Je du - American Mathematical Society
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number 1, August 1975
OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS HIROSHI ONOSE ABSTRACT.
linear assumed
criteria
y it) + a(î)/(y(0)
to be nonnegative
concerned
1.
Oscillation
equation
with
the
Introduction.
for all
interesting
Consider
are
given
■ 0, where large
recent
for the
the
values
ones
the second
second
coefficient of t.
These
of Wong's
paper.
order
order
a(i)
nonlinear
non-
is not
results
are
differential
equa-
tion
(1)
y"U) + a(t)f(y(t)) = 0,
and its
special
case
(2)
y"(t) + a(t)\y\y
where
a(t)
on U.,
large
sgn y = 0,
6 C[0, oo). We consider
oo). A solution
zeros.
Equation
only
of (1) is said
(1) is called
those
y > 0, solutions
of (1) which
to be oscillatory
oscillatory
if it has
if all such
exist
arbitrarily
solutions
are
oscillatory. For simplicity,
we mention
the conditions
(3)
f'(x) > k > 0
(4)
Ç°°-ÈL 0, fot some e > 0,
and
J—oo du -
0,
and Received
by the editors
February
25, 1974 and, in revised
AMS(MOS)subject classifications Key
words
and
phrases.
Second
form, May 21, 1974.
(1970). Primary 34C10, 34C15.
order
equations,
nonlinear,
oscillatory.
Copyright © 1975. American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
67
HIROSHI ONOSE
68
-e du /,10
lim
I
T-oo
J0
lim
KJ
— I
I a(s)dsdt
= + oo
J°
lim sup— j I a(s) ds dt = + °c, T-oo r •'o •'o
(9)
lim
inf I
X—»00
«(s) -À > -00, À > 0,
J 0
(10)
lim
inf j
T—.00
(11)
lim
Recently, provided
under
conditions [il].
tion of (1).
[6] proved
And also
In this
paper
study
ear case
(1).
proposed
have been
Proof.
(2)
cases
(2)
to [l]—[5],
(8) suffice
for y > 1 that
[7]—[9]
hypothesis
y > 0 and also
(7)
for (1).
Re-
for the oscilla-
conditions
(8) and
of (2). a theorem
of Wong's
of Wong's
result
type
for the sublinear
to the more general
Assume
for
case
superlin-
the contrary;
to be positive
(9), (11) and (12) hold.
for the behavior
Then
0 < y < 1.
then
there
exists
on [iQ, 00) for some
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
three
and
theorems.
is oscillatory
may be assumed
refer
problem
for (1) and
the weaker
(5) and
Theorem 1. Suppose that conditions equation
interesting criteria
obtained;
of (2) for all
Wong [12] proved
we prove
a very
is whether
conditions
the extension
Oscillation
«(s) ¿s = + 00.
Many oscillation
problem
for the oscillation
and also
2.
[12]
for the oscillation
Kamenev
(9) suffice
to it.
(6) or (10)
Now Wong's
is sufficient cently,
solution
a(s) ds dt = +00,
JO
sup I
J. S. W. Wong [12]
a partial
I
JO
lim
a(s)ds>0,
^ 0
sup I
7"—oo
(12)
and
e > 0,
a(s) ds = +oo,
T-oo r J°
(8)
for some
/(«)
(6)
n\
.
< oo
of y (t):
a solution
y(t)
which
tQ > 0. We distinguish
OSCILLATION CRITERIA FOR DIFFERENTIAL (i)
y'(t)
oscillatory
(ii)
y'(t)
> 0 on [z+, oo) for some
t^ > ZQ,
(iii)
y'(t)
< 0 on [z^, oo) for some
Z+ > /-.
Suppose such
that
grating
(13)
case
then
there
exists
—> oo. Dividing
a sequence
(2)
through
\t ; n = 1, 2, • • •{
by yy(t)
and
inte-
from Z, to t, we obtain
where
once more from
(14)-y1
and
t,
(14), we have
Next suppose (1) through
y'U.)
ds + A(t) = 0,
= 0 (k is some
A(t) m £
integer).
a(s) ds,
Integrating
(13)
to t, we obtain
k.
y(î)+yf
i-y
'(s)
+ y £-
ß = (y + l)/2
f
——
rf„rfS+ f
JlkJtk\yP(u))
a contradiction
that
by yy(t)
Jtk
to condition
y'(z) > 0 for Z > í
and integrating
A(S)^ =-i.
(11).
> ZQ; thus
from
i-y
y(t) > y(Zj).
Z„. (> t ) to
Dividing
Z, we obtain
'(s)
(15) where
t
69
ItQ, oo),
(i) holds;
y (t ) = 0 and
y~7U)y\t)
From
on
EQUATIONS
ds s);) ds y-y(t)y\t) + y r 2L_i ) ¿s + p «U Jt*\yß(s)J Jt* c, = y'U^/y^U^.).
Finally, can estimate
This
we assume (15)
that
as follows y\t)
(16)
->-{c. yV{t)If the integral
(17)
leads y'(t)
to a contradiction
< 0 for
(in this .
case
NN
(15) rt
to (12).
> Z . By condition is also
valid)
(y'(s))2
+ X) + y-ds. l h*y?+l(s)
in (16) is finite,
namely
n (y'(^))2 lim y I-ds
< M
i-oo
-
J',
Z> t
= c ,
yy+i(s)
(M is constant),
then we have
,.
. , 0. Taking first the case
k = 0, we
have ß - y = (l - y)/2 > 0, y(f) < 1 and
(I9)
^\_/^>V(^-^'A/(,)>0,
for sufficiently
large
Z. From
(16)
and
(19), we have
y'(t) y'U)
(20)
-—0—>->-c.-A,
yHt)
c.=-.
yy(t)
/ -c1 - A, i.e. X> -c..
If lim
inf y'(t)/yy(t)
pothesis have
of c case
then
So we must have
a contradiction
value this
(12).
> -oo,
by (15)
lim
^inf
to (21) because
is larger
than
X. Hence,
we have
a contradiction
y'(t)/y>'(t)
= -oo.
In this
that
to hy-
case
we can take
Z+ such
the absolute
we suppose
k > 0 and y'(z) < 0. In
we have
(22)
& < y( 0 and (16) we have -y'(t)/ky
>-y'(t)/yy(t)>~Cl-X,
so that this fact and (24) imply
(25) From Hence,
0>-Cj-A. (25)
and the argument
used previously,
we also
we suppose
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
have
a contradiction.
we
71
OSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIONS
lim
(26)
n
In this
case
we may choose
(y'(s))2
y I
t^oo
-ds
T > t^ such
that
(y'(s))2 rr vy \s). y I -ds J'*
For t > T., we multiply
y^
= 1 + c, + A.
+lf As)x
(16) through by
-y'ith |yWj.(CitAUr;.^!j and integrate
from
= oo.
Jt*yy+l(s)
yr(T)/y7(/).
It follows
from
(27)
that
y'(t)
< -yy(T)
< 0, which
contradicts
the assump-
tion that y(t) > 0.
Theorem 2. Suppose that (10) and (12) hold
Then, equation
(1) is
oscillatory. The proof of Theorem
Theorem
2 is obtained
by closely
looking
at the proof of
1.
Theorem 3. Suppose that (3), (4), (8) and (9) hold. Then, equation (1) is oscillatory. Proof.
Suppose
may be assumed through
that a solution
to be positive
by /(y(z))
y(i)
of (1) is nonoscillatory;
on [zQ, oo) for some
and integrating
it twice,
y(t) (1)
we obtain
y'U)
(28)
then
tQ > 1 > 0. Dividing
ds + A(t) = c.,
f(y(t))
and /t
y'(s) -ds
t0>
ds < t2N2
1).
Joo, then
Otherwise,
we arrive
at a contra-
by multiplying
(33) through
by
fiyit))y\t)
(c1 +A)+Jto f /'(yU))|^4 1 lfiyis))}
fiyit)) and by employing
y'(z) 0. Remark. valid,
For the equation
but Wong's
[12] theorem
y"(z) + ad^yU)^ cannot
be applied.
+ yit)) = 0, Theorem
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
3 is
OSCILLATION CRITERIA FOR DIFFERENTIAL For the case
without
condition
(3), we also
EQUATIONS
have,
73
by the argument
of
Theorem 3, Theorem
4.
Suppose
(1) is oscillatory,
rivative
y'(t)
being
for some
(A), (8) and
M \yit)\
(9) hold.
= 0, or lim inf
Then
every
x \yit)\
solution
= 0 with
of
its de-
oscillatory.
Acknowledgment. referee
that
or lim
The author
useful
wishes
to express
his thanks
to the
comments.
REFERENCES 1. N. P. Bhatia,
Some oscillation
theorems
for second
tions, J. Math. Anal. Appl. 15 (1966), 442-446. 2.
L. E. Bobisud,
Oscillation
of nonlinear
Math. Soc. 23 (1969), 501-505. 3.
W. J. Coles,
eguations, 4.
second
order
equations,
An oscillation
criterion
for second
order
linear
L. Erbe,
Oscillation
theorems
for second
order
P. Hartman,
On non-oscillatory
I. V. Kamenev,
Certain
linear
nonlinear
Amer.
differential
differential
W. Leighton,
specifically
The detection
H. Onose,
On oscillation
equa-
differential
equations
of second
order.
MR 14, 50. nonlinear
oscillation
of the oscillation
theorems.
of nonlinear
second
order
Mat.
MR 44 #4284.
of solutions
linear differential equation, Duke Math. J. 17 (1950), 57-61. 8.
Proc.
MR 40 #5973.
Zametki 10 (1971), 129-134 = Math. Notes 10 (1971), 502-505. 7.
equa-
Proc. Amer. Math. Soc. 19 (1968), 755-759-
Amer. J. Math. 74 (1952), 389-400. 6.
differential
MR 40 #448.
tions, Proc. Amer. Math. Soc. 24 (1970), 811-814. 5.
order
MR 34 #3017.
of a second
order
MR 11, 248; 11, 871. equations,
J. Math.
Anal.
Appl. 39(1972), 122-124. 9. P. Waltman,
An oscillation
criterion
J. Math. Anal. Appl. 10 (1965), 439-441. 10. A. Wintner, A criterion
for a nonlinear
second
order
equation,
MR 30 #3265-
of oscillatory
stability,
Quart. Appl. Math. 7 (1949),
115-117. MR 10, 456. 11. J. S. W. Wong, On two theorems of Waltman, SIAM J. Appl. Math. 14 (1966), 724-728. MR 34 #6228. 12. -,
A second
Soc. 40 (1973), 487-491.
order
nonlinear
oscillation
theorem,
Proc.
Amer.
Math.
MR 47 #7132.
DEPARTMENT OF MATHEMATICS, IBARAKI UNIVERSITY, MITO CITY, JAPAN 310
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS HIROSHI ONOSE ABSTRACT.
linear assumed
criteria
y it) + a(î)/(y(0)
to be nonnegative
concerned
1.
Oscillation
equation
with
the
Introduction.
for all
interesting
Consider
are
given
■ 0, where large
recent
for the
the
values
ones
the second
second
coefficient of t.
These
of Wong's
paper.
order
order
a(i)
nonlinear
non-
is not
results
are
differential
equa-
tion
(1)
y"U) + a(t)f(y(t)) = 0,
and its
special
case
(2)
y"(t) + a(t)\y\y
where
a(t)
on U.,
large
sgn y = 0,
6 C[0, oo). We consider
oo). A solution
zeros.
Equation
only
of (1) is said
(1) is called
those
y > 0, solutions
of (1) which
to be oscillatory
oscillatory
if it has
if all such
exist
arbitrarily
solutions
are
oscillatory. For simplicity,
we mention
the conditions
(3)
f'(x) > k > 0
(4)
Ç°°-ÈL 0, fot some e > 0,
and
J—oo du -
0,
and Received
by the editors
February
25, 1974 and, in revised
AMS(MOS)subject classifications Key
words
and
phrases.
Second
form, May 21, 1974.
(1970). Primary 34C10, 34C15.
order
equations,
nonlinear,
oscillatory.
Copyright © 1975. American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
67
HIROSHI ONOSE
68
-e du /,10
lim
I
T-oo
J0
lim
KJ
— I
I a(s)dsdt
= + oo
J°
lim sup— j I a(s) ds dt = + °c, T-oo r •'o •'o
(9)
lim
inf I
X—»00
«(s) -À > -00, À > 0,
J 0
(10)
lim
inf j
T—.00
(11)
lim
Recently, provided
under
conditions [il].
tion of (1).
[6] proved
And also
In this
paper
study
ear case
(1).
proposed
have been
Proof.
(2)
cases
(2)
to [l]—[5],
(8) suffice
for y > 1 that
[7]—[9]
hypothesis
y > 0 and also
(7)
for (1).
Re-
for the oscilla-
conditions
(8) and
of (2). a theorem
of Wong's
of Wong's
result
type
for the sublinear
to the more general
Assume
for
case
superlin-
the contrary;
to be positive
(9), (11) and (12) hold.
for the behavior
Then
0 < y < 1.
then
there
exists
on [iQ, 00) for some
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
three
and
theorems.
is oscillatory
may be assumed
refer
problem
for (1) and
the weaker
(5) and
Theorem 1. Suppose that conditions equation
interesting criteria
obtained;
of (2) for all
Wong [12] proved
we prove
a very
is whether
conditions
the extension
Oscillation
«(s) ¿s = + 00.
Many oscillation
problem
for the oscillation
and also
2.
[12]
for the oscillation
Kamenev
(9) suffice
to it.
(6) or (10)
Now Wong's
is sufficient cently,
solution
a(s) ds dt = +00,
JO
sup I
J. S. W. Wong [12]
a partial
I
JO
lim
a(s)ds>0,
^ 0
sup I
7"—oo
(12)
and
e > 0,
a(s) ds = +oo,
T-oo r J°
(8)
for some
/(«)
(6)
n\
.
< oo
of y (t):
a solution
y(t)
which
tQ > 0. We distinguish
OSCILLATION CRITERIA FOR DIFFERENTIAL (i)
y'(t)
oscillatory
(ii)
y'(t)
> 0 on [z+, oo) for some
t^ > ZQ,
(iii)
y'(t)
< 0 on [z^, oo) for some
Z+ > /-.
Suppose such
that
grating
(13)
case
then
there
exists
—> oo. Dividing
a sequence
(2)
through
\t ; n = 1, 2, • • •{
by yy(t)
and
inte-
from Z, to t, we obtain
where
once more from
(14)-y1
and
t,
(14), we have
Next suppose (1) through
y'U.)
ds + A(t) = 0,
= 0 (k is some
A(t) m £
integer).
a(s) ds,
Integrating
(13)
to t, we obtain
k.
y(î)+yf
i-y
'(s)
+ y £-
ß = (y + l)/2
f
——
rf„rfS+ f
JlkJtk\yP(u))
a contradiction
that
by yy(t)
Jtk
to condition
y'(z) > 0 for Z > í
and integrating
A(S)^ =-i.
(11).
> ZQ; thus
from
i-y
y(t) > y(Zj).
Z„. (> t ) to
Dividing
Z, we obtain
'(s)
(15) where
t
69
ItQ, oo),
(i) holds;
y (t ) = 0 and
y~7U)y\t)
From
on
EQUATIONS
ds s);) ds y-y(t)y\t) + y r 2L_i ) ¿s + p «U Jt*\yß(s)J Jt* c, = y'U^/y^U^.).
Finally, can estimate
This
we assume (15)
that
as follows y\t)
(16)
->-{c. yV{t)If the integral
(17)
leads y'(t)
to a contradiction
< 0 for
(in this .
case
NN
(15) rt
to (12).
> Z . By condition is also
valid)
(y'(s))2
+ X) + y-ds. l h*y?+l(s)
in (16) is finite,
namely
n (y'(^))2 lim y I-ds
< M
i-oo
-
J',
Z> t
= c ,
yy+i(s)
(M is constant),
then we have
,.
. , 0. Taking first the case
k = 0, we
have ß - y = (l - y)/2 > 0, y(f) < 1 and
(I9)
^\_/^>V(^-^'A/(,)>0,
for sufficiently
large
Z. From
(16)
and
(19), we have
y'(t) y'U)
(20)
-—0—>->-c.-A,
yHt)
c.=-.
yy(t)
/ -c1 - A, i.e. X> -c..
If lim
inf y'(t)/yy(t)
pothesis have
of c case
then
So we must have
a contradiction
value this
(12).
> -oo,
by (15)
lim
^inf
to (21) because
is larger
than
X. Hence,
we have
a contradiction
y'(t)/y>'(t)
= -oo.
In this
that
to hy-
case
we can take
Z+ such
the absolute
we suppose
k > 0 and y'(z) < 0. In
we have
(22)
& < y( 0 and (16) we have -y'(t)/ky
>-y'(t)/yy(t)>~Cl-X,
so that this fact and (24) imply
(25) From Hence,
0>-Cj-A. (25)
and the argument
used previously,
we also
we suppose
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
have
a contradiction.
we
71
OSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIONS
lim
(26)
n
In this
case
we may choose
(y'(s))2
y I
t^oo
-ds
T > t^ such
that
(y'(s))2 rr vy \s). y I -ds J'*
For t > T., we multiply
y^
= 1 + c, + A.
+lf As)x
(16) through by
-y'ith |yWj.(CitAUr;.^!j and integrate
from
= oo.
Jt*yy+l(s)
yr(T)/y7(/).
It follows
from
(27)
that
y'(t)
< -yy(T)
< 0, which
contradicts
the assump-
tion that y(t) > 0.
Theorem 2. Suppose that (10) and (12) hold
Then, equation
(1) is
oscillatory. The proof of Theorem
Theorem
2 is obtained
by closely
looking
at the proof of
1.
Theorem 3. Suppose that (3), (4), (8) and (9) hold. Then, equation (1) is oscillatory. Proof.
Suppose
may be assumed through
that a solution
to be positive
by /(y(z))
y(i)
of (1) is nonoscillatory;
on [zQ, oo) for some
and integrating
it twice,
y(t) (1)
we obtain
y'U)
(28)
then
tQ > 1 > 0. Dividing
ds + A(t) = c.,
f(y(t))
and /t
y'(s) -ds
t0>
ds < t2N2
1).
Joo, then
Otherwise,
we arrive
at a contra-
by multiplying
(33) through
by
fiyit))y\t)
(c1 +A)+Jto f /'(yU))|^4 1 lfiyis))}
fiyit)) and by employing
y'(z) 0. Remark. valid,
For the equation
but Wong's
[12] theorem
y"(z) + ad^yU)^ cannot
be applied.
+ yit)) = 0, Theorem
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
3 is
OSCILLATION CRITERIA FOR DIFFERENTIAL For the case
without
condition
(3), we also
EQUATIONS
have,
73
by the argument
of
Theorem 3, Theorem
4.
Suppose
(1) is oscillatory,
rivative
y'(t)
being
for some
(A), (8) and
M \yit)\
(9) hold.
= 0, or lim inf
Then
every
x \yit)\
solution
= 0 with
of
its de-
oscillatory.
Acknowledgment. referee
that
or lim
The author
useful
wishes
to express
his thanks
to the
comments.
REFERENCES 1. N. P. Bhatia,
Some oscillation
theorems
for second
tions, J. Math. Anal. Appl. 15 (1966), 442-446. 2.
L. E. Bobisud,
Oscillation
of nonlinear
Math. Soc. 23 (1969), 501-505. 3.
W. J. Coles,
eguations, 4.
second
order
equations,
An oscillation
criterion
for second
order
linear
L. Erbe,
Oscillation
theorems
for second
order
P. Hartman,
On non-oscillatory
I. V. Kamenev,
Certain
linear
nonlinear
Amer.
differential
differential
W. Leighton,
specifically
The detection
H. Onose,
On oscillation
equa-
differential
equations
of second
order.
MR 14, 50. nonlinear
oscillation
of the oscillation
theorems.
of nonlinear
second
order
Mat.
MR 44 #4284.
of solutions
linear differential equation, Duke Math. J. 17 (1950), 57-61. 8.
Proc.
MR 40 #5973.
Zametki 10 (1971), 129-134 = Math. Notes 10 (1971), 502-505. 7.
equa-
Proc. Amer. Math. Soc. 19 (1968), 755-759-
Amer. J. Math. 74 (1952), 389-400. 6.
differential
MR 40 #448.
tions, Proc. Amer. Math. Soc. 24 (1970), 811-814. 5.
order
MR 34 #3017.
of a second
order
MR 11, 248; 11, 871. equations,
J. Math.
Anal.
Appl. 39(1972), 122-124. 9. P. Waltman,
An oscillation
criterion
J. Math. Anal. Appl. 10 (1965), 439-441. 10. A. Wintner, A criterion
for a nonlinear
second
order
equation,
MR 30 #3265-
of oscillatory
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DEPARTMENT OF MATHEMATICS, IBARAKI UNIVERSITY, MITO CITY, JAPAN 310
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