Some Nonoscillation Theorems For Second Order Differential Equations
(有關於二階微分方程的非振動理論)
作者：高春玉
0. Introduction
In this paper, we discuss the nonoscillatory property of the solutions of the second
order liner differential equation
(1)
[r (t )u (t )]  c(t )u (t )  0
and the second order half-liner differential equation
(2)
{r (t ) [u (t )]}  c(t ) [u (t )]  0
where
(i) r, c  C ([t 0 , ), R : (, )) and r (t )  0 on [t 0 , ) for some t 0  0 ;
(ii)  (u ) | u | p 2 u for some fixed number p  1 .
Clearly, if p  2 then (2) reduces to (1). By a solution of (2) will be meant a real-valued
function u (t ) which is not identically zero on [t 0 , ) and satisfies (2).
Equations (1) or (2) is said to be nonoscillatory in [t 0 , ) if no solution of equations (1)
or (2) vanishes more than once in this interval. The equation (1) or (2) will be said to be
oscillatory if one (and therefore all) of its solutions have an infinite number of zeros on
[t 0 , ) .
Our main concern will be to obtain nonoscillatory (or oscillatory) criteria for equation (1)
or (2), that is, conditions on the functions r (t ) , c(t ) and  from which conclusions may be
drawn as to the nonoscillatory (or oscillatory) character of equation (1) or (2). There
exists an entersive literature on this subject, see, for example, [1-14]. In [8], Li and Yeh
obtained some nonoscillatory criteria of the second order differential equation (1) by
using the subsituation w(t ) 
u (t )
a(t )
. In this note, we first will use another method for
equation (1). Using this result, we improve some results in [3,4,8,10]. In the Second
section, we extend a Leighton oscillatory criteria from equation (1) to the second order
half-liner differential equation (2).
1
1. Oscillation Criteria For Equation (1)
Let u (t ) be a solution of (1). According to the Kummer transformation (see Kwong
and Zettl [5] or Willett [14] ), we define
w(t ) 
u (t )
on
a(t )
[t 0 , ) ,
where a(t )  C 2 ([t 0 , ), (0, )) is a given function. Then (1) is transformed into
(a(t )r (t ) w(t ))   (t ) w(t )  0
(3)
where  (t ) : a(t )[c(t )  r (t ) f 2 (t )  (r (t ) f (t ))] and f (t ) : 
a (t )
. Hence, equation (1), (3) and
2a(t )
the following differential equation are equivalent:
(4)
(a1 (t )a(t )r (t )v(t ))  a1 (t )[ (t )  a(t )r (t ) g 2 (t )  (a(t )r (t ) g (t ))]v(t )  0
where a1 (t )  C 2 ([t 0 , ), (0, )) and g (t )  
a1 (t )
on [t 0 , ) .
2a1 (t )
Using these equivalent relations, Li and Yeh [8] established the following
nonoscillatory characterization for equation (1) as follows:
Theorem A. Equation (1) is nonoscillatory if and only if one of the following conditions
holds:
(a) there exists a function f  C ([T , ), R) for some T  t 0 such that
on [t 0 , ) .
c(t )  r (t ) f 2 (t )  (r (t ) f (t ))  0
1
(b) there exists a function v  C ([T , ), R) for some T  t 0 such that
c(t )  r (t )v 2 (t )   (t )  (a(t )r (t )v(t ))  0 , t  T .
where a(t )  C 2 ([t 0 , ), (0, )) is a given function and  (t )  a(t )[c(t )  r (t ) f 2 (t )  (r (t ) f (t ))] .
Cleraly, condition (b) is condition (a) if a(t )  1 . We also have the following
observation:
If c(t )  0 for t large enough, then equation (1) is nonoscillatory. Suppose that “ c(t )  0
for t large enough” does not hold. If we can find a , a1  C 2 ([t 0 , ), (0, )) such that the
coefficient of w(t ) and v(t ) in (3) or (4) is nonpositive, then equation (1) is nonoscillatory.
Using Theorem A Li and Yeh [8] obtained many nonoscillatory criteria for equation
(1). In this section, we use another method to derive Theorem A. Using this result, we
establish some nonoscillatory criteria which generalize some results of Hille [3], Kneser
[4] and Li-Yeh [8]. An alternative proof of the Sturm comparison theorem [10] is also
given. For other related results, we refer to [2,6,10].
Throughout this section, we assumed that a(t )  C 2 ([t 0 , ), (0, )) is a given functio,
a (t )
v 2 (t )
and  (t ) : a(t )[c(t )  r (t ) f 2 (t )  (r (t ) f (t ))]  a(t )(c(t ) 
 v(t )) . Hence,
2a(t )
r (t )
v(t ) : r (t ) f (t ) .
f (t ) : 
2
Now, we can state and prove our main result as follows:
Theorem 1. The following three statements are equivalent:
(a) Equation (1) is nonoscillatory.
(b) There is a function v(t )  C 1 ([T , ), R) such that
v(t )   (t ) 
v 2 (t )
 0, t T
a(t )r (t )
for some T  t 0 .
(c) There is a function v(t )  C 1 ([T , ), R) such that
v(t )   (t ) 
(5)
v 2 (t )
 0, t T
a(t )r (t )
for some T  t 0 .
Proof. (a)  (b)：If (1) is nonoscillatory and u (x) is a solution of (1) on [t 0 , ) , then there
is a number T  t 0 such that u ( x)  0 on [T ,  ) . Let
v(t )  a(t )r (t )(
u (t )
 f (t )),
u (t )
t T .
Then
r (t )u (t )
 a(t )r (t ) f (t )]
u (t )
(r (t )u (t ))
r (t )u (t ) a(t )r (t )(u (t )) 2
 a(t )
 a (t )

 a (t )r (t ) f (t )  a(t )( r (t ) f (t ))
u (t )
u (t )
u 2 (t )
r (t )u (t )
1
a(t )r (t )u (t ) 2
 a(t )c(t )  2a(t ) f (t )

(
)  2a(t )r (t ) f 2 (t )  a(t )( r (t ) f (t ))
u (t )
a(t )r (t )
u (t )
v (t )  [a(t )
1
a(t )r (t )u (t ) 2
u (t )
[(
)  2a 2 (t )r 2 (t )
 (a(t )r (t ) f (t )) 2 ]
a(t )r (t )
u (t )
u (t )
1
a(t )r (t )u (t )
 a(t )[c(t )  (r (t ) f (t ))  r (t ) f 2 (t )] 
[(
)  a(t )r (t ) f (t )] 2
a(t )r (t )
u (t )
1
 a(t )[c(t )  (r (t ) f (t ))  r (t ) f 2 (t )] 
v 2 (t ) ,
a(t )r (t )
 a(t )[c(t )  (r (t ) f (t ))  r (t ) f 2 (t )] 
which implies
v (t )  a(t )[c(t )  r (t ) f 2 (t )  (r (t ) f (t ))] 
1
v 2 (t )  0
a(t )r (t )
for t  T . Hence
v(t )   (t ) 
v 2 (t )
 0, t T .
a(t )r (t )
(b)  (c)：It is clear.
(c)  (a)：If there exists a function v(t ) satisfying
3
v 2 (t )
 1 (t ) : v(t ) 
  (t )
a(t )r (t )
(6)
for t  T ,
then
v( s )
ds)
T a( s)r ( s)
w(t )  exp( 
(7)
t
satisfies
(a(t )r (t )w(t ))  1 (t )w(t )  0 ,
t T .
In fact,
w(t )  w(t )
v(t )
a(t )r (t )
which implies
(a(t )r (t ) w(t ))  ( w(t )v(t ))  w(t )v(t )  w(t )v(t ) 
w(t )v 2 (t )
v 2 (t )
 w(t )( 1 (t ) 
).
a(t )r (t )
a(t )r (t )
Thus, (7) is a nonoscillatory solution of
(8)
(a(t )r (t )w(t ))  1 (t )w(t )  0 , t  T .
It follows from (6), (8) and the Sturm Comparison Theorem that equation (3) is
nonoscillatory, and hence, equation (1) is nonoscillatory. This completes our proof.
Taking v(t )  a(t )r (t ) w(t ) in Theorem 1, our Theorem 1 reduces to condition (b) of
Theorem A.
Corollary 2. If (a(t )r (t ))  0 for t large enough and
(9)
lim sup
t 
t 2 (t )
1
 ,
a(t )r (t ) 4
then equation (1) is nonoscillatory.
Proof. It follows from (9) that there exist two numbers T  t 0 and  
 (t ) 
r (t )a (t )
t2
for
t T .
Let
v(t )  a (t )r (t )h(t ) ,
where h(t ) 
1
. Then, for t  T ,
2t
v 2 (t )
v(t )   (t ) 
a(t )r (t )
 (a(t )r (t ))h(t )  a (t )r (t )(
 a (t )r (t )(
1
)   (t )  a (t )r (t )h 2 (t )
2t 2
1 
1
 2  2 )  (a (t )r (t ))h(t )
2
2t
t
4t
4
1
such that
4
 a(t )r (t )(
4  1
)
4t 2
0
.
This and Theorem 1 imply (1) is nonoscillatory.
Remark 3.
(a) Let a(t )  1 , then  (t )  c(t ) . Thus our Corollary 2 reduces to Theorem 3.5 in Li
and Yeh [8].
(b) Let a(t )  r (t )  1 . Then Corollary 2 reduces to the result of Hille-Kneser [3,4].
Corollary 4. If (a(t )r (t ))  0 for t large enough and
(10)
lim sup t 2 log 2 t (
t 
 (t )
a(t )r (t )

1
1
) ,
2
4
4t
then equation (1) is nonoscillatory.
Proof. It follows from (10) that there exist two numbers T  t 0 and  
 (t )  a(t )r (t )(
1

 2
) for
2
4t
t log 2 t
1
such that
4
t T .
Let
v(t )  a (t )r (t )h(t ) ,
where
h(t ) 
1 1
1
( 
),
2 t t log t
h (t ) 
1 1
1
1
( 2  2
 2
).
2 t
t log t t log 2 t
then, for t  T ,
So, for t  T ,
v 2 (t )
a(t )r (t )
 (a(t )r (t ))h(t )  a(t )r (t )h(t )   (t )  a(t )r (t )h 2 (t )
1 1
1
1
1

1 1
2
1
 a(t )r (t )[(  )( 2  2
 2
)]  a(t )r (t )( 2  2
)  a(t )r (t )( )( 2  2
 2
)
2
2
2 t
4 t
t log t t log t
4t
t log t
t log t t log 2 t
4  1
 a (t )r (t )( 2
)
4t log 2 t
v(t )   (t ) 
0
.
Thus, by Theorem 1, equation (1) is nonoscillatory.
5
Remark 5.
(a) Let a(t )  1 , then  (t )  c(t ) . Thus our Corollary 4 reduces to Theorem 3.6 in Li
and Yeh [8].
(b) Let a(t )  r (t )  1 . Then Corollary 4 reduces to the result of Hille-Kneser [3],
Kneser [4].
Next, we will give an another proof of the Sturm comparison theorem and the HilleWintner comparison theorem by using Theorem 1. To do this, we consider the following
second order differential equation
(11)
(r1 (t )u(t ))  c1 (t )u(t )  0
where r1 , c1  C ([t 0 , ), R) with r1 (t )  0 on [t 0 , ) . For a given function a1  C 2 ([t 0 , ), (0, )) ,
(11) is equivalent to the following second order linear differential equation
(12)
(a1 (t )r1 (t )w(t ))  1 (t )w(t )  0
where 1 (t ) : a1 (t )[c1 (t )  r1 (t ) f1 2 (t )  (r1 (t ) f1 (t ))] and f1 (t ) : 
a1 (t )
.
2a1 (t )
We next prove that Theorem 1 and Corollary 1 in Moore [9] can be used to answer,
in part, an open question in Theorem 2 of Taam [12], which is a generalization of the
Hille-Wintner comparison theorem (Hille [3], Wintner [14]).
Theorem 6. Let a(t )r (t )  a1 (t )r1 (t ) and
(13)


t
t
0    (s)ds   1 (s)ds  
for t  t 0 . If equation (11) is nonoscillatory, then equation (1) is nonoscillatory. That is,
If equation (1) is oscillatory, then equation (11) is oscillatory.
Proof. It will be convenient to separate the proof into two cases :
(i)
(ii)

1
ds   .
a1 ( s )r1 ( s )

1
 a1 (s)r1 (s) ds   .

Case (i). If equation (11) is nonoscillatory, then equation (12) is nonoscillatory. Thus, as
in the proof of Theorem 1, there exists a function v  C 1 ([T , ), R) for some T  t 0 such
that
v (t ) 
v 2 (t )
 1 (t )  0 ,
a1 (t )r1 (t )
t T .
Integrating it from t to  (t   )
(14)


t
t
v( )  v(t )   1 ( s)ds  
v 2 ( s)
ds  0 .
a1 ( s)r1 ( s)
6
We can prove that


t
v 2 (s)
ds   , t  T and lim v( )  0 . Letting    in (14),
 
a1 ( s )r1 ( s )


t
t
v(t )   1 ( s)ds  
v 2 ( s)
ds , t  T .
a1 ( s)r1 ( s)
Let

y(t ) : v(t )   (1 (s)ds   (s))ds , t  T .
t
This and (13) imply v(t )  y (t )  0 . Moreover,
y(t )  v(t )  1 (t )   (t )
v 2 (t )

 1 (t )  1 (t )   (t )
a1 (t )r1 (t )
for t  T . Thus
v 2 (t )
  (t )  0 ,
t T .
a1 (t )r1 (t )
1
1

It follows from v(t )  y (t )  0 and
that
a(t )r (t ) a1 (t )r1 (t )
y (t ) 
y (t ) 
y 2 (t )
  (t )  0 ,
a(t )r (t )
t T .
Hence, by Theorem 1, equation (1) is nonoscillatory.
Case (ii). Since




1
ds  
a1 ( s )r1 ( s )

and

t
1 (s)ds   ,
imply
1
ds  
a( s)r ( s)
and


t
 (s)ds   .
Thus, it follows from Corollary 1 of Moore [9] that equation (1) is nonoscillatory.
Letting a(t )  a1 (t )  1 in Theorem 6, we have the following corollary.
Corollary 7. Let r (t )  r1 (t ) and


t
t
0   c(s)ds   c1 (s)ds   ,
on [t 0 , ) . If equation (11) is nonoscillatory, then equation (1) is nonoscillatory.
7
2. Leighton Oscillation Criteria
In 1950, Leighton [6] showed the following oscillation criterion:
Leighton’s Oscillation Theorem. If



1
dt   c(t )dt   ,
r (t )
then equation (1) is nonoscillatory.
In this section, we will extend Leighton’s Oscillation Theorem to the second order halflinear ordinary differential equation (2) by using the Coles’ technique [1].
Theorem 8. If

where


c(t )dt   r 1 q (t )dt   ,
1 1
  1 , then equation (2) is oscillatory.
p q
Proof. Suppose not. Then (2) has a nonoscillatory solution u (t )  0 on [T ,  ) for some
T  t 0 . Define
v(t ) 
r (t ) (u (t ))
,
 (u (t ))
Then, for t  T ,
v (t )  c(t ) 
t T .
r (t ) (u (t )) (u (t ))
 2 (u (t ))
r (t )u (t ) | u (t ) | p 2 ( p  1)u p  2 (t )u (t )
 c(t ) 
u 2 p 2 (t )
( p  1)r (t ) | u (t ) | p
 c(t ) 
u p (t )
 c(t )  ( p  1)r
1
p
p 1
r (t ) | u (t ) | p 1 p 1
(t )[
]
u p 1 (t )
p
 c(t )  ( p  1)r 1q (t ) | v(t ) | q
Thus, for t  T ,
v(t )  c(t )  ( p  1)r 1q (t ) | v(t ) | q  0 .
(15)
It follows from (15) that, for t  T ,
t
t
t0
t0
v(t )  v(t 0 )   c( s )ds   ( p  1)r 1 q ( s ) | v( s ) | q ds .
Since


c (t )dt   , we can always find t1  t 0 such that
t
v(t 0 )   c( s)ds  0
t0
for all t  [t1 , ) . Thus
8
t
v(t )   ( p  1)r 1 q ( s ) | v( s ) | q ds
t0
for all t  t1 . Let
t
R(t ) :  ( p  1)r 1 q ( s ) | v( s) | q ds ,
t0
then R(t )  0 , | v(t ) |  R (t ) and
q
q
R(t )  ( p  1)r 1q (t ) | v(t ) | q  ( p  1)r 1q (t ) R q (t ) ,
for t  t1  t 0 . Thus,
R (t )
 ( p  1)r 1 q (t ) .
q
R (t )
Integrating it from t1 to t , we have
t dR ( s )
t
 R1q (t1 )
1

( R1q (t )  R1q (t1 ))   q
  ( p  1)r 1q ( s) R q ( s)ds .
t1 R ( s )
t1
1 q
1 q
Letting t   ,


 R1q (t1 )
 ( p  1)  r 1q (s)ds   ,
t1
1 q
which is a contradiction. Thus (2) is oscillatory.
Remark 9. Let p  2 . Then Theorem 8 reduces to Leighton’s Oscillatory Theorem.
Using Leighton’s Oscillatory Theorem, we have the following:
Corollory 10. Let a , a1  C 2 ([t 0 , ), (0, )) . If either



1
dt    (t )dt   ,
a(t )r (t )



1
dt   a(t )[ (t )  a(t )r (t ) g 2 (t )  (a(t )r (t ) g (t ))]dt  
a1 (t )a(t )r1 (t )
or
where  (t ) and g (t ) are defined as in section 1, then equation (1) is oscillatory.
Remark 11. In [2], Harris used a very complicated transformation which transformed
equation (1) into a Riccati integral equation and then to show Corollary 10 ( Theorem 1
in [2]) holds.
References
[1] W. J. Coles, Asimple proof of a well-known oscillation theorem, Proc. Amer. Math. Soc. 19 (1968),
507.
9
[2]
B. J. Harris. On the oscillation of solutions of linear differential equations, Mathematika 31 (1984),
214-226.
[3] E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234-252.
[4] A. Kneser, Untersuchungen über die reelen Nullstellen der Integrale linearer Differential
gleichungen, Math. Ann. 42 (1893), 409-435.
[5] M. K. Kwong and A. Zettl, Integral inequalities and second order linear differential equation, J.
Diff. Equs. 45 (1982), 16-33.
[6] W. Leighton, The detection of the oscillation of solutions of a second order linear differential
equation, Duke J. Diff. Math. 17 (1950), 57-62.
[7] W. Leighton, Comparison theorems for linear differential equations of second order, Proc. Amer.
Math. Soc. 13 (1962), 603-610.
[8] H. J. Li and C. C. Yeh, On the nonoscillatory behavior of solutions of a second order linear
differential equation, Math. Nachr. 182 (1996), 295-315.
[9] R. A. Moore, The behavior of solutions of a linear differential equation of second order, Pacific J.
Math. 5 (1955), 125-145.
[10] C. Sturm, Sur les équation differentielles lińeaires du second order, J. Math. Pures Appl. 1 (1836),
106-186.
[11] C. Swanson, “Comparison and oscillation theorey of linear differential equations”, New York and
London, Academic Press (1968).
[12] C. T. Taam, Nonoscillatory differential equations, Duke Math. J. 19 (1952), 493-497.
[13] D. Willett, On the oscillatory behavior of the solutions of second order linear differential equations,
Ann. Polon. Math. 21 (1969), 175-194.
[14] A. Wintner, On the comparison theorem of Kneser-Hille, Math. Scand. 5 (1957), 255-260.
10