º¹
As a third example, we
consider a particular
example
for the » case# when $
. If
8
y
k=lrmo
q
y
¢
¼ ½
has the form of (1.3) (i.e.,
),
, and
then it is known that
§
o #
o .
¿
¿
equation (3.2) is oscillatory if ¾ :
and is nonoscillatory if ¾
Since in this case (2.3)
holds
trivially,
it
follows
from
Theorem
2.2
that
all
nonoscillatory
solutions
of (1.1) satisfy
§ o ©
@B
«¬
À ¿
t
t
.
R EMARK 3.1. From Theoremz 4.64! in [4] (Leighton–Wintner Theorem) it follows that
equation (1.5) is oscillatory for all :
if
q
S
(3.5)
S
g
>R,/.
Since the second condition in (3.5) implies that (2.3) holds, Theorem 2.4 implies that all
solutions of the Emden–Fowler equation (2.5) are oscillatory. That is, the Leighton–Wintner
Theorem is valid for (2.5) and more generally
for (1.1) if (1.2) holds. We
note again that
"¹
there are no explicit sign conditions on
. For the special case when $
and (1.1) is
kÁ
(3.6)
where Ã
HÅÄ
l
l
kPÂ7moy"!#
lnmo
, it follows that (3.6) is oscillatory if
Æ
(3.7)
R,/.
l
lnÇo
That is (3.7) implies that the linear equation
¶kE
(3.8)
z
Sz+
l
l
lnmo
/!
!
is oscillatory for all :
and z
soÈ
oscillation
of (3.6) is a consequence of Theorem 2.4. If
qr#
we consider equation (3.8) with6q
then Theorem
4.51 of [4] (see also [18]) implies that
6
É
(3.6) is oscillatory if for any É
there exists É o
such that
Æ
l
@T
Ê Ç
l
±
©
Ê
6>qM.
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RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
55
Consequently, by Corollary 2.3, all nonoscillatory solutions of (3.6) satisfy
@B
§"qM.
l
t
l
t
4. Nonlinear Dynamic Equation with Positive
. In this section we shall consider
the
nonlinear
dynamic
equation
(1.1)
under
some
different
hypotheses.
We assume that both
#¥
¨0
are positive, real–valued right–dense continuous functions, and
is continuous
&'1Ë&
and satisfies (1.4), and we shall again consider the two cases (2.1) and (2.2).
InzvDošlý
and Hilger [8], the authors consider the second order linear dynamic equation
Ìq
(1.5)
and give necessary and sufficient conditions for oscillation of all solutions on
unbounded time scales. Often, however, the oscillation criteria require additional assumptions
on the unknown solutions, which may not be easy to check.
same
equation
and suppose
that there
In Erbe
and
Peterson
[18], the authors consider the
H
#
#L,S
v@TACED
0rH
#L,SLK
!#
:
$
e
{
exists
such that
is bounded above on {
,
and showed via Riccati techniques that
=
>,v.
Í
#L,S].
implies that every solution
is oscillatory on { <Î! It is clear
that the results given in
[18]
<ÏÁÐ9Ñ
qr.
cannot be applied when is unbounded, e
and
when Ò :
The
papers [18] and [9] also give additional linear oscillation criteria, and also treat more general
situations.
Recently Bohner and Saker [6] considered (1.1) and used Riccati techniques to give some
sufficient conditions for oscillation when (2.1) or (2.2) hold. They obtain some sufficient
conditions which guarantee that every solution oscillates or converges to zero.
We use a generalized Riccati transformation technique to obtain several oscillation criteria for (1.1) when (2.1) or (2.2) holds. Our results in this section improve the results given in
Došlý and Hilger [8] and Erbe and Peterson [18] and complement the results in Bohner and
Saker [6]. Applications to equations to which previously known criteria for oscillation are not
applicable are given. In section 6, we will apply our results to linear or nonlinear dynamic
equations of the form
(4.1)
g
Ó
Ò
Á
/!
¾
to give some sufficient conditions for oscillation of all their solutions.ÖT#
We shall first need to briefly discuss the exponential function Ô¥Õ
to be the unique solution of the IVP
'
#a
]#
which is defined
²qr#
where it is assumed that
×HØÙ0W²DGi0
is rd-continuous and regressive
$<1Ë&
KM.
We define
ØZmS0W²DGiHZØÙ0q
=8
!#H
KM.
:
e
$
For properties of this exponential function, see Bohner and Peterson [4]. One such property
that we will use is the formula
Ô Õ
?
#
q
{
e
]#=
Ô Õ
.
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56
L. ERBE AND A. PETERSON
hHXØ
]#LFG
<H¨Ø
, then ÔÁÕ
is real-valued and nonzero on $ . If
Also if
is always positive. The following results in this section
appear in [19].
L EMMA
4.1.
Assume
that
(2.1)
holds,
and
solves
(1.1) with
V+
.
Define Ú
Then we have
#
m
!
then ÔÁÕ
for all
:
]#
:
.
(4.2)
Q'!
!¶§
and
Ú
§
#a
Ú
:
½
Õ ¼ ­
­
£ Í
and
!§
(4.3)
q
§
#a
HØ
, assume that
Ö
is a differentiable function, and define the auxiliary
Û
³
]#
b
b
Ü
]#
³
Ü
y0W²q
e
70
ÔGc
]#=
o
0
e
Û
]#=
ÔGc
½
Õ ¼ ­
­
q
¤Ýu¨
q
o
=V
k
#
Û
2à
ß
Û
Þ
#
b
³
#
?
³
#
£ Í
.
½
Õ ¼ ­
­
£ Í
Next suppose Û
functions
:
q
]#
0WÈf
Û
e
Û
#
Û
b
for
.
:
We also introduce the following condition
á[
There exists
!
such that
:
µ
Û
]#
ã7§
Ôâc
µ
for all large
].
Our first oscillation result in this section is
T HEOREM
4.2. Assume
that (1.4), (2.1), and (A) hold. Furthermore,
assume that there
HvØ
×Ö
6
m
Û
Û
exists
such
that
is
differentiable
and
such
that
for
any
there exists a
o
:
so that
@B
(4.4)
(=)*
FGFI>,/#
©ä
where
³
ä
for
]#=
Ü
k
8f
ß
ä
.
#],.
³
o
b
#
Then equation (1.1) is oscillatory on {
³
From Theorem 4.2, we can obtain different
sufficient conditions
for oscillation
of all
.
^å!#
^æ!#
solutions
of
(1.1)
by
different
choices
of
For
instance,
let
then
Û
Û
]#=
Èqr#
/uX
Ü
Ô c
and
and we get the following well–known result.
C OROLLARY 4.3 (Leighton–Wintner Theorem). Assume that (1.4) and (2.1) hold. If
:
(4.5)
then equation (1.1) is oscillatory on
S
FGF¤>,v#
#L,S].
{
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57
RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
]#
o
, then Ô c
Í
and it follows that condition (A) holds, provided is
If Û
bounded above, and so Theorem 4.2 yields the following result:
C OROLLARY 4.4. Assume is bounded above, that (1.4) and (2.1) hold, and for any
6
:
there is a o
such that
@T
(4.6)
¯
(=)*
FG
?
á
FG
°
F k
ß
FG
k
f
F
Þ
©ç
FG
u¨FG
FGâè
ßMé
F¥
b
FGêè
FI",/#
b
ç
where
¯
fRq
q
áFG70
F
q
F
FG8f
FG
F
FG °
e
#
FG
e
F k FG
FG70
é
b
b
.
#L,S].
Then (1.1)
is oscillatory
on {
²q
8
9
'
If
and
, then equation (1.1) reduces to the linear dynamic equation
g
(4.7)
H
v!+#
#],.
for
{
From Theorem 4.2 we have the following oscillation criterion for equation
(4.7) which improves some of the results in Bohner and Saker [6] and Erbe
and Peterson
[14].
6
:
there is a o
C OROLLARY 4.5. Assume that (1.4) and (2.1) hold and for any
such that
@T
(4.8)
(=)*
¯
Ý
q
°S
Þ
F
?
k
f
FG
F k
ß
©
á
q
Ý FG8f
FG
?
o
b
FG
o
b
FG à
ßMé
o
o
F¥
FG
FI",/#
à
where
¯
fRq
á
q
F
F
e
F¥ °
o
b
FG
e
FfX
FG
FG
o
FG8f
FG
o
b
F7
é
q
FG
o
#
e
F k
o
b
FG
²q
.
#],.
Then equation (4.7) is oscillatory on {
E XAMPLE 4.6. Consider the Euler–Cauchy dynamic equation
¦
g<
(4.9)
for
"!#
H
#L,S].
{
Here
(4.10)
@T
()*
If $
&
equation
#
ë
.
¼ ½
Then (4.8) in Corollary 4.5 reads
¦
q
ÝìÝ
?
f
F
Þ
FG
?
F
ß
F k
©
á
k
b
f
o
FG à
F¥
o
b
ßMé
o
o
FG
F¥
F¤>,/.
à
then the dynamic equation (4.9) is the second order Euler–Cauchy differential
¦
í í
(4.11)
and in this case
rewritten as
"!#g6"q
k
FGJî!#
e
?
FGîFr#
b
o
F¥Jåq
and
á
o
FGJ!.
Therefore (4.10) can be
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58
L. ERBE AND A. PETERSON
@B
¦
()+*
q
Ý
F
F
f
F[
F
Þ
à
F k
ß
@T
¿
Ý
©
¦Zf
(=)*
F[,v.
©
.
¦
o
à
F
#
¦
o
o
provided that : ¿ Hence every solution of (4.11) oscillates if : ¿ which agrees with
the well–known
oscillatory behavior of (4.11), (see Li [29]).
"¹
If $
, then (4.11) is the second order discrete Euler–Cauchy difference equation
¦
¶kÁ
(4.12)
FG
and we have e
²qM#
FG
"FvqM#
?
Ð
FG
o
b
v!#g
mo
~/q¥
­
Ð
qM# Þ
Í
m o
Í
#Á.T.B.
#
­
á
k
é
k
F¥
o
FG
o
F k
.
F/q¥ÁFªfX
/q¥ k
Therefore (4.10) can be rewritten as
@B
ÝÝ
@T
f
©
o
F k
ß
F
F7/q¥Fyfh
ÁFªfX
/q¥ à
q
F[,v.
F
Þ
F
ß
à
©
¦
q
f
F
k
f
à
F¥ï
ß
¦
Ý
fSq
F
Þ
(=)*
F k
F
q
¦
()+*
.
¦
o
#
provided that : ¿ Hence every solution of (4.12) oscillates if : ¿ which agrees with
the well–known
oscillatory behavior of (4.12). It is known in Zhang and Cheng [32] that
§\qGð ß
when e
, (4.12) has a nonoscillatory solution. Hence, Theorem 4.2 and Corollary 4.5
are sharp. Note that the results in Došlý and Hilger [8] and Erbe and Peterson [18] cannot
be applied to (4.12).
T HEOREM
4.7.
Assume that
(1.4) and (2.1) hold. Furthermore,
assume
that there
exists
HØ
ñÖ
6
m
a function Û
such that
there is a o :
Û is differentiable and given any
such that
³
q
@T
(4.13)
()*
Â
©
FG
³
FG
o
b
F[,v#
FG
à
is a positive integer. Assume further that
¯
(4.14)
k
F¥gf
ß
where Ã
Ý Ü
8f<FGÂ
°
Â
Â
q
FM#
Ô c
ã5FG
~FG
Ð
o
Æ
Û
ò Ç
©
?
FG8fX ò
FfXÂ
Ð
ò Ð
oÁF
#L,S
is bounded above. H
Then
every solution
of equation (1.1) is oscillatory
on {
.
Ø
y§!#
"!#
m
Note that if Û
and Û
then (4.14) holds. When Û
then (4.13) reduces
to
q
(4.15)
@T
()*
8f<FG Â
Â
F¥=F¤>,v#
©
which can be considered as an extension of Kamenev type oscillation criteria for second order
differential equations,
(see Kamenev [24]).
& , then (4.15) becomes
When $
q
(4.16)
@T
(=)*
fFG Â
Â
©
FGìFI",/#
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RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
and when $
/¹
59
, then (4.15) becomes
Ð
q
@T
(4.17)
(=)*
o
Æ
fF¥ÂIFG
Â
Ç
>,/#
©
­
We next give some sufficient conditions for the case when (2.2) holds, which guarantee
#],.
that every solution of the dynamic equation (1.1) oscillates
or converges to zero on {
The next result removes a monotonicity assumption on in Bohner and Saker [6].H¨Ø
m
T HEOREM 4.8. Assume that (1.4) and (2.2) hold and assume there exists Û
such
Ö
that Û is differentiable and such that (4.4) holds. Furthermore, assume
q
(4.18)
FGFG
>,/.
and let (A)#L,S
hold.
Then every solution of equation (1.1) is either oscillatory or converges to
zero on {
.
In a similar manner, one may establish the following theorem.
T HEOREM 4.9. Let all of the conditions of Theorem 4.8 hold with condition (4.13)
replacing
(4.4). Then every solution of equation (1.1) is oscillatory or converges to zero on
#],.
{
5. Application to equations with damping. Our aim is to apply the results in section
4, to give some sufficient conditions for oscillation of all solutions of the dynamic equation
(4.1) with damping terms. We note that all of the results in section 4, are true in the linear
case. Before stating our main results in this section we will need the following Lemmas, (see
Bohner and Peterson [4]).
#
H
b cPd and
L EMMA 5.1. If Ò ¾
qyf
(5.1)
k
e
Ò
I/
;
!#aH
e
$
then the second order dynamic equation (4.1) with
form (1.5), where
(5.2)
]#=
Ô ë
#ó
with
8
9
v
q
Ò
qyf
f
e
e
¾
e
can be written in the self-adjoint
¦8 {
¦g
(5.3)
#
¾
.
¾
k
Ò
e
¾
L8EMMA
5.2. If Ò is a regressive function, then the second order dynamic equation (4.1)
/
can be written in the self-adjoint form
(5.4)
= V
"!#
where
(5.5)
]#=
Ô Ñ
and
¾
Now, by using the results in section 3 and Lemma 5.1 we have the following results
immediately. The following
results appear in [19].
#=
T HEOREM 5.3. Let
be defined as in (5.5) and ssume that (2.1) holds. Furthermore,
HØ
assume that there exists a Û
with Û differentiable such that (4.4) holds with
q
(5.6)
Ü
Ô c
?
]#
¤Ý
= Þ
Û
k
.
Û
ß
b
à
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60
L. ERBE AND A. PETERSON
8
/
#],.
is oscillatory on {
Then equation (4.1) with
C OROLLARY 5.4. Assume8that
(2.1) and (3.5) hold, where
and are as defined in
/
#],.
(5.5). Then equation (4.1) with
is oscillatory on {
u¨
C OROLLARY
5.5.
Assume
that
(2.1)
and (4.6) hold except that the term 8
²
is
re]#
placed by
where
and
are
as
defined
in
(5.5).
Then
equation
(4.1)
with
is
#L,S].
oscillatory on {
H<Ø
T HEOREM 5.6. Assume that (2.1) holds. Furthermore,
assume that there exists Û
#Á
Ü
with Û differentiable such that (4.10) holds, where 8
and
are as defined by (5.5)
and (5.6)
v
#L,S].
respectively, and à is odd integer. Then (4.1) with
is oscillatory on {
T HEOREM 5.7. Assume that all the assumption of Theorem 5.3 hold except that the
condition
(2.1) is replaced by (2.2). If (4.18) holds,#L,S
then
every solution of equation (4.1) with
8
9
'
].
is oscillatory or converges to zero on {
T HEOREM 5.8. Assume that all the assumption of Theorem 5.6 hold except that the
condition
(1.3) is replaced by (2.2). If (4.18) holds,#L,S
then
every solution of equation (4.1) with
8
9
'
].
is oscillatory or converges to zero on {
Oscillation criteria for equation (4.1) are now elementary consequences of the oscillation
results in Theorems 5.3-5.8. The details are left to the reader.
6. Linear Second Order Dynamic Equations. In this section we shall be interested in
obtaining comparison theorems for the second order linear equations
RôV
~
(6.1)
"!#
{
(6.2)
(6.3)
Iô<
Ú
õ {
{
!
5
Ú
õ
~
"!#
"!#
#¥
.
:
where
and
, are right-dense continuous on $
D
EFINITION 6.1. We say that a solution of (6.1) has a generalized zero at in case
"!.
]#
ªQS!
!
?
:
We say has a generalized zero in ?
in case
and e
.
#=r
We say that (6.1) is disconjugate on the interval
{ ö
,
if
there
is
no
nontrivial
solution
of
#Pn.
(6.1) with two (or more) generalized zeros in { ö
#L,S
D EFINITION
6.2. Equation (6.1) is said to be nonoscillatory
on { ÷
if there exists
H
#L,S
#=r
:
ö
{ ÷
{ ö
ö . In the opposite
such that this equation is disconjugate
on
for
every
#L,S
. Oscillation of (6.1) may equivalently be defined
case (6.1) is said to be oscillatory on { ÷
as follows. A nontrivial solution
Ú
of
(6.1)
is called oscillatory if it has infinitely many
#L,S
(isolated) generalized zeros in { ÷
. By the Sturm type separation theorem, one solution of
(6.1) is (non)oscillatory iff every solution of (6.1) is (non)oscillatory. Hence we can speak
about oscillation or nonoscillation of equation (6.1).
Basic oscillatory properties of (6.1) are described by the so-called Reid Roundabout
Theorem which is proved e.g. in [4, Theorem 4.53, Theorem 4.57].
T HEOREM 6.3 (Reid Roundabout Theorem). The following statements are equivalent:
#=r
(i) Equation (6.1) is disconjugate on { ö
.
#Pn
(ii) Equation (6.1) has a solution without generalized zeros on { ö
.
(iii) The Riccati dynamic equation
(6.4)
ø
k
[V
/!
ø
e
!
:
has
a solution ø with
for
e
ø
is left-dense and right-scattered at which
ø
ªH
#Pnù
(except for the case when
may be nonpositive).
{ ö
e5ø
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61
RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
ú
(iv) The quadratic functional
ûü
#=ì
w ûnI x k
fhûn~=kÿ[
ýþ
û¶H
is positive definite for
d
ö
#P
, where
#P
ö
o
²DEûH
ö
b
#=rJ0gû
vû
/!K.
{ ö
Õ
ö
This result makes it therefore clear that there are at least two methods of investigation of
(non)oscillation of (6.1). The first one – the variational method – is based on the equivalence
of (i) and (iv) and its basic statement can be reformulated
as#],
follows:
H
!S;
ûXH
{ ÷
|
L EMMA 6.4 (Variational method). If for any |
there exists ¢
,
where
g²DEû¶H
o
|
b
Õ
#L,S^0gû
/!
{ |
ú
ú
ûü
#],g
û#
#
for
H
#
{ ÷
|
and
#
|
such that
|
:
?
û
"!
]#
|
for
H
{
?
|
#L,SLK#
7§'!
?
|
|
|
such that
, then (6.1) is oscillatory.
Another method of investigation for the oscillation theory of (6.1) is based on the equivalence of (i) and (iii) in Proposition 6.3. This is usually referred to as the Riccati technique
and by virtue of the Sturm Comparison Theorem implies that for nonoscillation of (6.1), it is
sufficient to find a solution of the Riccati-type inequality as given in the next lemma. A proof
may be found in [12] or [4].
L EMMA
6.5 (Riccati technique). Equation (6.1) is nonoscillatory if and only if there
H
#],
exists |
and a function ø satisfying the Riccati dynamic inequality
{ ÷
k
V
ø
§S!
ø
e
with
~
!
H
ø
#],
{ |
for
.
For completeness, we recall the following (see [4])
L EMMA 6.6 (Sturm-Picone Comparison Theorem). Consider the equation
e
ø
:
where
§
Rã<
(6.5)
5
/!+#
{
§Ù
and satisfy
the same assumptions as and . Suppose that
and
#L,S
#L,S
on { |
for
all
large
| . Then (6.5) is nonoscillatory on { ÷
implies
(6.1)
is
#],.
nonoscillatory on { ÷
We mention first a few background details which serve to motivate
the results in this
!#LR,S
section which may be found in [20]. Suppose that $ is the real interval {
so that (6.1)
becomes
4ô4V
"!#
(6.6)
{
!#LR,S
where
is continuous and positive and
is continuous on
. It was shown
in
{
[9]
that
if
(6.6)
is
oscillatory,
then
multiplying
the
coefficient
by
a
function
where
6q
4 and
is nonincreasing preserves oscillation; i.e.,
(6.7)
{
4 4
/!#
is also oscillatory. Of course, if
is nonnegative, these
results follow immediately from
the usual Sturm-Picone Comparison Theorem, but when
changes sign on each half line,
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62
L. ERBE AND A. PETERSON
obvious
if (6.6) is oscillatory. One may also notice that if (6.6) is
oscillation of (6.7) is
not
pFE\z6\q
ö
oscillatory and if
then oscillation
of (6.7) follows immediately
from
z
the Sturm-Picone Theorem by dividing the equation by (for the case when
may change
sign). This result, i.e., the statement that says that if (6.6) is oscillatory, then so is
4T4Sz+
Ú
/!
Ú
z/6îq
for any constant
, was also observed by Fink and St.Mary [21]. Kwong in
[26] then
showed that the result of [9] may be strengthened
to
a
larger
class
of
functions
by relax
4 ing somewhat the monotonicity assumption on
as given in [9]. We present below
three different comparison theorems along with their corresponding corollaries, and show by
examples, that they are all independent. In addition to extending the results of [26] and [9] in
the case of equations (6.6) and (6.7) in the continuous case, the results we obtain are new in
the discrete case and the more general time scales case. It should also be noted that because
of the techniques of proof used, both (6.2) and
(6.3) may
be viewed as the time-scales
ex
obtained
when
multiplying
by
(which
is
the
same
as
when
tensions
of
(6.1),
$
& .)
is more positive than negative, then the
Our first result
shows that if, “on average”,
assumptions on
are quite mild. To be precise, we have
H
o
b cLd and
T HEOREM 6.7. Assume
B
@
B
@
A
C
¥
F
=
J
F
6
!
;
!
(i)
£ ¡
but ¢
for all large | ,
F¤>,/#
o
½
(ii) £ ´ Õ ¼
!Q
­ 7§"qM#
y§!+.
(iii)
Then (6.1) is nonoscillatory implies (6.3) is nonoscillatory.
The corresponding “oscillation” result is
H
o
C OROLLARY 6.8. Assume
b cPd and
B
@
B
@
A
C
G
F
G
F
F
6
!
;
!
£ ¡
(i)
but ¢
for all large | ,
F¤>,/#
o
½
(ii) £ ´ Õ ¼
­ qr#
6
76!.
(iii)
Then
(6.1) is oscillatory implies (6.3) is oscillatory. Ifwe
strengthen the assumptions on
somewhat, then we may relax the assumptions on
and in this case, we consider the
relation between (6.1) and (6.2).
For
convenience, we state first the “oscillation” result.
H
o
b cLd and
T HEOREM 6.9. Assume
6qr#
(i) 76S!+#
(ii) e
w x §'!.
(iii)
Then (6.1) is oscillatory implies (6.2) is oscillatory.
In this case, the analogous “nonoscillation” result becomes
o H
#
o
C OROLLARY 6.10. If
b cPd
!Q
7§"qM#
(i) 7§S!+#
(ii) e
¯
(iii)
o
½
¼ °
§'!.
Then (6.1) is nonoscillatory on implies (6.2) is nonoscillatory.
In the following theorem we let denote the characteristic function of the set of rightscattered points $ defined by
$
0²DE
H
0
$
e
!K.
:
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RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
That is,
qr#OH
y0W
$
!+#ORðH
$
H
63
.
#
o
T HEOREM
6.11.
Assume
and that the following conditions hold:
b cLd
!
~
7§
Þ
Þ
:
and
e
!
H
#
o e
:
(ii)
for! some o :
and for all
$
(iii) there is an :
such that the function
70
Í
x k
58f
e
w
Iã=x w
Þ
f
for all large , where
@T
(=)*
8f
!#
:
f[,/#
e
FGF
6S!
:
£ ´
(iv)
!
ã7§
#
H
.
(v) there exists a constant µ :
such that
for
µe
$
Then (6.1) is nonoscillatory implies (6.2) is nonoscillatory.
Again, we have a corresponding “oscillation”
result:
w o x H
o
C OROLLARY 6.12. Assume
b cLd and that the following conditions hold:
¼ ½
Ó
k
(i)
(ii) there is an
(iii) there exists
¼ ½
¼ ½
o
:
6>q
!
!
:
for all large ,
H
o for
:
such that
e
such that the function
Í
70
Þ
for all large , where
8f
e
@T
!#
k
#
F¥FGF
°
§!#
q
fg
¯
and where
:
0
8f
e
ä
,
$
f[,
(=)* :
£ ´
(iv)
, !
ã7§
(v) there is an µ :
such that
µe
for all large .
Then (6.1) is oscillatory, implies that (6.2) is oscillatory.
We notice in the last two results how the graininess function is!+involved
in thecriteria
#LR,S
!
¢
for oscillation/nonoscillation. In particular, for the case when $
, then e
,
{
so conditions (ii), (iv), and (v) of Corollary 6.12 hold trivially, and it may be shown that (iii)
reduces to the condition of Kwong in [26]. The nonoscillation result Theorem 6.11 is new in
all cases. It turns out that Lemma 2.1 is useful in proving the above results.
7. Examples and Remarks. We begin this section with several examples showing the
independence of the above criteria.
q
E XAMPLE 7.1. Let Û : . Consider the time scale
Í
$
]#
\
±
0
Û
0
Û
fq¥
?
Û
e
Û
In this case,
for all
Í
scale Û
is called an r-difference equation. Let
q
H
#
$
¦
#
v
k
H×Ä
É
and
Û
fq¥
.
and any dynamic equation on the time
A
Û
A 7
"!
zfRq¥
A
Û
Ay
¼ ½
#
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64
L. ERBE AND A. PETERSON
¦8#Pz
are real constants and
where
zh"
;
!
Ay
of one sign for
. Since
as
Ay=ð
$%
q
¯
Û
F
o
Ay
is not eventually
,
A
1
Û
('
q
fSqG
F[
FG
&
H×Ä
. Observe that
, it follows that we have
c
½
o ã
¼ c Ð
and so (2.1) holds. Further,
1
A
Û
FI"
o
,
0
q
#
°
È
/q¥ k !
:
Û
*) +
¶H
$ , so condition (iii) of Corollary 6.10 fails to hold. Note that this condition is not
for
v]Ð
!
even satisfied for any
, : . On the other hand, we have
for
I x w
Þ
gf
!¶Q
e
§"qâð
Þ
{
fq¥
Iã x k
w
f
vqG k
Û
qyf
¿¥¿
Þ
{
fq¥
Û
Û
6!
, and condition (iii) of Theorem 6.11 is satisfied, and
Û
vq
f
Q'!#
Û
k ï
Û
!Q
QqGð
fqG7§
yQh§
o
so (iii)
of Theorem 6.7 holds. If
, then o e
for
Û
µ
µe
7H
all
$ , so conditions (ii) and (v) of Theorem 6.11 hold. Breaking up the integral and using
¼ ½ 8FGF[
=8
the identity £
we get
e
¦
/
FGFI
Ay
fq¥=zfRq¥
Û
!
q
¼ ½
q
Ý
q
f
Ay
A
fS.E.Á.
A
Û
Û
Hence
¦Zf
fq¥=z×Q
Ay+S
Û
FGFJQ¦"
fSqGz.
Û
In [30] it was proved that equation (6.1) is nonoscillatory provided
(7.1)
-,
f
o
£ ´
and
o
½
Õ ¼
e
@T
Õ ¼
"!
F
½
­
q
Q
ß
@B@TAC
7§
/.
@T
(=)*
.
Q
#
ß
where
¯
.
0
¯
q
FE°
FG
o
We have
o
½
Õ ¼
e
£ o
o
Õ ¼
­
½
F
A
Û
Ay
FGFE°.
k .
à
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RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
65
and so condition (7.1) is satisfied. Further,
fSq
fSq
¦Zf'
Û
A
fq¥zQ
{
.
Û
Û
Set
¦8
Û
Ò
A
{
fq¥=z.
Û
Û
z
fq¥ k
and
A
Û
!
¦/
Û
fq¥
6
yQ
Û
.
A
¾
Û
QqGð ß
:
¾
¾
If Ò
and Ò
, then (6.1) is nonoscillatory and Theorem 6.7 or Theorem
6.11
!Q
Q²f
¾
Ò
can bef appliedf toð show that (6.3) and (6.2), respectively, are nonoscillatory. If
ß
:
¾
and Ò
, then (2.3) fails to hold, equation (6.1) is nonoscillatory and in this case,
only Theorem 6.11 can be applied.
E XAMPLE
7.2. ÈqGð v¹
]
~vq
(i) Let $
,
and
. Then condition (v) of Theorem 6.11
fails to hold since
is unbounded. Theorem 6.7 (for
satisfying (2.3)) or Corollary 6.12
can be applied since
,
10
20
q
Æ
0
Æ
Ç
Ç
´
´
0
3
and
¯
¯
54 0
q
"
°°
0
0
5
0
8f
q
",/#
5/q
5/q
QS!
vqG
0 0
~/q
" Ð
¹
_ Þ /qG Ð
k
(ii) Let $
,
and
!
1
fails to hold since
as 1
lary 6.12 can be applied since
q
Æ
Ç
´
/!+.
Þ
k
,v#
´
fRq7f
o
Þ 5/q¥
Ç
0 *6
. Thencondition
(ii) of Theorem 6.11
. Theorem 6.7 (for
satisfying (2.3)) or Corol-
,
Æ
5/qyf
Q!
5/q¥ k
and
¯
¯
q
=
w Þ 5/q¥ Ð
°°">
on Þ ~/q¥x[/!+.
-¹
R2¦Ð
¦
q
I-z
zH>!#Eq¥
(iii) Let $
,
, :
and
,
. Then
condition
(ii) of
!
,
Theorem 6.11 and condition (ii) of Theorem 6.7 fail to hold since
as 1
and
1
q
Æ
Æ
z
Ç
´
Ç
´
Ð
>,/#
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66
L. ERBE AND A. PETERSON
respectively.
On the other hand, the assumptions of Corollary 6.12 are satisfied provided
¦~zÅHh!#Eq
since we have
Ð
Èqyf¨¦~¦
Ð
o
Q!
and
¯
¯
q
°°"\ã¦Zfq¥Áã¦~zfq¥¦5z
=
Q'!.
Notice that only Corollary 6.12 can be applied in this case.
Following the idea of the above examples, it is not difficult to find examples showing the
independence of Theorem 6.9
and Corollary 6.12.
"¹R#]
²q
E XAMPLE 7.3. Let $
,
q
It is easy to see that
f
Þ 5
Þ
~/q¥
zh/
;
!
changes sign for
and
Þ ~/fRq¥
zfRq¥
¦
.
and
Þ
fRqG
§'!.
f'fRq¥ Þ /fRq¥ Þ
It can also be shown that conditions (iii) from Corollary 6.12 and (iii) from Theorem 6.11 fail
to hold since
¯
¯
q
°ª°/
=
ß
fRq¥
Þ
qªf
8fv
k
k
and
is equal to a fraction with a positive denominator
and
k
a numerator such that the coefficient in the numerator of the highest power changes sign.
Further we have
¦f<ziQ
Æ
FG7QV¦z.
Ç
­
¦26¸z
!
¦¨zÈQjqâð ß
:
Hence, if
and
, then equation (6.1) is nonoscillatory and (2.3)
holds, so only Theorem 6.7 can be applied. We may
obtain
the same conclusion
for the
[
¦×fz
qâð ß
Þ 8fRq¥
:
corresponding
oscillatory
counterparts
provided
and
with
z
!
:
.
& ) (i) In this case, with the assumption that the expression
R4 EMARK
7.4. (Case $
is differentiable, condition (iii) of Theorem 6.11 is equivalent to
Þ
4 4
fñ
4 =kJ6!#
while condition (iii) of Corollary 6.12 takes the form
Á
4 = 4
f
Þ 4 =kJ6!.
This shows that Corollary 6.12 is a consequence
of Theorem 6.11 in this case. This remark
holds also for the oscillatory counterparts if $
& , see [26].
(ii) Theorem 6.9 (and Corollary 6.12)do
not require to
be nondecreasing
(resp.
¶åqIfqGðâ
f"qG k
nonincreasing) on $
and
we have an
& . Indeed, with
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67
RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
, where Corollary 6.12 can be applied. This, however, has no
example of an increasing
“discrete” counterpart
since
conditions
(ii) from Corollary 6.12 and (v) from Theorem6.11
2¹
fail to hold when $
. Note that in Theorem 6.7 (and Corollary 6.8) the function
is
required to be nonincreasing (resp. nondecreasing) on any time scale.
R EMARK 7.5. (Repeated application)
(i) A repeated application of Theorem 6.9 (resp. Corollary 6.12) gives the following more
general
result:
Let equation
(6.1) be oscillatory (resp. nonoscillatory), and let the functions
#
]#E.Á.E.Á#
satisfy
the assumptions of Theorem 6.9 (resp. of Corollary 6.12) and
o
k
l
l
[
Ço
let
. Then equation (6.2) is oscillatory
(resp.
nonscillatory).
It is easy
to
¹
7
7LÐ o
7
see
that
this
result
is
indeed
more
general;
e.g.,
let
,
and
$
o
k
q
#
. The functions o
satisfy
all the assumptions of Corollary 6.12, but condition
k
h
LÐ k
(iii) fails to hold for
. Therefore, an iteration (repeated application) gives a
better
result. Note that the (weaker) assumption (iii) of Theorem 6.11 is satisfied directly for
[2LÐ k
, however, but to apply this theorem directly, one needs an additional restriction
.
on
(ii) Theorem 6.11 can also be applied repeatedly for monotonic functions
but we
must show that
9
87 9
@B
(=)*
+
FG=¶F
f[,
´
@B
implies
:
(=)*
+
FGF
f[,/.
:
´
By the time scale
version
of the second mean value theorem of integral calculus, see [30],
H
|
$
there exists |
such that
@B
(=)*
+
6
@T
¡
()*
FGF
´
÷
FG=¶F7
+
´
ç
¼ ½
FGF
¡
.
è
¼ ½
f[,
The expression on the right-hand side is greater than
since
is bounded and both
integrals are of the same type as that in the assumptions.
Additional comments may be made to extend the results to dynamic equations of the
form
(7.2)
(7.3)
(7.4)
:
:
:
<;
ãõI
;
Ú
õ5
/!#
;
Ú
/!#
/!.
Ú
Ú
We leave this to the interested reader.
8. Euler–Cauchy Dynamic Equation. In this section we are concerned with the socalled Euler–Cauchy dynamic equation
(8.1)
?
g
L
Åv!#
on a time scale $ (see [23] for these results), where we assume
assume throughout the regressivity condition
(8.2)
?
gf
S
e
kM["
;
!
e
Y@TAC
!
$
:
. We will
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68
for
L. ERBE AND A. PETERSON
H
ù.
$
The equation
z
(8.3)
k
zSyv!
is called the characteristic equation of the Euler–Cauchy dynamic equation (8.1) and the roots
of (8.3) are called the characteristic roots of (8.1).
We have then
z
#Lz
z
;
T HEOREM 8.1. Assume o k are solutions of the chacteristic equation (8.3). If o
z
k , then
is a general solution of (8.1). If
=>
ö o Ô
z
=>
ö o Ô
>z
o
]#
©
k
]#=
ö k Ô
]#
«
, then
?= >
©
=>
ö k Ô
]#=
©
q
F
FSz
Í
FG
o e
is a general solution of (8.1).
Next we would like to show that if our characteristic roots are complex, then there is a
nice form for all real-valued solutions of the Euler–Cauchy dynamic equation in terms of the
generalized exponential and trigonometric functions. Even in the difference equations case
the complex roots are not considered (see Kelley and Peterson [27]).
z
T HEOREM 8.2. Assume that the characteristic roots of (8.1) are complex, that is o k
Ñ #
»
!+#
HñØi.
:
and m Ñ ¼ ½
Then
Ò
¾ , where ¾
BA
?C >
ö o Ô
]#=
EDGF
@
>JI CGH KGL >NM
(
]#=
C>
ö k Ô
]#
>JI CGH KGL >NM
@BA
(
]#=
is a general solution of the Euler–Cauchy dynamic equation (8.1).
In the remainder of this section we will be concerned with the oscillation of the Euler–
Cauchy dynamic equation (8.1). We assume throughout this section that $ is now unbounded
above. We now show if the characteristic roots of (8.1) are complex how a general solution can be written in terms of the classical exponential function and classical trigonometric
functions.
z
!#
:
L EMMA 8.3. If the characteristic roots are complex, that is o k
Ò
¾ , where ¾
then
O@
"á~
where
é
(
Í
Ô
¯
=
¯
û
´
5#
é
#
Ò
¼ ´ ½
Í
@TA
(
ö k
> T
IXW H
RQ > ?S KGLVU M C U Z
Y á
é
DF
ö o
PA °°¨
¾
÷
÷
is a general solution of the
Euler–Cauchy dynamic equation (8.1). If, in addition, every point
H
in $ is isolated, then for
$ ,
á
\[]
¼ ½
q
´ Ç
Í
÷
3
=
÷
e
÷
Ò
k
¾
k
e
k
#
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RECENT RESULTS CONCERNING DYNAMIC EQUATIONS ON TIME SCALES
é
Æ
[
¼ ½
´ Ç
¯
á
p
Ûâö
Í
¾~e
°
÷
÷
Òe
.
÷
D EFINITION 8.4. If the characteristic roots of (8.1)
are complex, then we say the Euler–
é
Cauchy dynamic equation (8.1) is oscillatory iff
is unbounded.
qr#L,S
As a well-known example note that if $ is the real interval {
and the Euler–Cauchy
equation has complex roots, then the Euler–Cauchy equation is oscillatory. This follows from
what we said here because in this case by (1.1)
é
¾
Ï
Í
÷
o
F1^
q
¾
÷
q
é
which is unbounded. If $
, where :
, then one can
again
show that
is
Í
unbounded and hence the Euler–Cauchy dynamic
equation
on
is
oscillatory
when
$
/Ä
the characteristuc roots are complex. If $
, then
Ð
é
Æ
±
o
¯
á
Ûâö
p
°#
¾
É
Ço
Ò
which
can be shown to be unbounded and hence the Euler–Cauchy dynamic equation on
æÄ
$
is oscillatory when the characteristuc roots are complex. These last two examples
were shown in Bohner and Saker [6], Erbe, Peterson, and Saker [19], and Erbe and Peterson
[14], but here we established these results directly. 0D¥ # #ÁÖEÖÁÖ]K
0WDGF
#LF
#ÁÖEÖÁÖK
o
o
T HEOREM
8.5
(Comparison
Theorem). Let $ o
and $ k
,
D¥
K
DâF
K
,
where l and l are strictly increasing sequences of positive
numbers
with
limit
.
If
¬
§
f
Q
pS6²!
¬
¼­ ¬ ½
¼ ¬ ½ , for
the Euler–Cauchy equation (8.1) on $ o is oscillatory and Ò
,
­
then the Euler–Cauchy equation (8.1) on $ k is oscillatory.
@B
¼ ½
T HEOREM 8.6. Assume every point in the time scale $ is isolated and
exists as a finite number, then the Euler–Cauchy equation in the complex characteristic roots
case is oscillatory on $ .
,
@T
Theorem 8.6 does not cover the case when $ is a time scale where
.
¼ ½
>,
@T
The next theorem considers a time scale where
.
¼ ½
\qr#=
i6v!
0WÈDE
0M
"
K.
o #p<HiÄ
l
n
l
m
o
l
¬
T HEOREM 8.7. Let
and let $8Õ
In the
complex characteristic roots case, the Euler–Cauchy dynamic equation (8.1) is oscillatory on
$ Õ .
One might think that one could use the argument in theD¥ proof
of Theorem
8.7 to
show
K
o
, then
that if there is an increasing unbounded sequence of points Ê in $ with e Ê
the Euler–Cauchy equation (8.1) is oscillatory on $ in the complex characteristic roots case.
The following example shows that the same type of argument does not work.
E XAMPLE 8.8. Assume that
the Euler–Cauchy
dynamic equation (8.1) has complex
!
0
pIfñqG k ¨qM#p k #
ln
:
¾ , ¾
Ço {
characteristic roots Ò
and $
then (8.1) is oscillatory.
J_
_`
PA 2a
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Kent State University
etna@mcs.kent.edu
70
L. ERBE AND A. PETERSON
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