Internat. J. Math. & Math. Sci.
(1986) 105-109
105
Vol. 9 No.
COMPARISON THEOREMS FOR FOURTH ORDER
DIFFERENTIAL EQUATIONS
GARRET J. ETGEN
Department of Mathematics
University Park
University of Houston
Houston, Texas 77004
WILLIE E. TAYLOR, JR.
Department of Mathematics
Texas Southern University
Houston, Texas 77004
(Received April 23, 1984 and in revised form August I, 1985)
ABSTRACT.
This paper establishes an apparently overlooked relationship between the
iv
line,r equations y
0 and
p(x)y
+ p(x)y 0, where
yiV
p’air of fourth order
p
,
is
positive, continuous function defined on
[O,m).
It is shown that if all
solutions of the first equation are nonoscillatory, then all solutions of the second
equation must be nonoscillatory as well.
An oscillation criterion for these equations
is also given.
KEY WORDS AND PHRASES.
Oscillatory solutions; nonoscillatory solutions; fourth order
linear differential equations.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
34CI0
INTRODUCTION.
This note is concerned with the pair of fourth order linear differential equations
y
y
where
p(x)
p
iv
iv
p(x)y
0
(I.I)
+ p(x)y
0
(1.2)
is a positive, continuous function on
[0,).
relate the oscillatory character of (i.I) with that of (1.2).
Recall that a linear differential equation is said to be
val
I.
I
if it has a nontrivial solution
Also, such a solution
y
y
y(x)
The objective is to
oscillatorx
on an inter-
which has infinitely many zeros on
is called oscillatory.
On the other hand, if no non-
solutions
trivial solution of the equation is oscillatory, that is if all nontrivial
Hereafter,
I.
on
nonoscillatory
be
are nonoscillatory, then the equation is said to
the term
"solution" shall be interpreted to mean "nontrivial solution".
This study is motivated by several recent results concerning pairs of equations
of the form
(k)
p(x)z
0
(1.3)
z(k)
+ p(x)z
0
(1.4)
z
G.J. ETGEN and W. E. TAYLOR JR.
106
See, for example, D. L. Lovelady [i] and W. E. Taylor, Jr., [2].
k,
When the order,
of the equations is odd, each equation is the adjoint of the other and it is natural
In particular, (1.4) is oscil-
that their oscillatory character should be related.
latory i and only if (1.3) is, when
In contrast, (i.I) and (].2) are each
is odd.
k
self-adjoint and the current literature suggests that the two equations are unrelated
Indeed, the two equations have always been studSee the works of W. Leighton and Z. Nehari [3], M. Keener [4], and
S. Ahmad [5] for detailed investigations of these equations. The fact that ether all
with respect to oscillatory behavior.
ied separately.
the solutions of (1.2) are oscillatory or they are all nonoscillatory, while (].l) al-
ways ias nonoscillatory solutions, provides a demonstration of the apparent independence of the two equations in terms of oscillation. However, contrary to this evidence,
(1.1) and (1.2) that involves the
we shall show that there is a relationship between
oscillatory behavior of the solutions.
2.
MAIN RESULTS.
A proof of this result appears in
Our work requires the following general lemma.
tle paper [6] by G. D. Jones.
LEMMA.
f(i)(x)
If the function
e O,
i
0,|
k,
f(x)
f
f(x)
>
f(i) (x)
on
(k+l)-times differentiable and satisfies
is
f(k+l)(x)
and
on an interval
< 0
(k-i)!
k!
(x-a)i,
[a,oo),
then
i < k-l,
[a,o).
Both Ahmad [5] and Keener [4] have shown that (1.1) always has a pair of solutions
y
and
w
satisfying
y(x)
O,
[b,),
on some half-line
b
w(x) > O,
on
on
[0,o).
[b,),
y’(x)
>
a O,
and
w’(x)
O,
O,
y"(x)
0,
y"’(x) > 0
(2.1)
w"(x)
O,
w"’(x)
(2.2)
< 0
Solutions of (i.i) which satisfy (2.1) are said to be strongly increasing
while solutions satisfying (2.2) are termed
stron_ decreasing.
Ahmad
proved that (i.i) is oscillatory if and only if all nonoscillatory, eventually positive
solutions satisfy either
(2.1) or (2.2).
This fact serves as the basis for our main
tleorem.
THEOREM i.
PROOF.
If (i.i). is nonoscillatory, then (1.2) is nonoscillatory.
Assume that (1.1) is nonoscillatory.
It follows from the remarks above
that (l.1) has an eventually positive solution which fails to satisfy either (2.1) or
(2.2).
It is easy to show that such a solution, say
y(x)
on
[a,)
for some
a >
O,
y’(x)
0.
Let
equation
z
,nd
z
satisfies
z(x)
O,
>
0,
y’(x).
....
p(x)y(x)
y’(x)
z’(x)
O,
z
z"(x)
[a,oo)
< 0
(2.3)
0
0
must satisfy
is a solution of the third order
z
Then
y(x),
y"’(x)
y"(x) > O,
z
clude that (2.3) is (],2)-disconjugate o
y
on
[a,oo).
From this we can con-
and hence nonoscillatoy (see [2]).
COMPARISON THEOREMS FOR FOURTH ORDER DIFFERENTIAL EQUATIONS
107
Using the Lemma, we have
y(x)
x-a
y’ (x)
2
Thus, by a well-known comparison theorem (see, for exa,nple, Nehari [7]), the third
order equation
(x-a)
z"’
p(x)z
2
(2.4)
0
[a,oo). Therefore, (2.4) has a soluu’(x) > O, u"(x) < 0 on [a,==). It follows
0 and u’(x)
a 0 as x.Subs:ituti=g
is also nonoscillatory and (l,2)-disconjugate on
tion
u
u(x)
.
u(x) > O,
satisfying
u"(x)
from these inequalities that
nto (2.4) yields
(-a)p(x) u(x)
2
u"’(x)
We integrate this eqation from
u"(s)
Now, letting
s
x
I
u"(x)
s
fs.x
(t-a)p(t)u(t)dt,
and using the fact that
-u"(x)
Integrate again from
u’(x)
to obtain
to
x
u’(s)
s,
to
x
u"(s)
s
i
f
we have
(t-a)p(t)u(t)dt.
(r-a)p(r)u(r)d
l(s-x)
0,
using integration by parts.
x
2
/
x > a
The result is
dt
(t-a)p(t)u(t)dt
+ ;s
(t-x)(t-a)p(t)u(t)dt
Is
(t-x)(t-a)p(t)u(t)dt
i
x
or
u’(x)
u’(s) +
g(s-x)
s (t-a)p(t)u(t)dt
+ I
x
Thus, we can conclude that
u’ (x)
on
[a,).
i (t_x)2p(t)u(t)dt
_>
x
Furthermore, since
u(a) + |x u (s)ds
a
u(x)
wu halve
u(x)
u(a) +
-
l;x I (t-s)2p(t)u(t)dtds
a
Now, by standard iteration techniques, there
fined on
[a,)
such that
w(x)
Differentiating w,
w(a) +
is a nonnegative function
lx[ (t_s)2p(t)w(t)dtds
a
we find that
w(x) > 0.
w’(x)
>
0. w"(x)
>
O. w"’(x)
< 0
w
w(x)
de--
I08
G.J. ETGEN and W. E. TAYLOR JR.
and
w
Thus,
w
(1.2)
,are
iv
-p (x)w(x)
(x)
(1.2) which implies that all solutions of
is a nonoscillatory solution of
[3]).
nonoscillatory (see
If (1.2) is oscillatory, then (i.i) is oscillatory.
COROLLARY.
REMARK.
Euler equations can be used to show that the converse of the Theorem
-4
iv
For .:.:ample, the equation y
does not hold.
0 is oscillatory while
x y
V
iv
+
x- 4y
0
is nonoscillatory.
We conclude this note with an oscillation criterion for (1.2) and hence for (i.i).
"conditionally" oscillatory second
This result involves a comparison of (1.2) with a
order differential eq,ation.
_
THEOREM 2.
for arbitrary
Let
a
v(x)
v
be a function such that
2
(x-a)
3!v(x)
lira inf
X
0
->-
and suppose that the differential equation
z" + cv(x)p(x)z
is oscillatory for some
c
e (0, I).
Then (1.2) is oscillatory.
Suppose that (1.2) is nonoscillatory.
PROOF.
[a,oo).
some half-line
y’(x)
y(x) > O,
of (1.2) satisfying
From the
Then there is a solution
>
O,
y"’(x)
> 0
for all
y(x)
y
x
on
it follows that
2
Note that (1.2) can be written as
[a,o).
z" +
z
y"(x)
0,
(x-a)
_>
y"(x)
where
>
l, enma,
_X_()
on
0
y"(x).
y" (x)
is nonoscillatory.
t
a,
X
Using the Sturm comparison theorem, we conclude that
z" +
there exists
0
Z
lim inf
3!v(x)
x
such that
Since
a
b
(x-a)2p(x)
(x-a)2 >_ 1,
corresponding to each
b c
(0,1)
2
(x-a)
3!v(x)
Therefore bv(x)
<
(x-a)
3!
> b
for
x _>
on
It b,)
from which it follows that
z" + bv(x)p(x)z
is nonoscillatory.
b.
2
0
This contradicts the hypothesis of the theorem and the result is
established.
While the techniques in this note are similar to those used in
are new.
[6],
the results
COMPARISON THEOREMS FOR WOURTH ORDER DIFFERENTIAL EQUATIONS
109
REFERENCES
Asymptotic Analysis of Two Fourth Order Linear Differential Equations, Annales Polonici Math. 3--8 (1980), 109-119.
I.
Lovelady, D.L.
2.
Taylor, W.E., Jr.
3.
Leighton, W. and Nehari, Z. On the Oscillation of Solutions of Self-adjoint Linear
Differential Equations of the Fourth Order, Trans. Amer. Math. Soc. 89 (1958),
Oscillation and Asymptotic Behavior in Certain Differential
Equations of Odd Order, Rocky Mountain J. Math. 12 (1982), 97-;02
325-377
On the Solutions of Certain Self-adjoint Differential Equations of
Fourth Order, J. Math. Anal. and Appl. 33 (1971), 278-304.
4.
Keener, M.S.
5.
Ahmad, S.
6.
Jones, G.D.
7.
Nelari, Z.
On the Oscillation of Solutions of a Class of Linear Fourth Order Differential Equations, Pacific J. Math. 34 (1970), 289-299.
An Ordering of Oscillation Types for
Anal. 112 (1981), 72-77.
53-76.
Green’s
y
(n) + py
Function and Disconjugacy, Archive for
O, SlAM J. Math.
Rat. Mech. 62 (1976),