Asymptotic and oscillatory solutions of N-th order linear differential equations
by Gerald Edwin Bendixen
A Thesis submitted to the Graduate Faculty in partial sulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Mathematics
Montana State University
© Copyright by Gerald Edwin Bendixen (1973)
Abstract:
In this thesis asymptotic solutions of the n-th order, linear, homogeneous differential equation (x^n)(t)
+ A1(t)(x^(n-1))(t)+...+An(t)x(t)=0 are obtained, where the Aj(t) are continuous complex-valued
functions on [a ,∞). The solutions are found by transforming the given equation into a vector-matrix
differential system for which the result of N. Levinson, Duke Math. J. 15 (1948), pp. 111 - 126, is
applicable. To apply his result, asymptotic estimates for the zeros of related characteristic polynomials
are obtained. The special case where n = 3 is treated in detail. The oscillatory nature and boundedness
properties of solutions of the given equation are also investigated. The results generalize those of G. W.
Pfeiffer, J. Diff. Equations 11 (1972), pp. 138 - 144 and pp. 145 - 155. ASYMP T O T I C A N D O S C I L L A T O R Y SOLUTIONS O F N - T H O R D E R
L I N E A R D I F F E R E N T I A L EQUATIONS
by
GERALD E D W I N BE N D I X E N
A Thesis s u b m itted to the Gr a d u a t e Fac u l t y in partial
sulfillment of the requirements for the degree
of
D O C T O R OF P H I L O S O P H Y
in
Mathematics
Approved:
"EJU-f P
F/»y,
H e a d , M a j o r Department
airman,
E x a m i n i n g Committee
Graduate^ Dean
M O N T A N A STATE U N I V E R S I T Y
Bozeman, M o n t a n a
June,
1973
iii
ACKNOWLEDGMENT
The author w ishes
m a j o r professor, Dr.
to express his sincere a p p r e c i a t i o n to his
Louis C. Barrett,
for his c o n t i n u e d i n terest and
h e l p f u l suggestions d uring the p r e p a r a t i o n of this dissertation.
iv
TABLE' O F CONTENTS
VITA
..............
ACKNOWLEDGMENT
ii
............
^
^
. iii
A B S T R A C T ................................... ■........... ..
INTRO D U C T I O N
v
. . ....................................
CHAPTER I
FUNDAM E N T A L L E M M A S
vi
. . ..........................
I
\
CHAPTER 2 .
A S Y M P T O T I C S O L U T I O N S ......... '.................. 21
C H APTER 3
O S C I L L A T O R Y AND B O U N D E D N E S S PROPER T I E S
. O F SOLUTIONS
CHAPTER 4
................. .
S U MMARY AND E X A M P L E S
. '. . . . . .
. . . . , . • ......... .
. .
46
53
A P P E N D I X I ...................................... ■ . . . . ■ ............. 61
A P P E N D I X 2 . ............ ..
REFERE N C E S
. ...................................... •.
........................................
63
68
V
ABSTEACT
In this thesis asymptot i c solutions of the n - t h o r d e r , linear,
homogeneous d i fferential equation
(t) + A 1 C O x ^ n 1 ^Ct) +.. .+ A
(t)x(t) = O
are obtained, w h e r e the A . (t) are continuous co m p l e x - v a l u e d functions
on [a ,co) . The solutions ^re found b y transforming the g i v e n equa­
tion into a v e c t o r - m a t r i x di f f e r e n t i a l s y s t e m for w h i c h the result
of -N. Levinson, D u k e Math. J. 15 (1948), p p . Ill - 126, is appli­
cable.
To apply his result, a s y m p t o t i c estimates for the zeros of
related chara c t e r i s t i c p o lyn o m i a l s are obtained.
Th e special case
w h e r e n = 3 is treated in detail.
T he osci l l a t o r y n a t u r e and
bounde d n e s s properties of solutions of the g i v e n e q u a t i o n are also
investigated.
The r e sults g e n e r a l i z e those of G. W. Pfeiffer,
J. D i f f . Equations 11 (1972), pp. 138 - 144 and pp. 145 - 155. '
vi
INTRODUCTION
In this thesis w e inves t i g a t e the a symptotic b e h a v i o r of
solutions of ordinary linear h o m o g e n e o u s d i f f e r e n t i a l equations
(t) +
(t)x^n
0
(t) '+...+ A n (t)x(t)
(I)
w h e r e the A^ (t) are continuous c o m p l e x ^ v a l u e d functions on fa.,ro ).
Several authors have considered p a r t i c u l a r cases of this problem.
Coppel
[2] determ i n e d asymptotic solutions of
c e rtain restrictions on p and q.
( r y !)' + py = .0 under
To o b tain his s o l u t i o n s , he trans­
f o r m e d b o t h the dependent and the inde p e n d e n t v a r i a b l e s so that the
resulting equation could be compared w i t h either z(t) + z(t) = 0 or
z(t) - z(t) = 0.
He used a fund a m e n t a l result of L e v i n s o n
[53 to
d e t e r m i n e the asymptotic solutions.
In a similar fashion, H i n t o n
d i f ferential e q u ation
[4j studied the t w o - t e r m
(ry^m ^ ) ^ ^ + qy = 0.
His c o nclusions are
b a s e d u p o n a c omparison of the trans f o r m e d system w i t h a system
w i t h constant coefficients w h o s e charac t e r i s t i c p o l y n o m i a l is given
b y either A 0I + X = 0 or An - A = 0.
tions of y ' '' +
Pfeiffer
[6] d i s c u s s e s solu­
py' + qy = 0 by f o l l owing the same p r o c e d u r e s ’and
u l t i m a t e l y comparing the tra nsformed e q uation w i t h o ne w h o s e
c h a r a cteristic p o l y n o m i a l is either A
3
+ A = O
or A
3
- A = 0 .
vii
To study the solutions of e q u a t i o n
techniques
( I ) , w e e m ploy the same
as those of the p r e v i o u s l y m e n t i o n e d authors.
However,
w e a l l o w the transformed equ a t i o n to-be compared w i t h a system
w h o s e c h a r a cteristic p o l y n o m i a l is given by Xn + a X^
w h e r e the a. are comple x - v a l u e d constants.
+. . .+ a ^ ,
In order to apply the
f u n d a m e n t a l r esult of Levinson, w e d e rive a symptotic estimates for
the zeros of the c h a r acteris t i c p o l y n o m i a l .
of Chapter I i n c lude these estimates.
L e m m a 1.2 and L e m m a 1.5
T he other lemmas of Chapter I
are u s e d to p r o v e the central theorems of Chapter 2.
These
theorems p r e s e n t specific as y m p t o t i c solutions of e q u a t i o n
The case n = 3 is treated in d e tail
(I).
b e c a u s e the al g e b r a i c cal c u l a ­
tions are not so tedious.
Chapter 3 contains several resu l t s descr i b i n g the oscil l a t o r y
n a t u r e and bounde d n e s s p r o p e r t i e s of solutions of
(I).
illus t r a t i v e examples of the theory appear in Chap t e r 4.
Some
CHAPTER I
FUN D A M E N T A L L E MMAS
T h e p r i m a r y p u rpose of this thesis is to i n v e s t i g a t e the asymp­
totic b e h a v i o r of solutions of ordinary, linear, h o m o g e n e o u s diff e r ­
ential equations.
the t r a n s formations
T h e p r e se n t
chapter is devoted to a deriva t i o n of
and lemmas used to p r o v e the Central theorems
given in Chapter 2.
W e b e g i n w i t h the n - t h order eq u a t i o n
(t) + A 1 C t y x cn
and assume that
continuous
(t) +. . .+ A n (t)x(t)
= 0
the coefficient functions A_. (t) j = l , 2 ,...,n,
(I)
are
comple x - v a l u e d functions of the real v a r i a b l e t on an
i n t erv al a < t < oo .
Equation
(I) m a y b e replaced b y the equivalent,
first order v e c t o r - m a t r i x d i f f e r e n t i a l s y s t e m
(2)
y' (t) =. a y ( t )
w h e r e y(t)
=
[x, x ' , . . . , x Cn
and
0
I
0
0
0
0
1
0
0
0
0
I
A
A-I A-2
1
2
U p o n changing the dependent v a r i a b l e fr o m y to z(t) , b y setting
z(t)
= T ( t ) y ( t ) , w h e r e T is the n o n s i n g u l a r diagonal m a t r i x
T = dia[cp(t) , I, Cp \t),.... , Cp^ n (t)],
continuously d i f f erentiable function
and cp(t) is any positive,
on a _< t < oo , s y stem (2)
becomes
z'(t)
[T CL T-1 + T ' T ™ 1 ] z(t).
=
W e t r a n s f o r m this system,
(3)
in turn, b y intro d u c i n g a n e w inde­
pendent v a r iable s g iven by
t
s (t) = / 'Cp(T) dr
a
(4)
To ensure that the function s(t) has a u n i q u e continuous inverse
t =
iJ
j(s )
, defined on 0
s < co , w e further require cp(t) to b e non-
integrable on a _< t < °o .
In terms of the n e w v a r i a b l e s,
(3) may
b e w r i t t e n as
z(*(s))
w h e r e B(ip(s)) =
[cp 1 T Q T
"*" + cp 1T 1T ^ ] .
(5)
It is i m p o rtant to note
-I
that ip(s), and n o t s, is the argument of each of the functions'Cp ,
T , G,
T
and T'
o c c u r i n g in this expression.
ture of the m a t r i x B is indi c a t e d by
T h e e x p l i c i t struc­
■3 '
. —2
cp'cp
I
O
O
. O
O
O
.
/
. . .
O
I
. . .
O
, -2
-cp’cp
. . .
O
(6)
. . .
O
O
- A cp"11
n
. . . - A 1Cp- 1 + ( 2-n)cp'cp-2
I
A - Z n
- A n-l<pl" n
I
The central theorems of C h a p t e r 2 are applications of a funda­
m e n t a l result of L e v i n s o n
[5] to s y s t e m (5).
F or c o n venience of
r e f e rence this r e s u l t , taken w i t h o u t c h ange of n o t a t i o n fr o m
C o d d i n g t o n and L e v i n s o n
T h e o r e m 1.1.
Cp 1 J T , G
(5).
j
[1], p a g e 92,
is r e p r o d u c e d n e x t as
O b s e r v e that the d e p e ndent v a r i a b l e z , as we l l as
T ^
and 'T' , are all c o m p o s i t e functions of s in system
O n the o ther hand,
the dependent v a r i a b l e x and the n x n
m a t r i c e s V and R, in s y s t e m (7) h a v e the i n d ependent v a r i a b l e t as
their arguments.
The n o t a t i o n
[v|
and E is the n x n i d e ntity matrix.
is u s e d to denote the n o r m of V,
4
T h e o r e m 1.1
Consi d e r the linear system
x ' (t) = (A + V(t) + R ( t ) ) x(t)
L e t A b e a constant m a t r i x w i t h c h a r a c t e r i s t i c roots
j=l,2,...,n,
all of w h i c h are distinct.
(7)
,
L e t the m a t r i x V b e
d i f f e r e n t i a b l e and satisfy
CO ■
f |v' (t) I dt < CO
G
and let V(t) ->0 as t -> co .
Let the m a t r i x R be integr.able and let
CO
/ |'R(t) I dt < co
0
L e t the roots of d e t (A + V(t) - XE) = 0 b e denoted by
X .(t) , j=l,2,...,n.
J
I i m X . (t). = y ..
t ->oo
Clearly, by reord e r i n g
the y. if necessary,
J
For a g i v e n k, let
D k;.(t) = R e ( X k (t) - X_.(t))
Suppose all j , I j< j ^ n, fall into one of two classes
j e
if / D
(t ) dr
n, Kj
It.
j
e 1 2 if f
*2
D
co as t
D
co and
.(t ) dr > - K
kj
(t ) dr < K
and
'
(t0 > t
' 2 - 1
(t2 >. t1 ^
0)
> 0)
5
w h e r e k is fixed and K is a constant.v e c t o r of A associated w i t h
L et
be a charac t e r i s t i c
80 that
Then there is a solution (j)^ of
(7) and a t^, 0 <_
< co , such that
t
Iim <f>k (t) exp
t-5-OO
[.- /
Xk Cr) dr]
o
If the hypoth e s i s is satisfi e d for all k, I <_ k <_ n , and if $ is the
m a t r i x w i t h columns <j> ,
• I
2.
<j) , t h e n
is a f u n d a m e n t a l m a t r i x
n
b e c a u s e det$(t) f 0 for large t since the p_. are independent.
The l e ngthy form of the h y p o t h e s i s of T h e o r e m 1.1,
concerning
the real p a r t of the d i f f e r e n c e of the e i g e n v a l u e s , s u ggests the
f o l l owing convention.
W e shall,
henceforth,
say that afu n c t i o n D ( t ) ,
d e fined on a <_ t < co , satisfies C o n d i t i o n I iff either
CO
(i)
'
J
D(t)
dt - co
and
a
f
2 D(t) dt > - K , a < t]_ < tg
f
2 D(t) dt < K , a
for some constant K
or else
(ii)
for some c onstant K
6
By the same t o k e n , a f u n c t i o n D(s) , defined on 0 <_ s < co , satisfies
C o n d i t i o n I iff
(i) or
(ii) is fu l f i l l e d w i t h a = 0 and t = s.
Once
the d omain of D is understood, w e shall s i mply say that D satisfies
C o n d ition I.
To m o t i v a t e m u c h of wha t follows,
T h e o r e m 1.1.
let us e x a m i n e , for a moment,
Since X .(t) denotes a zero of an n - t h d e g r e e p o l y n o m i a l
3
in X, if n > 5 it w i l l u s u a l l y be i m p o s s i b l e to express X ^ (t)
explic i t l y in terms of the coef f i c i e n t s of the polynomial.
nately,
Fortu­
exact expres s i o n s for these zeros are not n e e d e d to m a k e use
of T h e o r e m 1.1.
F o r , suppose each X .(t) admits of an appro x i m a t e
3
r e p r e s e n t a t i o n of the form X^. (t) = X^ (t)
i ntegra ble on
[0,oo) .
R e t n 1 (t) - n.(t)].
k
J
+ ry (t) , w h e r e ry(t) is
T h e n Re[X^.(t) - X^ (t) ] = Re[X^.(t) - X^ (t) ] +
It follow s that D 1 .(t) = R e t X 1 (t) - X .(t) ]
kj
k
J
.
satisfies C o n d ition I , w i t h k fixed,
.
• ~
and for e very j , iff
(t) =
R e [X^(t) - X _ (t)] does also.
As for the concl u s i o n of the theorem,
relation
it entails the asymp t o t i c
t
I i m cp (t) exp f
- X 1 (t ) dr = P 1 w h i c h beco m e s
t -> co k
.
k
k
M • Iim cp (t) exp
t -> co k
B u t , a nonzero
f
-X
k
(x) dx = p
k
w h e r e M is a n o n z e r o
constant.
constant times a c h a r a c t e r i s t i c v e c t o r is also a
7
c h a r a c t e r i s t i c vector.
Hence,
the c o n c l u s i o n of the t h e o r e m remains
v a l i d if X.('t) is replaced b y any a p p r o x i m a t i o n
d i f f e r e n c e A_. (t) - X_. (t) is integrable,
Now,
on
X . (t) su c h that the
3
[ t ^ ,c°) .
let us r e t u r n to the m a t r i x B(^(s))
of s y s t e m
(5).
Sup­
p o s e its charac t e r i s t i c p o l y n o m i a l is g i v e n by
P(X) = Xn + B ^ ( s ) X n ^ + . . . + B^(s)
zeros.
Suppose that the
Iim B .(s) = a.,
g _>oo I
I
constants.
h a v i n g X ^ ( s ) , k-l , 2 , . . . , n ,
as its
(s) are c ontinuous and that
for j=l,2,...,n, w h e r e the a. are comple x - v a l u e d
^
Set p (X) = Xn + a^Xn ^ + . . . + a^ and i m p o s e the condition
that the zeros p^, k=l,2,... , n ,
of this p o l y n o m i a l shall b e distinct.
By R o u c h e 's T h e o r e m , and a r e o r d e r i n g of the indices if n ecessary,
w e have that X^(s)
is contin u o u s on
[O ,o°) , and w e m a y take
Iim X..(s) = V1 , k=l,2,...,n.
bo
K
CO K .
S3
T h e r a t e at w h i c h e a c h d i f f e r e n c e X^(s) - v^. c o n v e r g e s to zero
is d e p e n d e n t upon h o w fast the d i f f e r e n c e s Ey(s) - a_. of the r e s p e c ­
tive c o efficients in P(X)
assumptions,
B
and p(X)
con v e r g e to z e r o .
U n d e r our.,
(s ) - a^ and X^(s) - v^ b e l o n g to the class of com­
p l e x - v a l u e d f u n c tions f (s) that a re c ontinuous on IO ,co) , and w h i c h
converge to zero as s b e come s infinite.
that
. It can be r e a d i l y ve r i f i e d
(Lemma I, A p p e n d i x I) any su c h f u n c t i o n that is a-iiiember of
[O , co) is also a m e m b e r of L r [O,00) f or r j>_ q.
and g(s)
are two such funct i o n s w i t h f(s)
F u r t h ermore,
e L ^ f 0,=”) and
if f(s)
8
g(s)
e L r [0,oo) , an applicat i o n of H o l d e r ’s inequ a l i t y
A p p e n d i x I) shows
that the pro d u c t
m e m b e r of L m [0,co) w h e r e m = m a x
As has b e e n mentioned,
f ( s ) *g(s)
of these functions is a
(I, qr/ (q+r)) .
exact formulas for the zeros of an
arbitrary n - t h degree polyn o m i a l cannot be found,
it b e comes n e c e s s a r y to resort
to such zeros.
(Lemma 2,
in general.
Thus,
to meth o d s that y i e l d approximations
W e n o w p r o c e e d to e s t a b l i s h five im p o r t a n t lemmas.
Two of these give useful
set forth sufficient
ap p r oximations
to A1 (s) .'
K
conditions for Re[A^(s)
The other three
- A (s) ], defined on
[0,c?), to s a tisfy C o n d ition I.
L e m m a 1.2
—--- ----- •
—
Let P(A) = An + B n (S) An ^ +...■+ B (s) w h e r e the B .(s)
. . .
i ■
n
j
are continuous
compel ex-va l u e d functions defined o n 0 < s < co .
S u p p o s e that
(i)
limB.(s)
g
J
= a..
J
j=l,2,...,n, w h e r e the a. are complexI
v a l u e d constants,
(ii)
p (A) = An +
a^An
+. . .+ a^ has distinct zeros p^ ,
I I , 2 , . . . ,n,
e\
(iii)
(a ■- B^ (s))
e L [0,oo), j=l,2,...,n.
9
Then the zeros
A^(s-) , k = l , 2 ,. . ,n, of P(X)
are given by
n
Ik (S) - Vk + (P1(Vk))-1 J h a j - B ^ ( s ) ] p ^ ^ + nk (s) w h e r e nk (s) is
some integrable function on
P(X) w h i c h converges
to
ambiguity of notation,
Xfc(s) = ( P 1 (O))
Proof:
[0,co), and w h e r e X^(s)
as s bec o m e s infinite.
To avoid
1 Ean - B n (s)l + nk (s>, i f Mfc = 0. .
As has b e e n p o i n t e d out,
as s becomes infinite.
I i m o (s) = 0.
that zero of
let it be u n d e r s t o o d that
hypothesis
ensure that X^(s) - Mk is continuous on
S —> OO pC
is
(i) m a k e s it p o ssible to
[0,oo) and converges to zero
Hence, w e m a y set X^(g) = Mk + ^k (s) w h e r e
Since X (s) is a zero of P(X), P ( m i + a . (s)) - 0.
K.
K - K
t
The left m e m b e r of this e q u a t i o n m a y b e v i e w e d as a p o l y n o m i a l of
degree n in ak (s)-
W h e n w r i t t e n as such, w e o b t a i n
P 1 1 (Vk )
2
P ( V k ) + P 1 (Vlt)Ok + — j;
ak +.. .+ «k - 0
■
w h i c h gives P ( M k ) + P 1 (Tjk ) ak Il + o(l) ] = O', w h e r e o(l)
function of s that converges
for a
to zero as s becomes infinite.
and n o t i n g that 1/[1 + o(l)] =
[I + o(l)],.we ha v e
P(Pk)
\ (s)
designates a
P U T T ft + 0(1)1
K
Solving
10
Now,
p (M^) = 0, and P'(p^)
(s) - a
j=l,2,...,n,
=
[pT (y^.) + o(l) ], b e c a u s e the differences
in P t (Pfc) - p '(Pfc) =
n-1
Y ji (n-j) [ B . (s) - a .]p^ ^ ^ are all continuous on [0 ,°°) and converge
j=l
3
3
k
to zero as s becomes infinite.
[I
+
o(l) ]/[p'
since, b y
(Pfc) +
Moreover,
o(l>] = [I + O(I)IZpt(Pfc).,
( i i ) , the pfc are distinct.
and p ' ( p fc)
+ 0
Hence
-P(Plr) + P ( P lr)
”k<s) = —
t o o —
where efc(s) = (Pt(Pfc))
Y
[1 + 0<1)} = £k<s) + pk (s) ’
-J
ta-j ~ Bj
j=l
(Pt(Pv ))
-I
[p(p, ) - P(p,)] and
in the coefficients of P ( p fc).
Iim rp(s) = 0 .
S -voo
Note that s appears
S i n c e Afc(s) = Pfc + efc(s) + pfc( s ) ,
it is evident that nfc(s) has the fo r m g iven in the c o n c l u s i o n of the
theorem.
W e next p rove that nfc(s) is integrable.
S t a r t i n g w i t h the
e q u a t i o n P (pfc + efc(s) + nfc(s)) = 0,. w e express its left h a n d m e m b e r
as a p o l y n o m i a l of degree n in PfcCs ) •
This gives
P ( p fc + efc) + P ' ( P fc + efc) fc + . . . + nfc = 0, w h i c h m a y be expressed as
P ( p fc + efc) + P ' ( p fc + Efc) fc[l + o ( l ) ] = 0.
F o r rea s o n s that are
11
already familiar, w e h a v e P'Cp
+ e^). = p' (y^) + o(l) .
Thus, w e find
that P ( p k + Efc) + p'(yk )nk [l + 6(1)] = 0.
Now,
e xpand P (pk +
ek ) as a p o l y n o m i a l in ek , to o b t a i n
P ( U k ) + P t (uk >ek +. . .+
T h e n u s e the d e f i n i t i o n of ek (s)
e”
+ P 1 (uk )nk [l + o-(l) ] = 0
(8)
to v e r i f y that
P(Uk) + P'(Wt)Ek = [P'(uk) - p'(%%)]=&
Equation
(8) becomes
[p'(pk) - p'<pk)]=k + i r''(;k)«k + - " + Ek “ -p' V \ [1 + 0(1,1
E a c h term on the left h a n d side of this l a s t equ a t i o n consists of
b o u n d e d functions
times prod u c t s of two dr mo r e factors of the type
(s) - a ^ , j - 1 , 2 , . . . ,n.
From hypothesis
( i i i ) , it follows that
each term is an integrable fu n c t i o n on [0,°°) so that Pk Il + o(l) ]
is integrable.
Therefore,
nk (s) is integrable,
and the proof is
complete.
In stating our n e x t lemma,
define
(s) = R e f i t s )
specifies sufficient
-
it w i l l b,e convenient if w e first
(s) ], on 0 _< s < °° .
conditions for e a c h D
Condit ion I w h e n k is fixed,
KJ
T he l emma
(s) to satisfy
and j=l,2,(. . . ,n.
12
L e m m a 1.3
----------
Let P(A)
are continuous,
- A
+ B, (s)A
k /■
^ +...+ B
n
(s) w h e r e the B .(s)
j
c o m p lex-valu e d functions defined on 0 <_ s < 03 .
Suppos e that the zeros of P(A)
are d e n o t e d . b y A ^ ( s ) , k=l,2,...,n,
and that
(i)
Iim B^(s)
,n, w h e r e the a. are c o m p Iex-
1
v a l u e d constants,
(ii)
■
the zeros y^ , j=l , 2 ,...,n,
of p(A)
= An + a ^ A n ^ + . . . + a^
are distinct,
(iii)
a. - B .(s) , j=l,2 , . . . , n ,
J
J
are members of L
2
[0,°°),
n
(iv)
for fixed k, Re
fa
m=l ■
- B
(s)I
■
satisfies Conditi o n I if j is an index such that
Re[p
- p .] = 0.
3
K
Then, D
KJ
(s) = Re[A
K
(s) - A .( s ) ] s a t isfies C o n d ition I for the
J
fixed k and all j-l,2,...,n.
/
I
13
Proof:
This
l e m m a w i l l be p r o v e d b y considering the s eparate cases
w h e n R e [ y ^ - y^
^
O and w h e n R e f y ^ - y^ ] = 0 .
k is fixed and j is an index for w h i c h R e [ y
Iim Re[A, (s) - I .(s) ] = L ^ O
S —S - O O ic
on
J
[0,oo) and converge to y
^c
(s) satisfies part
(ii)
- y .} >
k .
j
b e c a u s e A (s)
K
0.
and A . (s)
J
and y ., respectively.
I
(i) of C o n d i t i o n I.
of Condition I holds.
F i r s t , suppose that
Then
are continuous
If L is positive,
If L is negative,
part
Consequently, Re[A (s) - A .( s ) ] satisfies
''R , • ■ j
C o n d ition I for fixed k, and any j su c h that R e t y 1 - y .] ^ 0.
k
j
Next,
assume that for some i ndex j , R e [ y
k
- y .] = 0.
J
Then, b y
L e m m a 1.2, Re [A (s) - A . (s) ] is equal to
KJ
n
Ke 2
m=l
( P 1 (Mj))
[am " Bm (a) 1
1 Mj m ] + R e t n k ( S ) - P j C s ) ],
w h e r e b o t h P1 and p . are integ r a b l e on 0 <_ s < co .
expre s s i o n v a n i s h e d if j=k,
jfk.)
so there is no n e e d to assume that
In v i r t u e of hypothes i s
C o n d ition I.
Hence,
(Note that this
( i v ) , R e t A k (s) - A ^ ( s ) ] fulfills
the conclusion of the lemma is established.
'
A n alternative,
is given by:
and mor e easi l y applied v e r s i o n of L e m m a 1.3
14
L e m m a 1.4
W i t h P(A), B .(s) , arid A, (s)
J
■. K
the same as in L e m m a 1.3,
suppose that
(i)
Iim B.(s)
s-> oo J
= a., j=l,2,...,n, w h e r e the a. are complexJ
J
valued constants,
(ii)
the zeros p ^ , j=l,2,...,n,
of p(A)
= An + a ^ A n
a
n
are distinct,
(iii)
a. - B .(s) , j=l,2,...,n,
3
3
are integrable on 0 <; s < oo ,
-
Then Re[A^.(s) - A .(s) ] satisfies C o n d i t i o n I for k = l ,2,. . .,n , and
J 1,2,'. ..,n.
P r o o f : . Since,
the a^ - B^ (s) are continuous on
zero as s becomes infinite,
[0,°°), and tend to
2
they are all members of L |0,oo).
Hence.
.
the zeros A^(s)
k = l ,2,...,n , m a y be r e p r e s e n t e d as in L e m m a 1.2.
F o r any p a r t i c u l a r k, Re[A^(s)
Re
- A ^ ( s ) ] = R e [ p ^ - y^] +
E [ a - B (s)][(p'(p )) 1 P ^ m m
m
k'
m=l
(p'(Pj))
As in the p r o o f of L e m m a 1.3, w e
Re[yk -
3"
-I n -m
"p"
3
+ R e [ n k ( s ) - n j ( s)]
consider w h e n
^ 0, and when Re[pk - p^ ] - 0.
If -Re.[pk - y ] f 0,
it follows as b e f o r e that R e [ A1 (s) - A .(s)] satisfies C o n d ition I.
• k
3
In case R e I p 1
] = 0,
(iii)
and L e m m a 1.2 ensure t h a t
Re[A, (s) - A . ( s ) ] is equal to a .finite sum,
&
3
each term of w h i c h is
15
inte'grable on
[0,00)'.
Since any i n t e g r a b l e function satisfies part
(ii) o f C o n d ition I, Re[A^(s)
-
k=l,2,...,n,
as wa s
and j=l,2,...,n,
( s ) ] satisfies Co n d i t i o n I for
to b e shown.
To i l l u s t r a t e h o w L e m m a 1.2 m a y b e rep l a c e d by an analogous
result,
i n v o l v i n g a m o r e general r e q u i r e m e n t than w e p r e s e n t the
following lemma.
F u r t h e r e x t e n s i o n s •are p o s s i b l e at the e x p e n s e of
considerable additional alg e b r a i c m a n i p u l a t i o n s
and a t t e nding anal­
y tical difficulties.
L e m m a 1.5
Let P(A)
■
are continuous
(i)
= An + B 1 (s) An
•
. I ■
+ . . . + 'B (s) w h e r e the B .(s)
n
j
complex-valu e d functions on 0 _< s < 00 .
Iim B (s) = a , j=l,2
s ->-co J
J
n, w h e r e
the a.
Suppose that
are complex­
v a l u e d constants.
(ii)
p(A)
= An +
S1 An " 1 + . . . + a h as d istinct zeros p .,
n
3
j—I ,2,.•.,n,
(iii)
Then,
a
J
- B
J
(s) , j=l,2
the z e r o s ' A ^ ( s ) , k=l,2 ,...,n,
of P(A)
are given by
Ak (s) = Pk + ek (.s) + Pk (S)
w h e r e nk (s) is integrable on 0 < s <
00
, and
16
-I
G^(S)
Zt^
(?'(%%))
+
m=l
ZU
( p ' (;%))
*m=l
I
Proof:
tti
m
,Z [am- Bm (s) ](n-m)yj~in'1
- Bm (s)]yJ m
m
+
m=l
Z i am - V
<p'<\»'3 p " ( V
m=l
s n a k"
The p r o o f of this l e m m a follows along the same lines as that
of L e m m a 1.2.
Set A^(s)
= y^ + e^(s) + n^(s), w h e r e
e^fs)
function already d e fined in the s t a tement of the lemma.
X^.(s) satisfies the e q u a t i o n P(A^)
P(y^ +
+ n^) = 0.
=0,
is that
Since
for e v e r y s, w e have
T h e left side of this eq u a t i o n m a y b e v i e w e d
as a p o l y n o m i a l in n ^ ( s ) , so that w e have
P ( y k + ek ) + B 1 Cyk + Efc) nk +•..+■ nj = 0.
the term e^(s)
as s tends
By hypotheses
(i)
and ( i i ) ,
and the differ e n c e A^(s) - y^ b o t h converge to zero
to infinity.
Hence,
q^(s)
also converges
to zero,
and so
P ( y k + ek > + p t Cyk + eJcU k Il + o C l ) ] - 0.
As s b e c o m e s infinite, E 1 Cyk + ek ) converges to the n o n z e r o value
p'(y ).
U s i n g this fact, w e o b t a i n
p Cyk + Ek ) + P 1 Cufc) nk [l + o(l)] = 0.
17
E x p a n s i o n of P (p^ +
e^), as a p o l y n o m i a l in
yields
■
P(Pk) + pt(Pk)Ek +.••+ Ek + p'(Pk)nk [l + O (I)] = 0
Hypotheses
(i)
and
(iii) , ensure that
Therefore,
e ™ (s ) is i ntegrab l e on 0 _< s <
'(Lemma 2, A p p e n d i x I).
Equation
P(Pk) + P t(Pk) Ek + Y P''(Pk) 4
e^.(s) is a m e m b e r of L
+ fk (s) + P t(Pk)Pk El + o(l)] = O
P(X)
- p (X) , w e
P(Pk ) +
tran s f o r m e q u a t i o n
Ept(Pk ) + P ( P k )]
[0,co) .
(9) m a y b e w r i t t e n as
is some i ntegra b l e f unction on fO,co)«'
= P(X)
3
, for 3 _< m _< n.
w h e r e f (s)
K
(9)
(10)
Defining
into.
+ -| P M (Pk )£k + §k (s) -
- P t (Pk ) Ok El + O-(I)J
w h e r e g^(s)
= f^(s)r
together w i t h
also.
(10)
I ^
+ -^P
2
ek "
(11)
~
F r o m t^ e defin i t i o n of P(A) ,
. ■
3
(iii) , it follows that P 1Cp ) is a m e m b e r of L £0,°°)
- t.' -... - 1 ' ■
3
Consequently, P t (Pfc), and P tt(Pfc),. are m e m b e r s of L EO,oo) .
T h e . f u n c t i o n gfc(s)
is therefore i n t e g r a b l e on 0 < s < °° .
Re p l a c i n g
Efc by its expression
£k = “ (P'(pk))-1 F(Pk) + (Pt(Pk)) 2 p (Pk) pt(pk) +
^ (Pt(Pk))""^ Cp(Rk))2 Ptt(Rk) ,
w e conclude that the left side of e q u a t i o n
H e n c e h 1 is an integrable function,
k.
(11) is integrable.
and the l e m m a holds,
'18
In order to introduce our final l e m m a , w e m a k e two important
restrictions in the h ypothe s e s
(i)
and
(ii) of L e m m a 1.5.
r e quire that all of the coefficients a. of p(A) be real,
3
allowing them to b e complex-valued.
be distinct,
gate pairs,
First, we
instead of
Since all of the p. are still to
J
c o m p l ex-valued zeros of p(A) m u s t h o w o c c u r in conju­
if they occur at all.
Secondly, w e i m pose the condition
that no m o r e than two of the y. are to h a v e real parts equal
3
same real number.
I n particular,
gate pairs of complex zeros,
if p (A) admits ‘of d i s t i n c t conju­
the real part of one p a i r is n ot equal
to the real part of any oth e r pair.
the real part of a complex z e r o .
l e a d to the following result.
to the
Neither.is
a real ,zero to equal
T h e s e alterations in L e m m a 1.5,
19
L e m m a 1.6
.
Let P(A)
= An + B_,(s)A^ ^ +. . .+ B (s) w h e r e the B .(s)
x
n
j
are c o n t i n u o u s , complex-val u e d functions defi n e d on 0 _£ s <
the zeros of P(A)
(i)
be denoted b y A ^ ( s ) , k=l,2,...,n,
co . L e t
and suppose that
Iim B .(s) = a., j - 1 , 2 , . . . ,n^ w h e r e the a. are reals _> co 3
I
3
v a l u e d constants,
(ii)
the zeros p ^ , j=l,2,...,n,
are distinct;
of p(A)
furthermore. Rep.
= An + a ^An ^ + . . . + a^
= Rep,
implies
that
= Uk ’
(iv)
for fixed k,
n-m
-
(s) ]lt
p
m=l
n
'(Pk )
n
(n - m ) p 2 ™ - r - l
Z • Z l m ^ am - %
m = l r=l
n
(s))(a
- B
(s)) ]lm
(P'(Pk » 2
n
Z
E l m ^ a m - Bm ( s ))(ar - B r (s))]lm
m = l r=l
p'v
t r *
3
( P i(Pk ))1
satisfies Condition I for any index j such that
Re [p^ —
Then,
Re[A,(s)
• . k.
J I »2,. . . 5n.
P^] = O'.
-
- A .( s ) ] satisfies C o n d i t i o n I for the fixed, k and all
J
20
Proof:
As in the p r o o f of L e m m a 1.3, w e
w h e n R e [ p ^ - p ] f 0, and w h e n R e [
for w h i c h Re[.y^ - y ] ^ 0,
-
consider the se p a r a t e cases:
y -] = 0 .
If j is an index
the proof that Re,[A^ - A ] satisfies Con­
dition I is just the same as in the p r o o f of L e m m a 1.3.
So suppose j is an index such that R e [ y ^ - y^ J = 0.
since the c o e fficients of p(A)
implies
that
and A .(s) = y. +
J
and e^(s)
lemma.
Thus, w e h a v e A^(s)
e . (s) + n .(s) , w h e r e n,(s)
l
J
= y^ +
and y .(s)
R
s^(s) + n^.(s)
are integrable,
J
and s_. (s) have the f o r m given in the c o n c l u s i o n of that
U s i n g these expressions,
w e o b t a i n Re[A
a nd the fact that R e [ y ^ - y_.J = 0,
(s) - A .(s) ] = R e t e 1 (s) -
As w a s inferred earlier, w h e n n^(s)
[Q, oo) , Re[A^(s)
e .(s) ] + R e J n 1 (s) - n.-(s)].
and n (s)
are i n t e g r a b l e on
- A^ (s)] satisfies C o n d i t i o n I iff
Re[e, (s) - e .(s) ] does.
K
the complex conjugate of y .
restrictions on the a^ and the 'zeros of p ( A ) ,
L e m m a 1.5 still applies.
J
are real n u m b e r s , h y p o t h e s i s "(ii)
, w h e r e y^ denotes
U n d e r our present
Then,
J
Hence,
o ur p r o o f wi l l b e c o m p l e t e if w e
can show that R e t e 1 (s) - e ( s ) ] satisfies C o n dition I w i t h y.
K
To this end,
conjugates of e
R
v e rify
that
= y ..
kz
denote the i m a g inary unit b y i and the complex
and e . b y e
J
R
that R e f e 1 - e .] =
R J
p ’ Cbj) = P 1 (yk >.
j
and e ., respectively.
J
A l s o observe
(1/2) fe, + E1 - e . - e . ], and that
R
R
J
J
B y u s i n g familiar conjugation laws, w e m a y then
J
21
n-m
'k
n
- e = 21
k
Jja
- B (S)Jlm
m
m
_p'<Y
m =l
n
2i E
(n - .m) u
E f am -
+
2n-m-r-l
+
B m .(s)][ar .- B r (S)Jl
m = l r=l
n
i
Finally,
n
\_2n-m-r-
E
E f a m - Bm fs^ f ar " B ^ ( S ) J l
m = l r=l
3
.
(P'(Y)
add to the forgoing e x p r e s s i o n its complex conjugate.
E x c e p t for a"factor of 4,
hypothesis
clude that,
this results in the e x p r e s s i o n given in
( i v ) , w h i c h is p r e s u m e d to fulfill Co n d i t i o n I .
for fixed k, Re[A^(s)
- A y (s)J
satisfies Co n d i t i o n I
in case j is any index such that R e I p ^ - y J = 0 .
the proof of the lemma.
W e con­
This
concludes
CHAPTER 2
A S Y M P T O T I C SOLUTIONS
In this chapter w e p r es e n t fi v e theorems and one c o r ollary d e s ­
cribing a symptotic solutions of l i near d i f f e r e n t i a l equations.
The
proofs of these results are b a s e d o n the p r e l i m i n a r y lemmas and
T h e o r e m 1.1,
of Chapter I.
final theorem,
The first two theorems,
as w e l l as the
are expressed in r a t h e r general terms.
they a ppear to be quite complicated.
Therefore,
T he other theorems deal W i t h a
third order l inear differ e n t i a l equation.
In the case of the third order equation,
it is p o s s i b l e to m a k e
a n u m b e r of s i m p l i fications in the g e n e r a l theory.
F or then, the
order of the d i f f e r e n t i a l equ a t i o n is no longer an i n d e f i n i t e fixed
"n".
Thus, w i t h c o nsiderabl y less effort, we. are ab l e to o b tain
sharper results w h e n n=3.
.Oscillatory and bound e d n e s s p r o p e r t i e s of our a s y m p t o t i c so l u ­
tions are examined in Chapter 3.
23
T h e o r e m '2.1
Consider the linear d i f f e r e n t i a l equ a t i o n
x (n)(t) + A 1 ( t ) x (n“ 1 ) (t) + . . . + A (t)x(t) = 0
a.
n
(i)
Suppose that
(i)
A
( t ) , j = l s2,...,n,
are continuous comple x - v a l u e d
functions defin e d on
(ii)
cp(t) is a positive,
d e f ined on
(iii)
[a ,<x>) ,
c o n t i n u o u s l y d i f f e r e n t i a b l e fun c t i o n
[a,oo) ,
f cp(t) dt =
CO
,
a
(iv)
I i m cp’ (t)cp -^(t) = 0,
t->oo
(v)
Iim- A. (t)cp ^ (t) = a . ,
t-^oo 3 c
3
complex v a l u e d c o n s t a n t s ,
(vi)
p (X) = Xn +
P
(vii)
J
^ +....+ a^ has di s t i n c t zeros
, j - 1 , 2 , . . . ,n,
(cp' cP
’ and
(A^cp^)',
j=l,2,...,n,
are integ r a b l e on
[ a ,oo) ,
(viii)
cp'cp
and [a. - A.cp J ] cp
3
3
2
bers of L [ a,°°) ,
, j=l,2,...,n,
a re all m e m -
i;
24
(ix)
For fixed k, .
n
R e ^ g [a
.m=l
-
m
Cnr H + .l)„ ^ 3 ( 11 ^ 1 1
2
[(P1 (Pfc))
m -
A
rp-™"^'. _ A qf™]
m-lY
Y ■
mY
(Pf Cyj ))
* 9
satisfies Condit i o n I if j is an index for w h i c h
Reu.. = R e u fc.
(See P a g e 5 f or the d e f i n i t i o n of
C o n d i t i o n I ) .Then,
a c t
there exists a solut i o n x fc(t) of
-O
< co
(I)' and a n u m b e r t^,
such that
x fc(t) = Cp
(t)[ exp
/ [y
+ Efc(T) lcp(t) dr] [I + o(l) ]
(12)
where
-m
EfcCt) =
2 (p'(Pfc))
- A* cp
-
m=l
(n-irri-1) -3(h-m+l)
2
Furthermore,
, -m-1.
"n-m
m-lV 9
J ‘ \
the first n - 1 deriv a t i v e s of x fc(t) ha v e t he asympt o t i c
form
x ^ 3 ) (t ) = Cp3. 1 (t)[exp
for j = l , 2 , . . . ,n-1.
J
iUfc + Efc(T) ]cp(x) dx] [ufc +
o(l)]
25
If hypothesis
(ix)
is s a t i sfied for k=l,2,...,n,
then there
w i l l exist n l i n early independent s o l u tions of the type ju s t d es­
cribed.
Proof:
In the previ o u s chapter, w e transformed E q u a t i o n
(I) into
the system
dZ~~d s ^ ^
The argument
(5)
z(\p(s))
ip(s) w a s obtain e d as the inve r s e of 'the f u nction
t
f (p(T)dT .
s(t) =
Expression
(6), P a g e 2, indicates the explicit
a
f o r m of the m a t r i x B(tp(s)).
and
(iii)
ensure that
elements of
Thus,
It w a s n o t e d that h y p o t h e s i s
(i) , (ii) ,
t = \p(s) is u n i q u e l y defined, and that the
B(\p(s)) , as w e l l as i^(s) ,. are continuous on 0
the t r a n s f o r m a t i o n of (I) into
(5) is possible.
s < co .
M o r e over, t .
tends to infinity w i t h s .
W i t h the addition of h y p o t h e s i s
(iv) and
( v ) , it follows that
B(\jj(s)) converges to the constant m a t r i x C w h o s e e lements are identi/
fied b y
26
O
I
O
C
O
O
O '
l
0
0
0
0
. ari
an -l
an-2
A direct e x p a n s i o n of det[S(^)(s))
0.
' - '
3 I.
- AEJ s reveals that the coeffi­
cients of the c h a r acteristic p o l y n o m i a l
P(A) - An + B ^ ( s ) A n
+. . .+ B^(s),
assoc i a t e d w i t h
B(i/;(s) ) , m a y b e
w r i t t e n as
B j (S) = A j V j +
w h e r e i n ^(s)
A j_ 1cp > " j " 1 + b.(s)
is the argument of A., cp, and A.
of terms e a c h involving
(13)
and b .(s) is a sum
t -2
cp cp . r a i s e d to a power m > 2, and A
= I. '
0
A g a i n , let A.(s)
J
p.(A) = A +
a An
I
and y. deno t e the zeros of P(X)
and
J
+ . . . + a , respectively.
n
p o i n t e d but on p a g e 7 , w e m a y a s sume that
As has alre a d y be e n
I im A.(s) = p . ,
S-> OD J
j=l,2,...,n.
B e c a u s e of
t h e r e f o r e , p'(p^)
4- 0.
J
(vi) , ea c h P j is a simple zero of p(A');
The rest of our proof consists of a detailed
explan a t i o n of h o w T h e o r e m 1.1 m a y b e applied to s y s t e m
(5).
W i t h s , r a t h e r than t, v i e w e d as the indep e n d e n t v a r i a b l e , set
A = C ,
V(s) = B (,xJj(s )) - C, and R(s) = 0.
H ypothesis
(vii)
then
27
OO
guarantees-that
f
00
|V 1 ( s ) |ds =
[B 1( t ) |dt < co
)
a
0
Of c o u r s e ,
J
'f |.R(s)|ds = 0.
Hence,
I im V(s)
s' -> oo
= 0.
all of the h y p o t h e s e s of
0
T h e o r e m 1.1 w i l l be satisfied, p r o v i d e d D
(s)' = Re[A^.(s) .- A_. (s) ]
fulfills Condi t i o n I w i t h k fixed and for all j=l,2,...,n.
To a s c e rtain the b e h a v i o r of the D^. (s) , w e sh o w that the
hypoth e s e s of L e m m a 1.3 are fulfilled.
and using
(iv) and
(v) w e first v e r i f y that
has a l ready b e e n mentio n e d ,
transformation t =
thesis
(viii) .
R e t u r n i n g to E q u a t i o n
the
Iim B.'(s) = a..
s—>• oo d
d
are distinct.
'As
N e x t , employ the
ip(s) to change f r o m the v a r i a b l e t to s in h y p o ­
This reveals that cp'(i|;(s)) cp ^ O K s ) )
[a_. - A j (Tf)(S)) Cp ^ (Tf)(s)) ] are m e m b e r s of
are also continuous on
b e c omes infinite.
(13)
0
[0,oo) .
and
T h e s e functions
_< s- < oo , and they c o n v e r g e to zero as s
Consequently, b . (s) is i n t e g r a b l e o n
J
2
a. - B .(s) is a m e m b e r of L [ O,00) for q=l,2,...,n.
3
3
that every hypothesis of L e m m a 1.3 holds,
it also h o l d s , in v i r t u e of
(iv).
[0 ,co) .
W e h a v e shown
except the last one.
J = l , 2 ,. .. ,n.
But,
Therefore, L e m m a 1.3 is in force.
It follows that, for fixed k, C o n d i t i o n I is satisfied b y each
(s)
Thus,
28
W i t h this r esult established.
T h e o r e m 1.1 m a y b e applied.
conclude that there exists a solution z^C^Cs))
number s ,
o
0
< s
. -
o
< oo
We
system (5), and a
such that
Zk (Ks) = Iexp" J lk (5)d5][p'k + o(l)]
(14)
So
w h e r e p k is the e igenvector of C co r r e s p o n d i n g to the e i g e n v a l u e pk .
The symbolism
o(l) denotes a v e c t o r w h o s e components tend to zero
(but n o t n e c e s s a r i l y at the same rate)
L e t us center our attention,
as s bec o m e s infinite.
for a m o m e n t , on E q u a t i o n
(14).
B y L e m m a 1.2, w e have
n
.V
s) ■ 1V
w h e r e nk (s) is integr a b l e on
lent e xpres s i o n
JJ l
[0 ,co).
IX
- By(s)]pk
2 + nk (s)
R e p l a c i n g B_. (s) b y its equiva-
(13), we obta i n
A (s) = Wk +
^ 1 I [aj " a JcP ' ~
- b.(s))p”-j + nk(s)
w h e r e b . (s) is integr a b l e on
3
[O,00).
C ollecting i n t e g r a b l e terms,
w e m a y w r i t e this e q u ation as Ak (s) = Pk + E ^ ( K s ) ) + ^ ( s )
where
S
29
£k = (P'
2' U,j=l
3
A.(p"j
a
.
'Cp - ^ 1 J u
and
;(s) = nk (s) -
is integr a b l e on
C p tCyk ))
^
(s)yk
3
[0 ,oo) .
Using this r e s u l t , in (14), w e have
zk (ifj(s)) =
[exp / [yk + e £ G K £ ) ) - ] d £ H e x p
N o w set dk = exp
exp
J
J
/
nk (C)dgJIpk + G(I) ]
rik (s)ds, a nd o b s e r v e that
pk ( £)dg = dk exp[o(l)j =
^[1
dk [l + o ( I ) ] [pk + O (I)] = IdfcP k +
+ o(l)]»
0 (1 )]
W h a t is more.
; consequently,
Zjc(Ks)) = [exp / Lvk +. Ek(^CO) dCHdkPk + o(l)]
6O
30
Note that
is a scalar, w h e r e a s
is an n - c o m p o n e n t v e c t o r .
R e c a l l i n g that t = ^(s), and that t becomes i nfinite w i t h s, we
r e w r i t e this result as
t
zk (t) =
[exp
J [ +
ek (T)]cp(T)dT][dk p k + O (I)J
tO
w h e r e ek (t) is the same fun c t i o n of t that e* is of ^ ( s ) .
Without
loss of generality, w e m a y take dk pk = [l, .p^, p ^ , . . . , p ^
For, w i t h ^k Pk
thus defined,
it follows b y direct expansion,
the fact that P(Ijk ) = 0, that [C - Pk E] ^k Pk = 0.
z(t)
=
and
S ince
T (t)y(t) , w e have
cp(t) Xk Ct)
I
:
1 V)
+ O(I)
Cpk + 0(1)
*i«>
<P
T.
t .
(t)
= [exp
Cpk + 0(1)
/ [pk + ek (T)Jcp(T)dT]
t
O
<p2‘ n (t) X ^ - U (t)
From this w e v e r i f y
CPk " 1+ 0(1)
that x k (t)
and its
first n-1 deri v a t i v e s have
the properties d e s c r i b e d in the c o n c l u s i o n of our theorem.
hypothesis
(ix)
is s a t i sfie d for k = l , 2 ,...,n,
If
then b y T h e o r e m 1.1,
w e h a v e n linearly independ e n t s o l u tions of the type d e s c r i b e d above
31
The only p u r p o s e
for w h i c h h y p o t h e s i s
p r o o f of the p r e c e d i n g theorem, wa s
of L e m m a 1.3.
fact,
it is
(ix) w a s u s e d , in the
to confirm the final hypot h e s i s
The conclusi o n of T h e o r e m 1.1 is still.true,
true for k = l , 2 ,...,n,
n ative hypoth e s i s of our nex t
if (ix)
theorem..
and, in
is rep l a c e d b y the alter­
W h e n applicable.
T h e o r e m 2.2
is g e n e r a l l y e a s i e r to apply than T h e o r e m 2.1.
T h e o rem 2.2
Suppose h ypoth e s e s
continue to hold.
In addition,
(i)
through
(viii)
of T h e o r e m 2.1
let
[A.(t)cp"j (t) +
Y
-<Kt>
be int egr a b l e on [ a, oo) for j=?l,2 ,. . . ,n.
Then,
k=l,
there exist, n l i n e a r l y i n d e p e n d e n t solutions x ^ ( t ) ,
2 ,..
. ,n, of
(I)
and a n u m b e r t , a <_ t^ <
-I
x k (t) = Cp [exp
co , su c h that
t / Pk Cp(T)dr] [I + o(l) ].
tO
Moreover,
the first n-1 derivatives of x ^ t )
ha v e the a s y m p t o t i c
form
x^^(t)
= cp^
1 [exp
/
pk cp(T)dT] [p.^ + o(I) ]
tO
j = l , 2 ,...,n.
32
Proof:
This
theorem can-be e s t a b l i s h e d b y repeating that part of
the p roof given for T h eorem 1.2 w h i c h used h ypotheses
(viii).
H a v i n g found b.(s)
J
to b e integrable,
(i)
through
continue the proof
b y showing that a. - B . (s) is also i n t e g r a b l e on [0 ,oo) , u n d e r the
j
I
additional h ypothesis of the p r e s e n t theorem.
L e m m a 1.4,
forms
i n stead of L e m m a 1.3,
to Condi t i o n I.
Finally,
that for k = l , 2 ,...,n,
on
0
X^(s)
Then,
invoice
to v e r i f y that every
(s)
con­
apply T h e o r e m 1.1 u s i n g the.fact
= U1
nf*(s) w h e r e p*"
n**(s)
Jji
s. + n**(s)
is integrable
_< s < oo .
• As w i t h the foregoing theorem,
the rest- of the theorems in this
chapter are o b t a i n e d by m a k i n g a v a r i e t y of changes in the h y p o ­
thesis of T h e o r e m 2.1.
Thr e e of the ne x t four theorems
for the third o r d e r differe n t i a l equation,
general e q u atiqn of order " n " .
are stated
r a ther than for the
U n d e r this restr i c t i o n on order,
is m u c h e a sier to find explicit formulas
the characteristic polyno m i a l P(A).
for the coefficients of
Consequently,
there is no ne e d
to a p proximate the
( s ) , j=l,2,3.
again concerned w i t h
the n = t h o r d e r d i f f e rential equation.
O u r initial theorem,
pertaining
comes about w h e n h y p o t h e s e s
In o ur fourth theorem, w e are
to the third o r d e r equation,
( i v ) , ( v ) , ( v i i i ) , and (ix)
T h e o r e m 2.1 are revised as follows.
allowed to converge to any Constant
of
The p r o d u c t cp’cp ^ in (iv)
c;
it
the a. of
(v)
is
are r e q u i r e d
33
to be r e a l ; the n e w functions d e s c r i b e d by
(viii) n e e d only be
3
m e m b e r s of L
[0,oo) ; and finally,
since the a^ are all real n u m ­
bers , w e m a y algebr a i c a l l y si m p l i f y hypoth e s i s
Specifically,
T h e o r e m 2.3
(ix).
this yields:
C o n s i d e r the l i near d i f f e r e n t i a l e q uation
x " '(t) + A^Ct)
x" (t) + A 2 (t) x ’.(t) + A 3 Ct) x(t)
= 0
(15)
Suppose
(i)
A . ( t ) , j =1,2,3,
J
are continuous
c o m p l e x - v a l u e d functions
d e fined on a < t < co,
(ii)
cp(t) is a positive,
con t i n u o u s l y d i f f e r e n t i a b l e function
d e f i n e d on a < t < oo ,
(iff)
(iv)
(v)
(00Ct)dt =
a
po ,
-2
Iim cp cp
t -> CO
=c,
w h e r e c is a constant,
D e f i n e . B 1 Ct) = A^(t)cp ^(t)
ggCt)
= A 2 (t)qr 2 (t) - A^(t ) ? ' ( t ) 9 ~ 3 ( t )
-
(cp'(t)(p~2 (t ))2
B 3 (t) = A 3 (t)cp
3 (t)
A^Ct)cp' ('t)cp
4 (t)
Iim B .(t) = a., j=l,2,3, w h e r e the a. are real- v a l u e d
t ->oo 1
3
I
constants,
34
(vi)
3
p (A) = A
+ a^A
2
+ a^A +
a^ has dis t i n c t zeros
Pi s P 2 ’ ^ 3 » and R e Pj = R e p k implies
(vii)
(cp'cp ^)'
and
(Ayp
j=l,2,3,
(viii)
[a. - . B . ( t ) ] c p ^ ^ \ t ) ,
J
I
(ix)
F o r fixed k,
p^ = Pfc,.
are i n t e g r a b l e on [a ,oo) ,
j=l,2,3,
are m e m b e r s of L^[a,oo) ,
3-m
CpTlmB^
P 1(Pk )
1=1
^
3
r
(3-m)pf-m-r
+
m = l r=l
[
?
(p 'C p , ) ) -
3
P-(Pk )
i
<PZ
2 M ( %
m = l r=l
- U
K
- aj]
P
^
Im
(p'(pk))^
satisfies Condi t i o n I if j is an index for w h i c h
R e [ p k - P j ] = 0.
if pk =
Then there exists
It is to be u n d e r s t o o d that pP = I
k
0.
a solution x JcCt ) of
(15)
and a n u m b e r tQ , such
that
x k (t) = cp
1 Ct)
[exp
f Jpk + Ek Cx) ]cp(x)dT][l + o(l) J
t
where
O
.35
3
-i
=k =
^ a_ - BraIy
Lm=l
3-m
m
+
2- m
( p ' ( Y ) " 2[ S am
V ^ k -I f Z [a„ - Bm ](3 J P=I
-Itt=Fl
k
i 3-m
% (P'(P_))""V(^)
Icv I ^ i“m - Bm Iuk
itt-1
Furthermore, the first and second derivatives of x^(t) have the
asymptotic forms
t
x^.(t) = [exp /
[Uk + Efc(T) ]cp(T)dT [Pk ~ c + o(l)]
tO
and
x£(t) = cp(t) [exp / Iuk + Ek (T)J Cp(T) dr ][uk - UfcC + o(l) ]
tO
(16)
36'
Proof:
The p r o o f of this
for T h e o r e m 2.1.
formed into
theo r e m follows the line of p r o o f given
Hypotheses
(i ) , ( i i ) , and (ill)
a l l o w to be trans­
the system
(17)
where
H
CO
Cd
, - 2.
cp cp
0
-3
-A^P
and
I
0
0
I
. m -1
- A 11P
-p
-2
-Ag?
z = [cpx, x 1 , <P "*"x"]T
t
T h e argument ^(s)
i.s o b t a i n e d b y l e t t i n g s(t)
d e n oti ng the inverse t r a n sfo r m a t i o n b y
B ( 1K s ) ) ,
and
1
I1(S)
as s tends
F r o m hy p o t h e s i s
to infinity,
/
t = ^(s).
are continuous o n 0
to i n f inity w i t h s.
=
s <
(iv)
oo .
and
p
(t )
Th e elements of
Mo r e o v e r ,
( v ) , w e observe that
or e q u i v a l e n t l y as t tends
to infinity,
B ( 1K s ) ) approaches the constant m a t r i x C displayed below.
0
C
w h e r e b^, b ? , b^ are constants.
I
t tends
37
B y direct e x p a nsion of det [ B( i|j(s ) ) - AE ], w e find that
the coeffi­
cients of the charac t e r i s t i c p o l y n o m i a l
P(A)
= A^ + B ^ ( ^ ( s ) ) A ^ + . B g ( ^ ( S ) ) A + .B^(^(s))
B(^(s))
are those given in h y p o t h e s i s
replaced by
Again,
lK s ) .
Let
A^ (s)
associated with
(iv) w i t h the argument t
denote the zeros of P(A).
u t i l i z i n g R o u c h e 1s T h e o r e m and renaming subscripts if
necessary, w e m a y take
Iim A .(s) = y., j = l ,2 ,3, in v i r t u e of ( v i ) .
S ->03
J
3' ■
W e shall n o w apply T h e o r e m 1.1 to s y s t e m (17).
L e t t i n g A = C,
and V(s)
that / |v'(s) |ds. < co
9
=
B(\jj(s)) - C, h y p o t h e s i s (vi) shows
furthermore,
Q
I i m V(s)
= 0.
In o ur case,
S -> OO
the m a t r i x R(s) m e n t i o n e d in T h e o r e m 1.1, is. id e n t i c a l l y zero.
Hence,
the hypoth e s e s of T h e o r e m 1 . 1 w i l l b e s a t i s f i e d p r o v i d e d the
real p a r t of
A (s) - A .(s) s a t isfies C o n d ition I.
k
j
Hypothesis
(viii)
3
are m e mbers of L (0,co).
Re[A
k
implies
that the functions
(a. - B .(^(s))
J
J
N o w ap p l y i n g L e m m a 1.6, w e m a y v e r i f y that
(s) - A . ( s ) ] satisfies C o n d i t i o n I.
' I
Hence, w e m a y
conclude b y
T h e o r e m 1.1 that there exists a s o l u t i o n z (i|r(s)) and a n u m b e r s
K
such that
s
Zk ( M s ) )
[exp
f \ ( 0
s
O
d^ H P k + o(l)],
O
...
where
is the e i genvector of C corresp o n d i n g
N o loss in
Pk =
38 .
IX
generality results
Pk -C,
2'
■ T
Pk - Pk c ] >
to the e igenvalue
in assuming, that
Since z(t)
is of the form
= T(t) y(t) , and s(t)
=
^
J Cp(T)dT,
a
w e m a y retrace the transfor m a t i o n s b a c k to the v a r i a b l e s x and t ,
obtaining
-I
t
X fcCt).= cp (t)Iexp / Xfc(s(T))cp(T)dT]Il + o(l)].
ON o w using
the asymptotic f o r m of Afc(s) given b y L e m m a 1.5, w e
clude that the asymptotic form of x fc(t)
sion of the theorem.
as that given in the conclu­
I n a s i m i l a r fashion, w e t r a n s f o r m the' other
two components of the a symp t o t i c f o r m of zfc(^(s))
expressions
for the drivatives of X1 (t)
to o b t a i n the
given in t he c o n l c u s i o n of
this theorem.
C o r o l l a r y 2.4
S u p p o s e that h y p o t h e s e s
T h e o r e m 2.3 h o l d and, in addition,
j =1,2,3,
(i)
through
(viii)
s u p p o s e that Im[A^ cp ^
are integrable on a _< t < oo .
T h e o r e m 2.3 holds.
con­
of
],
Th e n the c o n c l u s i o n of
39'
Proof:
If the Im[A.cp
J
Im.[B cp] j = l , 2 ,3..
infinity,
i=l,2,3,
Since the
and are continuous,
are integrable,
then so are
] converge to zero as s tends to
p roducts of them w h e n m u l t i p l i e d by
Cp(t) are also i ntegrable functions o f t on a _< t <. oo .
hypot h e s i s
(ix) of T h e o r e m 2.3 holds
■ L o o k i n g back,
H ence
and the corollary is proved.
the addit i o n a l const r a i n t in h y p o t h esis.(vi)
T h e o r e m 2.3 w h i c h is n o t present in h y p o t h e s i s
(vi)
of
of T h e o r e m 2.1
was u s e d to show that the real part of the d ifference of A -(s) and
Ay (s) m a y b e e x p r e s s e d as the f u nction given in h y p o t h e s i s
of T h e o r e m 2.3
with
(except for a m u l t i p l i e r of cp) .
this additional
constraint,
cation for R e [A^(s) - A (;s)].
x i m a t e expressions
theore m illustrates
(ix)
If on e dispenses
there is u s u a l l y no su c h s implifi­
It is then n e c e s s a r y to use the appro­
for A^ and A
this change.
given b y L e m m a 1.5.
The p r o o f of this
that of T h e o r e m 2.-1 and is not given here.
T h e following
theorem follows
' 40
T h e o r e m 2.5
C o n sider the linear d i f f e rential equ a t i o n
x"'(t) + A 1 Ct) x"(t) + AgCt) x'(t) + Ag(t)
x(t') = 0
(15)
Suppose
(i)
A^ ( t ) , j = l , 2 ,3,
are c ontinuous
defined on the i n terval
(ii)
cp(t) is a positive,
defined on a < t <
(iii)
c o m p l e x - v a l u e d functions
[a,oo)}
cont i n u o u s l y d i f f e r e n t i a b l e function
oo ,
0,
y cp(t)dt = oo ,
a
. (iv)
(v)
Iim cp'cp ^
t-^oo
= c, w h e r e c is a constant,
.D e f i n e . B 1 Ct) = A 1 (t)cp
1 Ct)
( B 2 Ct) = A 2 (t)cp
2 (t)
'(t)cp
3 (t)
- A 2 Ct)(PrCt)Cp
4 (t)
- A 1 (t)cp
-
(q/(t)9~2(t))2
B 3 Ct) = A 3 (t)cp
and suppose
limB.(t)
t_»co d
3 (t)
= a., j = l ,2 ,3, w h e r e the a. are
3
I
c o m p l e x - v a l u e d constants,
(vi)
p(A)
3
2
= A + S3 A + a2A + a 3 has d istinct zeros
'
M1 , M2 , h3»
2) ' and (AjCp ^)', j=l,2,3, are i n t e g r a b l e on {a,00) ,
(vii)
Ccp1Cp
(viii)
[a. - B . ( t ) ] c p ^ ^ 3\ t ) ,
J
I
j:rl,2j3, are m e m b e r s of L 3 [a,oo)
,:
41
(ix)
Define P (A) = (B^-a^)A
+
A +
(B^ -a ) and
suppose the re a l part of
P ( P fc)
P ( P 1)
+
P ( P k) P ' (pk ) ' P ( P 1) P t (P1)
+
P ' ( Y
(p'(p k ))2
( P t(Pj))Z
i ?"(%%)( K y ) ^
i P--(P1) ( P ( P 1 ))Z
^
^
(p'(w%))3
multiplied by
(p'(hj))^
cp(t) , s a t i sfies Co n d i t i o n I .
Then there exists a solut i o n x k (t) of
(15)
and a n u m b e r t , such
that
^(t)
= cp
(t)fexp
f [pk + Ek (T) ]cp(T)dT]Il + o(l) ]
tO
where
P(Mk )
P ( M k ) P 1 (Mk )
E1 = --- TT— r + ------ r
"k
' P'(*k>
( P t(Pk ))Z
Furthermore,
i P " ( M k ) (P(Mk ))'
Z
(p'(Y)3
the first and seco n d derivatives of % k (t) h a v e the
asympt otic forms
k(t) =
and
[exp /
[pk + Ek (T) ]cp(T)dT
[pv - c + o(l) ]
42
*£(t)
=
J [pk + Efc(T) ]q)(T)aT][pk - PfcC + O ( I ) J 1
C|Xt) [exp
tO
We
conclude this
chapter w i t h one mo r e theorem.
as w e l l as T h e o r e m 2.1,
This
theorem,
is used in C h a p t e r 3, to d e r i v e osci l l a t o r y
and b o u n d e d n e s s properties, of s o l u tions of n - t h o r d e r linear
differential equations.
F o r these purposes,
n e c e s s a r y to h a v e asympt o t i c e x p ressions
it is not
always
available for the charac­
teristic v alues of the r e l a t e d d i f f e r e n t i a l system.
This
reflected in the hypoth e s e s of the e n s u i n g theorem.
T h e y are suf­
ficiently s trong that T h e o r e m 1.1 m a y be applied,
fact is
e v e n in the
absence of a s y m p t o t i c r e p r e s e n t a t i o n s of the c h a r a c t e r i s t i c values.
43
T h e o r e m 2.6
Consider the l i n e a r d i f f e r e n t i a l e q uation
(t) + A 1 (t) x ^ n ^ \ t )
I
+...+ A
n
(t) x (t) = 0.
(I)
Suppose that
(i)
(ii)
(iii)
A . ( t ) , j = 1 , 2 , . . . ,n, are c o n t i n u o u s , r e a l - v a l u e d f u nction
J
d e f i n e d on fa,oo) ,
,
cp(t) is a positive,
cont i n u o u s l y differentiable,
function d e f i n e d on
[a,co) ,
.
■
/ cp(t)dt = oo ,
a
-2
. (iv)
(v)
Iim qj'cp
t -> CO
= .0 ,
Iim A.(t)cp ^ (t) = a., j = l , 2 ,. . . ,n, w h e r e the a. are ■
t ->co 3J
3
constants,
(vi)
the zeros' p^ ,, j - 1 ,2 , . . . ,n, of p(:A) = An + a^.An
+...'+ a^
are distinct and Rep, = R e p . implies p, = P . ,
R
3
R
3
(vii)
(cp'cp 2 ) ' and
[a,oo) .
( A 1Cpj )', j = l , 2 ,...,n,
3
are integral Ie on
44
D efine P(A)
= detfB(t)
- AEj w h e r e
B ( t ) has the form
t “2
cp cp
I
O
...
O
O
O
I
. . .
O
O
O
, -2
-cp cp
. .
O
.O
O
O
...
I
B(t) =
. -n
K
t
A
l~n
-VicP
-An_2 cp^ n -. . . • -A^cp
+(2-n)cp'cp
Then there exist n linearly independent solutions
k=l,2,.. • s s O.f (I) and a number to , a —< to -
x, (t) = r cp 1 Iexp
, such that
Jr Ak (T)Cp(T) ’d T ] [I + o(I) ]
O
and
(t) = cp^- 1 [exp
t
J A k (T)Cp(T) dT ] [pk + o(l) ] ,
j = l , 2 , . .. ,n-1
tQ
w h e r e Afc(P) is that zero of P(A) w h i c h converges
infinite.
to pfc as t becomes
45
Proof:
The proof of this
of T h e o r e m 2.1.
The m o s t important d i f f e r e n c e is that all
coefficients of P(A)
valued.
and p(A)
This observation,
to v erify that R e [ A ^ - A^]
or P
(x)
(vi),
satisfies Co n d i t i o n I (as
f 0, the c o n d it i o n h o l d s as before.
then R e f P 1 - P .j = 0
k
J
o ccur as
together w i t h h y p o t h e s i s
functions of s) , for any indices k and j, I _< k, j _<. n.
Refp^ -
the
are real- v a l u e d ins t e a d of complex­
H e n c e , any complex - v a l u e d zeros of P(A)
conjugate p a i r s .
allow us
theo r e m is e s s e n t i a l l y the same as that
i d enti c a l l y for all s.
For, if
'If R e [ p ^ - p
= 0,
CHAPTER 3
O S C I L L A T O R Y A N D B OU N D E D N E S S P R O P E R T I E S OF SOLUTIONS
In this
chapter w e study the o s c i l l a t o r y and boun d e d n e s s
p roperties of solutions of the dif f e r e n t i a l equ a t i o n
(t) + A ^ ( t ) x ^ n
(t) +. . .+ A^(t)x(t)
w h e r e the coefficients A ^ ( t ) , j = l , 2 ,...,n,
uous r eal-valued functions defined on
n o n t r i v i a l solution x(t)
of
= 0
(I)
are assumed to be contin­
[a,oo) .
W e shall call a
(I) o s c i l l a t o r y if and only if it has
infinite n u m b e r of zeros in the i n t e r v a l
it n o n o s c i l l a t o r y otherwise.
Equation
an
[a,co) , and w e shall call
(I) w i l l b e c a lled o s c i l l a t o r y
if and only if it admits of some n o n t r i v i a l o s c i l l a t o r y solution.
Otherwise,
it w i l l h e
called nonoscillatory.
O u r results
are b a s e d
upon the theorems developed in C h a p t e r 2 concerning a symptotic
solutions of ( I ) .
A l t h o u g h a s y m p t o t i c solutions only provide i n f o r m a t i o n on some
interval
[t^,co) w h e r e tQ-_>
as is done in P f e i f f e r [7],
iff it is o s c i l l a t o r y on
a;
nevertheless,
is read i l y verified,
that a s o l u t i o n is o s c i l l a t o r y on
[a,co)
[t^,co).
H a v i n g d e rived asympto t i c forms for the deriv a t i v e s of solu­
tions of
(I)
in C h a pter 2, w e also p r e s e n t b o u n d e d n e s s p roperties
for these derivatives by em p l o y i n g o ut earlier results.
47
In addition to' the assum p t i o n that the coefficients A. (t)
J
are
all real-valued, w e suppose that all the h y p o t h e s e s o f - T h e o r e m
2.1
hold.
U n d e r these assumptions,
each a^ is n e c e s s a r i l y real.
■
T he
following corollaries of T h e o r e m 2.1 d e s c r i b e the o s c i l l a t o r y natu r e
and b o undedness proper t i e s of s o l u tions of E q u a t i o n
( I).
$ 0, then E q u a t i o n (I) is oscillatory.
C o r o llary 3.1
If I m p
"I K.
C o r o l l a r y 3.2
If for k = l,2 , . . . , n , h y p o t h e s i s
is s a t i sfied and all the p^ are real,
(ix)
of T h e o r e m 2.1
then E q u a t i o n (I)
is n on-
oscillatory.
C o r o llary 3.3
solution x^(t)
If R e p ^ < 0, then the m o d u l u s of the asymp t o t i c
of
(I), d e s c r i b e d i n the c onclusion of T h e o r e m 2.1,
increases w i t h o u t b o u n d as t beco m e s infinite.
T h e same is true of
the first n -1 derivatives of x ^ ( t ) .
C o r o llary 3.4
s o l ution x^(t)
converges
If R e p ^ < 0, then the m o d u l u s of the asymptotic
of (I),
desc ribed in the conclusion of T h e o r e m 2.1,
to zero as t beco m e s infinite.
The same is true of the
first n -1 d erivatives of x^(t).
oo
C o r o l l a r y 3.5
If R e p ^ = 0 and
_2
f R e [ e ^ - cp’cp
] dt < oo , then the
a
m o d u l u s of the solution des c r i b e d b y the conclusion of T h e o r e m 2.1.
is b o u n d e d on [a,oo).
48
Corollary 3.6
If
OO
= 0 and ■J R e [ e ^ 'va
m o dulus of the s o l u t i o n of
_2
cp’cp
J dt = oo , then the
( I ) , d e s c r i b e d b y the conclu s i o n of
T h e o r e m 2.1 is u n b o unded on [a, oo) .
To illustrate h o w these Corol l a r i e s m a y be established, we
shall prove the first three.
P r o o f of C o r o llary 3 . 1 :
B o t h the r e a l pa r t and the i m a g i n a r y part
of the a s y m p t o t i c s o l u t i o n x ^ ( t ) , given b y E q u a t i o n
Theore m 2.1,
are solutions of (I).
(12)
on
T a king the real p a r t , w e have
the solution
x^Ct)
- [exp f■R e [Pk
= cp
+ EfcJcp dr] [cos
tO
Since Efc = o(l)
I m p fc f 0,
f
, Im [ p fc +
EfcJcp dr [I + o(l) ]
tO
and 'ep(t) • is n o n i n t e g r a b l e , it is clear that if
the absolute valu e of t he a r gument of the cosine
increases w i t h o u t b o u n d as
t becomes infinite.
oscillatory a n d , t h u s , equation
function
H e n c e , XfcCt) is
(I) is oscillatory.
-4
49
Proof of C o r o llary 3 . 2 :
satisfied,
If for k = l ,2,...,n , h y p o t h e s i s
and all the
are real,
independent solutions of (I)
and the factor
positive.
is
then there are n li n e a r l y
s p e c i f i e d by
Each of these is nonoscillatory.
(ix)
(12)
F or the e (t)
[l + o ( l ) ] of the s o lution x^(t)
of T h e o r e m 2.1.
are n o w real-valued,
becomes
and remains
W e - still m u s t show that e v e r y s o lution is nonoscillatory.
n
To this end,
let
solution of (I).
which
c^ f 0.
= Yi- c, x, (t)
. k=l
x(t)
L e t a denote the i ndex of the g r e a t e s t zero
for
Note that
-i
Cp
x I (t)
Iim
denote an ar b i t r a r y n o n t r i v i a l
[exp
«
J [p
r
^ y-y =
a
+ e ]cp dr] [I + o(l) ]
t
H m
— ---- ---- 1------- ------ :
--------------- :
^
Cp 1 Jexp / [p
t
+ s -]cp dr] [I + o(l) ]
^
O
■ t.
-=
Iim [exp / [y
t - > CO
t
K
-p' +e - EJcp dr J[I + o(l) J = 0
a
k
«
o
if k Tt ot .
Therefore,
x(t)
t^, x(t)
0
^
= c^ x
on Jt 15Oo).
(t)[I + o ( l ) ] a n d , for s u f f i c i e n t l y large
50
P r o o f of Corollary 3 . 3 :
os T h e o r e m 2.1 has
The m o d u l u s of the s olution given by (12)
the form
t
Ixk Ct) I =
[exp
J
_2
ReIpk +
ek - cp'cp
]cp dx][l + o(l) J
tO
R e c a lling
that cp'cp ^ = o(l) , ek (t) = o(l) , and cp(t) is n o n i n t e g r a b l e
on [a ,oo) , it is clear that the m o d u l u s 'from h y p o t h e s i s
Ix k (t) I of the s o l ution
infinite,
if R e p k >
(iv)
in c r e a s e s w i t h o u t b o u n d as t becomes
0.
To draw the same concl u s i o n for the d e r i v a t i v e s , express
modulus
of the j-th
Ix£j ) (t) I = Iexp
the
deriva t i v e of x JcCt) .as
/
R e IPk + ek +
(j- 1 ) cp'cp ^Jcp dr] Jpk .+ o(l) I
tO
for j = l , 2 , . . . ,n-1.
F o r the same rea s o n s
as before,
the modu l u s
of the j - t h deriva t i v e of the s o l u t i o n x JcCt ) increases w i t h o u t
b o u n d as t b e c o m e s infinite. ■
The last three corollaries of the f o r e going list m a y be esta­
b l i s h e d b y m a k i n g only slight m o d i f i c a t i o n s in the above proof.
51
By' using T h e o r e m 2.5,
I n s t e a d of The o r e m 2.1,
it is p o ssible
to state and prove another set of corollaries p e r t a i n i n g
to the
o s cillatory n a ture and b oun d e d n e s s p r o p e r t i e s of solutions of the
third order l inear differential equation.
I n this instance,
the
oscill a t o r y n a t u r e m a y be ch a r a c t e r i z e d in terms of the b e h a v i o r of
the d i s c r i m i n a n t of the poly n o m i a l
p(A)
3
= A +
2
a^x
+ ’a^A + a^.
The. following corollaries are o b t a i n e d w h e n T h e o r e m 2.6 applies
T h e proof of these corollaries m a y be constructed' u s i n g the same
techniques emplo y e d in the proofs of the p r evious
corollaries of
this chapter.
C o r o llary 3.7
g i n a r y part,
Corollary 3.8
If one of the zeros
then equation
of p(A) has n o n z e r o ima­
(I) is oscillatory.
If all of the zeros of p(A)
are real,
then e q u a t i o n
(l) is n o n o s d i l a t o r y .
C o r o llary 3.9
If R e p ^ > 0, then t he m o d u l u s of the so l u t i o n x^(t)
of (I) d e s c r i b e d b y
the con c l u s i o n of T h e o r e m 2.6 in c r e a s e s w i t h o u t ,
b o u n d as t b e c omes infinite.
d e rivatives of X1 (t).
The same is true for the first n-1
: 52
C o r o llary 3.10
If
< 0, then the m o d u l u s of the sol u t i o n
/• '
x ^ X t)
described b y
as t becomes
the conclusion of T h e o r e m 2.6 converges
infinite.
tives of x^(t).
to zero
The same is true for the first' n - 1 deriva­
CHAPTER, 4
SUMM A R Y A ND E XAMPLES
I n this chapter w e discuss b r i e f l y the results of the first
three chapters of this
thesis.
C h a p t e r I contains
the p r e l i m i n a r y
lemmas u s e d in the proof of the theorems given in C h a p t e r 2.
In
C h a p t e r 2, several theorems w e r e p r e s e n t e d w h i c h gave the asymptotic
form of solutions of certai n l i n e a r differential equations.
These
theorems are applications of T h e o r e m 1.1 to a first o r d e r differ­
ential system.
The differences b e t w e e n the v a r i o u s
theorems have
already b e e n p o i n t e d out.
In each of these theorems: an u n s p e c i f i e d function cp(t) appears.
One m a y quest i o n w h e t h e r a m e t h o d exists w h i c h w i l l d e t e r m i n e an
applicable function cp(t).
unresolved.
However,
for this function.
IA_.(t)
The answer to this q u e s t i o n is still
there are c e r t a i n l y some likely candidates
A m o n g these candi d a t e s
, IReAj (t) j
J', and
|lmAj (t) 1
are the functions
, j=l,2,...,n.
any such choice m u s t still b e g o v e r n e d b y the r e q u i rements
cp(t) b e continuously d i f f e r e n t i a b l e , n o n i n t e g r a b l e on
Of
course,
that
[a,Co) and
satisfy the other relevant h y p o t h e s e s of the a p p l i c a b l e theorem.
U s i n g two of these s u g g e s t e d possibi l i t i e s , w e g e n e r a l i z e the results
of P f e i f f e r
[6 ].
54
■ Let n = 3,
= 0, A^ = q, A^ = r,
and cp(t) = r
T h e o r e m 2.1 generalizes T h e o r e m 4 of Pfeiffer.
1/3
In this
.
Then
c a s e , the
hypotheses of T h e o r e m 2.1 are less r e s t r i c t i v e than t hose of
Pfeiffer's
theorem.
F o r instance,
T h e o r e m 2.1 does n o t require A^
3
to b e a r e a l - v a l u e d function n or p (A) to be
m a y have
complex-valued
coefficients.
X
+1.
It is w o r t h m e n t i o n i n g that
T h e o r e m 2.1 corrects an o v e r s i g h t b y Pfeiffer.
inclu d e d the requirement that Im[q r
his other hypotheses.
I n fact, p(A)
-1/3
H e s h ould have
] satisfy C o n d i t i o n I among
It then b e c o m e s pos s i b l e
to p r o v e his
theorem
b y constructing a p roof simi l a r to that of T h e o r e m 2.1.
If n = 3, A 1 = •0, A^ = q, A^ = r , and cp(t) = q
generalizes Theorems
6
and
8
of P f e i f f e r
6
1/2
, T h e o r e m 2.1
.
1/n
I f A 1 = A_ =
JL
generalizes
z
... = A
= 0, A
Ti“ x
n
a c o r o llary of H i n t o n
= q, Cp = q
T
, T h e o r e m 2.1
[4S p . 5 9 4 J.
I
At the outset of Chapt e r 3 a n u m b e r of coro l l a r i e s of T h e o r e m
T h e o r e m 2.1 are given w h i c h describe o s c i l l a t i o n and b o u n d e d n e s s
properties of solutions of l i n e a r d i f f e rentail equations.
corollaries generalize Theorems 4 and 5 of P f e i f f e r
These
[7j.
T h e o r e m 2.1 m a y not b e a p p l i c a b l e to a d i f f e r e n t i a l equation of
the f o r m [r
derivative.
_
q
^ u n less r possesses a continuous k- t h
This e q u a t i o n is treated by H i n t o n 14], u s i n g methods
55
analogous
to those of Pfeiffer.
k - t h derivative,
H i n ton's
If 'r does n ot ha v e a continuous'
t r a n s f o r m a t i o n coupled w i t h the a symptotic
form of the zeros of the charac t e r i s t i c p olynomial found in Chapter
I, may be e m p loyed to genera l i z e his results.
Pfeiffer
y
+
[6 ] also considers an e q u a t i o n of t he fo r m
a 2 t^ 2y ' + a 2 t^2y = 0
as an example.
He obt a i n e d results
d e s c r i b i n g the asymptotic form of its solutions for a b r o a d range of
values of
and g 2 .
( 2 / 5 ) (2g 2 + 1 )
>
However,
if
> ( 1 / 2 ) (B 2 - I),. he could dr a w no
conclusions
using his
theorems.
W e shall de s c r i b e the a s y m p t o t i c f o r m of the
solutions
for values of gj and g 2 w h i c h inc l u d e those that satisfy
the p r e c e d i n g inequalities. . This example,
also illustrates
stat e d as T h e o r e m 4.1,
the m a n n e r in w h i c h one m a y apply T h e o r e m 1.1
w i t h o u t determ i n i n g close a p p r o x i m a t i o n s of the ch a r a c t e r i s t i c
values of the system.
56
Theorem 4.I
C o n s i d e r the e q u a t i o n
x"'(t) +
w h e r e a^,
a2 ,
8 %,
(Sg
a ^ t ^ x ' (t) +
are teal
o ^ t 2x (t) =
constants.
(18)
0
Supp o s e one of the
following h y p o t h e s e s holds.
3
U2
(i)
^ o, Sz > -3, and gg > - U 1.
3
A.3/-P1 ^ - A 2 , p 2 = A 2
(i) '
A
f 0,
S 1 > -2, and
^ 2 ~ y^ l
yI =
If
«2
is equal to zero,
theses c o n c e r n i n g
8%
i5
I +
2
S1 > ^
/3" i
Sz-
y3 =
D e f i n e cp(t) = t(G2/3),
- A i
, P3 = ./U2
2
-
D e f i n e cp(t)
W
2K
i
then that part of t h e r e l e v a n t hy p o ­
is assu m e d to hold.
Similarlyif
Ot2 = 0,
that part of the hypoth e s i s c o n c e r n i n g Sg is assumed to hold.
Then
there exist three solutions of (18) and a n u m b e r t^ su c h that
-I
t
x k (t) = cp (t) [exp /.^.(T)Cp(T)
t
O
w h e r e !^(t)
. A3
+
dr] [I + o(l) ]
k = l ,2,3
is that root of
[Ult6V 2(I)
-
(cp' (t)q)~2(t))2]A
+ a2t3^p“3(t) -
U lt 6 lCp' (t)cp ^(t) =
w h i c h converges to
as t tends to infinity.
Mo r e o v e r ,
the first
0
57
two d erivatives of
(t ) have the f o l lowing a s y m p t o t i c forms:
t
x'(t)
=
[exp
J A (t )<p (t ) dij [p
+ o(l) ]
and
x" (t) = cp(t) [exp f A (T)cp(t ) dx] [p
+ o(l) ]
62
R egion I
-4 -3 -2 -I
2
R e g i o n II
Given that Oi1 f 0, and a 2 f 0, R e g i o n I of the above d i a g r a m
depicts
those values of B 1 and g 2 w h e r e h y p o t h e s i s
R egion II depicts
holds.
His
(i) holds.
those values of B 1 and. B 2 w h e r e h y p o t h e s i s
The u n l i n e d region has b e e n c o n s i d e r e d b y G h i z z e t t i
results m a y be
found in Coppel
[2, P a g e 92].
(i)'
[3].
58
Proof:
F o r brevity, w e shall m e r e l y sketch the p r o o f of this
theorem.
Suppose hypothesis
(i) holds.
B y following the m e t h o d of proof
of T h e o r e m 2.3, w e obtain a s y s t e m m u c h like (17).
the m a t r i x has only real elements.
n o m i a l has real coefficients
and
case,
H e n c e the charac t e r i s t i c po l y ­
X^(ip(s)) = A ^ ( 1K s ) ) •
d ifference of any two zeros of P(A)
C o n d ition I.
In this
Sq the real
is easily s h o w n to satisfy
N o w apply T h e o r e m 1.1 to the system.
T h e re a m i n d e r
of the p r o o f consists of tra n s f o r m i n g the result to the v a r i ables
t and x.
S u ppose hypoth e s i s
( i ) ' holds.
B y again f o l l owing the m e t h o d
of the p r o o f of T h e o r e m 2.3, w e o b t a i n a s y stem m u c h like
. r
^
c h a r a cteristic polyno m i a l has on l y real coefficients.
(17).
The
If cij < 0,
the l imit of the zeros of the ch a r a c t e r i s t i c p o l y n o m i a l are real
numbers.
So the real part of the d i f f e r e n c e of any two
tic v alues satisfies Condit i o n I.
calculation,
Ai(C) =
±
If
characteris­
> 0,. w e h a v e b y direct
that
E-
and that ReA^(I)
CXl
+ A l t-(B l + 2) / 2 ][i + o(l)]
^
= R e A ^ (t) = - ( 1 / 2 ) A ^ ( t ) .
term is d r o p p e d from the above expression,
Obs e r v e that if one o(l)
the d i f f e r e n c e b e t w e e n
• the resulting approx i m a t e value of A ^ ( ^(s)) and the exact value
59
of
(^j(s ) ) m a y not b e integrable.
In this respect, this applica/
tion of T h e o r e m 1.1 is different than the theorems o f Chap t e r 2.
Now,
use the n o n o t o n e prope r t i e s of the e x p r e s s i o n above to show
that in terms of s, R e [ A
(ij;(s)) -
'
for any values of k and j-.
K
A.(t|>(s ))] satisfies C o n d ition I
J
T h e o r e m 1. 1
then applies and the proof
of this t h e o r e m follows.
O u r final example is an a p p l i c a t i o n of T h e o r e m 2.1 to a
d i fferential equation w h i c h is not
trea t e d b y any o t h e r known
theorem.
E x amp le 4.2
Let cp = I, and apply T h e o r e m 2.1 to the equ a t i o n .
x IM(t) + t
x* (t) +
(I + i t "*")x(t) =
T h e n there are t h r e e solutions x^(t)
h a v i n g the a s y m p t o t i c repre­
sentations
w h e r e p^, k=l,2,3,
0
are the three roots of -I.
60
APPENDICES
APPENDIX I
In this
appendix w e state and p rove two lemmas
concerning,
i n t e g r a b i l i t y p roperties of the class of comple x - v a l u e d functions
of the real v a r i a b l e s that are continuous on
converge to zero as s becom e s infinite.
application in the m a i n b o d y o f this
Lemma I
0
_< s < oo , and w h i c h
T h e s e lemmas
find frequent
t h e s i s , and in A p p e n d i x 2.
Suppose that
(i)
f(s)
is a continuous c o m p l e x - v a l u e d function,
v a r i a b l e s, def i n e d on
(ii)
Iim f(s)
S
(iii)
=
0
of a real
< s < °° ,
0,
CO
f(s)
is a m e m b e r of L^ l O ,oo) , I _< q < oo s
Then f(s). is a m e m b e r of L r [0 ,oo) for all r such that q < r < oo .
Proof:
By hypot h e s e s
(i)
and ( i i ) , w e m a y choose an s , 0 < S <00
—
o
\
such that
I
f (s)
I<
I
. I —
I on s o
OO
/
S
and the
< s <
—
.
00
Hence,
OO
:|,f(s) |r ds <_ / If (s) Iq ds
O
S
O
conclusion follows from (iii).
62
Lemma. 2
Suppose that
(i)
f(s)
and g(s)
are c ontinuous
c o m p lex-valued functions of
v
a real v a r i a b l e s on
(ii)
Iim f(s)
s ->oo
(iii)
=
0
and
s
j< s < co ,
0
Ii m g(s) =
>-oo
0,
f(s) e L q [ 0 ,oo) and g(s) e L r [0 ,co) w h e r e I £
Then the p r o d u c t function f (s) •g(s)
q, f <
oo ,
is a m e m b e r of L m [0 ,°°) w h e r e
m = max(l,qr/(q+r)) .
Proof:
Let
a = (q+r)/ r and g = (q+r)/q.
Applying Holder s
I n e q u a l i t y w e have
/Vsl
qr/(,+r> ds ll7[|f|qr/<q+r,j“dsj
1/c,l/”o |g|qT/q+r]6dBj 1/13
0
o
Since the r i g h t h a n d side o f the i n e q u a l i t y is finite, b y hypoth e s i s
( i i i ) , so is the left h a n d side.
Thus,
the l e m m a is p r o v e d if
qr/(q+r) _> I.
O n the other hand,
1f •g|
cIr / (Q+r)
if qr/(q+r)
< I, let s
< i w h e n S q _< s < oo .
/ [f'gjds <
/ |f-g|
be chosen so that
Since
qr/(q+r)
ds
So
w e ha ve that the p r o d u c t f u nction is a memb e r of L
concludes
the p r o o f of the lemma*
I
'
f0-,po) .
This
APPENDIX 2
The p u r pose of this a p pendix is to present a m e t h o d of g e n e r a ­
ting a p p r o x i m a t e formulas for the zeros of a p o l y n o m i a l w h o s e
coefficients are functions of a real variable.
The m e t h o d is one
w h i c h m a y b e i t e r a t e d to achieve c l o s e r a p p r o x i m a t i o n s
at ea c h suc­
cessive stage o f the process.
Let P(I)
tinuous
= Xn + B 1 (s) Xn ^
I -
comple x - v a l u e d functions of a r e a l ■va r i a b l e s defi n e d on the
interval 0 _< s < oo .
w h e r e the a.
J
n-2
a X
Supp o s e that
I i m B ;(s) = a.., j=l ,
t -> oo -I
I
are c o m p lex- v a l u e d constants.
+...+ a
n
2
B (s) w h e r e the B . (s) are conn
]
= Xn +
and r e qui r e the zeros y , k=l , 2 , . . . , n ,
k
n o m i a l to be distinct.
r
U s i n g R o u c h e 's Theorem,
t h a t the zeros X ^ ( s ) , k = l , 2 , . . . ,n, of P(X)
of s .
L e t p(X)
2
n,
a Xn ^ +
'I
of this poly-
it can be v e r i f i e d
are c ontinuous
functions
By reorde r i n g the indices if neces s a r y , w e m a y assume that
Iim X1 (s) = ]t , j “ l,2,...,n.
s ->oo k
k
The rate at w h i c h each d i f f e r e n c e X^(s) is d e p e n d e n t u p o n h o w fast the d i f ferences
tive coefficients in p(X)
and P(X)
converges to zero
a. - B . (s) of the respec1
3
converge to zero.
case of an n > 5, it is u s u a l l y i m p o s s i b l e to expr e s s
citly as a function of th e c o e f ficients B ^ (s ) .
I n the gen e r a l
X^(s)
T herefore,
expli­
an exact
d e t e r m i n a t i o n of X (s) - y, , in terms of a. - B .(s) , j = l , 2 ,. . . ,n,
k
k
J
3
64
w i l l usually be out of the question. ' A s uitable ap p r o x i m a t i o n to
A^(s) m a y be readily obtained, however.
an a p proximate zero A^(s)
A^(s) - A^(s)
of P(A)
is integr a b l e on
To this end,
set A^(s)
is a zero of P(A),
such that the differ e n c e
[0,co).
=
+ a^.(s).
R e m e m b e r i n g that A^(s)
and d r o p p i n g the s u b s cript
h a v e P(y + a) = 0 .
degree n in a.
F or our purpose, w e seek
for convenience, we
Now, P(y + a) m a y b e v i ewed as a p o l y n o m i a l of
W r i t i n g it as such, w e o b t a i n
P (y) + P ’ (y)a +. . .+ a 11 = 0.
b e c o m e s infinite,
Since a(s)
converges to zero as s
this equa t i o n m a y be e x p r e s s e d as
P(y) + P
1(y)«[l
+ o(l)| = 0
(19)
w h e r e o(I)
designates a function w h i c h v a nishes as s. beco m e s infinite
infinite.
As s b e comes infinite
n—I
P
1(y)
= nyn
1
+
^
j=l
converges
Sq
s <
to p * (y) .
00
(s) (n-j)y 11 ^
W e m a y assume that P
and p T (y) ^
1
3
1(y )
, w h e n S q is s u f f i c i e n t l y large,
are distinct,
is n o n z e r o on
for the zeros of p ( A )
0.
W i t h the u n d e r s t a n d i n g that s
for a.
B
Sq , e q u a t i o n
This yields
(19) m a y be solved
^
a =
P(U)
P1(U)
[I + o(l)]
(20)
65
To m a k e the dependence of
explicit,
recall that p(p)
a upon t he differences a. - B .(s) m o r e
J
3
=
0,
and w r i t e equ a t i o n
« - _ P(p) ~ .P(h)
P f (y)
( P ’ (y))
-I
""
^
J
Since P f (y) = p'(p)[l + o(l)]j
alternative form of e q u a t i o n
„ _
F r o m equations
A(s) = y +
a(s)
(20) and
as
(21)
[I + O (I) ]
[a. - B
j=l
(19)
(s)]yn j
[I + o(I) ]
J
and I/[I + o ( l ) ] = Jl + o ( l ) J , an
(20 ) is
' P(y) [I + O(I)]
p'(y)
(22)
(229, it is e v i d e n t that an approxi m a t i o n to
is g i v e n b y either
y -
P(y)
P ' (y)
or
y
P(y)
p' (y)
T h e s econd f o r m w o u l d appear to b e p r e f e r a b l e to the first, beca u s e
its denominator is independ e n t of the var i a b l e s.
shall see,
However,
as we
the first f o r m has some advantages w h e n it comes to
d e t e r m i n i n g succes s i v e l y b e t t e r approx i m a t i o n s to A.
IiL
«
66
Let us examine how close the appr o x i m a t e zero
%(s)
is to the exact zero A(s)
- A(s)
= B(s)
F o r simplicity,
e(s) =
• P(U)
of P(A).
W e shall show that if a^ A(s)
M -
(23)
P'(w)
Set A(s) = p -
(s) is a m e m b e r of
P t(P)
+
(Hs;
[O,00) , then
is a m e m b e r of L r [0,oo), w h e r e r = mhx(l,q/2).
let
P(p) _ p(p) - P(p)
P'(p)
P'(p)
(P'W)
-I
Z [a, - B , ( s ) ] p n j ,
i=l
J
'J
Again, starting with P(A) = 0, and employing the same kind of
reasoning as before, we develop the succession of equations:
P(p) = 0
,
r
P(p+ e +B) = 0
PCy + e) + P 1(y + e)B [I + o(l) ] = 0
(24)
P (p + e) + P 1(p) B [I + o (I) ] = 0
P(p) + P t(p)
+ (l/2)P"(p)e2 +...+ En = -P1(P)B [I + o(l)J
U s i n g the d efinition of
(l/ 2) P " (p)s 2 +
B y supposition,
(IZB)P i
e, w e o b t a i n
m
(P)E 3
U
(25)
+ o(l)J
the a^ - B^ (s) are continuous on I 0 ,oo), converge to
zero as s becomes infinite,
L e m m a 2,
E n = -p'( p ) B
it follows
and are m e m b e r s of L^[0,°°) .
that the left h a n d side of e q u a t i o n
m e m b e r of L r [0 ,co) , w h e r e r = max(l,q/2).
Fr o m
(25)
As a special case,
is a
if
tj
67
q = .2, w e have shown that the d i f f e r e n c e 8(s) = A(s) - A ( s ) , w i t h
A (s) = y -
is integrable on 10 ,op) „
To continue the procedure, w e
could use e q u a t i o n
(24) n e x t to
define
6(s)
P(p.+ e)
P'(U + e)
,
where
e
P(u)
P'(h)
It can be v e r i f i e d that if a. - B .(s) is a m e m b e r of L q l0,oo) , then
A - p - e - S i s
a m e m b e r of L l0,oo) w h e r e r = max(l,q/3) ,
This
should suffice to indicate h o w t h e s e ap p r o x i m a t i o n s m a y be extended.
68
REFERENCES
1.
C o d d i n g t o n , E . A,-, and L e v i nson, N.
Theory
o f Ordinary Differ­
ential E q u a t i o n s . N e w York:
McG r a w - H i l l , 1955.
2.
C o p p e l , W . A.
S t a b ility arid A s y m p t o t i c B e h a v i o r of D i f f e r e n t i a l
E q u a t i o n s . Boston:
D.C. H e a t h and C o . , 1965.
3.
Ghizzetti, A.
"Un t e or e m a sul c o m p o r t a m e n t o asint o t i c o degli
e q u a z i o n i d i f ferenziali l i n e a r ! o m o g e n e e " , Rend; Mat.' e 'A p p l .
(5) 8 (1949), 28-42.
4.
Hinton,
D.B.
(ry^)/^
" A s y m p t o t i c B e h a v i o r of Solutions of
+ qy =
0",
J. D i f f. E q u a t i o n s 4 (1968),
590-596.
5.
Levinson, N.
"The A s y m p t o t i c N a t u r e of S o l utions of L i n e a r
Systems of D i f f e r e n t i a l Equa t i o n s " ,
D u k e Math. J ; 15 (1948),
111-126..
6.
Pfeiffer, G.W.
" A s y m p t o t i c S o l u tions of y ’ ’ 1+ q y T + ry = 0",
J. D i f f . E q u a tions 11 (1972) , 145-155.
7.
Pfeiffer, G.W.
"The O s c i l l a t o r y N a t u r e of the E q u a t i o n
y' 1 1+ qy* + r y <= 0",
J. D i f f . E q u a t i o n s 11 (1972), 138-144.