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Electronic Transactions on Numerical Analysis.
Volume 27, pp. 1-12, 2007.
Copyright 2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
OSCILLATION CRITERIA FOR A CERTAIN CLASS OF SECOND ORDER
EMDEN–FOWLER DYNAMIC EQUATIONS
ELVAN AKIN–BOHNER , MARTIN BOHNER , AND SAMIR H. SAKER
Abstract. By means of Riccati transformation techniques we establish some oscillation criteria for the second
order Emden–Fowler dynamic equation on a time scale. Such equations contain the classical Emden–Fowler equation as well as their discrete counterparts. The classical oscillation results of Atkinson (in the superlinear case) and
Belohorec (in the sublinear case) are extended in this paper to Emden–Fowler dynamic equations on any time scale.
Key words. oscillation, dynamic equation, time scale, Riccati transformation technique
AMS subject classifications. 34K11, 39A10, 39A99, 34C10, 39A11
1. Introduction. In this paper we shall consider the second order Emden–Fowler dynamic equation
(1.1)
for
"! #%$'&)(
on a time scale, where and are positive, real-valued rd-continuous functions, and * is an
odd positive integer. We shall also consider the two cases
+,
(1.2)
-.
21
0
/
and
+3,
(1.3)
- .
16
054
/
In the case of *87:9 , (1.1) is the prototype of a wide class of nonlinear dynamic equations
called Emden–Fowler superlinear dynamic equations, and if 4 * 4 9 , then (1.1) is the
prototype of dynamic equations called Emden–Fowler sublinear dynamic
equations. It is in
teresting to study (1.1) because the continuous version, i.e., when is a continuous variable,
has several physical applications, see e.g., [20] and when is a discrete variable it is a difference equation of Emden–Fowler type and also is important in applications. By
a;<
solution
>=?;8#
of (1.1) we
mean
a
nontrivial
real-valued
function
satisfying
equation
(1.1)
for
=@;#
for some
. A solution of (1.1) is said to be oscillatory if it is neither eventually
7
positive nor eventually negative; otherwise it is called nonoscillatory. Equation (1.1) is said
to be oscillatory if all its solutions are oscillatory.
Our attention is0restricted
to=Mthose
solutions
! AB$C1
L
of
(1.1)
which
exist
on
some
half
line
and
satisfy
for any
DEGFHJI
IK
7
7
=N;3>A
.
Much recent attention has been given to dynamic equations on time scales (or measure
chains), and we refer the reader to the landmark paper of Hilger [20] for a comprehensive
treatment of the subject. Since then several authors have expounded on various aspects of
this new theory; see the survey paper by Agarwal, Bohner, O’Regan, and Peterson [1] and
the references cited therein. Two books on the subject of time scales, by Bohner and Peterson
[9, 10], summarize and organize much of the time scale calculus.
O
Received December 1, 2003. Accepted for publication March 8, 2004. Recommended by A. Ruffing.
Department of Mathematics and Statistics, University of Missouri–Rolla, Rolla, MO 65409 (akine,
bohner@umr.edu).
Mansoura University, Department of Mathematics, Mansoura 35516, Egypt (shsaker@mans.edu.eg).
1
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2
E. AKIN–BOHNER, M. BOHNER, AND S. SAKER
In recent years there has been much research activity concerning the oscillation and
nonoscillation of solutions of dynamic equations on time scales. We refer the reader to the
papers [2, 3, 4, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23].
In [3], Akın-Bohner and Hoffacker considered the second order dynamic equation
(1.4)
PQ
and
gave necessary and sufficient conditions for oscillation of all solutions
when *R7S9 and
4
4
*
9 . Their results cannot be applied in the case when *
9
and applied only to
discrete time scales.
In this paper we use the Riccati transformation technique to obtain some oscillation
cri
teria for (1.1) when (1.2) or (1.3) holds. Our results can be applied in the case *
9 and also
for any time scale. So our results extend and improve the results established by Akın-Bohner
and Hoffacker [3]. The paper is organized as follows. In the next section we present some
basic definitions concerning the calculus on time scales. In Section 3 we develop a Riccati
transformation technique and give some fundamental lemmas. These lemmas are used to
superlinear
case, i.e.,
give sufficient
conditions for oscillation of all solutions of (1.1) in the
;
2PB$
when *
(see Section 5).
9 (in Section 4) and in the sublinear case, i.e., when *
9
Our results when (1.2) holds are sufficient for oscillation of all solutions of (1.1), and when
(1.3) holds our results ensure that all solutions are either oscillatory or converge to zero. For
the superlinear case we present an extension of the classical Atkinson [5] result, and for the
sublinear case we present an extension of the classical Belohorec [6] result.
Since we are interested in oscillatory behavior, we suppose that the time
scale under
! #%$)13
consideration is not bounded above, i.e., it is a time scale interval of the form
.
2. Some Preliminaries on Time Scales. A time scale T is an arbitrary nonempty closed
subset of the real numbers U . On any time scale T we define the forward and backward jump
operators by
WVYXZ
K
b
)L
H\[
and
T<K][N7
^
K
)La6
DEBF_H`[
VcXBZ
4
T3K][
de
d2
A point
with
is said to be left-dense
if ^
and right-dense if
,
T
7
T
4
left-scattered if ^
and
right-scattered
if
.
The
graininess
function
for
a
time
7
f
g h
scale T is defined by f
. For a function i8K0TkjlU (the range U of i may
K
actually be replaced by any Banach space) the (delta) derivative is defined by
Pmh
a
(2.1)
i
mhn
i
i
if i is continuous at and is right-scattered. If is not right-scattered, then the derivative is
defined by
mh
oepYVcq
(2.2)
i
i
mh
-
r>s
i
th
kpYVcq
[
i
0h
-
rs
[
i
[
[
! #$)&'(
provided this limit exists. A function i:K
juU
is said to be rd-continuous if it is
continuous at each right-dense point and there exists a finite left limit at all left-dense points,
and i is said to be differentiable if its derivative exists. A useful formula is
o
(2.3)
i
i
f
C6
i
We will make use of the product
and quotient rules for the derivative of the product
8y
the quotient ixw\v (where vyvz|{
of two differentiable functions i and v
(2.4)
<
iv
i
v
i
3
z
v
n
iv
i
v
z
and
i
h
i
v
iv
}
va~
vyv
z
6
iv
and
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3
EMDEN–FOWLER DYNAMIC EQUATIONS
For
#$)&
and a differentiable function i , the Cauchy integral of i
T
+
o
&th
i
is defined by
#C6
i
i
/
An integration by parts formula reads
+
i
o!
v
(
i
+
h
v
/
i
C$
v
/
and improper integrals are defined as
+3,
Note that in the case T
8C$
o\
9
$
a
C6
i
/
+
C$
i
Q
,
/
^
and in the case T
s
we have
U
o
+
apcVYq
i
i
+
o
i
>C$
i
/
we have
o8Mh
$
^
o
9
a
i
\
i
h
i
9
+
C$
i
x
c
i
/
a
>C6
i
/
3. A Riccati Transformation.
Crucial for our calculations is the0following
lemma.
Q
d
L EMMA 3.1. If and are differentiable on a time scale T with
for all
{
then we have
(3.1)
}
d
hG
o
d
zn
T
,
6
~
Proof. Using (2.3), (2.4), and
RG
}
y
z
a
a
yz
P
, we obtain
fx
~
P
h
fx
¢h
¡
y
zn
z
£G
hk
G
z
¢h
f
hn
z
z
z
6
This shows that (3.1) holds.
Using Lemma 3.1, we now derive the following result.
0
¤
{
T HEOREM 3.2. Suppose solves (1.1) such that
for all
differentiable function and define ¥ by the Riccati substitution
(3.2)
¥
y
6
N
T
. Let
be any
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4
E. AKIN–BOHNER, M. BOHNER, AND S. SAKER
Then we have
G
h
(3.3)
d
¥
z
G
h
z
P
6
Proof. We use again (2.3) and (2.4) to find
h
<h
¥
hW¦
z
y
}
y
cG
}
~
G
y
@§
~
Q
h¨G
z
}
G
Q
y
~
h
z
}
Now applying formula (3.1) from Lemma 3.1 (with
6
~
replaced by
), we arrive at (3.3).
We will use the above Theorem 3.2 several times in the remainder of this paper, in conjunction with the formula
(3.4)
+
*
! ©G
h¡©>B( Q
z
=
J©x$
9
which is a simple consequence of Keller’s chain rule [9, Theorem 1.90].
The following result is used frequently in the
remainder of this paper.
t
L
EMMA 3.3. Assume that (1.2) holds. If is a solution of (1.1) such that
;<=
all
, then
%_
;3
(3.5)
;< =
for
7
for
6
Proof. In view of (1.1) we have
ªG%
>_kh>G
(3.6)
4
; =
<G
for all
, and so «|K
is an eventually
decreasing
function.
We first show that « is
;e =
4
«
eventuallynonnegative.
Suppose
there
exists
such
that
. Then from (3.6)
;<
. Hence
4
«
we have «
for
¬
«
0
$
which implies by (1.2) that
0d¬<0
(3.7)
x
t
«
+
-P­
[
t
/ [
h®1
as
j
5;g =
1$
j
@¯0
7
«
contradicting the fact that
for all
. Hence
=
nonnegative. Therefore, we see that there exists some such that
t
B$°
7
Hence (3.5) holds.
;WB$
d;<B$
«
«
4
for
; =
6
is eventually
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5
EMDEN–FOWLER DYNAMIC EQUATIONS
4. Oscillation Criteria
in the Superlinear Case. In this section we give some oscilla²±
is odd.
tion criteria when *
First we consider the case when (1.2) holds.
T HEOREM 4.1. Assume that (1.2) holds. Furthermore, assume that there exists a differentiable function such that
(4.1)
+
pcVYq
- s
DEBF
,
´³
µ
holds for all constants
[
z
hµ
[
x 0
[
21
¶
[
[
/
. Then every solution of (1.1) is oscillatory on
7
! #%$C1
.
loss of
Proof. Suppose to the contrary
that is a nonoscillatory solution of (1.1). Without
0
7
generality,
we
may
assume
that
is
an
eventually
positive
solution
of
(1.1)
such
that
;3=
#
¸h
7
for all
. We shall consider only this case, since the substitution ·
transforms
equation (1.1) into
an equation of the same form. By Lemma 3.3 we obtain that (3.5) holds.
;
Now note that *
9 and (3.4) imply
+
(3.4)
! ©G
=
+3
;
*
xy©
9
h"©%t( ! ©G0
x
y©
9
=
h"©%t( z
*
t
*
(3.5)
;
t=£ x
*
:
¹
=\3;
¥
- .
[
[
/
+
- .
[
h¸t
[
t
z
+
[
- .¿
[
P0
¿
- .
+
;
[
h¸t
[
[
z
[
h
¹
1
[
/
[
t
®
[
À
[
[
[
as
18$
j
[
/
[
Â
[
/
j
-
- .bÁ
(4.1)
[
/
À
[
[
[
[
®
[
z
z
[
[
[
+
! =$C1
¥
-
+
lh
¹
. Note N;¾7 = . Now define the function ¥ on
for all
. Therefore, using (3.3) from Theorem
¥
=£mh
x
½;
¥
;
x>¼ t=£
º» where we put
K
*
by (3.2). Then (3.5) implies
3.2, we obtain
¥
$
9
¹
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6
E. AKIN–BOHNER, M. BOHNER, AND S. SAKER
which is impossible. The proof is complete.
C OROLLARY 4.2. Assume that (1.2) holds. Furthermore, assume that there exists a
positive differentiable function à such that
(4.2)
+
pcVYq
- s
DEBF
,
à z
[
ÅÄÆ
mh"µ
t
[
È
[
Ã
µ
Ã
bÇ
[
CËÌ
È
[
à z
21
[
ÊÉ
[
/
holds for all constants
eÍ
Proof. Define
. Then every solution of (1.1) is oscillatory on
and note that [9]
7
Ã
Í
.
6
Ã
! #$C1
Í
Ã
à z
Since (4.2) holds for à , (4.1) holds for . So the claim follows by Theorem 4.1.
From Theorem 4.1 and Corollary 4.2 we can obtain different conditions for oscillation
of all solutions of (1.1) bydifferent
choices
of à . For instance, we obtain the following two
_Î
_¯
corollaries if we choose Ã
, respectively. The first choice confirms that
9 and Ã
the Leighton–Wintner theorem is valid for Emden–Fowler dynamic equations.
C OROLLARY 4.3 (Leighton–Wintner). Assume
+,
(4.3)
+,
21
and
0
/
a216
/
! #%$)13
Then every solution of (1.1) is oscillatory on
.
C OROLLARY 4.4. Assume that (1.2) holds. Furthermore, assume that
(4.4)
+
pYVcq
- s
˪Ð
DEGF
,
µ
[
ÄÏ
h"µ
x 0
[
Ì
9
[
Æ
È
Í
[
[
Q1
[
/
! #$C1
holds for all constants
7
. Then every solution of (1.1) is oscillatory on
.
0ÑÎ
The next result is the same as [3, Theorem 5] when
9 . But we note that [3,
Theorem 5] cannot be applied in the case when * and also not for the second order
9
Emden–Fowler differential equation, i.e., when T
U . So the following result extends and
improves in various ways the results established in [3]. Its classical version was given in 1955
by Atkinson [5].
T HEOREM 4.5. Assume that (1.2) holds. Define
+
Ò
6
[
t
/ [
If
(4.5)
+
pcVYq
Ò
- s
DEBF
,
>
[
218$
[
[
/
! #$C1
then every solution of (1.1) is oscillatory on
.
t
_;¾>=
Proof. Again we suppose is a solutionofÍ (1.1) such that
for all
. By
7
Ò
Lemma 3.3 we obtain (3.5). Now we let
and define the Riccati substitution ¥ by
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7
EMDEN–FOWLER DYNAMIC EQUATIONS
(3.2). Using the product rule from (2.4), we calculate
¥
9 G
¿
8
Ò
h
h
C
G
Ò
z
J$
z
Ó
(3.4)
+
h
9
! ©G
h
+<
*
Q
J©
z
9
z
J©
(
h¡©>
z
*
h¡©>0(
9
! ©G
=
h
9
due to (3.4) and because
Q
=
9
Ó
¬W
z
*
¬Ô
*
where the last inequality is true because
Ò
z
Ò
h
9
À
Ò
z
z
C
¬
6
Upon integration we arrive at
+
Ò
P
[
- .
[
+
¬
[
/
h
[
ah
)
*
= )
/
= =£
h
¥
9
[
*
h
¬
À
¥
9
)
h
h
¿
- .
C
¥
9
*
=\6
h
¥
*
9
This contradicts
(4.5) and finishes
the proof.
0aÎ
a
Ò
9 , i.e.,
Putting
in Theorem 4.5, we obtain the following corollary.
taÎ
9 . If
C OROLLARY 4.6. Assume
(4.6)
+
pYVcq
- s
DEBF
,
>
[
218$
[
[
/
! #%$C1
then every solution of (1.1) is oscillatory on
.
E XAMPLE 4.7. Consider the dynamic equation
(4.7)
Here
taÎ
9
o
and
z
» - ¼
9
>
z
;
for
6
9
. Using [8, Theorem 5.11], we find
+,
[
- .
[
+3,
e186
[
[
- .
/ [
/
!
$)13
Hence, by Corollary 4.6, equation (4.7) is oscillatory on 9
.
T HEOREM 4.8. Assume
that
(1.2)
holds.
If
there
exists
a
positive
differentiable function
²±
à and an odd integer Õ
such that
(4.8)
+
pYVYq
- s
DEBF
,
µ
Ö
0h
9
>Ö
[
Ã
Æ
ÄÏ
[
>
[
È
ah
Ã
[
Ã
[
È
Ã
t
[
P
[
˪Ð
Ì
Q1
[
/
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8
E. AKIN–BOHNER, M. BOHNER, AND S. SAKER
µ
! #%$)13
7
, then every solution of (1.1) oscillates on
.
holds for all
Proof. The proof is similar to [23, Theorem 3.2] and hence is omitted.
oÎ
9 , then (4.8) reduces to
Note that when Ã
+
pYVcq
DEBF
,
- s
Ö
mh
9
Ö
>
[
218$
[
[
/
which can be considered as an extension of Kamenev type oscillation criteria for second order
differential equations (see [22]).
Next we consider the case when (1.3) holds. Now we give some sufficient conditions
when
(1.3) holds, which guarantee that every solution of (1.1) oscillates or converges to zero
! #%$)13
on
.
T HEOREM 4.9. Assume that (1.3) holds. Furthermore, assume that there exists a positive
function such that (4.1) holds, and
+
+
,
(4.9)
9
t
[
ae186
[
/
/
! #%$)13
Then every solution of equation (1.1) is oscillatory or converges to zero on
.
T HEOREM 4.10. Assume that (1.3) holds. Furthermore, assume that there exists a
(4.9) hold. Then every solution of equation (1.1) is
positive function à such that (4.8) and
! #%$)13
oscillatory or converges to zero on
.
5. Oscillation Criteria in ¡
thePBSublinear
Case. In this section we give some new oscil$
lation criteria for (1.1) when *
is a quotient of odd positive integers.
9
First we consider the case when (1.2) holds.
T HEOREM 5.1. Assume that (1.2) holds and suppose that is differentiable and nondecreasing. Furthermore, assume that there exists a differentiable function such that
(5.1)
+
pYVYq
- s
DEBF
,
³
[
z
[
h"µ|
C
\0
[
[
®
e1
¶
[
[
/
µ
! #$C1
holds for all constants
. Then every solution of (1.1) is oscillatory on
.
7
Proof. Suppose to the contrary
that is a nonoscillatory solution of (1.1). Without
loss of
0
generality,
we
may
assume
that
is
an
eventually
positive
solution
of
(1.1)
such
that
7
;3=
#
for all
. Hence, by Lemma 3.3, we obtain (3.5), which implies
7
×;k
aR
Ñ
>
z
so
that, again by using (3.5),
! =M$)13
, and therefore we obtain
²¬Ø
. Hence
is nonincreasing on
+
0o8t=£x
¨;Ù=
for all
-.
¬<Ú²¡ÛC$
[
[
/
Ún0=£Bh¢=
where
we find that
(5.2)
=£
and
Û8
td¬
=`
Ü
. By putting Ü
for all
;<
Ú
I
6
¨Û
I
and
;3q5Ý£Þ
=Ê$
H
L
9
,
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9
EMDEN–FOWLER DYNAMIC EQUATIONS
"G$
Now note that *
and (3.4) imply
9
+
(3.4)
! ©B
+
*
! ©B
(5.2)
;
Ü
x
*
¹
¹
x
Ü
_;Q
º» 3;
¥
¥
mh
! =$)13
[
/
[
-
-P­
[
h¸t
[
t
z
¿
+
$
. Note b;Q7 = . Now define the function ¥ on
by
for all
. Therefore, using (3.3) from Theorem 3.2,
[
x
¥
-
- ­
[
[
0
>
[
¿
- ­
h¸t
[
[
À
[
[
/
[
[
z
[
[
[
/
[
À
[
[
[
/
+
;
z
[
[
+
;
+
J©
-
- ­
z
¥
+
lh
( h¡©%
9
¹
x>¼
J©
x
z
*
where we put
K
*
(3.2). Then (3.5) implies ¥
we obtain
Q
z
=
9
=
;
h¡©%0
( Q
z
*
[
- ­
Á
z
[
h
¹
P
)
\0
[
®
[
Â
[
[
/
(5.1)
1
as
j
18$
j
which is impossible. The proof is complete.
The next result follows as in the proof of Corollary 4.2.
C OROLLARY 5.2. Assume that (1.2) holds and suppose that is differentiable and
nondecreasing. Furthermore, assume that there exists a positive differentiable function Ã
such that
(5.3)
+
pcVYq
- s
DEBF
,
ÄÆ
[
mh"µ
à z
t
[
P
[
)
È
[
Ã
µ
Ã
0
Ç
[
[
ËÌ
È
à z
21
[
[
É
/
! #%$C1
holds for all constants
. Then every solution of (1.1) is oscillatory on
.
7
From Theorem 5.1 and Corollary 5.2 we can obtain different conditions for oscillation
of all solutions of (1.1) bydifferent
choices
of à . For instance, we obtain the following two
Î
a8
corollaries if we choose Ã
, respectively.
9 and Ã
C OROLLARY 5.3 (Leighton–Wintner). Suppose is differentiable and nondecreasing
and assume
+3,
(5.4)
21
0
/
+,
and
/
ae16
ETNA
Kent State University
etna@mcs.kent.edu
10
E. AKIN–BOHNER, M. BOHNER, AND S. SAKER
! #%$)13
.
Then every solution of (1.1) is oscillatory on
C OROLLARY 5.4. Assume that (1.2) holds and suppose that
nondecreasing. Furthermore, assume that
(5.5)
- s
is differentiable and
+
pYVcq
˪Ð
DEBF
,
>
ÄÏ
ahµ
[
P
)
[
t
[
Æ
È
Í
µ
Ì
9
[
[
[
e1
[
/
! #$C1
7
holds for all constants
. Then every solution of (1.1) is oscillatory on
.
The next result gives a condition for oscillation in the sublinear case. Its classical version
was given in 1961 by Belohorec [6].
T HEOREM 5.5. Assume that (1.2) holds. If
(5.6)
+3,
}
oe18$
0
~
/
! #$C1
then every solution of (1.1) is oscillatory on
.
t
Proof.
We
suppose
that
is
a
solution
of
(1.1)
satisfying
G . Then by Lemma 3.3 we obtain that «
and «
«
7
observe that
;8>=
for all
for all
4
+
0o0=£
7
®;e=
_
;<_mh"=`
[
- .
;
_
[
/
for all
;3
if
£ =
7
«
C
«
. Next note that
Ó
(3.4)
+
h
9
*
;Ô
h
*
! ©
Q
6
t
®
«
_
«
;
«
~
6
Ct
}
9
h
}
0
}
)
*
for all
«
z
«
;
J©
«
«
R
«
(
h¡©%
Using these two inequalities, we obtain after dividing (1.1) by
]
J©
«
9
*
(
h¡©%
9
«
=
h
9
Q
z
«
+
9
Ó
! ©
=
~
~
Upon integration we arrive at
+
-ß
[
}
0
[
+
¬
[
[
~
)
*
)
h
)
[
/
mh
«
«
[
*
h
9
9
¬
C
«
*
h
9
h
-ß
/
9
6
«
*
;3
,
and let
. First
ETNA
Kent State University
etna@mcs.kent.edu
11
EMDEN–FOWLER DYNAMIC EQUATIONS
This contradicts
(5.6) and finishes the proof.
0aÎ
9 in Theorem 5.5, we obtain the following corollary.
Putting
taÎ
9 . If
C OROLLARY 5.6. Assume that
+3,
(5.7)
y
oQ1$
/
! #%$C1
then every solution of (1.1) is oscillatory on
.
E XAMPLE 5.7. Consider the dynamic equation
(5.8)
tdÎ
º'á
9
à '
º á
Q
for
z
;
6
9
à º'á
d
9 and
9\w
Here
. As in Example 4.7 it follows from Corollary 5.6 that (5.8)
!
$C1
is oscillatory on 9
.
T HEOREM 5.8. Assume that is differentiable and nondecreasing and suppose
that (1.2)
ѱ
such that
holds. If there exists a positive differentiable function à and an odd integer Õ
(5.9)
+
pYVcq
- s
DEGF
,
Ö
P
[
µ
µ
Ö
mh
9
Ã
ÄÏ
x ah
[
[
È
Æ
[
Ã
C
[
Ã
È
0
[
P
Ã
˪Ð
Ì
21
[
[
[
/
! #%$)13
7
for all constants
, then every solution of (1.1) oscillates on
.
Proof. The proof is similar to [23, Theorem 3.2] and hence is omitted.
Next we consider the case when (1.3) holds. Now we give some sufficient conditions
when
(1.3) holds, which guarantee that every solution of (1.1) oscillates or converges to zero
! #%$)13
on
.
b;W
T HEOREM 5.9. Assume that
and that (1.3) holds. Furthermore, assume that
there exists a positive function such that (5.1) holds, and
+
+
,
(5.10)
9
0
o216
[
/
[
/
! #%$)13
Then every solution of equation (1.1) is oscillatory
or converges to zero on
.
;
T HEOREM 5.10. Assume that
and that (1.3) holds. Furthermore, assume
that there exists a positive function à such that (5.9) ! and
(5.10) hold. Then every solution of
#%$)13
equation (1.1) is oscillatory or converges to zero on
.
R EMARK 5.11. Note that our results also can be extended to the more general equation
0%"
where *n7
t
I
I
XNP0PQ
Dâ
for
¡! #%$)&'($
to cover the case when * is even.
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