Internat. J. Math. Math. Si.
No. 2 (1980) 275-292
275
Vol.
WRONSKIANS AND SUBSPACES OF CERTAIN FOURTH
ORDER DIFFERENTIAL EQUATIONS
G.J. ETGEN
Department of
_Mathematics
University of Houston
Houston, Texas 77004 U.S.A.
W.E. TAYLOR, JR.
Department of General Academics
Texas A & M University at Galveston
Mitchell Campus, P.O. Box 1675
Galveston, Texas 77553
(Received August 21, 1978)
ABSTRACT.
The objectives of the paper are to study the behavior of Wronskians of
solutions of the fourth order differential equations and to relate this behavior
with the oscillations of these equations, as well as to the structure of the sub-
spaces of the solution spaces of the equations.
KEY WORDS AND PHRASES. Ordinary Differential Equations, Wronsian, Oscillatory or
non-oscillatory Soldtions
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
34CI0.
276
i.
G.J. ETGEN AND W. E. TAYLOR
INTRODUCTION.
This paper is concerned with the fourth order differential equation
(p(x)y")"
q(x)y"
r(x)y
(L)
0
and its adjoint
q(x)y]"
[p(x)y"
r(x)y
(L)
0
where p, q and r are assumed to be continuous real-valued functions on the interval
+
R
+,
R
[0, ).
In addition, it is assumed throughout that p > 0, q >- 0 and r > 0 on
with r not identically zero on any subinterval.
It is clear that if q is a
(L) is non-selfadjoint.
constant, then (L) is selfadjoint; otherwise
The objectives of the paper are to study the behavior of Wronskians of solutions
of (L) and (L*), and to relate this behavior with the oscillation of (L) and (L*),
as well as to the structure of the subspaces of the solution spaces of (L) and (L*).
A nontrivial solution y of (L) {(L*)} is oscillatory if the set of zeros of y is not
bounded above.
If the set of zeros of y is bounded above, implying that y has only
+,
finitely many zeros on R
then y is nonoscillatory.
Equation (L)
atory if it has at least one nontrivial oscillatory solution.
(L*)}
is oscill-
For convenience here-
after, the term "solution" shall be interpreted to mean "nontrivlal solution."
Various special cases of
(L) have been studied in detail.
In particular, we
refer to the fundamental work of W. Leighton and Z. Nehari [10,Part I] on the self-
adj oint equation
(p(x)y")
r(x)y
0
(I.I)
M. Keener [7,Part I] continued the investigation of (i.i), concentrating on the
oscillatory behavior of solutions.
y
(4)
S. Ahmad [1] considered the selfadjoint equation
r(x)y
0
(1.2)
277
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
and gave a necessary and sufficient condition for the existence of a linearly inde-
In [12] and [13] V. Pudei investigated the
pendent pair of oscillatory solutions.
behavior of solutions of the equation
y(4)
q(x)y"
r(x)y
(1.3)
0
Finally, we refer to the authors’ work in
where sufficient conditions for the
5
oscillation of (L) and (L*) are given, and where the behavior of both oscillatory
and nonoscillatory solutions is studied.
2.
PRELIMINARY RESULTS.
As a notational convenience in treating the solutions of equations (L) and (L*),
we introduce the following differential operators.
D0Y(X)
D0Y(X
D2Y(X)
p(x)y"(x),
D2Y(X
p(x)y"(x)
,
D3Y(X)
[p (x) y" (x)
q(x)y(x),
,
D4Y(X)
y’(x),
DlY(X)
DlY(X)
y(x),
D4Y(X)
,
[p(x)y"(x)]
[p(x)y"(x)
D3Y(X)
q(x)y(x) ]
[p(x)y"(x)
Our first result is essential in the work which follows.
for equations (I.I), (1.2), and
q(x)y(x)]
(1.3), are given
in
Corresponding results
[7], [i] and [13], respectively.
The proof is a straightforward modification of the proofs of Lemmas 2.1 and 2.2 in
[Io].
,
THEOREM 2.1.
{Diy(a) _> 0},
one i, then
Di
t
If y is a solution of (L)
0, 1, 2, 3, for some a e
Y > 0
lim
x--
{Di
Y > 0}, i
DiY(X
{(L*) } such that
x-m
DiY(X)
If z is a solution of (L) {(L*)} such that
> 0
with strict inequality for at least
0, I, 2, 3,
{lim
DiY(a)
(-l)iD
D4Y
_>
}, i
0
{D4y _> 0}
on
(a,,) and
*
> 0}
0, 2
z(b) > 0 {(-i) i
Diz(b
G. J. ETGEN AND W. E. TAYLOR
278
i
O, I, 2, 3, for some b
then (-I)
iDi
z > 0 {
Dz
0},
>
+, b
R
0, with strict inequality for at least one i,
>
0, I, 2, 3, and
i
D4z
>_
0 {
Dz
> 0} on
[O,b).
It is clear from Theorem 2.1 that (L) and (L*) each have unbounded nonoscil-
i
+,
In fact for any point a
latory solutions.
R
the four solutions
Yi(x,a),
0, I, 2, 3, of (L) {(L*)} determined by the initial conditions
DiY i(a,a)
i, j--0,
I, 2, 3, where
ij
[a, )), unbounded solutions
canonical basis.
DiY(X)
> 0,
,
{DjY i(a,a)
ij
ij }
is the Kronecker delta, are monotone increasing
(L) ((L*)}; a so-called
which form a solution basis for
{(L*)} satisfying
In general, a nonoscillatory, solution y of (L)
DiY(X)>
0},i=0, i,2,3, on (b,), for some b
+,
R
(on
is strongly increasing.
The next theorem provides the existence of a bounded nonoscillatory solution.
The technique employed in establishing this result is well-known (see [7, Theorem
i.I], or [I, Theorem 2]) and, consequently, the proof is omitted.
THEOREM 2.2.
There exists a solution w of (L) {(L*)} such that
3
3
(+/-)
n
i--0
(ii)
Dow
n
{
lira
x->o
Diw(x)
i=0
D2w #
{sgn D0w sgn D2w
sgn
on
(iii)
# 0
Diw(x)
sgn
sgn
DlW
sgn
DlW
+
R
DlW(X)
lim
0} on R’
sgn
sgn
i
0, I, 2, 3, on
D3w}
,
0
lwCx)
{lim
k
>_
Diw(x)
0
Diw(x)
is said to be strongly decreasing.
each of (L) and (L*) has such a solution.
0, I, 2, 3,
0}, i
A solution w of (L) { (L*) }which satisfies (-I) i
+
R
D3w
> 0
{(-i)
i
*
Diw(x)
>
0}.
Of course, by Theorem 2.2,
279
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
Pudei
[13, Theorem 55] has shown that equation (1.3)
only if its adjoint is oscillatory.
is oscillatory if and
With obvious modifications, his proof can
be extended to the case of (L) and (L*).
An alternative proof of this fact can
th
be accomplished by showing that the n--conjugate point of a with respect to (L)
th
coincides with the n---conjugate point of a with respect to
applying Leighton and Nehari’s result [i0, Theorem 3.8].
(L*), and then
The final theorem in
this section gives a necessary and sufficient condition for the existence of
oscillatory solutions of (L) and (L*).
The authors established this result for
(L) (see [5, Theorem 4.1]) using the approach developed by Ahmad
in [i].
This
approach can also be used to establish the result for (L*).
THEOREM 2.3.
The following two statements are equivalent:
(a)
Equation (L)
(b)
If y is a nonoscillatory, eventually positive solution of (L)
(L*)
is oscillatory
(L*)},
then y is either strongly increasing or strongly decreasing.
3.
WRONSKIANS OF SOLUTIONS.
Let $ and
*
denote the space of solutions of (L) and (L*) respectively.
The theorems of the previous section suggest the identification of the following
subsets of $, and of $*:
I
$
either y or -y is strongly
increasing}
{w e $
either w or-w is strongly
decreasing}
{y e
O= {z S
Let I*,
D*
and
z,S uD}
0* be the corresponding subsets of S*.
specify that none of these subsets is empty.
(L) {(L*)}
Theorems 2.1, 2.2 and 2.3
Moreover, according to Theorem 2.3,
is oscillatory if and only if every solution in
0 (*)
is oscillatory.
In this section we study the Wronskians of solutions of (L) and (L*), we
give some basic identities satisfied by these Wronskians, and we give a neces-
sary and sufficient condition for the nonoscillation of (L) and (L*) in terms of
G. J. ETGEN AND W. E. TAYLOR
280
these Wronskians.
S
Let u, v, y, z
Then
u
W2 (u,v)
u
v
,
,
,
W2(u,v), W3(u,
W4(u,
v, y) and
=-
v, y, z)
basis for
+,
k} on R
(L) {(L )}.
and k
# 0
v, y, z) of solutions u, v, y, z
Of course
of (L) are defined in a similar manner.
{W(u,
z
DlU DlV DIY DlZ
D2u D2v D2Y D2z
D3u D3v D3Y D3z
,
W4(u,v,y,z
The Wronskians
y
v
u
DlU DlV DIY
D2u D2v D2Y
Y)
W3(u,v,
Dlu Dlv
y
v
if and only if
v, y, z)--k (constant)
W4(u
{u, v, y, z}
is a solution
Also, it is well-known (and easy to verify by direct
calculation) that if u, v, y are any three linearly independent solutions of
(L)
{(L*)}
then W
3(u,
v, y)
The linear operators
L
{W(u,
and
L
v
y)}
is a solution of
associated with equations
(L*
(L) }.
(L) and (L),
respectively, are defined by
L[y]
[p(x)y"]
[y]
[p(x)y"
q(x)y"
r(x)y,
q(x)y]
r(x)y
A function y defined on R+ is said to be admissable for
py" {py’
qy} is twice differentiable on R/"
admissable for
L
/_
{*}
if each of y and
If y is admissable for /_ and z is
then they satisfy the Lagrange identity.
z/_[y]
yL [z]
{y;z}’,
(3.1)
281
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
where
{y;z}
.
+ (pz"- qz)y
i *
(-I) Diz
DB_iY
z (py")
3
z(py")
(pz"-qz) y
(3.2)
i--0
implies {y;z}
/
,
then {y;z}
0, which
Thus the relation (3.2) determines a
k (constant) on
}: S x
function
S
S and z
If follows from (3.1) that if y
R (the reals), and it is easy to verify that this
function is linear in each of its arguments.
This function can also be used to
In
express a relationship between certain Wronskians of orders 3 and 4.
W3(u,
v, y)
S
Then
and
{z;
Similarly, if v, y, z
W3(v
S.
S be linearly independent, and let z
particular, let u, v, y
E
v, y)}
W3(u
S’are
W4(u,
v, y, z)
linearly independent and u
(3.3)
S
then
y, z) e S and
{W3(v,
y, z); u}
W4(u,
v, y, z)
(3.4)
These identities are an extension of the ideas introduced by J. M. Dolan in [4].
They can be verified by expanding W
4
first column.
,
along its last column and W
4
along its
It is clear from identities (3.3) and (3.4) that {u, v, y, z}
forms a solution basis for S (S) if and only if
{z;W3(u,
v, y)}
k
({W3(v,
# 0
y, z); u}
k
# 0).
Our first two results establish a connection between two and three
dimensional subspaces of S
(S*)
and certain second and third order Wronskians.
These results are related to the Wronskian identities established by W. J. Kim
in [8].
THEOREM 3.1.
defined by
,
Let y
(
S (S) and let
,
Sy (Sy)
be the subset of
S
,
(S)
G. J. ETGEN AND W. E. TAYLOR
282
’,.*
O}
{y;z}
{z
Y
{u e S
(Sy
{u;y}
O)
Then
(i)
The set
Let u
(ii)
S*Y (Sy)
S (S*).
is a three dimensional subspace of
Then
S*Y S* (Sy
S
(S).
S u) if and only if y
U
ku for
some nonzero constant k.
Let u, v, z be a basis for S
(iii)
(W3(u,
v, z)
W3(u,
v, z)
v, z)
my
In fact, the basis u, v, z
(W3(u
y
W3(u
Then
my) for some nonzero constant m.
may be chosen such that
NOTE.
(Sy).
Y
y)
v, z)
this section will be given in terms of elements of
The proofs in
S
It will be clear that the same arguments apply equally as well for
only.
elements of
S
Part (i) follows from the fact that {y;z}
PROOF.
0 essentially defines
a third order, linear, homogeneous differential equation.
Consider part (ii).
easy to see that if y
S
assume that y, u
subspace of S
Since
{. ,.} is linear in each of its arguments, it is
ku, then S y
u
are linearly independent
Su).
To show the converse
Since
S
is a three dimensional
Y
+
it follows that if a is any point in R
and i, j are any two
distinct integers, 0 < i, j < 3, then there exists a solution z
Diz(a)
Djz(a)
0.
+.
easy to verify, using the Lagrange identity, that S
satisfying
D0z(a)
D2z(a)
Dlz(a)
0
u(a) # 0.
u are linearly independent
# k, 0
ztS.
U
i
From part (i) we can assume that y(a)
the least integer such that
j
D3z(a)
0, y(a) # 0.
The same argument applies if u(a)
< j, k < 2 and j
DiY(a)
#
i, k # 3
Y
such that
Then it is
contains the solution z
and z
Su
Thus S
S
y # u
Now suppose y(a) # 0 and
u(a).
Let i, i
_<
i
<_ 3
be
Such an integer exists since y and
Diu(a).
Now choose z
# 3
0, u(a) # 0.
Suppose y(a)
R
Fix any a
S*y
S
i.
Y
such that
D.z(a)3 DkZ(a)
0,
Then from the Lagrange identify
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
For (iii), let {u, v, z} be a basis for S
{W;v}
from (3.4), {W;u}
W
i*
*i
THEOREM 3.2.
(i)
(ii)
S*y
N
,
,
Sy,
0, which implies SW
{W;z}
ira_W,
v,
S*
(Sy
z
S z)
N
,
kW
2(u,
,
{W
2(y,
v)
v,
S(S
Let y,z
v, z}.
Then,
and so
v,
Then:
be linearly independent.
S * (S).
is a two dimensional subspace of
If {u,v} is a basis for
W2(y, z)
Y
W--W3(u
and let
The last part of (iii) follows from the fact that
my from (ii).
v,
,
,
283
z)
S*y n S*z (Sy
kW
n S
then
z
v)} for some nonzero
2(u,
Moreover u andv may be chosen so that W2(y, z)
,
W
2(u,
,
{W
constant k.
2(y,
v)
z)
W
2(u,
v)}.
Part (i) follows from the fact that the intersection of two
PROOF.
distinct three dimensional subspaces of a four dimensional vector space has
dimension two.
To prove part (ii), assume that y, z
basis for
S*
y
N
S*z
S are linearly independent and {u,v}
is a
Then u and v satisfy the "third" order equations
,
D0Y D3
and
D0z
D *3qc
,
DIY D2
,
DlZ D2c
+
D
,
,
D
D2Y I- D3Y 0=
,
,
+ D2z D1- D3 z D0q:
0
0
Multiplying these two equations by y and z, respectively, and subtracting, yields
the second order equation
W2(Y,
Now
D2
p
z) D2
q
W2(Y,
p(yz
I+ [y D3z
zy )D
z
D3Y]=
0
so that this equation can be written.
z)
W2(Y
z) + [y
D3z
z
D3Y
q]
0
P
Since u and v satisfy this equation, we have, by Abel’s identity,
W2(y, z)
k W
,
2 (u, v),
for some nonzero constant k, on any interval on which
W2(Y
z) # 0.
The
G. J. ETGEN AND W. E. TAYLOR
284
continuity of
W2,
+.
holds on R
W
2
and their derivatives imply that this equation actually
,
Finally, since k
W2(u
,
v)
W2(ku,
v), we have (ii) in the case
y, z
We conclude this section with a necessary and sufficient condition that
,
each of (L) and (L) be nonoscillatory.
This condition is stated in terms of
the nonoscillation of second and third order Wronskians of linearly independent
In particular, the Wronskian W
solutions.
independent solutions of
such that W
i
..q (S)*
is
nonoscillator
Equations (L) and
if there is a number b
S
{S*}
W2,
W
{W2, W 3}
3
is oscillatory.
of linearly independent
is nonoscillatory.
Assume that all second and third order Wronskians of linearly
independent solutions of (L) are nonoscillatory.
order Wronskians W
,
..q
(Wi)
+
R
(L) are nonoscillatory if and only if
every second and third order Wronskian
PROOF.
2 or 3, of linearly
i
(W i) is nonzero on [b, ); otherwise W
i
THEOREM 3.3.
solutions of
(Wi)
i
3
are nonoscillatory.
Then, in particular, all third
By Theorem 3.1 (iii) every solution of
is the Wronskian of three linearly independent solutions of
S.
Thus we can
conclude that (L), and hence (L), are both nonoscillatory.
Now assume that (L) is nonoscillatory.
Then Theorem 3.1 (iii) and the fact
,
that (L) is also nonoscillatory implies that all third order Wronskians of
linearly independent solutions of (L) are nonoscillatory.
examine the second order Wronskians.
Thus, it remains to
Let y, z
S be linearly independent and
+.
It was shown in [5, Lemma 3.1]
assume y > O, z > 0 on [b, =) for some b e R
that i u is any nonoscillatory solution of (L), then
2
D.u # 0
on
[c, =) for some
+
R
c
i=O
Therefore, by taking b large enough, we may assume y
D.zl
# 0,
i
i, 2, on [b, ).
Suppose
W2(Y
>
O, z
>
O,
z) is oscillatory.
DiY
O,
Recall the
285
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
subsets
I, D
and 0 of S.
(or vice versa).
impossible.
Since
I,
We show first that we cannot have y
z)
W2(Y
yz
zy
Suppose, therefore that y e
z
it is clear that y e
0.
and z
,
D u 0
z
is
There are three possible
cases for the signs of z and its "derivatives".
(I)
z > 0, z
>
(II)
z > 0, z
< 0,
(III)
z > 0, z
> 0,
0,
D2z
D2z
D2z
on
[b, ),
> 0
on
[c, =),
c
< 0
on
[c, ),
c > b.
< 0
>
0,
>
0,
D3z
D3z
>- b,
Since y > 0 and y is strongly increasing, we may assume that
i
i, 2, 3, on [b, =).
satisfies
(I), then
W2(Y
If z satisfies (II), then
W2(Y
z)
yz
DiY
>
0,
z) < 0 on [c, =).
If z
[b, =) which implies that W 2 has
Finally, suppose z satisfies (III).
zy
< 0 on
constant sign on [d, ) for some d > b.
Then it is easy to verify that
y
(PW2)
D2Y D2z
Z
D3Y D3z
and as a result of the signs of y, z and their "derivatives", we find that the
first term on the right hand side is monotone decreasing and the second term has
limit
-.
and so W
2
Thus lim pW
-,.
2
is eventually of one sign.
oscillates, then either y, z
Assume y, z
and
W2(Y
are linearly independent.
{y, z, u}
This implies that W
I,
< 0 on
[d, ,) for some d >- b
We can now conclude that if
or y, z
W2(Y
Choose u
0 such that y, z,
z, u) # 0 on [a, ) for some a
W3(y,
is a solution basis for the third order differential equation
y
z
u
DIY DlZ DlU
p"
(P)
z)
D u 0.
z) oscillates.
Then
2
D2Y D2z D2u
D3Y D3z D3u
+,
R
and
u
G. J. ETGEN AND W. E. TAYLOR
286
which can be written in the form
WB(Y
WB(Y
z, u)(p
Now, the fact that
z, u)(p
+ f(x) + g(x)
(3.5)
O.
W2(Y, u) are nonoscillatory implies that there
0 on [b, ). Therefore, according
and W2(Y, u)
u and
b > a such that u
0
exists
to
Ahmad [2, p292] (see also, G. Polya .F12]),(8) is disconjugate on Fb, ).
Thls
implies that the adjoint of equation (3.5) is disconjugate on [b, )
(see J. H. Barrett [3]).
But (x)=
W2(Y, z)/W3(Y,
z, u) is an oscillatory
solution of the adjolnt equation, and we have a contradiction.
of proof can be used if we assume that y, z
4.
D
The same method
u 0 and W 2(y, z) oscillates.
SUBSPACES OF S AND S:
In this section we consider the structure of the three dimensional
subspaces of S and
S*,
and as a corollary, we also identify certain two
We will be making use of the subsets
dimensional sbspaces.
*
S and I *
and
O*
*
of
In this regard
D
and 0 of
defined in Section 2, and we shall also be concerned
with the "complimentary" subspaces
S (S).
,
S
Y
recall that if
(Sy)
y
determined by the solutions y of
S (S)
then S
y
(Sy)
is the three
dimensional space of solutions of the third order equation
,
D0Y D3
,
,
,
DlY D2+ D2Y DI- D3Y D0
D0Y D3 DlY D2 + D2Y DI D3Y DO
(4.1)
0
0}
In [6], M. Hanan defined two classes of third order linear differential
equations.
In particular, a third order equation is in class C I on [a, )
if amy soltion,wlth
it is in class
CII
on
D double
[a, =) if any solution with
nonzero on (b, ).
THEOREM 4.1.
zero t some point b > a is nonzero on [a, b), and
Let y e S.
a double zero at b
>- a,
is
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
(1)
z
I
E
I,
If y
u, v
equation
*
0* W2(u
E
(9)
D,
E
[a, ) for some
C
on
[z, u, v] (= the space spanned by z, u, v), where
Y
CII
then S
on
,
v) # 0 on
is in class
(ii) If y
I
S
then
on
*
[’a,
) for some a > 0, and the third order
[a, ).
[w, u v], where w
y
If y
PROOF.
E
0,
Suppose y
eventually positive.
[a, ).
from (9)
then
E
S*Y
[z, w, u], where z
[, and assume,
0
Y
n S
S *
y
,
DiY
D*
w E
Diy
and u
0*
E
E
S.
>
0, i
0, i, 2, 3,
D
*
E
[
and equation (4.1) is in class
{w;u}
> 0"on
D,
[0, ).
0* Finally,
,
W (u, v)
k
y
on
W2(Y
2
[a, ).
n S
If
w
0*
Thus
from Theorem 3.2 (ll), there is a
w)
0 on [a, ).
Therefore
is complete.
and assume, without loss of generality, that
Choose any b > 0 and let z be a solution of (4.1) such
that z has a double zero at b.
,
S
, CII
0 from which it follows that u
[u, v], where u, v
Suppose y
We may assume,
with at least one inequality being strict,
and consider the two dimensional subspace
then {y;u}
w
Then it is easy to verify
have opposite sign.
[z, u, v] and the proof of Dart (i)
(ll).
I*
without loss of generality, that y is
,
nonzero constant k such that
i
S
E
such that z has a double zero at b.
D2z(b) and D3z(b) cannot
,
that D2z(b) > 0, D3z(b) >_ 0
,
Now choose w
uS*
y
E
,
,
and so, by Theorem 2.1, z
(-i)
*
O* W2(u,v)
(4.1) is in class
Then there exists a > 0 such that
that_
therefore,
Y
E
Choose any b > a and let z be a solution of (9), i.e. let z be an
element of S
S
u, v
As in Section 3, we will prove the theorem only for the case y
(1).
,
P*
a > 0, and the third order equation
The corresponding statements hold if y
S
E
[0, m).
(ill)
on
287
Then it is easy to see from (4.1) that D
,
,
,
2
z(b)
z(b) must have opposite sign, say D z(b) > 0, D z(b) < 0. Thus, by
2
3
i
Theorem 2.1, we have (-i) D * z > 0, i
on [0, b) and equation (4.1)
0,
2,
3,
i,
i
and D
3
is in class C
I
on
[0, ). Now, for each positive integer n, let z be a solution
n
288
G. J. ETGEN AND W. E. TAYLOR
of (4.1) which satisfies
Zn(n)
by a standard
D
Zn(n)
I
0,
D2Zn(n) > 0, D3zn(n)
< 0.
"sequence argument" (see A. C. Lazer [9, Theorem 1.17
Then
or Ahmad
[17),
we can construct a solution w of (4.1) such that
sg w
sgn
D2w #
sgn
I and
choose any y e
sgn
DlW
assume D
D3w
> 0
iY
[0, =), i.e. such that w
on
(iii).
and
0
u, v
Suppose y e 0.
W2(u,v)
because {z;u}
{w;u}
S*
[a, ).
on
Choose any z e
is a one dimensional subspace of
0*
# 0
I
and any w e
and if u e
zero of y.
,W3(z I,
W3(z I,
Then
3)
z3)(b)
z
z2,
z 2,
S*z
* n
y
.
n
Then S
*
S.,
w
3}
Then by
y
S
0
n S
z
w
then u must be in
Y
suppose that y is oscillatory, and let {z I, z 2, z
Theorem 3.1 (iii),
Now
[w, u, v],
Y
Thus S * contains an element u in
0.
D
0, i, 2, 3 on [a, =), a e 0.
i
using the same argument as in part (i), we can conclude that S
where w
e
0*
be a basis for S
my for some nonzero constant m.
,
Now,
Then, by
Y
Let b be a
0 which implies that there exists a
nontrivial linear combination z of the solutions z I, z 2, z such that z has a
3
triple zero at b.
* > 0, i
that D.z
such that
(-i)
i
> 0 and
conclude, by Theorem 2.1,
Let {b
n
be the sequence
Then for each positive integer n there is a solution z
Diz n (bn)
* > 0, i
D.z
i n
argument"
D3z(b)
0, I, 2, 3, on (b,=) so that z e
1
of zeros of y.
We may assume that
0
0
i
i
D3Zn(bn)
2 and
< 0.
Thus
n
of (4.1)
by Theorem 2.1
0, i, 2, 3 on [0, b ), and so, by using the "sequence
n
cited above we can construct a solution w in
Finally, it is easy to verify that S
Y
S * such that w
Y
[z, w, u].
Now suppose that y is nonoscillatory.
As observed in the proof of
Theorem 3.3, y and its derivatives must satisfy one of the following three sets
of inequalities on an interval [a, =).
(I)
y > 0, y
>
0,
(II)
y > 0, y
<
0,
(III)
y > 0, y
>
0,
D2Y
D2Y
D2y
< 0
>
0,
>
0,
D3Y
D3Y
> 0
< 0
WRONSKIANS AND SUBSPEACES OF DIFFERENTIAL EQUATIONS
289
In each case y has two consecutive derivatives D iY, D i+lY, 0 -< i _< 2 which have
the same sign on [a, ,), and two consecutive derivatives
which have opposite sign on
[a, ).
Choose any b > a.
Let z be a solution of
m
DiY(b)
a coefficient of either
or
It then follows that the remaining
Di+lY(b).
Assutng that at least
two "derivatives" of z at b cannot have opposite sign.
I
one of these "derivatives" is positive, we have z e
,.
increasing sequence in [a, ,) with limb n
n--t
n
n)
be a solution of (4.1) such that D z _(b
m n
property that
DmZn(bn)
0 -< j -< 2
0 for the two m’s with the property that D z(b) is not
(4.1) such that Dmz(b)
z
Djy, Dj+lY,
Now let {b
n
be an
For each positive integer n, let
0 for the two m’s with the
is not a coefficient of either
Djy(bn)
Dj+lY(bn).
or
Then we can conclude from equation (9) that the retning o "derivatives" of
z
n
at b
n
must have opposite sign.
Thus the consecutive "derivatives" of z
must have opposite sign on [0, n), and the
construct a solution w of
"sequence argument" allows
(4.1) such that w
n
us to
Finally, it is easy to
verify that the three solutions, u, z, nd w are linearly independent, and this
completes the proof of the theorem.
There are a variety of consequences of Theorem 4.1 which describe the
structure of the two and three dimensional subspaces of
S and
S*.
We list these
results in the following corollaries.
COROLLARY:
(i)
Every three dimensional subspace of S($
has a non-
oscillatory solution.
(2)
(L)}
Equation (L)
is oscillatory if and only if every three dimensional
subspace of S (S) has an oscillatory solution.
(3)
Every three dimensional subspace of S (S) contains a pair of solutions
whose Wronskian is nonzero on [a, ) for some a >- 0.
PROOF.
{Yl’ Y2’ Y3 }
Let
T
be any three dimensional subspace of .q (q), and let
be a basis for
T.
Then y
W(Yl, Y2’ Y3
{y
W
(Yl’ Y2’ Y3 )}
is
G. J. ETGEN AND W. E. TAYLOR
290
S*
an element of
(S)
T
and
*
y).
Sy (T
and S
We now consider the two dimensional subspaces of
T
corresponding to any two dimensional subspace
,
T of S
Similarly, a two dimensional subspace
T
dimensional subspace
there is a unique two
,
determines a unique two
S.
of
Let T be a two dimensional subspace of S.
COROLLARY 2.
We note that
S * n S * of S * where {y, z} is any basis for T.
z
y
T*
dimensional subspace
of
,
Then
T
satisfies
exactly one of the following:
(i)
T
[y,u], y
(2)
T
[y,w], y
E
I,
u E
0,
and
I,
w
D,
and
T*
v
in 0*
[u,
and every combination of u and v is
(3)
T
[w u]
w E
D,
u E
(4)
T
[u v]
u E
O*
v
E
and
T*
E
I*
[y w], y
0,
and
O*
T*
[z, vl
w
E
subspace of
T*
S,
then
T
o (0 u
D)
E
D
@
and
T
T
(0 u I) # @.
0
O*
is any two dimensional
This fact can be
T n S and T n Sw’ where y
y
T specified
in cases (i)
(3) is easy to verify.
E
0*
must contain either an element of
I,
structure specified in (4) follows from the fact that if y
dimensional subspace of S
T
0*
E
I*
and using Theorem 4.1.
The structure of
P.
v E
is a two dimensional subspace
The four cases follow from the fact that if
established by considering the intersections
w
D*
E
D*
,
PROOF.
z
v
0*
O*
v
and every combination of u and v is in
Clearly, the corresponding statements hold if
of S
u E
I*,
O*
[z, vl, z
Thus
T*
[y w!
Y
cannot contain an element of
y
E
I
w
O*
The
then any two
or an element of
from which it follows that
0
We conclude by noting that if the equations (L) and
(L*)
are oscillatory,
then the structure of a two dimensional subspace can be determined by
WRONSKIANS AND SUBSPACES OF DIFFERENTIAL EQUATIONS
looking at the Wronsklan of a basis for the subspace.
(L*)
are oscillatory, then every solution fn
COROLLARY 3.
and let W(x)
(i)
Recall that if (L) and
fs oscillatory.
Assume that equation (L), and hence (L), are oscillatory.
Let T be a two dimensional subspace of
T,
0 and 0
W2(Yl,
S,
v
{YI’ Y2
let
be a solution basis for
y2 ).
If W is oscillatory, then either
[z, v], z
0
or
T
T
[y, u], y
P,
[w, u], w
u
yD,vO.
(2)
T
If W is nonoscillatory, then either
[u, v], u, v
0
[y, w
7,
E
u
0 and
E
0 and T
_,
y
[y, v],
0,
w
every linear combination of u and v is in 0
zeros of two independent solutions separate on
versa, i.e.
291
[u, v], u, v
0,
etc., and
and
and the
O, or vice
[a, ) for some a
[y, w], y
P
w
REFERENCES
i.
Ahmad, S., "On the oscillation of solutions of a class of fourth order
differential equations", Pacific J. Math., 34 (1970), 289-299.
2.
Ahmad, S., "Oscillation properties of third order linear differential
equations and their adjoints", Journal of Math. and Phys. Sciences, VIII
(1974), 291-298.
3.
Barrett, J. H., "Oscillation theory of ordinary linear differential
equations", Advances in Mathematics, 3, Academic Press, New York,
(1969), 415-509.
4.
Dolan, J. M., "On the relationship between the oscillatory behavior of a
linear third order differential equation and its adjoint",
J. Differential Equations, 7 (1970, 367-388.
5.
Etgen, G. J., and W. E. Taylor, Jr., "On the oscillation of a class of fourth
order differential equations", Rocky Mtn J. Math., 6 (1976), 71-84.
6.
Hanan, M., "Oscillation criteria for third-order linear differential
equations", Pacific J. Math., ii (1961), 919-944.
7.
Keener, M. S., "On solutions of certain self-adjoint differential equations
of fourth order", J. Math. Anal. and Appl., 33 (1971), 278-304.
G. J. ETGEN AND W. E. TAYLOR
292
8.
Kim, W. J.,
"A
relation satisfied by solutions of the adjoint equation",
Kyungpook Mathematical J., 13 (1973), 127-131.
9.
Lazer, A. C., "The behavior of solutions of the differential equation
y’’’ + p(x)y’ + q(x)y 0" Pacific J Math 17 (1966), 435-466
i0.
Leithton, W. and Z. Nehari, "On the oscillation of solutions of self-adjoint
linear differential equations of the fourth order", Trans. Am. Math. Soc.,
89 (1958), 325-377.
ii.
Polya, G., "On the mean-value theorem corresponding to a given linear
homogeneous differential equation", Trans. Am. Math. Soc., 24 (1922),
312-324.
12.
+ P(t)Y"
Pudei, V., 0 vlastnostech reseni diferencialni rovnice
q(t)y
0, Casopis pro Pestorani Matematik, 93 (1968), 201-216
13.
Pudei, V., Uber die Eigenschaften der Losungen linearer Defferential
Gleicheingen gerader Ordnung", Casopis pro Pestovani Matematik, 94
(1969), 401-425.
y(4)
+