Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
989 Angelo B. Mingarelli !E
Ser. y
14~
, "%~
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
989 Angelo B. Mingarelli !E
Ser. y
14~
, "%~
Cat.
Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions ETHICS ETH-BIB
IIIIIIIlUUIIIIIIIIIIIIIIIIIII 00 ] 0 0 0 0 0 3 8 ] 261
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author Angelo B. Mingarelli Department of Mathematics, University of Ottawa 585 King-Edward Avenue, Ottawa, Ontario, Canada K1N 9B4
AMS Subject Classifications (1980): Primary: 45 J 05, 45 D 05, 47 A 99 Secondary: 34 B25, 34 C10, 39A10, 39A12, 47 B50 ISBN 3-540-12294-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12294-X Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the materia is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Quest'
opera
ai m i e i Oliviana
cari
~ umilmente genitori
e al m i o
dedicata
Giosafat
fratello
Marco
A.M.D.G.
e
PREFACE
The aim of these notes a d o p t e d by m a n y authors F.V. Atkinson, qualitative
W.T.
is to pursue a line of r e s e a r c h
(W. Feller,
Reid,
M.G.
among others)
Krein,
I.S.
in order to d e v e l o p a
and spectral theory of V o l t e r r a - S t i e l t j e s
e q u a t i o n s with
specific a p p l i c a t i o n s
ential and d i f f e r e n c e
equations
(comparison theorem,
general
setting.
of the second order.
of such e q u a t i o n s
separation
In c h a p t e r and,
theorem)
to this more
3,4,5,
certain g e n e r a l i z e d o r d i n a r y d i f f e r e n t i a l
theory
apply some aspects of
it to the study of the s p e c t r u m of the o p e r a t o r s
g e n e r a t e d by
expressions
associated
integral equations.
In order to make these notes dices have been added w h i c h m a i n text.
results of
2 we study the o s c i l l a t i o n
in C h a p t e r s
with the a b o v e - m e n t i o n e d
integral
to real o r d i n a r y differ-
We begin by an e x t e n s i o n of the c l a s s i c a l Sturm
Kac,
self-contained
include results
some appen-
fundamental
to the
Care has been taken to give due c r e d i t to those
r e s e a r c h e r s who have c o n t r i b u t e d
to the d e v e l o p m e n t of the theory
p r e s e n t e d h e r e i n - any o m i s s i o n s
or errors are the author's
sole
responsibility. I am g r e a t l y w h o s e hands
indebted to P r o f e s s o r F.V. A t k i n s o n at
I learned the s u b j e c t and I also take this o p p o r t u n i t y
to a c k n o w l e d g e w i t h thanks the a s s i s t a n c e of the N a t u r a l
Sciences
and E n g i n e e r i n g R e s e a r c h C o u n c i l of C a n a d a
finan-
cial support.
My sincere
thanks go to Mrs.
for c o n t i n u e d
Frances M i t c h e l l
VI
for her expert typing of the m a n u s c r i p t . Finally,
I am deeply g r a t e f u l
Jean for her c o n s t a n t e n c o u r a g e m e n t
to my wife Leslie
and p a t i e n c e
and I also
wish to thank P r o f e s s o r A. Dold for the p o s s i b i l i t y p u b l i s h the m a n u s c r i p t
to
in the Lecture Note series.
A n g e l o B. M i n g a r e l l i Ottawa, A p r i l
1980.
TABLE
OF
CONTENTS
Page INTRODUCTION
CHAPTER
..........................................
1 Introduction i.i.
CHAPTER
1
oo,o,o,,oooooooo°.oooo,oooo°oooo,°
Comparison Theorems for Differential Equations
i°2.
Separation
1.3.
The
Green's
Theorems
Stieltjes Integro..................
4
.....................
20
....................
25
Function
2 Introduction 2.1.
..................................
Non-Oscillation Criteria for Linear Volterra-Stieltjes Integral Equations
2.1A.
Applications
to
Differential
2.1B.
Applications
to
Difference
2.2.
Oscillation
2.2A.
Applications
to
Differential
2.2B.
Applications
to
Difference
2.3.
An Oscillation Theorem in t h e N o n l i n e a r Case ....................................
Addenda
CHAPTER
x
Criteria
28
...
29
..
52
....
60
Equations Equations
.................... Equations Equations
74 ..
80
....
82
.......................................
87 113
3 Introduction 3.1.
..................................
Generalized
Derivatives
.................
118 120
VIII
Page CHAPTER
3
(continued)
3.2.
Generalized Differential Expressions of the Second Order ........................
123
3.3.
The
129
3.4.
Applications
3.5.
Limit-Point
3.6.
J-Self-Adjointness of G e n e r a l i z e d Differential Operators ..................
156
Dirichlet Integrals Associated with Generalized Differential Expressions
....
180
for Three-Term .....................
183
3.7.
3.8.
CHAPTER
Weyl
and
Limit-Circle
Criteria
143
....
147
4
4.1.
4.2.
...................................
197
Sturm-Liouville Difference Equations with an I n d e f i n i t e W e i g h t - F u n c t i o n ...........
199
Sturm-Liouville Differential Equations with an Indefinite Weight-Function ......
212
5 Introduction 5.1.
5.2.
APPENDIX
.................
............................
Dirichlet Conditions Recurrence Relations
Introduction
CHAPTER
Classification
o
o
o
The Discrete Differential
o
o
o
o
,
o
o
o
°
o
o
o
o
o
o
o
o
o
,
o
o
o
o
°
o
o
o
.
.
o
.
Spectrum of Generalized Operators ..................
The Continuous Spectrum Differential Operators
of Generalized ..................
225
226
242
I I.l.
Functions
1.2.
The
1.3.
G e n e r a l T h e o r y of V o l t e r r a - S t i e l t j e s Integral Equations ......................
264
Construction
273
1.4.
of B o u n d e d
Variation
..........
256
Riemann-Stieltjes
Integral
..........
258
of
the
Green's
Function
....
IX
Page APPENDIX
II II.1.
APPENDIX
in
LP
and
Other
Spaces
..
280
III III.l.
Eigenvalues Equations
of G e n e r a l i z e d Differential ............................
296
....
299
Linear
Operators
in
III.3.
Linear
Operators
in a K r e i n
III.4.
Formally Self-Adjoint Even Order Differential Equations with an Indefinite Weight-Function ...........
Index
a Hilbert
Space
Space
292
..
III.2.
BIBLIOGRAPHY
Subject
Compactness
303
.........................................
309
........................................
318
INTRODUCTION
Let
p,q:
Lebesgue
measure)
Consider
the
I÷
]19, p(t)
and
i/p,
formally
a solution
of
- q(t)y
(i.e.,
py'c
AC(I)
and
Then
a quadrature
p ( t ) y ' (t) where
B =
(PY') (7).
o(t)
=
/t q ( s ) d s a
will
be
a solution
Stieltjes
Since
exists
where
the
sense.
whenever
o e BV(I)
is c o n t i n u o u s used
to d e a l
need
not (2)
discrete term
be
to b e
On
on
be
recurrence
(I)
y: I)
I ÷ C
such
I.
,
that
Let
y e I.
equation
of
and
difference as w e l l
in t h e
variation
on
equations
the
so
equations as
Riemann-
I)
and
y
(2) m a y
Moreover
require
(2) c a n
(2)
a meaning
form
(i).
as w e
a
t c I,
has
of
y
form
also
(as l o n g I)
,
Hence
satisfies
the
say,
(2)
equations
I
on
relations)
if y(t)
hand
integral
~ e AC(I).
only
bounded
differential on
indefinite
ft y ( s ) d o ( s ) Y
other
e.g.,
its
interpreted,
Hence
continuous
problems,
on
t e I and
8 +
the
continuous
on
a.e.
c L(I)
if a n d
(i.e.,
I.
with
=
may
Stieltjes
of
q
integro-differential
integral
c IR
t ~ I,
for
(i)
p ( t ) y ' (t)
[a,b3
t ~ + / y(s)q(s)ds Y
=
of
of
equation
a function
(I)
for
I =
sense
t c I.
continuous
satisfies
gives,
where
differential
mean
absolutely
y(t)
(in t h e
= 0,
(i) w e w i l l
y c AC(I),
a.e.
q c L(I)
symmetric
(p(t)y')' By
> 0
be
a
a solution
used
to
(or t h r e e -
continuous
be
problems
treat
XI
as w e h a v e 0 c o r r e s p o n d s
in g e n e r a l ,
c a s e of u n r e s t r i c t e d In t h e
former
expression
the latter
r(t)
to
~(t)
symmetric
in t h e w e i g h t e d
space,
signed measure.
t f y(s)do(s)}
(t) -
derivative
appearing
derivative.
non-decreasing
corresponds
case the operator
(Pontrjagin)
y,
a Radon-Nikodym
case the operator
is f o r m a l l y
restrictions)
{p(t)
(under
The
a n d the
by t h e d i f f e r e n t i a l suitable
space
is J - s y m m e t r i c
since the measure
on
to ~(t) e BV(I).
defined
Hilbert
(5)
domain
L 2 (I,d~).
In
in a K r e i n
induced
by ~(t)
is a
XIV
Expressions W.
Feller
o(t~
e
on
BY(I)
monotone,
form
(5) w e r e
[683,[693,[703,[713,[723,[733
- constant
function
of the
I, was
cf.,
on
I, p(t)
(cf.,
also
treated
- i, a n d Langer
by I.S.
[46,p.49].
in t h e
~ a given
[41]). Kac
first
considered case
by
when
non-decreasing
The more
[353,[36],[373
general when
case
~ is
CHAPTER
1
INTRODUCTION: In t h i s Stieltjes of
the
chapter
we
shall
integro-differential
study
equations;
defined valued
on a f i n i t e
> 0
Historical
what
the
of
is,
of
equations
functions
1836.
[a , b]
and
of b o u n d e d
p , o
are
variation
real
on
I
there.
comparison we call
scalar
first
(i.0.0)
y(s)do(s)
I =
and
the
separation
Sturmian
theorems
theory.
of
Sturm
Comparison
com-
theorems
equation
Ip(t)y'(t)l'
were
that
theory
Background:
The
for
interval
right-continuous
p(t)
prise
Sturmian
form
p ( t ) y ' (t) = c +
and
the
obtained In t h a t
by
paper
- q(t)y(t)
Sturm Sturm
[58,
p.
135]
considered
(i.o.i)
= 0
in h i s the
famous
memoir
equations
!
(K~y')
- G~y
= 0
(1.0.2)
2
(K2z')
on a f i n i t e G2 ~ Gi , then
interval equality
between
is at l e a s t result
any one
usually
proof
depended
coefficients to
K2
and
then
as the valid
he
[K2Yz
upon
'
- KlY
'
= 0
that
(1.0.3)
if
0 < K2 ~ K1 ,
everywhere
some
on the
solution
solution
of
of
the
introduction
GI
to the
him
as
location
varied.
This
Theorem.
the
parameter zeros
depended
is the Sturm's
from
was
of the
upon
there
in the
continuously
of the
It a l s o
(1.0.2)
of a p a r a m e t e r
to p a s s
G2 ,
interval,
(1.0.3).
as the Sturm-Comparison
allowed
studied
all
of
of a n y
known
parameter for
zeros
zero
from
showed
not holding
two
which
and
and
- G2z
the
KI
increased, solutions identity
tI , t2 • I ,
t2 z]tl
=
f t2 (G 2 tl
G1)YZ
-
dt + f t2 (K 2 - K l ) y ' z ' tI
dt (1.0.4)
which [13,
c a n be o b t a i n e d p.
case
comparison
[58,
of a t h r e e - t e r m equation
A discrete
analog
[21,
of S t u r m equations.
p.
p.
by
that first
recurrence though
of the
] whose
applied
186]
theorem
difference
Fort
of G r e e n ' s
theorem
291]. It s e e m s
of the
b y an a p p l i c a t i o n
the
to d i f f e r e n c e
having
relation latter
comparison
method
Sturm
came
shown or
result
theorem
of p r o o f
was,
equations
to the c o n c l u s i o n it t r u e
second was was
the
order
not
published.
published
in e s s e n c e ,
instead
for
by
that
of d i f f e r e n t i a ]
In 1909 Picone
[48, p. 18]
proof of the c o m p a r i s o n
theorem
gave by far the simplest
in the continuous
case.
He
made use of the formula
t2
[z
(K2YZ'
KlY'Z)
.t2
t 2
1 f tl
tl
+
dt t 2
It I
(G 2- G1)y2 dt
2
(1.o.5)
-
commonly allows
known
as the Picone
an immediate
[33, p. 226].
One important
a variational Q[y]
y e C 1 (a, b) termed
and
(1.0.2-3) = y(b) For such
Q[y]
acting
had the property
of
(1.0.3) would have
Swanson Q[y]
~ 0
= 0
that
Q[y]
to vanish
the solutions
y 7 0
(such functions
were
y ,
< 0
(1.0.6)
admissible
then every
at some point
Leighton's
reaching
in
functional"
df = fa% (K2Y ,2+ G2y2) dt
[59, p. 3] w e a k e n e d for
the theorem
on functions
The main result was that if some non-trivial y
theorem was
He made use of a "quadratic
y(a)
'admissible').
Theorem
of the c o m p a r i s o n
[42, p. 604] who interpreted
with
(1.0.5)
[74]).
extension
setting:
associated
The use of
proof of the Sturm C o m p a r i s o n
(cf., also
that of Leighton
Identity.
real solution in
(a, b)
condition
Q[y]
the same conclusion
were not constant m u l t i p l e s
of
function
y
.
< 0
provided
to
4
Sturm-Separation
The linearly
independent
separate
one
recurrence known
solutions
another.
relations
in the
of,
A similar and
latter
In s e c t i o n
theorem s t a t e s t h a t the z e r o s of
in f a c t
case. 1 we
say,
(1.0.2)
result
holds
a more
general
(See s e c t i o n
shall
give
an e x t e n s i o n
"Leighton-Swanson
Theorem"
equations
(i.0.0)
as c o r o l l a r i e s ,
continuous
and d i s c r e t e
In s e c t i o n Theorem
for
(i.0.0)
and
and
chapter
a study
with
problem of
the
§l.1
finding
functions Pi(t)
Pi(t)
functions
t e
are
ing a f i n i t e simplicity can
,
of b o u n d e d
> 0 ,
FOR
~i(t)
,
variation
[a, b]
,
number
be o m i t t e d ,
in m o s t We will,
We
(See
its
STIELTJES
over
on
theorems,
integral
corresponding theorem. Separation
this
for b o u n d a r y
for
the
to the solution
3).
INTEGRO-DIFFERENTIAL
,
be
and
real
valued
We
assume
that
there.
without assume
all
with
chapters
in g e n e r a l ,
of
application
[a , b]
following
afore-
to b o t h
[a, b]
i= 1 , 2 ,
of t h e
conclude
section
i : 1 , 2
is
Sturm
function and
result
comparison
of the
of d i s c o n t i n u i t i e s
In the
the
representation
problem.
right-continuous
only.
conclusions.)
Green's
three-term
class
applications
(i.0.0)
COMPARISON THEOREMS EQUATIONS : Let
some
an e x p l i c i t
non-homogeneous
of t h e
equations.
of t h e with
to t h e
a proof
give
difference
associated of
versions
2 we give
differential
problems
give,
for
or
21).
mentioned
and
interlace
(This
possessis for
hypothesis
affecting that
four
each
this
that
all
the these
functions
are
lim
exists
o(t)
continous
Consider
at
a ,b
as
t ÷
the
equations
pl(t)u'(t)
= c +
and
,
i
if
b = ~
,
then
t u(s)dol(s)
(i.i.o)
v(s)do2(s)
(l.i.l)
a
P 2 ( t ) v ' (t)
where
a solution of
by
u(t)
e AC[a,
each
point
b]
with
t c
DQ
=
f2
(I.i.0),
say,
pl(t)u'(t)
we mean
6 BV(a, b)
a function
satisfying
(i.i.0)
at
[a , b]
Associated functional
= c' +
Q[u]
with with
{u : u E A C [ a ,
the
pair
domain
b]
(i.i.0-i)
is
the
quadratic
DQ
, p2 u ' ~ B V ( a
b)
, u(a)
= u(b)
= 0} (i.i.2)
and where,
for
U
e DQ
Q[u]
We
can
Swanson
result.
THEOREM
i.i.0:
Let
Pi
now
,
=
state
' o_ 1
,
'i~
(P2u'2dt+
u2d°2)
and
an
extension
be
defined
prove
i= 1, 2
,
(1.1.3)
of
as
the
above
Leighton-
and
let
u E DQ
,
u ~ 0
,
be
such
that
Q[u]
Then multiple
of
(1.1.4)
every
solution
of
(i.i.i)
u(t)
vanishes
at
least
Proof:
Assume,
(a, b)
and
and
< 0
on
let
the
contrary,
a < s < t < b
once
that .
which
v
is n o t
in
(a , b)
does p 2 v '/v
Then
a constant
not
vanish
in
c B V I o c (a , b)
so
(1.1.5)
exists.
Case
i:
v(a)
~ 0 ,
For
u e DQ
S
u2 d
v(b)
~ 0
satisfying
=
s
u2
=
in p a s s i n g
(i.i.i).
dp2v,
+ p2v' dv -I
s
=
where
(1.1.4),
from
Integrating
dP2V'
(1.1.6)
+ u P2V
u2d~ 9 - 2
p2tT
(1.1.6)
to
(1.1.7)
(1.1.5)
by
parts
we
we
~
used
find
dt
the
that
(1.1.7)
equation
u2d
=
P2
-
s
Combining
(i.i.7),
(i.i.8)
and
i to b o t h
sides
2
fs
v vuu
P2
(i.i.8)
s
we
adding
t
,2 P2 u
s
obtain,
(P2 u ' 2 d t +
u2dd2)
=
EP2 v'
+ S
i
+
t
p2 u
,2
t
ruv, 2
P2[--v--/
S
dt
- 2
It
s
, , u p2 v u v
s 2 t
t
2
= [~ v U I + f ~{u S
=
[p2 v ' u ~ I t
UV~v
S
+ IS" p2 v 21vU--}' 2
S
(1.1.9)
for
a < s < t < b
obtain,
.
Hence
if w e
since
v(a)
let ,
Q[u]
The
hypothesis
on
u
s ÷ a+ v(b)
0
,
t ÷ b-0
in
(1.1.9)
~ 0 ,
(i.i.i0)
u fa~ P 2 V 2 {v) '~=> 0
=
we
implies
Q[u]
= 0
but
since
v 7 0
,
! f
we must [a, b]
~
have
lul
which
we
= 0
or
excluded.
that This
u
is a m u l t i p l e
contradiction
of
shows
v that
on v
must
vanish
Case
2:
at l e a s t
v(a) To
once
= v(b)
settle
in
(a, b)
= 0 .
this
case
it s u f f i c e s
to s h o w
that
in
(1.1.9),
u
2
(t)P2 (t)v' (t)
lim t÷b-0
= 0
(l.l.ll)
= 0
(i.i.12)
v (t)
and u
2
(s)P2
lim s÷a+0
It is p o s s i b l e problem
to s h o w
(i.i.i),
v(a)
See A p p e n d i x
I and
v' (a)
(The p r i m e
~ 0
derivative
which
is c o n t i n u o u s since
v(b)
lim s÷a+0
provided implies Thus
the that
solutions
= cI ,
P2(a)v'(a)
341].
here
point
to the
Thus
usually
(two-sided)
~ 0
latter
limit
exists.
it is c o n t i n u o u s
P2(t)
right-neighborhood
The
in s o m e
a right-
derivative
lim s+a+0
if
°2
[3, p.
2 uv(s) (s)
hypothesis
348],
on
(1.1.13)
°2
right-neighborhood
is c o n t i n u o u s
in some,
Hence
unique:
Hence
in s u c h
a
value
= 0 ,
Similarly
is c o n t i n u o u s
of
are
v(a)
represents
in q u e s t i o n . )
P2(b)v'(b)
initial
= c2
since
(s)P2(S)V' (s) v(s) = P2(a)v' (a)
P2(t)v'(t)
Similarly
that
is an o r d i n a r y
= 0 , 2
v(s)
[3, p.
at the
u
s)v' (s)
v' (t)
of
a neighborhood. possibly
different,
is c o n t i n u o u s
(i.e.
a.
is
an o r d i n a r y
(a, a +
6)
,
In ordinary (a , a + we
can
derivative)
the
,
same
it c a n
be
shown
in
(a, a +
Q)
,
theorem
that
q > 0
2 u ( t ) u ' (t)
L'Hopital's
u' (t)
.
Thus
Since
to
the
u,
limit
v
is a n in
e AC[a,
in t h e
b] ,
right
of
to obtain
2 u (s) v(s)
lim s÷a+0
_
since,
as w e
exists
and
saw
is
above,
we
0
v' (a)
it c a n
(i.i.ii),
(1.1.9)
v' (s)
~ 0
Hence
the
limit
(1.1.12)
zero.
Similarly Combining
2 u ( s ) u ' (s)
lira s÷a+0
=
in
right-neighborhood
way
(u2(t)) ' =
apply
(1.1.13)
some
6 > 0
derivative
q)
in
be
shown
(1.1.12)
obtain
(I.i.i0)
v(a)
= 0
v(b)
This
case
and
that
(i.i.ii)
letting
again
and
holds.
s ÷ a+ thus
0
,
derive
t ÷ ba contra-
diction.
Case
3:
combination and
Associated with
domain
z 0
is e a s i l y
of C a s e s
(1.1.11-12).
Q' [u]
,
1 and
This
with DQ,
or
v(a)
disposed
2 leading
proves
the
(i.i.0)
is
of to
~ 0
,
v(b)
as
it
is
(i.i.i0)
= 0
simply via
a
(1.1.9)
theorem.
the
quadratic
functional
0
10 DQ, = {u : u e A C [ a , b]
, pl u' e B V ( a , b)
, u(a)
= u(b)
= 0}
(1.1.14) and Q' [u] = lab (PlU'2dt + u2dgl )
i.i.0:
COROLLARY
Let u(a)
=
u(b)
u =
(1.1.15)
(Swanson [59, p. 4], L e i g h t o n Cor. i]). be a n o n - t r i v i a l
solution
of
[42, p. 605,
(i.I.0) w i t h
0
Then every s o l u t i o n constant multiple
of
u
v(t)
of
(i.i.i) w h i c h
is not a
must v a n i s h at least once in
(a , b)
provided b ~ {(Pl - P2 )u'2dt + u2d(~l - ~2 ) } ~ 0
Proof:
Let
u
be a s o l u t i o n
of
(i.i.0),
u(a)
(1.1.16)
= u(b)
= 0
Then
ud(PlU')
Using the e q u a t i o n
=
b
[uPlU'] a -
(1.1.0)
plu'
in the l e f t - s i d e
2
dt .
of
(i.i.17)
(1.1.17)
we
find that Q' [u] =
( P l U ' 2 d t + u2dOl )
[UplU,l b a =
o
.
(1.1.18)
11
(1.i.16)
now
says
Q' [u]
that
- Q[u]
or,
> 0
because
of
(1.1.18),
Q[u]
Since
u
applies
is n o t and
Swanson's
a constant
hence
v(t)
extension
obtained
by
[59,
0
qi ,
E C[a, ql(t)
b]
Theorem) i = 1,
,
2
and
suppose
that
> q2 (t)
If
(plu')'
and
u(a)
for w h i c h which
= u(b) v(c)
is n o t
= 0 = 0
- qlu
= 0
(i.1.21)
(P2V') ' - q 2 v
= 0
(1.1.22)
,
is
then
whenever
a constant
there v
multiple
is of
at
least
a solution u
one of
c
E
(a , b)
(1.1.22)
12
Proof:
Let
o
follows
from
Corollary
1
(t)
is n o n - d e c r e a s i n g We
now
recurrence
a fixed
defined I.i.0
on
as
on
[a, b]
interpret
relation.
t_l
be
be
the
above for
of
the
.-.
interval
Let
be
an arbitrary
b 0 , b I , ..., bm_ 1
for
n=
continuous at
p(t)
= Cn_l(t n-
0 , 1 , 2 ....
the
function
now
- o2(t)
hypothesis. a three-term
[a , b]
a given
p(t)
result
o1(t)
< t m _ I < tm
be
a function
The
Let
c_l , c O , c I , ... , C m _ 1
define
that
results
= a < tO < tI
r = n
holding
> 0
and
for
every
A(Cn_iAUn_
bn n
=> q n .
I) - b n u n
for
If
=
U_l
0
n=
0 , 1 , ....
= um
0
m-i
and
(1.1.51)
,
19
then
there
is at
least
one
node
A(rn-iAVn-l)
in
of
- qn Vn
(1.1.52)
= 0
(a , b)
REMARK:
We lent of
to
note
u(a)
that
the
= u(b)
= 0
when
U_l
= um = 0
is c o n s i d e r e d
u
is e q u i v a a solution
(i.I.0). By
"polygonal the
a node
we mean
a point
curve"
defined
by
the
on
the abscissa where
finite
sequence
vn
the crosses
axis.
Proof:
The
implies
that
we
from
find
condition
c
Pl (t)
> P2 (t)
(1.1.45-6)
ol(tn)
Since n
condition
,
o I , 02 (1.1.53)
for
t e
that
Corollary
(i.i.i) to t h e
[a , b]
has
at
required
Moreover,
> 0
$ ol(t n-
0)
step-functions
implies
with
(1.1.47-8)
since
bn = > qn
that
- o 2 ( t n)
are
along
> r > 0 n = n
that
This,
of(t)
along
is a p p l i c a b l e
least
one
zero
conclusion.
on
in
the and
(a, b)
0) .
(tn_ 1 , t n)
- o2(t)
with
i.i.0
- o2(tn-
(1.1.53)
for
each
is n o n - d e c r e a s i n g
above hence
Remark, the
which
shows
equation
is e q u i v a l e n t
20
Note:
In g e n e r a l ,
a comparison
theorem
for
equations
of
the
- bn Yn
= 0
(1.1.54)
r n Z n + l + r n - i Z n - i - q n Zn = 0
(1.1.55)
form
Cn Yn+l + Cn-i Yn-i
under
For
the
assumptions
example
let
rn = qn = c/2 that
in t h i s will
§1.2
on
solutions, case
for
known at t h e
to
this the as
of
If
n=
but
has
large
section
we
separation
all
n
no
,
is n o t
and
b
available.
n
nodes
= 3c We
0 , 1 , ... , m - i
a simple
computation
,
see
shows
then that,
eventually
while
(1.1.55)
classical
Sturm
separation
n
> 0
prove of
p.
186]
the
zeros
of
differences
[58,
of his
n
for ,
b n => q n
,
THEOREMS:
finite
c
n
a consequence
Sturm
end
each
b n > qn
SEPARATION
theorem,
= c > 0
(1.1.54)
nodes
In
the
,
case,
have
n
for
> rn
cn
c
C n => r n
the
one
linearly
results
this
as
of
result can
independent
in
section
was
also
gather
from
i.
In
probably the
remarks
memoir. for
all
n
,
then
the
nodes
of
solutions
of
Cn Y n + l + C n - i
Yn-i - bn Yn
= 0
(1.2.0)
21
separate proof
The case
one
another
of t h i s
Sturm
result
separation
of a g e n e r a l
if t h e s e will
are
follow
theorem
linearly below.)
i~ not v a l i d
three-term
recurrence
[6, p.
solutions
of
176]
points
(1.2.1)
out
holds
P
for all false,
n
in the
in g e n e r a l ,
an e x a m p l e The
nodes
the
initial
Y0 = 6
do not One
if
note
that
the
condition
p(t)
> 0
for
the
n
Pn
p(t)
> 0
then
property
for
(1.2.2)
The
fails. =
= 0 ,
one
result
He g i v e s
1 ,
Qn
=
Rn
is h o w e v e r [6, p.
-i
=
solutions
Y0 = 1
and
177]
for
all
as n .
corresponding Y-I
= -i0
,
another.
separation given
(1.2.2)
property
of
by M o u l t o n
[45,
is the
analog
+ r(t)y
= 0
(1.2.1) p.
of the
under
137].
We
condition
equation
p(t) y" + q(t)y'
If
separation
> 0
independent
was
(1.2.1)
= 0
the
considered.
Y-I
of the
(1.2.2)
R
(1.2.2)
separate
proof
the h y p o t h e s i s
n
linearly
values
in the
if
case where
of the
to the
range
that
in g e n e r a l
relation
Pn Y n + l + Qn Yn + Rn Y n - i
Bocher
(Th e
independent.
the
zeros
of
linearly
.
independent
(1.2.3)
solutions
22
of
(1.2.3)
that
separate
(1.2.3)
can
one
then
another.
(One w a y
be t r a n s f o r m e d
into
of
seeing
this
an e q u a t i o n
is
of the
form
(P(t)y' 1 ' + Q ( t ) y
where
P(t)
property
THEOREM
of
> 0
and
the
the
zeros
result
of
separate
zeros
one
generate and
say,
find stant
the
separation
of
linearly
u
that
has the
two
solutions
= c +
y(s)d~(s)
linearly
independent
solution
Pl = P2 when
v
multiple
must of
u
Sturm
in
space
(1.1.16)
vanishes
vanish
of
(1.2.5)
[3, p. we
in b e t w e e n
348].
can
at t w o
solutions
apply
If w e
v
now
Corollary
consecutive since
u ,v
points
is n o t
set i.i.0 to
a con-
u
In p a r t i c u l a r the c l a s s i c a l
independent
another.
(1.2.5)
°l = °2 to,
from
(1.2.4).)
p(t)y'(t)
which
follows
1.2.0: The
Proof:
(1.2.4)
= 0
if
o E C' (a , b)
separation
theorem.
we
immediately
obtain
23
COROLLARY
1.2.0: If
then
the
o c C'(a,
zeros
of
b)
and
linearly
o'(t)
independent
(p(t)y'] ' - q(t)y
separate
each
other.
Porter
[49,
solutions the
of
limiting
differential
p.
= q(t)
55]
showed
(1.2.0)
generate
process
which
takes
t e
solutions
[a, b]
of
(1.2.6)
= 0
that
the
,
two
solution
linearly space
a difference
independent
and
considered
equation
to a
equation.
Defining
o , p
as
in
(1.1.24-25)
we
obtain
the
discrete
analog
COROLLARY
1.2.1: If
n=
c
n
> 0
,
0 , 1 , ... , m - I
n=
-i,
is a n y
0,
sequence
A ( C n _ i A Y n _ I)
then
the
nodes
of
..., m-I
linearly
and
b
,
n
and
(1.2.7)
- bnY n = 0
independent
solutions
separate
one
another.
As recurrence
an
application
relation
of C o r o l l a r y
(1.2.1)
we
state
1.2.1 the
to
the
following
[45, p.
137].
24
COROLLARY
1.2.2:
Let
P n Yn+l
for
n=
be
Pn ' Qn ' Rn
real
finite
sequences
+ Qn Yn + R n Yn-i
=
and
0
(1.2.8)
, m-i
(1.2.9)
0 , 1 , ... , m - 1
If
P
then
the
nodes
separate
each
n
idea
(1.2.9)
can
be
linearly
is
to
brought
C_l
(1.2.9)
implies
C_l
.
If
>
0,
1,
independent
now
...,
that
into
the
and
that we
show
solutions
(1.2.8) form
consider
Cn_ I ~
bn
n=
0 , 1 ....
of
(1.2.8)
under
(1.2.7)
the after
hypothesis which
we
1.2.1.
0
P n R n
Cn
for
n=
Corollary
Let
> 0
0
other.
The
apply
>
n
of
Proof:
simply
R
c
n
n=
> 0
the
recurrence
relation
0 , 1 , ... , m - i
for
n=
0 , 1 , ....
(1.2.10)
m-i
since
set
=
m-i
-c n
,
-
Cn_ 1
then
a
Cn Qn p n simple
(1.2.11)
computation
shows
that
25
with
the
substitutions
three-term
recurrence
(1.2.10-11), relation
(1.2.7)
(1.2.8).
reduces
Hence
the
to t h e result
followsl
§1.3.
The
GREEN'S
FUNCTION:
In A p p e n d i x tence
of a G r e e n ' s
~(t)
I to t h i s function
= e + B
work
for
we
the
~do
= u2~
=
shown
inhomogeneous
--+
u1~
have
the
exis-
problem
ds +
(1.3.0)
-P
(1.3.1)
0
where
Ui ~ =
2 ~ j=l
i=l,
IMij ~ (j-l) (a) + Nij p ( b ) ~ (j-l) (b)}
2 , (1.3.2)
and the that
M
(1.3.1)
homogeneous the
with
on
~' = q
.
constants,
with
If
(1.3.0)
In t h i s
is c o n t i n u o u s
case
and
p(t) with the
the b o u n d a r y we mean
boundary in
is of the
the h y p o t h e s i s
and
(By t h i s
f = 0
and
under
f = 0)
homogeneous
equation
then
(with
incompatible.
~ • C ' ( a , b)
[a, b]
(i.0.0)
is
solution.)
integral
If
real
problem
equation
zero
resulting
ous
are
the h o m o g e n e o u s
conditions
only
, N.. 13
l]
conditions
(1.21.0)
form
then
reduces
"derivative"
the G r e e n ' s
the has
the
(1.0.0).
is p o s i t i v e f = 0
that
function
and
continu-
to
(1.2.6)
appearing reduces
in to the
26
usual
one.
(See A p p e n d i x
On then
the o t h e r
(1.3.0)
with
f = 0
recurrence
relations.
difference
equations
seems
to h a v e
Another
been
t • same
p(t)
We
in A p p e n d i x
then
given
the
order
constructed
was
showed
case
of h i g h e r
first
,
o(t)
c a n be m a d e
In t h i s
unique
step-functions
to i n c l u d e
and, the
more
Green's
[3, p.
if
three-te@m
generally,
by Bocher
by A t k i n s o n I that
are
function [5, p.
of
83].
148].
(1.3.0-1)
solution
for
with
(1.3.0-1)
f = 0 is
by
~(x)
for
if
treatment
is i n c o m p a t i b l e given
hand
I, p. 278 .)
x •
[a , b]
[a, b] points
=
In the
and
f(t)
where
G ( x , t)df(t)
particular
as usual,
o(t)
the
case when
is a s t e p - f u n c t i o n has
its
fi = f(ti)
where,
(1.3.3)
t.
jumps
with
and
p(t)
= 1 ,
jumps
at t h e
if w e
denote
- f(ti-0)
represent
the
by
(1.3.4)
jump
points
of
f ,
1
then
a simple
computation
~n
- ~(tn)
shows
=
that
G(tn,
t)df(t)
m-i i=0
G ( t n ' ti) • If(t i) - f(t i - 0)) (1.3.5)
27
and if w e w r i t e
Gni
-= G ( t n , t i)
,
0 =< n ,
i =< m - 1
we
find
that
m-.l ~ G .f. i=O nl i
~n =
This
~n
then r e p r e s e n t s
inhomogeneous derived
difference
directly
for e x a m p l e see A p p e n d i x
149]
I, s e c t i o n
We note
the s o l u t i o n boundary
using methods
[3, p.
and
problem.
of f i n i t e
[5, p.
p(t)
of b o u n d e d
variation
appearing
in
is c o n t i n u o u s
Green's
I, the d i s c o n t i n u i t y function
(1.3.6)
For
is
(See
further details
on
are c o n t i n u o u s
[a , b]
t h e n the d e r i v a t i v e
everywhere
in the
and so,
first derivative
from of the
is g i v e n by
Gx(t+
which
Usually
differences.
84].)
, ~(t)
functions
Appendix
to the c o r r e s p o n d i n g
1.4.
that when
(i.0.0)
(1.3.6)
0 , t) - G x ( t -
is the u s u a l m e a s u r e
function
associated
equation
of the f o r m
with
0 , t)
p(t)
of d i s c o n t i n u i t y
a second-order
(1.2.6).
(1.3.7)
of the G r e e n ' s
linear
differential
CHAPTER
2
INTRODUCTION: There subject order
is a v e r y
of o s c i l l a t i o n
differential
[59]).
On
hand
establishing
criteria
behaviour
solutions
particular c a n be p.
found
425].
[12],
case in
[23,
Other
of
the o s c i l l a t o r y
of d i f f e r e n c e
pp.
known
less
second for e x a m p l e ,
non-oscillatory In t h e
relations
and more
some
recently
are m o r e
or
chapter
we
be c o n c e r n e d
the
about
equations.
recurrence
126-128]
and
results
oscillation
be n o t e d
that
y"
will
oscillatory
and
shall
non-oscillation
Stieltjes
the p o t e n t i a l
which
little
(see,
with
in
scattered:
results [32,
[21],
[20] .
non-linear will
is
dealing
of r e a l
on a half-axis
there
for
literature
non-oscillation
of t h r e e - t e r m
In t h i s some
and
equations
the o t h e r
of
extensive
integral
equations
if o n e m a k e s q
for
obtaining linear
on a h a l f - a x i s .
an h y p o t h e s i s
o n the
and It
integral
in
- q(t)y
guarantee
criteria
with
the
solutions,
= 0
existence
then
t e
[a, ~)
(2.0.0)
of o s c i l l a t o r y
a certain
discrete
or n o n -
analog
will
29
exist
for a t h r e e - t e r m In s e c t i o n
Stieltjes order
integral
difference
a result
sufficient
1 we
give
equations
equations.
o n the o s c i l l a t o r y extend
recurrence
condition
some and
their
75]
2 we
and
guarantees
state all
solutions
of a n o n - l i n e a r
equation
corollary
we
the
discrete
analog
Various
examples
are
theorem should
§2.1
shall
[2, p. help
obtain
643].
visualize
the
theorems
NON-OSCILLATION CRITERIA INTEGRAL EQUATIONS: In the
equations
following,
of t h e
we
give
and
that
are
criteria
applications
of s o l u t i o n s
[8, p.
which
non-oscillation
In s e c t i o n
behaviour
of B u t l e r
relation.
to some
for
second results
in s e c t i o n
3 we
a necessary
and
continuable
oscillatory.
As
a
of A t k i n s o n ' s
included
which
stated.
FOR LINEAR
shall
VOLTERRA-STIELTJES
usually
be c o n s i d e r i n g
form
y' (t) = c +
i
t t e
y(s)do(s)
[a , ~)
,
(2.1.0)
a
where
a
variation
is a r i g h t - c o n t i n u o u s
function
on
of
[a , ~)
assume,
in a d d i t i o n ,
remains
finite
can
also
Because that
in f i n i t e
be e x t e n d e d
the
the
number
intervals.
to e q u a t i o n s
p ( t ) y ' (t) = c +
of
locally
of b o u n d e d
applications
we
shall
of d i s c o n t i n u i t i e s The
theorems
the
form
y(s)do(s)
of
proved
here
(2.1.1)
30
in the
case when
p(t)
> 0 ,
(2.1.2) a
p
satisfying
every
the u s u a l
equation
(2.1.2), (2.1.0)
of
can be by
the
conditions
form
will
equation
[a , ~)
A
solution
to the r i g h t
zeros when
then
all
of
the
form
variable
~k
(2.1.3)
P
[0 , ~)
ISee A p p e n d i x
I,
of of
if
(2.1.0) a
,
there
is
said
an i n f i n i t e is
some
oscillatory
to b e
number
tO ~ ~
of
such
zeros
that
if it and
is
it has
no
t a tO
From that
into
satisfies
an equation
=
For
2.1.1:
non-oscillatory
see
T(t)
p
1.
(I.3.14) .1
DEFINITION
has,
take
into
in C h a p t e r
where
of i n d e p e n d e n t
t~-~
which
stated
(2.1.1),
transformed
the c h a n g e
p
the
if o n e
Sturm
solution
solutions Equation
oscillatory)
separation
are
is o s c i l l a t o r y
oscillatory
(2.1.0)
if all o f
theorem,
is
its
said
Theorem
1.2.0,
we
(non-oscillatory)
(non-oscillatory).
to be
solutions
oscillatory (non-
are
oscillatory (non-
oscillatory). Unless
otherwise
stated
we
shall,
in
the
following,
31
assume
that
O(t)
appearing
,
in
(2.1.0),
has
a limit
at
~
,
i.e. l i m o(t) t+~
exists
and
assume
i t is
has
the
(2.1.0)
is
same
zero
(for if w e
result
of H i l l e
tory behaviour
[31,
of
non-linear
THEOREM
2.1.1:
and that
~
the
condition
a solution,
~
for
= o(t)
T(~)
which
~(~)
we
- ~(~)
= 0
can
then
T
Moreover,
is r e p l a c e d
by
T )
of a w e l l - k n o w n
relates
the n o n - o s c i l l a of
solutions
of
a
equation.
and
with
~(~)
(2.1.0)
To
show
(2.1.4)
has
that
+
sufficiently
Proof:
(cf.,
the
a solution that
= o(t)
to the e x i s t e n c e
for
by
locally = 0
of b o u n d e d Then
a necessary
to b e n o n - o s c i l l a t o r y
is
equation
at infinity
implies
and
243]
(2.1.4)
integrable
then
T(t)
be r i g h t - c o n t i n u o u s
integral
limit
is an e x t e n s i o n
p.
v(t)
have
~
integral
satisfying
sufficient
this
let
if
(2.1.0)
certain
variation
as
unchanged
first
Let
Denoting
properties
remains The
theorem
finite.
(2.1.4)
v(t)
(2.1.5a)
v2(s)ds
large
t
,
which
is s q u a r e
[80]).
condition
is
v E L 2 (t O , ~)
sufficient ,
some
is r i g h t - c o n t i n u o u s ,
assume
tO locally
that
(2.1.4) of
32
bounded
variation
and
v(~)
= 0
Put
y(t)
Then
y(t)
is
locally
y' (t)
= exp
v(s)ds
absolutely
= v(t) e x p
(2.1.5b)
continuous
i
and
so
t v(s)ds
(2.1.6
to
everywhere,
as
jump
of
points
Letting
h
> 0
a two-sided v(t) ,
t
derivative,
which
are
the
except
same
as
possibly
those
of
the ~(t)
arbitrary,
exp y (t+h)-y(t) h
= y(t)
v(s) ds
-
[Jt
•
(2.1.7 h
Now
,!exr ){f [
~t
for
each
h
use
Theorem
> 0
v-i
,
= ~
fixed
t
H of Appendix
I to
i I t+h lira ~ h÷0 +
.
~t
v+2![3
Hence
we
find
that
v(s)ds
t
can
= v(t)
v
let
+ "'"
}
h ÷ 0+
(2.1.8
and
(2.1.9)
33
while
the o t h e r
continuity Hence
of
letting
terms
the
are
zero by v i r t u e
h ÷ 0+
the d e r i v a t i v e
derivative
which
(2.1.9)
and
the
integrals. in
(2.1~7)
y' (t)
where
of
is
we
obtain,
from
above,
= y(t) v(t)
is in g e n e r a l locally
(2.1.10)
understood
of b o u n d e d
as a r i g h t -
variation.
Thus
if
t > tO
y' (t)
- y' (to)
=
i
t dy' (s)
to
= ft 0
=
d(y(s)v(s) )
ft
vdy
+
ft
ydv
to
=
where
we h a v e
by
0
v dy +
y do -
to
0
equation
(2.1.10),
=
(vdy-
It 0
vy'ds)
+
ydo
0 Theorem (2.1.11)
K of A p p e n d i x vanishes
for
.
(2.1.11)
0
I now all
yv 2
implies t
that
and h e n c e
the
first
integral
in
34
y' (t) = y' (to)
it y d o
+
t > t =
0
to
so t h a t
y(t)
equation. for
is a p o s i t i v e
This
t ~ tO
and hence
To p r o v e non-oscillatory positive
for
For
implies
that
v(t)
(2.1.0)
of
the
has
above
a positive
the n e c e s s i t y solution
y(t)
we
suppose which
we
that
solution
can
(2.1.0) suppose
has
a
is
t ~ to
t => t O
we
set
is l o c a l l y
=
y' (t) y(t)
of b o u n d e d
(2.1.12)
variation
on
[t O , ~)
is r i g h t - c o n t i n u o u s .
Hence,
integral
is n o n - o s c i l l a t o r y .
v(t)
Then
solution
for
t > tO
v(t)
- v ( t 0) =
dv(s) 0
=
y(s) 0
dy'
(s)
ds
-
to
and
35
ft =
do(s)
-
v 2(s)ds
to
0
Hence
v(t)
= o(t)
- o(t0)
+ v(t0)
-
v
2
(2.1.13)
0
for
t ~ tO
that
the
Suppose,
Since
same
must
a(t)
be
true
if p o s s i b l e ,
square-integrable
has of
that
at
~
there
y' (t) is
a
< 0 t2
for such
v(~)
and
v(t2)
If w e
let
(2.1.13)
t , tO
(2.1.13)
shows
cannot
of
T => t 2
then by
T
be
that
(2.1.14)
because
- d(t2)
replaced
implies
v
= -~
when
+ a(T)
Then
(2.1.13)
t 3 = m a x { t 0 , t I , t2} with
~
= ~ ~ 0
so
t ~ tI that
at
v(t)
l i m v(t) t+~
Hence
a limit
(2.1.12).
Moreover
,
< -i
(2.1.15)
.
using , t3
(2.1.15)
in
respectively,
we
obtain
v(T)
< -i
+ ItT
y'(s) y(s)
v(s) ds
(2.1.16)
3 whenever Exercise
T => t 3 . i]
in
We
(2.1.16)
now
use
Gronwall's
to o b t a i n
inequality
[9, p.
37,
36
v(T)
t =
0
Hence
oo (Tx n) (t)
= 0 2 (t)
+ I
x 2 ds
t
n
~oo ÷ ~2 (t)
+
I x 2 ds t
df = (Tx) (t)
=
This
implies
n ÷ ~
that
ITx n - Txl 2 ÷ 0
,
each
0
t => t O ,
as
Moreover
I (Tx n) (t) - (Tx)(t) I 2 < 4v21
whenever
(2.1.29)
t > t
t ~ tO
Thus
the
Lebesgue
dominated
(2.1.30)
convergence
40
theorem
[24,
p.
ll0]
I
implies
- Txl 2 ÷ 0
~ITx
t
for
t
> t =
of
as
Hence
T
is
show
Appendix
II.
~ VI
and
that
TX
is
compact
(II.l.4)
is
satisfied
choose
EA
Corollary
since
if
x
II.1.2
e X
=
Vl 2
2.1.32
{t :
tO
-< A
-< t
large
< ~}
so
then
given
6
> 0
will
then
imply
,
we
that
2 vI < E
This
,
tO
sufficiently
A
use
,oo
tO
let
we
so
oo
we
(2.1.31)
continuous.
f Txt2< ] If
n ÷ ~
n
0
To
ITXl
that
(2.1.33)
that
oo
~I for To
all
x
prove
Since
c X
by
virtue
(II.i.6-7) vi
is
a
we
2
V 1 (t)
of
need
solution
(2.1.34)
I~ 1(t) 12
t __> t o
(2.1.35)
41
and
so
2 01
e L
(t O , ~)
By
the
same
argument,
v I (t)
and
=>
t > t = 0
vI
so
~oo
VI e L
(2.1.36)
(t O , ~)
t
The
following
If
f
theorem
L P [ t 0 , ~)
[24,
,
p ~
p.
1
,
] will
also
be
useful.
then
llf(x+~)-f(x)ll
÷
as
o
h ÷ 0
(2.1.37)
P
Since
o I e L2[t0
(~2 c L 2 [t O , ~)
, ~)
and
account
of
have
from
(2.1.19)
that
thus
Ilo2(t+h)-
on
we
o 2 ( t ) ll + 0
as
h +
0
(2.1.38)
(2.1.37).
Similarly
if we
set
oo
V(t)
:
i
v~ t
then llV(t+h)
-V(t)
ll ÷ 0
as
h ÷ 0
(2.1.39)
42 because
of
Thus if
(2.1.36).
x ~ X , e > 0
to+h
to+h (2.1.40)
to
if
lhl < 6 ,
=
to
by the c o n t i n u i t y
of the integral.
This proves
(II.l.6). For
x e X ,
e > 0
II(Tx) ( t + h )
- (Tx)(t)ll = lla2(t+h)
- ~ 2 ( t ) + it x2 as II t+h (2.1.41)
+ II it
=< 11~2(t+h)-~2(t)ll
t+h
From
(2.1.38)
we can choose
h
so that if
lla2(t+h )_~2(t)l I < !2 Similarly
there
is a
62 > 0
~211 lhl < 61
"
such that w h e n e v e r
llV(t+h) - V ( t ) ll < _g 2 "
then
(2.1.42)
lh] < 62
(2.1.43)
Thus
t 2 II IIitt+h x211 la 2(t) l
t__> t o
(2.1.45)
If
v l(t)
has
a solution
for
= a l(t)
t ~ tO
v(t)
(2.1.46)
+ It~v I 2 ds
then
= ± °2 (t) +
(2.1.47)
v 2 ds t
has
a solution
Proof:
This
THEOREM
2.1.3:
With
for
follows
a l ' °2
t ~ tO
immediately
as a b o v e
from
and
the
theorem.
44
a i(t) suppose
__> I~ 2(t) I
(2.1.48)
t__> t o ,
that
f
y' (t) = c i +
t y ( s ) d a i(s)
(2.1.49)
z(s)da2(s)
(2ol.50)
a
is n o n - o s c i l l a t o r y .
Then
z'(t)
= c 2 -+
f
t
a
is n o n - o s c i l l a t o r y .
Proof:
This
is i m m e d i a t e
from Corollary
2.1.2
and
Theorem
2.1.1.
THEOREM
2.1.4: Let
~ (t)
satisfy
the
conditions
of T h e o r e m
2.1.1.
If
1 tJa(t) J < ~
then
(2.1.0)
is n o n - o s c i l l a t o r y .
Proof:
Let
Theorem
2.1.3.
equivalent
(2.1.51)
t > to > 0
ai(t)
= 1/4 t
This
and
o2(t)
is p e r m i s s i b l e
=- o(t)
since
and
(2.1.49)
apply is
then
to
y"
1 + y = 0 4t 2
(2.1.52)
4~
which
is a n o n - o s c i l l a t o r y
result
now
COROLLARY
Euler
equation
[59, p.
45].
follows.
2.1.3: (2.1.0)
is n o n - o s c i l l a t o r y
if
l i m sup tla(t) I < ! 4 " t÷+~
Proof: t
This
is i m m e d i a t e
is s u f f i c i e n t l y
THEOREM
The
since
(2.1.53)
(2 1.53)
implies
(2.1.51)
if
large.
2.1.5: Let
o I , 02
be as in T h e o r e m
of(t)
>
la2(t) I
If
(2.1.49)
is n o n - o s c i l l a t o r y
of
(2.1.49)
there
corresponds
2.1.2
and
t > tO
then
(2.1.54)
to e v e r y
a solution
solution
z(t)
of
y(t)
(2.1.50)
such t h a t
z(t)
Proof:
We
account
of T h e o r e m
If
y(t)
either
first note
_ 0
(2.1.55)
is n o n - o s c i l l a t o r y
on
2.1.3.
is a n o n - o s c i l l a t o r y y(t)
t > t*
or
y(t)
< 0
solution for
of
t > tI
(2.1.49) If
then y(t)
> 0 ,
46
Theorem
2.1.1
implies
that
t ~ t* = m a x { t 0 , tl} for
t ~ t*
Hence
(because
2.1.2
z(t)
> 0
guarantees
w e can r e c o v e r
solution for
has
(2.1.47)
of C o r o l l a r y
some non-oscillatory suppose
(2.1.46)
vl(t)
has a s o l u t i o n
2.1.2)
z(t)
t ~ t*
a solution
which
v(t)
corresponds
of
(2.1.50).
Since
the p r o o f
for
to
We can of T h e o r e m
that
Iv(t) I < vl(t)
t > t*
(2.1.56)
the n o n - o s c i l l a t o r y
solutions
y ,z
to find
that
z(t)
If
z(t)
< 0
other hand the a b o v e
for
if
_< y(t)
t > t*
y(t)
argument
< 0
the l a s t line for
shows
t > t*
t ~ tI
that there
(2.1.57)
is clear.
then
-y(t)
is some
O n the > 0
solution
and z(t)
such that
z(t)
This
completes
THEOREM
< -y(t)
t > t*
(2.1.58)
the proof.
2.1.6: Let
~ (t)
satisfy
the h y p o t h e s e s
of T h e o r e m
2.1.1.
If oo
f
t
o2(s)ds
< llo(t) I
t > tO ,
(2.1.59)
47
then
(2.1.0)
Proof:
By
is n o n - o s c i l l a t o r y .
Theorem
2.1.I
it
a solution
for
use
of
the
Schauder
fixed
Let
X
be
a subset
of
X =
sufficiently
suffices large
point
L2(t0
t
to
show
.
We
that
shall
(2.1.4) again
has
make
theorem.
, ~)
defined
{V £ L 2 (t O , ~) : Iv(t) - O(t) I
to} (2.1.60)
For
v ~ X
we
define
a map
T
by
oo
(Tv) (t)
= o(t)
+ I
(2.1.61)
v 2 ds t
If
S e
[0 , i]
ISU+
and
u ,v e X
(l-s)V-O
I =
_
to
(2.1.62)
48 Hence co
i (TV) (t) - O(t) i = I
v 2 ds
t oo
< 4 ]
~2ds
t => t 0 ,
]~(t) l
t => t o ,
t
_-< 4 . ~
1
< i~(t) i
which
implies
in e x a c t l y make
use
that
the
of
TX c X
same w a y
(2.1.62)
instead
of
TX
to the
TX
by m a k i n g
procedure fore
is
can be
similar
omitted. of
a fixed
completes
the proof.
continuity
of
shown
to t h a t
point
2.1.2
by
T
is
shown
wherein
we
now
applying use
of
in T h e o r e m the
of
(2.1.63)
Iv(t) i ~ vl(t)
extensive
Consequently
existence
The
as in T h e o r e m
compactness set
.
t __> t O
v = Tv
Corollary (2.1.62).
2.1.2
Schauder of
The
and
theorem (2.1.61)
is
II.l.2 The there-
implies and
the
this
REMARK : We n o t e (2.1.49-50) priate
THEOREM
and
change
that the
in
~
can be
conclusion
replaced will
be
by
the
-~ same
in with
the
appro
(2.1.0).
2.1.7: Let
o(t)
satisfy
the h y p o t h e s e s
of T h e o r e m
2.1.1
and
49 (2.1.59) .
If
o(t)
> 0
then
z' (t) = c 2 +
will
be n o n - o s c i l l a t o r y
z(s)do2(s)
(2.1.64)
o2(s)ds
(2.1.65)
where
~oo
o2(t)
Proof:
Let
implies
that
o l(t)
= 4 J t
-- o(t)
Since
o2(t)
~i (t) > °2 (t)
Therefore Theorem and
Theorem
2.1.3
this
2.1.6
now implies
completes
COROLLARY
shows that
=> 0 ,
t > tO
that
(2.1.0)
(2.1.64)
(2.1.59)
(2.1.66)
is n o n - o s c i l l a t o r y .
is n o n - o s c i l l a t o r y
the proof.
2.1.4:
Let
o(t)
Then
(2.1.0)
has
for
t => t O ,
~ 0
satisfy
the h y p o t h e s e s
a non-oscillatory
ly(t) I =< exp
solution
{I2 t o(s)ds }
of T h e o r e m y(t)
such
2.1.7. that,
2 .i .67)
to
Proof:
The h y p o t h e s i s
implies
that
(2.1.0
is n o n - o s c i l l a t o r y .
50 The p r o o f solution
of T h e o r e m of
2.1.6
(2.1.4),
then i m p l i e s
such
(2.1.67)
and t h a t
o(t)
that
for
It is p o s s i b l e hold
implies
(2.1.62)
holds.
of This
v(t)
,
a
estimate
y
to r e m o v e
> 0
the e x i s t e n c e
the r e q u i r e m e n t s
in T h e o r e m
2.1.7
that
(2.1.59)
and then s t a t e
the
converse.
THEOREM
2.1.8:
Let With
~2
d(t)
defined
satisfy as in
the h y p o t h e s e s
(2.1.65)
suppose
of T h e o r e m that
2.1.1.
(2.1.64)
is non-
oscillatory. Then
(2.1.0)
solution
z
is n o n - o s c i l l a t o r y of
(2.1.64)
there
and for e a c h n o n - t r i v i a l is a s o l u t i o n
y
of
(2.1.0)
such that
0 < y(t)
< Iz(t)12exp
Id(S) Ids 1
for
t ~ tI
Proof: space
We use S c h a u d e r ' s L 2 ( t 0 , ~)
X = {v e
where
say.
v~ (t)
L2
fixed point
and a s u b s e t
X
theorem.
defined
by
i (t o , ~) : Iv(t) I < ~ vl(t) + Id(t)
is a s o l u t i o n
of the i n t e g r a l
Consider
, t > to}
equation
the
51
v(t)
= 4
c2(s)ds
+
v2(s)ds t
which fine
exists a map
by virtue T
of T h e o r e m
2.1.1.
For
v • X
we de-
by
oo
(Tv) (t) = c(t)
+ I
t > t
v2ds
=
t
As
in T h e o r e m
is c o n v e x .
If
v c X
2.1.2
a simple
0
calculation
shows
that
X
, t => t O ,
0o
f _< lo(t) l + f
l(Tvl (t) l _< l°(t) l +
v2 ds
t oo
1
2
{yv1+lol}
ds
t
=< 1~(t) i + 2 . ~ 1
r
Jt
v i2 d s
+ 2
o2ds
1
__< i~(t) 1 + y v i(t)
Hence TX
TX
c a n be
heavily
c X
The
shown
upon
This
which
means
T
and
as in T h e o r e m
1 Iv(t) l __< yv1(t)
Schauder's
v = Tv
analogously
of
the
compactness
2.1.2
of
relying
the e s t i m a t e
vEX:
Thus
continuity
theorem
implies
necessarily
that
(2.1.0)
+ la(t) i
that
satisfies
T the
has
t => t o
a fixed
latter
is n o n - o s c i l l a t o r y
point
inequality.
and we
can recover
52 an e v e n t u a l l y integral
positive
equation
such
0 < y(t)
solution
y(t)
of
(2.1.0)
f r o m the
that
0
[31, p.
The e x i s t e n c e
(2.1.72)
is r e m i n i s c e n t
243]
of a
of W i n t n e r ' s
375].
2.1.2A: Let
a(t)
, b(t)
IA(t) I >
be d e f i n e d
IB(t) I
as a b o v e
and s u p p o s e
that
t > tO
(2.1.73)
V~ ds
(2.1.74)
If co
V l(t)
=
IA(t) I + I t
has
a solution
for
t => t O
v(t)
a l s o has
a solution
for
then
= B(t)
+
t ~ tn
I
~ v2 ds t
(2.1.75)
54 Proof:
This
o i ( t ) E A(t)
COROLLARY
is i m m e d i a t e ,
~2(t)
from Theorem
2.1.2
upon
setting
~ B(t)
2.1.2A:
Let
A(t)
=> 0
A(t)
for
t => t O
=> IB(t) I
and
t => t O
(2.1.76)
v~ ds
(2.1.77)
If
v l(t)
has a s o l u t i o n
Proof:
Immediate
When by Hille
THEOREM (those
then
2.1.3A: stated
(2.1.75)
f r o m the
a(t)
[31, p.
> 0 ,
(2.1.70)
oscillatory.
+
has a s o l u t i o n .
theorem.
b(t)
Theorem
> 0
2 . 1 . 2 A was p r o v e n
245].
Let
a(t)
, b(t)
in the b e g i n n i n g
A(t)
and
= A(t)
__> [B(t) I
is n o n - o s c i l l a t o r y
satisfy
of this
the u s u a l
subsection).
t > to
then
(2.1.71)
conditions If
(2.1.78)
is a l s o n o n -
55
Proof: o2(t)
This H B(t)
As p.
he
(See a l s o
stated
~ B(t)
Hille's with
[31,
A(t)
p.
H A(t)
and
of
369,
theorem
exercise
general the
7.9].)
theorem
form
to T a a m
[60,
[63, p.
a(t)
~ 0
,
In fact,
than
(i.0.i).
to W i n t n e r
criterion
is due
the
The
257]
above
as
case
who
b(t)
~ 0
conditions.
If
extended along
~ B(t)
Let
a(t)
t
(2.1.70)
Proof:
245]
of(t)
2.1.4A:
THEOREM
then
more
is due
setting
2.1.3.
[25, p.
equations
~ 0
by
the p r e v i o u s
a slightly
considered
A(t)
verified
in T h e o r e m
it s t a n d s ,
495].
Taam
is r e a d i l y
This
setting
COROLLARY
o(t)
iit
satisfy
the
a(s)ds
< ~
usual
1
t > to > 0 ,
(2.1.79)
is n o n - o s c i l l a t o r y .
follows
immediately
from
Theorem
2.1.4
upon
- A(t)
2.1.3A : (2.1.70)
is n o n - o s c i l l a t o r y
l i m sup t÷~
t
a(s)ds t
if
< 1 4 "
(2.1.80)
56 Proof:
Set
o(t)
Wintner 2.1.4A
= A(t)
[62, p.
by r e p l a c i n g
Thus
he 3 -7
to
The
latter
number
2.1.4A
was
Corollary shown
that
[31, pp.
THEOREM
~ a(s)ds
lower
bound
seems
is b e s t
possible
by H i l l e can be
the b o u n d
t > to
(2.1.81)
1 -4
of
to be o p e n or not.
[31,
found
p.
in
appearing
in
as to w h e t h e r When
246].
[31,
in
appearing
p.
a(t)
For
a(t)
246]
where
(2.1.80)
is b e s t
(2.1.79) the
> 0
Theorem
> 0 , it is a l s o possible
2.1.5A:
the
a(t)
, b(t)
integrals
conditionally).
be
continuous
(2.1.68-69)
Suppose
A(t)
>
are
further
(2.1.70)
is n o n - o s c i l l a t o r y
of
(2.1.70)
there
corresponds
on
[a , ~)
convergent
and
suppose
(possibly
that
IB(t) I
If
such
< ~i
248-49].
Let that
theorem
with
question
2.1.3A
extends
t
the
proven
2.1.3.
essentially
I
=
improved
370]
(2.1.79)
3 < t 4
in C o r o l l a r y
t > tO
then
(2.1.82)
to e v e r y
a solution
solution
z(t)
of
y(t)
(2.1.71)
that
z(t)
=< ly(t) I
t > t*
(2.1.83)
57
Proof:
This
follows
in the p r o o f
The only
of T h e o r e m
conclusion
assume
sign.
The
Note
that
need
not be
(2.1.71)
latter
is due
the above where
a(t)
A(t) Thus
can be o b t a i n e d whenever
via
from
theorem
a(t)
to H a r t m a n
requires
so for
can be estimated.
the s u b s t i t u t i o n s
and W i n t n e r
(2.1.83), of
t
635].
b u t this
for solutions
under
of
the above
(2.1.70)
[26, p.
to
[26, p.
for large
estimates
(See for e x a m p l e
if we
is u n r e s t r i c t e d
> 0
the s o l u t i o n s
also holds
are k n o w n
or
636]).
2.1.6A: Let
the i n t e g r a l for
of
~ b(t)
(2.1.82)
2.1.5
2.1.3A.
a(t)
hypotheses,
THEOREM
from T h e o r e m
a(t)
be c o n t i n u o u s converges
A (t)
on
[a , ~)
(possibly
and suppose
conditionally).
that If
t => t O oo
I A21s)ds ¼1A(t l
(2.1.84)
t
then
(2.1.70)
Proof:
This
setting
o(t) When
[47, p. which
312]
states
is n o n - o s c i l l a t o r y .
follows
immediately
from T h e o r e m
2.1.6
upon
E A(t) A(t)
~ 0
the above
and e x t e n d e d that
(2.1.70)
a result
t h e o r e m was p r o v e n of W i n t n e r
is n o n - o s c i l l a t o r y
[62, p. if
by Opial 371]
58
Thus
in T h e o r e m
non-negative. the E u l e r
THEOREM
A 2(t)
=< ~1 a(t)
2.1.6A
A(t)
Equality
equation
in
t => t 0
is no l o n g e r
(2.1.84)
required
is a t t a i n e d
to be
in the case of
(2.1.52).
2.1.7A: Let
along with
a(t)
satisfy
(2.1.84).
If
the h y p o t h e s e s A(t)
> 0
of T h e o r e m
for l a r g e
t
2.1.6A
then
y" + 4A2(t) y = 0
(2.1.85)
is n o n - o s c i l l a t o r y .
Proof:
Refer
to T h e o r e m
Whether (2.1.85)
is,
we shall
discuss
COROLLARY
(2.1.68)
appears
2.1.7 w i t h
being
~ t)
-- A(t)
non-oscillatory
to be an o p e n q u e s t i o n
in s e c t i o n
implies
[59, p.
that
93] w h i c h
2.2.
2.1.4A:
Let
A(t)
Then
(2.1.68)
for
t => t O ,
has
> 0
and s u p p o s e
a non-oscillatory
ly(t) I __< exp
2
that
(2.1.84)
solution
A(s)ds
.
is s a t i s f i e d .
y(t)
such
that,
(2.1.86)
59 Proof:
This follows
from C o r o l l a r y
2.1.4 with
a(t)
H A(t)
T H E O R E M 2.1.8A: Let
A(t)
be d e f i n e d
as in
(2.1.68)
and suppose
z" + 4A2(t) z = 0
is n o n - o s c i l l a t o r y .
(2.1.87)
Then
(2.1.88)
y" + a(t)y = 0
is n o n - o s c i l l a t o r y (2.1.87)
that
and for each n o n - t r i v i a l
there is a solution
0 < y(t)
y(t)
of
< [z(t)12exp
solution
(2.1.88)
z(t)
of
such that
(2.1.89
[A(s) Ids l
for
t
sufficiently
Proof:
large,
say,
t -> t I
This is an a p p l i c a t i o n of T h e o r e m 2.1.8. The first part of the theorem is i d e n t i c a l w i t h a
t h e o r e m of H a r t m a n and W i n t n e r (2.1.89) where
is stronger
than the c o r r e s p o n d i n g
the absolute value
not appear.
[27, p. 216]
sign about
A(t)
though the e s t i m a t e estimate in
in
(2.1.89)
[27] does
Thus the first part of T h e o r e m 2.1.8 extends
Hartman-Wintner (2.1.0), while
r e s u l t cited
above
to e q u a t i o n s
the second p a r t extends
result only w h e n
o(t)
~ 0
of the type
the c o r r e s p o n d i n g
in T h e o r e m 2.1.8.
the
60 2.1B
APPLICATIONS
In 2.1
to
this
TO
DIFFERENCE
subsection
recurrence
relations
CnYn+ 1 +
where
cn
> 0
sequence,
n We
, =
we
n
=
EQUATIONS:
apply
of
the
the
Cn_lYn_
theorems
of
section
form
1 + bnY n
-1 , 0 , 1 , ....
=
(2.1.90)
0
(b n)
is
any
given
specified,
that
real
0 , 1 , ....
shall
assume,unless
otherwise
co
(2.1.91) 0 Cn-i
be
satisfied We
with
saw
jumps,
where
of
as
in
at
t_l
=
an
a
extra Chapter
fixed
a
and
tn
-
of
1
that
increasing
1 Cn_l
tn_ 1 -
upon if
the
o(t)
sequence
n=0
c
n
is
a
of
points
, 1 , ...
step-function t n)
,
2.1.92)
magnitude
~(tn)
for
condition
n= some
resulting [a , ~)
~(t n
0 , 1 , 2 , ... "extended" solution which
has
,
0)
then
recurrence is the
-b n
(2.1.0)
in
curve
that
if
2.1.93)
Cn-i
gives
relation
a polygonal property
n
rise the
y(t) we
write
to
sense
solutions that
defined Yn
the
on
H Y(tn)
'
61
then
the
sequence
currence
relation We
for m
given
> 0
(yn)
note
(2.1.90) that,
=
a solution for
n=
whenever
sequences
cn
, bn
the
three-term
re-
0 , 1 , ....
o(t)
is d e f i n e d
then
,
to
for
t c
by
(2.1.93)
[tm , t m + I)
,
,
(t) =
This
is
follows
from
o(a)
(2.1.93)
m - ~ 0
(2.1.94)
(b n + c n + Cn_ 1 )
and
the
relation
o(t n-
0)
O ( t n _ I)
THEOREM
2.1.1B: Let
~(~)
o(t)
exists
condition
for
and
be is
defined
zero.
(2.1.90)
as
Then
to b e
in
(2.1.94)
and
a necessary
and
non-oscillatory
is
assume
that
sufficient that
oo
v(t)
f
v 2 ds
(2.1.94),
have
= o(t)
+
(2.1.95)
t
where
o(t)
L2
infinity.
at
is
Proof:
This
results
of
REMARK
given
by
follows
Chapter
immediately
from
a solution
which
is
in
Theorem
2.1.1
and
the
is
to b e
oscillatory
i.
:
A solution
(yn)
of
(2.1.90)
said
62
if
the
and
sequence
exhibits
non-oscillatory
constant
sign.
theorem
shows
if,
The that
if
Moreover
the
transition
when
is
(2.1.0)
for
all
by
number
> N
,
the
version
solutions from
of
is
to
shows
(2.1.90)
with
defined
sequence Sturm
the
that
a
property.
a given and
retains
(non-
same
if
changes
separation
(2.1.90),
(non-oscillatory) of
sign
oscillatory
(2.1.0)
solution
of
the
inherit
(2.1.94),
is o s c i l l a t o r y
corresponding
n
a solution
then
given
infinite
discrete
oscillatory)
o
an
in
case
solution
only
is o s c i l l a t o r y
the
if
of
the
(non-
oscillatory). Thus,
y' (t)
~
= c +
f
as
in
(2.1.94),
t t c
y(s)do(s)
[a , ~)
,
(2.1.96)
a
is o s c i l l a t o r y
(non-oscillatory)
if a n d
only
CnYn+ 1 + en_lYn_ 1 + bnY n = 0
is o s c i l l a t o r y The Hille's
functions
latter
(2.1.97)
theorem
thus
gives
the
discrete
version
of
' gn
define
step
[31].
given °1
n = 0 , 1 , ...
(non-oscillatory).
theorem For
if
' ~2
sequences on
c
[a , ~)
n
> 0 by
,
b
n
we
setting
m 01 (t)
= a I (a)
- [
(b n + c n
÷
Cn_ 1
)
(2.1.98)
63
if
t e
[tm , tin+I)
,
m => 0
,
and
m ~2(t)
if
t ~
=
(~2(a)
[tm , tm+ l)
(2.1.92)
We
afortiori, With
°i
of T h e o r e m
Theorem
2.1.2B
latter
two
former
two,
omit
refer
to e i t h e r
corollary
we
shall
mean
THEOREM
2.1.3B: Let
both
exist
c
by
n
and
conditionally
and
denoted
respectively.
in the
same way
it s h a l l
2.1.2
satisfy
the d i s c r e t e
2.1.2
of T h e o r e m
lim m÷~
m [ 0
lim
m [
satisfy
be
2.1.2B
or its
by
Since as
the
the
understood or its
corollary
with
finite
further
(2.1.91).
Suppose
that
(b n + c n + Cn_ 1 )
(2.1.100)
(gn + c n + C n - i )
(2.1.101)
(so t h a t
convergent).
Suppose
also
n +
we o b t a i n
and
Theorem
cn
(2.1.98-99).
> 0
are
as
2.1.2B
them
we
given
÷ ~
n
stated
that when
o I , 02
the
and C o r o l l a r y
c a n be
shall
t
(2.1.99)
Cn_ 1 )
+
that
so d e f i n e d
2.1.2
results
(gn + c n
recall
and C o r o l l a r y
we
~ 0
so t h a t
' ~2
analogs
~
that
the
series
need
only
be
64 oo
oo
(c n + Cn_ 1 + bn)
m
for
m => m 0
>
(2.1.102)
[m (c n + Cn_ 1 + gn)
If
(2.1.103)
CnYn+ 1 + C n _ l Y n _ 1 + bnY n = 0
is n o n - o s c i l l a t o r y
then
(2.1.104)
CnZn+ 1 + C n _ l Z n _ 1 + gnZn = 0
is n o n - o s c i l l a t o r y .
Proof:
Define
(2.1.100-101) ~i(~)
~I
by
(2.1.98),
are then e q u i v a l e n t
, ~2 (~)
by an a d d i t i v e This
' ~2
exist
then i m p l i e s
to r e q u i r i n g
and be finite.
factor,
(2.1.99)
we can assume
respectively. that b o t h
Since we can alter that
a1(~)
these
= ~2(~) = 0
that
c~
Ol (a) = ~ (c n + C n _ 1 + b n)
(2.1.105)
0 oo
O2 (a) = ~ (c n + Cn_ 1 + g n )
(2.1.106)
0
Hence,
for
t e
[tm_ 1 , t m)
,
oo
Ol(t)
= [ (c n + C n _ 1 + b n) m
(2.1.107)
oo
02 (t) = ~ (C n + C n _ 1 + g n ) m
(2.1.108)
65
Thus
the r e q u i r e m e n t
is e q u i v a l e n t m
.
From
that
to the
the r e m a r k
oscillatory.
Hence
(2.1.48)
be
satisfied
for
large
t
requirement
that
(2.1.102)
hold
for
large
we
that
(2.1.49)
2.1.3
applies
see
Theorem
is n o n - o s c i l l a t o r y .
Consequently
oscillatory
completes
of
the
this
The
latter
theorem
Taam
result
[60].
extension [63]
and
of
the d i s c r e t e
and H i l l e
[31]
is
must
be n o n -
and hence
(2.1.104)
is a l s o
(2.1.50) non-
the proof. therefore
the d i s c r e t e
Simultaneously
it p r o v i d e s
version
theorem
(see T h e o r e m
of
the
2.1.3A).
Thus
analog an
of W i n t n e r for e x a m p l e ,
if
b
> 0 n =
n = 0 , 1 , ...
(2.1.109)
gn => 0
n = 0 , 1 ....
(2.1.ii0)
and
~bn= m
then be
(2.1.i04)
the
is n o n - o s c i l l a t o r y
formulation
of
(2.1.ili)
m > m=0 '
>~gn m
the d i s c r e t e
if
(2.1.103)
analog
is.
of H i l l e ' s
This
theorem
[31].
THEOREM
2.1.4B: Let
sequence
c
(b n)
n
> 0
and
assume
satisfy
that
(2.1.91).
(2.1.100)
For
exists.
a given If
would
66
1
li then
(2.1.90)
Proof: we
1 (c n + Cn_ 1 + b n)
We
shall
m > m0
m 0
-2
y(n+l)
0 , 1 , ...
if
(2.1.117)
n=0,1
....
in
(2.1.116).
Then ¥ A2yn_ 1 +
2 Yn
(2.1.118)
= 0
(n + i)
is n o n - o s c i l l a t o r y
if
1 y < ~
.
For
= m
mibn
~ m
Y (n+l) oo
< m y • Im x-2 dx
< =
my m+l
1 < -= 4
Consequently is
(2.1.117)
non-oscillatory
equation.
holds
where
m
with
1 y ~ ~
.
> 0 =
.
m0 = 0
and
thus
This
the
discrete
is
(2.1.118) Euler
88 Example
2:
Let
b n = y(-l)n/(n+l)
,
n=0
,1
Then A2 Yn-i
is n o n - o s c i l l a t o r y
V (-i) n ( n + l ) Yn = 0
+
(2 .i.119)
[y[ < ~1 .
if
co
For
[ bn
is c o n d i t i o n a l l y
m ~m bn
convergent
and
~m (n+-li (-i) n
=
mly]
N
, o2(t)
follows
satisfy
n
z n =< lynl
z(t n)
(2.1.121)
(2.1.100-101-102).
then to e v e r y
corresponds
Proof:
section
427]
is o s c i l l a t o r y .
non-oscillatory there
(bn)
in
[32, p.
to be a b s o l u t e l y
in the n e x t 1 ~
(2.1.117)).
2.1.5B: Let
Theorem
the r e s u l t s
of
and L e w i s
is r e q u i r e d
gent.
i.e.
of
[because
a r e s u l t of H i n t o n for
(2.1.121)
< 14
(2.1.55)
2.1.3B.
set to o b t a i n
(2.1.122) .
THEOREM
2.1.6B: Let
the s e q u e n c e s
c
n
, b
n
satisfy
the h y p o t h e s e s
of
70 Theorem 2.1.4B. If, for
i=m
m => m 0 ,
i
j= +i
Co-1
= ~
(c i + ci_ 1 + b i)
i=m+l
(2.1.123) then
(2.1.103)
Proof: for
is non-oscillatory.
We define
o
t • [tm_ 1 , t m)
as in the proof of Theorem 2.1.4B. ,
~
is given by
o2(s)ds = t
(2.1.113).
o (s)dsi= -i
Then
Consequently
o (s)ds
i
tm-i (2.1.124)
Since
o
is constant on each
[tm_ 1 , t m)
,
m = 0 , 1 , ...
we
obtain oo
I
oo
2 (S) ds =
t
i=m-1
(ti+l- ti){ j= ~ +i
- ( t - t m _ I)
~ j m
(c J + cj-i +bj)
(cj + C j _ l + b
j)
(2.1.125)
Co
=
I i=m-1
G i2+ l 1
-
(t-
t m - i ) G m2
where oo
G i = j=i ~
(cj + cj_ 1 + b j)
(2.1.126)
71
Since
t e
neglected
[tm_ 1 , t m) and
o
the
(s)ds
=
2
equality
in
(2.1.129)
when
m = 0 .
72
A simple
computation
corresponding the
lower
now
to the
shows
initial
that
values
(2.1.116)
THEOREM assume
solution
Y-I
= 0
,
of
(2.1.116)
Y0 = 1
admits
bound
Yn => n - i
Hence
the
is n o n - o s c i l l a t o r y .
2.1.7B: that
n => i
With
the
(2.1.123)
c
n
, b
n
is s a t i s f i e d
as in T h e o r e m for
large
m
2.1.4B
.
If
co
(t)
-
[ i=m
(C i + C i _ 1 + b i) > 0
,
t c
[tm_ I , t m)
(2.1.131) then
the
differential
equation
z" + 4~ 9 (t) z = 0
(2.1.132)
is n o n - o s c i l l a t o r y .
Proof:
This
COROLLARY
suppose
oscillatory
immediately
from
Theorem
2.1.7.
2.1.4B:
Let and
follows
c
n
that
, b
n
satisfy
(2.1.131)
solution
(yn)
the h y p o t h e s e s
holds. such
Then that
of T h e o r e m
(2.1.90)
for
n ~ N
has
2.1.7B
a non-
,
t lynl
< e x p { 2 IT n o ( s ) d s }
(2.1.133)
73 where
o
is as in
Proof:
Follows
THEOREM
2.1.8B: Let
(2.1.131).
from Corollary
o(t)
be as in T h e o r e m
n e e d n o t be n o n - n e g a t i v e .
Assume
z" + 4o
is n O n - o s c i l l a t o r y .
2.1.4.
2
2.1.7B
except
that
that
(t) z = 0
(2.1.134)
Then
(2.1.135)
CnYn+ 1 + Cn_lYn_ 1 + bnY n = 0
is n o n - o s c i l l a t o r y (2.1.134)
there
and for e a c h n o n - t r i v i a l
is a s o l u t i o n
(yn)
of
solution
(2.1.135)
z
such
of that
t 0 < Yn
0
t => t O
(2.2.1)
to(t)
1 < ~
t > tO
(2.2.2)
If
then
If
(2.1.0)
e > 0
is n o n - o s c i l l a t o r y .
and
to(t)
1 > ~ + e
t > tO
(2.2.3)
75 then
(2.1.0)
Proof:
The
is o s c i l l a t o r y .
first part
p a r t we w r i t e
~i (t) -- ~(t)
then i m p l i e s
2.1.4.
To p r o v e
the s e c o n d
and
ill 1
a2(t)
(2.2.3)
is T h e o r e m
- ~ + ~
(2.2.4)
t __> t o
that
a 1(t)
> a 2 (t)
z'(t)
= c +
t > tO
(2.2.5)
z(s)dJ2(s)
(2.2.6)
Furthermore t
s
a
is o s c i l l a t o r y
since
it is e q u i v a l e n t
to
(2.2.7)
and
the l a t t e r
Since
(2.1.0)
equation must
is o s c i l l a t o r y
either
c a n n o t be n o n - o s c i l l a t o r y that
(2.2.6)
is o s c i l l a t o r y
THEOREM
2.2.2:
Let
be o s c i l l a t o r y
a
satisfy
which
e > 0
[59, p.
45].
or n o n - o s c i l l a t o r y
for then T h e o r e m
is n o n - o s c i l l a t o r y
(2.1.0)
for
2.1.3 w o u l d
is i m p o s s i b l e .
a n d the t h e o r e m
is p r o v e d .
the h y p o t h e s e s
of T h e o r e m
imply Thus
2.1.1
and
it
76
assume
that
(2.2.1)
holds.
If
o2(s)ds
__< ~ o(t)
t > tO
(2.2.8)
t
then
If
(2.1.0)
~ > 0
is n o n - o s c i l l a t o r y .
and
It 2
then
(2.1.0)
Proof:
The
(s)ds
>
first
part
under
is T h e o r e m
(2.2.9)
it is n o n - o s c i l l a t o r y . such
Then
then
we
2.1.6.
To p r o v e
assume,
there
on
exists
the
that
contrary,
a solution
(2.1.0) that
y(t)
of
that
y(t)
This
(2.2.9)
is o s c i l l a t o r y .
is o s c i l l a t o r y
(2.1.0)
t > tO ,
+ c o(t)
implies
that
> 0
the
t __> T
integral
.
(2.2.10)
equation
~oo
v(t)
admits
a solution
non-negative
for
= ~(t)
+ I t
v 2 ds
v(t)
for
t > T
t > T
and
v(t)
v(t)
=> o(t)
t > T
This + 0
t __> T
as
solution t ÷ ~
(2.2.11)
v(t)
is
Moreover
(2.2.12)
77 Set
(2.2.9)
then implies that
oo
I
g2(s)ds > ~ ( t )
t => t O
(2.2.13)
t NOW,
oo
v(t) => g(t) + I
°2(s)ds
t => T
(2.2.14)
t
> (l+e)g(t)
- elo(t)
t > T .
(2.2.15)
We may take it that oo
I 2(s)d s < t since
v E L9
and
Using
(2.2.15)
in
v > g
for large
t
(2.2.11) we obtain
oo
v(t)
> g(t) + e~ I
gg(s)ds
(2.2.16)
t
2 > g(t) + 6~i~g(t) 2 > (i +a1~)O(t)
~2o(t)
t ~ T
(2.2.17)
Repeating the above process we find that
v(t)
> e d(t) =
n
t > T =
(2.2.18)
78
where 2 i + aen_ I
an
is i n d e p e n d e n t (en) must
For
of
t
is i n c r e a s i n g be
if
A
n => 2
simple
and hence
induction
tends
B < ~
then,
1 e > ~
since
letting
n ÷ ~
bounded
at every
by
solution
shows
can
such
B
(2.2.18)
point of
v
we
limit
shows ~
that
which
t
where
on
finite
must
theorems for
of T h e o r e m
v(t)
B = ~ must
~ 0
This This
Thus
be
un-
contradicts contra-
a non-oscillatory
be oscillatory•
therefore
we
hold
in T h e o r e m
2.1.7
therefore
complement
Theorem
in T h e o r e m
that
have
(2.1.0)
2.2.2
Hence
intervals.
(2.1.0)
conditions
(2.2.20)
o(t)
and hence
previous
2
find
cannot
because
THEOREM
to s o m e
can e x i s t .
(2.1.0)
sufficient"
(2.1.59)
argument
(2.2.19),
that
The
Thus
no in
the b o u n d e d n e s s diction
(2.2 19)
infinity.
= 1 + aB
and
,
give
"e-necessary
to b e n o n - o s c i l l a t o r y .
see
that
is n o t
too
2.1.8
the
condition
restrictive.
with
the
result
that We stated
2.1.7.
2.2.3: Let
o
be
right-continuous
and
and
locally
of bounded
79
variation
on
[a , ~)
If
lim t+~
then
(2.1.0)
Proof:
t => t O 2.1.1
on
the
solution If
we
(2.2.21)
= -~
is o s c i l l a t o r y .
Suppose,
oscillatory
o(t)
v(t)
contrary,
of
(2.1.0)
= y' ( t ) / y ( t )
that
with ,
y(t)
say
t >
is
y(t)
tO
,
a non-
> 0
then
for from
Theorem
have
v(t)
= o(t)
- ~ ( t 0)
+ v ( t 0)
.t v 2 ds
-
t > tO
to (2.2.22)
< ~(t)
We
can
then
obtain
that
- o ( t 0)
proceed
to
the
v(t)
Arguing
then
diction
(2.1.17).
and
this
as
proves
During
in
the
+ v ( t 0)
limit
t > tO
as
÷ -~
t + ~
in
(2.2.23)
t + ~
Theorem
2.1.i,
Thus
non-oscillatory
no
(2.2.23)
(2.2.24)
to
(2.2.24)
leads
to
solution
the
contra
can
exist
theorem.
the writing
of these
notes
there
appeared
a
80 paper of Reid
[50, p. 801] who also,
independently,
proved
T h e o r e m 2.2.3.
2.2A
APPLICATIONS
TO D I F F E R E N T I A L EQUATIONS:
T H E O R E M 2.2.1A: Let and suppose
a(t) that
satisfy the c o n d i t i o n s A(t)
~ 0
(where
A(t)
of T h e o r e m 2.1.6A is d e f i n e d in
(2.1.68)>. If tA(t)
then
If
(2.1.70)
E > 0
(2.2.25)
is fixed and
(2.1.70)
Proof:
t => t O
is n o n - o s c i l l a t o r y .
tA(t)
then
1 =< ~
1 => ~ + s
t > tO
(2.2.26)
is oscillatory.
This is a c o n s e q u e n c e
of T h e o r e m 2.2.1 where
o(t)
A(t) The first part of this t h e o r e m is due to W i n t n e r p. 260]
and the second part follows a l m o s t i m m e d i a t e l y
this r e s u l t
(see
[44, p. 131],
[63, p. 259]).
[63,
from
81
THEOREM
2.2.2A: Let
and
suppose
a(t) that
satisfy A(t)
the
conditions
of T h e o r e m
2.1.6A
=> 0
If A 2 (s)ds
< ~ A(t)
t > to
(2.2.27)
t
then
If
(2.1.70)
is n o n - o s c i l l a t o r y .
~ > 0 ,
5
A2(s)ds
=>
+~
t
then
(2.1.70)
Proof:
This The
THEOREM
A(t)
t > tO
(2.2.28)
is o s c i l l a t o r y .
follows above
from Theorem
theorem
is d u e
2.2.2. to O p i a l
[47,
p.
309].
2.2.3A: Let
(2.1.68)
1
a (t)
be c o n t i n u o u s
on
[a i ~)
and
suppose
that
exists.
If co
I
a(s)ds
a
then
(2.1.70)
is o s c i l l a t o r y .
= ~
(2.2.29)
82
Proof:
We
let
The the p.
case 115]
§2.2B
latter
when for
= -
a(s)ds
theorem
a(t)
> 0
general
APPLICATIONS
THEOREM
was
in T h e o r e m
proven
and w a s
2.2.3.
by F i t e
extended
[19, p.
by Wintner
347]
in
[61,
a(t)
TO D I F F E R E N C E
EQUATIONS:
2.2.1B: Let
2.1.4B
o(t)
and
of T h e o r e m
the
c
assume
, b
n
satisfy
n
further
2.1.6B,
that
G
the h y p o t h e s e s ,
m
is n o n - n e g a t i v e
defined
for
of T h e o r e m
in the p r o o f
m ~ m0
If
i }Z
(C n + C
0 Cn-i
then
(2.1.90)
1 m 0
(2.2.30)
is n o n - o s c i l l a t o r y .
If
1 0 Cn-I
• m
1 => ~ + e
(c n + en_ 1 + b n)
m >m =
0
(2.2.31)
where
Proof: An
e > 0
The
argument
latter
is
first
fixed,
part
similar
theorem
shows
then
(2.1.90)
is a c o n s e q u e n c e
to the one that
used
(2.2.3)
is o s c i l l a t o r y .
of T h e o r e m
in the p r o o f
is e q u i v a l e n t
of to
2.1.4B. the (2.2.31).
83
As
COROLLARY
a consequence
this w e
obtain
in p a r t i c u l a r ,
2.2.1B:
Let convergent
of
be a n y
(bn)
sequence
whose
series
is c o n d i t i o n a l l y
and co
b n => 0
(2.2.32)
m => m 0
m
If co
m
< 1 [ bn = ~
m > m0
(2.2.33)
m
then
If
(2.1.116)
s > 0
is
is n o n - o s c i l l a t o r y .
fixed
and
CO
m
[ b n => ~ +1s
m>
m0
(2.2.34)
m
then
(2.1.116)
Proof: 2.1.4B
Example
This
is o s c i l l a t o r y .
follows
as a p p l i e d
l:
from
the
to T h e o r e m
The d i s c r e t e
discussion
Euler
equation,
Y
+
=
2 Yn
0
(n + i)
is o s c i l l a t o r y
whenever
Theorem
2.2.1B.
Z~2
Yn-i
following
y > ! 4
This
is b e c a u s e
(2.2.35)
84 oo
oo
m [ m
Y (n+l)
>my 2 =
> =
x
m m+l
y
1 > ~-+
for
e > 0
some
Consequently
if
the
m
is
above
2 dx
+1
e
(2.2.36)
sufficiently
corollary
large
implies
that
1 y > ~
since
(2.2.35)
.
is
oscillatory. Using comparison (b n)
is
the
discrete
equation
we
a positive
(2.1.116)
2b
b n => gn
=> 0
to a c o n t r a d i c t i o n Similarly
(2.1.116)
deduce such
and if it
the
=
(2.2.35)
following
that,
> 1 n = 4 + s
for
n => n O
For
if w e
as
(2.1.116) can be
2.1.3B was
shown
< 1 = ~
is n o n - o s c i l l a t o r y .
If
fixed
s > 0
,
(2.2.37)
,
let
(2.2.38)
would
assumed
that
a
result.
[i + e] ( n + l ) - 2
Theorem
(n+l)2bn
then
equation
is o s c i l l a t o r y .
gn
then
can
sequence
(n+l)
then
Euler
if
n > nO
lead
immediately
non-oscillatory. b n =>
0
and
(2.2.39)
85 THEOREM
2.2.2B: Let
(c n)
satisfy
, (b n)
the
hypotheses
of
Theorem
2.1.4B.
If
for
m => m 0
, co
1 G2 c-- i+l i=m 1
then If
(2.1.103)
for
(2.2.40)
is n o n - o s c i l l a t o r y .
s > 0
fixed
and
(2.1.103)
m ~ m0 ,
I----G2 c. i+l z
i=m
then
1 < 4 Gm+l
is o s c i l l a t o r y
> =
+ s Gm
,
(2.2.41)
where
co
Gm = i=m[
Proof:
The
follows
from
holds
first the
whenever
(2.1.126)
we
part proof
t ~
(ci + c i - I
is T h e o r e m of
the
+ bi)
2.1.6B
latter
[tm_ 1 , t m)
(2.2.42)
while
theorem.
Thus
letting
find
co
oo
I 21slds > t
= m-i
i
i+l
m oo
=fiG
c.
m
1
2
i+l
i
G2
Cm_ 1
m
the
second
For
(2.1.126)
t ÷ tm - 0
part
in
86
since
o(t)
= Gm
when
t e
[tm_ 1 , tm)
Thus co
t E [tm_ 1 , t m) t and thus T h e o r e m
2.2.2
applies.
Consequently
(2.1.103)
is
the discrete
analog of Opial's
oscillatory. The latter
theorem gives
theorem
(see Theorem
THEOREM
2.2.3B: Let the
Theorem
(cn)
2.2.2A).
, (bn)
satisfy
the hypotheses
of
2.1.4B.
If oo
(2.2.43)
(Cn + C n _ 1 + b n) = co 0
then
(2.1.103)
Proof:
is oscillatory.
We define
o(t)
as in
is a s t e p - f u n c t i o n
with
find that
is then e q u i v a l e n t
Theorem
(2.2.43)
2.2.3 applies
oscillatory.
jumps
(2.1.93) at the
and c o n s e q u e n t l y
remembering (tn)
to
From
(2.2.21).
(2.1.103)
is
that
o(t)
(2.1.94) Hence
we
87
The p.
426]
theorem
and extended,
by R e i d
§2.3
latter
in the
shown
same
In this gives
equation
THEOREM
section
we
a necessary
continuable
solutions
extend
and
of
t e
class
[a , ~) tion
of a s e c o n d
the p r o o f
the
[a , ~) of all
whose
on
[32, 2.2.3,
in
[8]
first
for
nonlinear shall
to e q u a t i o n s
[8, p.
differential
be m a i n l y of
of
the
an
form
derivatives
(2.3.1)
(2.3.1)
absolutely are
75]
all
f(y(s))dq(s)
A solution locally
of B u t l e r
condition
order
Our proof
CASE:
is a g a i n
continuous locally
sought
functions
of b o u n d e d
in
on
varia-
[a , ~) We
case
& Lewis
as T h e o r e m
a result
sufficient
y' (t) = c -
the
direction
IN THE N O N L I N E A R
to be o s c i l l a t o r y .
adaptation
where
by H i n t o n
[50].
AN OSCILLATION
which
was
of
shall
(2.3.1) :
be m a i n l y
concerned
It is c h a r a c t e r i z e d
with by
the the
"superlinear" convergence
of
integral
I Equations
of
"Emden-Fowler
the
form
type",
-+~
(2.3.1) i.e.
dt f (t)
include
those
(2.3.2)
those
equations
which
with
are of
88
In
the
in
f(y)
= y
case
of
given
as
(2.3.3)
first
characterization
2n+l
n=
ordinary
the
result of
1 , 2 .....
differential
equations
of A t k i n s o n
oscillatory
(2.3.3)
[2, p. 643]
solutions
of
with gave
f the
the
equation
y"
when
p(t)
tion [8]
on
> 0
the
+ p(t)y 2n+l
and
= 0
continuous,
coefficient.
This
n=l
in has
, 2 , ...
terms
of
,
an
recently
(2.3.4)
integral
been
condi-
generalized
to e q u a t i o n s
y"
where into prove
p(t)
is u n r e s t r i c t e d
a "superlinear" later
sufficient
give,
condition
for
to be
oscillatory.
crete
analog
the
+ bnf(Yn)
As
for
42 Yn-i
k
sign The
(2.3.5)
and
= 0
theorem
f
result
difference
a corollary
> 1
= 0
in p a r t i c u l a r ,
of A t k i n s o n ' s
then,
to
equation.
on will
2 A Yn-i
positive
+ p(t) f(y)
which
a necessary
= 0
shall and
0 , 1 , ...
shall
[2],
i.e.
obtain If
,
~ 2k+l + mnYn
we
(2.3.5)
equation
n=
we
turns
n=0
, 1 , ...
(2.3.6)
the (bn)
disis
89
has
a non-oscillatory
solution
if a n d o n l y
if
oo
< co [nb n 0
In the so
that
f oy
of
(2.3.1),
absolutely
following
continuous
has meaning
THEOREM
2.3.1: o
such
y
is a s o l u t i o n
that
case
the
f
be o n l y
integral
integral.
function,
locally
yf(y)
exists,
further
> 0
that
for all
y ~ 0
f' (y) > 0
i
dt f(t)
-i
dt (t----~ f < ~
--co
b)
l i r a inf T+co
c)
I
T | P(s)ds Jt
P2(r)drdst
s
>-~
< ~
of
(2.3.8)
and
i
in
that
Suppose
and
f ~ C' (-~ , co)
when
In a n y
f
infinite.
y ~ 0
76].
T - lira do(s) T÷~ t
f e C' (-co , ~) if
[8, p.
that
to r e q u i r e
as a S t i e l t j e s
and
P(t)
a)
continuous
be a right-continuous
variation
and may be
assume
it is p o s s i b l e
(2.3.1)
bounded
shall
is a b s o l u t e l y
though
Let
we
(2.3.7)
for all
where
P_(t) +
t
= max{~P(t)
, 0}
90
Then a necessary (2.3.1),
and sufficient
continuable
over
condition
the half-axis,
for all solutions to be o s c i l l a t o r y
of is
that P(s)
+
P2(r)dr
t
Note:
We shall prove
showing
that
ds
= +~
(2.3.9)
s
(2.3.9)
the sufficiency with
P2(r)
of
(2.3.9)
replaced
by
by first
P+2(r)
will
F
imply
that all solutions
oscillate
Isince
(2.3.9)
and
(c)
k
along with
the relation
I t Proof:
p
+
p2 (t)
2(r)dr ds = + ~
Assume,
some n o n - o s c i l l a t o r y Isince
+
imply
(2.3.10)
@
s
(Sufficiency)
positive
2 = P+(t)
P2(t)
solution
-y(t)
on the contrary, y(t)
that there is
which we can take to be
is also a solution I .
Thus y(t)
> 0
t > tO
(2.3.11)
We let !
g(t)
where
the prime
Then
g(t)
variation
= f~y(t) (t)]
represents
t > tO
in general
shall be r i g h t - c o n t i n u o u s on
(2.3.12)
a right-derivative. and locally
of b o u n d e d
(t o , ~)
An integration
by parts
shows
that,
for
tO ~ t ~ T ,
91
IT i tfl.y,s,, [ ~ J)l dy' (s) = g(T)
ds (y')2 + ]t f' (y) f(y) 2
- g(t)
(2.3.13)
where we have omitted simplicity. (2.3.1)
the v a r i a b l e s
Moreover,
shows
an a p p l i c a t i o n
T 1 t f(Y)
combining
g(t)
of the i n t e g r a l
for equation
that
I Hence
in the i n t e g r a n d
(2.3.13-14)
= g(T)
+ O(T)
IT dy'
=-
we
t
d~
(2.3.14)
.
find
- o(t)
+
j.T f'
< (y)if(y)!
2 ds
(2.3.15)
t t O =< t < T .
whenever
Our basic
assumption
leads
us to two
cases:
Case t > t =
I)
lim sup
P(s)ds
= +~
some
II)
lim sup
P(s)ds
< ~
all
I:
(2.3.16) If there
implies
that
is a s e q u e n c e
g(Tn)
then for
n
sufficiently
t => t O
the r e l a t i o n
0
T
n
+ ~
t => t O
is v a l i d
such
(2.3.17)
for all
that
> 0
l a r g e we
(2.3.16)
(2.3.18)
s h a l l have,
for
g2
t O t I
(2.3.22)
-
do
(2.3.23)
that
P(t)
> 0
all
such)
so
(2.3.15)
now
implies
g(T)
Moreover large of
(2.3.20)
f' (y)
t
(2.3.16) t
some
=< g(t)
implies
(not n e c e s s a r i l y t2 ~ ti
such
for
which
arbitrarily
shows
the
existence
that
.T do
> 0
T >
t2
(2.3.24)
t 2
if w e
assume
that
P(t)
< ~
for
large
t
.
We
note
that
93
P(t)
= ~
for
some
P(t)
= ~
for
all
g(t)
÷ -~
proceed
as
t
if a n d
larger
t ÷ ~
t
only
.
if
This
because
of
~(~)
would
= ~
then
(2.3.15).
and
thus
imply
that
And
we
can
then
as b e l o w . Hence
by
(2.3.23)
g(T)
Replacing
t
by
t2
for
T > t2 ,
< g(t2)
in
- -K
(2.3.15)
and
(2.3.25)
< 0
using
( 2 . 3 . 2 4 - 25)
we
obtain
g(T)
< -K
+ IT
=
f' (y) IY' I g ds .
t
(2.3.26)
f(Y)
2 If w e
write
t => t 9
and
~(t)
=
f' [y(t)] IY' (t) [ flY(t)]
~(t)
> 0
for
so
g(T)
|,T + I ~(s)g(s)ds
t 2
(2.3.29)
i.e.
y' (T)
and
the
latter
0
I T P(s)ds t
We
now
such
proceed
that
as
holds
that
[ < Mt
in C a s e
(2.3.18)
such
I. for
If
T > t
there
large
n
.
(2.3.31)
is
a sequence
we
find
T
from
n
+
(2.3.20)
that
g
Now
either
i)
ii)
i)
Let
y(t)
2
2 > P+(t)
(t)
y(t)
=> 6 > 0
there
is
y ( t n)
+ 0
=> ~ > 0
t > t3 .
,
t => t 3 ,
(tn)
such
t => t 3
that
then
implies
or
tn
co
f
Since
C = i n f { f ' (u) : 6 < u < ~}
(2.3.20)
(2.3.32)
> 0
(2.3.33)
that
oo
g(t)
> P(t)
+ c I
g2 ds t
(2.3.34)
95
g(t)
Integrating T ÷ ~
sides
+ c It P+2 (r) dr
over
[t, T)
w e get a c o n t r a d i c t i o n
integral
ii)
both
> P(t)
of the r i g h t
Let
t
For large
n
+ ~
n
side of
be s u c h
and
t
because
of
(t)
(a)
y(t
n
lim sups
s i n c e by h y p o t h e s i s
as the
is d i v e r g e n t .
) + 0
t => t 3 ,
t
>
f(s)
(2.3.201),
and t a k i n g
(2.3.35)
that
fixed,
Y (t n) 0 > ly - ds =
to
(2.3.35)
e
ftn[ f P(s)
+
f' ( y ) P
dr ds
s
in the limit,
and
(2.3.32).
Thus Y (t n ) fy
(t)
tn ds f(s)
> ft =
(2.3.36)
P(s)ds
so t h a t
n lim inf n+~
as
f(s)
}
P(s)ds
=
(t)
n +~
and so
0
T}
by
JIy[EB : llyli +
where
I II~
is
the
usual
uniform
norm.
Then
B
a subset
B
is
a Banach
space. For
Bn =
n = 1 , 2 , ...
y c B :
0 =< y(t)
we
define
=< 2
,
t =>T
I]y ' Q -i][
,
of
n
=< 1
B
by
,
I Y'(t 2) -Y'(ti) l < alo(t 2) -o(ti) l + It2 - t i] , t i , t 2 ~ i T , T + n ) " and
For
each
fact
n
Bn
by
B
= constant
n
is
compact.
continue
II, as
on
a closed The
in A p p e n d i x We
n
,
is
presented
B
y(t)
in
[T+n
convex
proof Lemma [8]
of
, ~)
subset
the
of
latter
B
and
result
in
is
II.l.l.
and
define
an o p e r a t o r
A
n
on
*
99 oo
oo
t e t
(2.3.41)
[T , T + n]
s
(ANY) (t) = 1 -
For
y e Bn
each
t
,
t c
t > T+n
f (y) do ds T+n
.
(2.3.42)
s
[T , T +n)
,
AnY
has
a right-derivative
at
g i v e n by
oo
(ANY) '(t) = f
f(y(s))do(s)
t
If
y e B
,
n
then
f(y(s))do(s)
integration
= f(y(t)]
by p a r t s
do +
t
shows
do
that
f' (y)y'ds
.
t
Hence
f t
Thus
f(y)do
= f(y(t))P(t)
+
i
P(s)f' (y(s))y' (s)ds
t
for
T -< t -< T + n
,
I (ANY)' (t) I < alP(t) ] + b It IP(s)l{aIP(s) ] + 2ab
and p r o c e e d
to show as in
[8] that
] (ANY)' (t) I < Q(t)
if
T
is so large
that
s p2 ds
t e
[T , T + n )
(2.3.43)
100 oo
2b I
IP(s) lds
< 1
t => T
t
If
t => T + n
,
(ANY) ' (t) : 0
If, in addition,
we require
hence
An(Bn)
c B
T so large that
oo
I
Q(s)ds
< 1
t > T
(2.3.44)
t
then 0 =< (AnY) (t) =< 2
t => T ,
since we can estimate
the inner integrals
(2.3.43)
then gives
and
also follows
(2.3.44) from
(2.3.42)
T =< t I < t 2 < T + n
in (2.3.41 -42)
(2.3.45).
For
y c B
n
by it
that
t > T+n
(ANY) (t) = constant
If
(2.3.45)
.
, t2
I (ANY)' (t 2) - (ANY) ' (t l) I =
It
f (Y(S) ]d°(s) 1 1
t2
T
Js Substituting
this
I ( A N Y ) ' (t 2)
Thus
A n (Bn)
be
tions
done in
the
-(ANY)'
former
equation
we
(tl) ] =< a l o ( t 2) - o
obtain
tl) I +
It2 - tll
c Bn
There can
in
remains
as
the
in
to
[8, p.
definitions
show
that
A
is
n
82]
with
the
of
a(6)
, b(6)
continuous:
appropriate
This
modifica-
there.
i.e.
a(6)
: suP{if(y)
b(6)
= sup{if'
From
the
b(~)
÷ 0
above
given
s ,
6
we
c(6)
lY-xl
< 6}
0 < x , y < 2 , lY-x]
see
that
as
6 ÷ 0
< 6}
both
f E C' is
chosen
sufficiently
small
so
that,
fix - Yll < 6 ,
2 c(~)
where
0 < x ,y < 2 ,
(y) - f'(x) I :
definitions
since If
- f(x) I :
= max{a(6)
, 2ab(6)
IIAn y
N =
and is t h a t
=+oo
(2.3.51)
i
f r o m the p r e c e d i n g
which
> 0
we have
theorem with
a suit-
and the i n t e g r a l
(2.3.51).
.
Then
a necessary
and
105
sufficient
condition
for
1 m=n
Proof:
[
Pm_l
follows
(2.3.51 - 5 2 )
THEOREM
+
Cm_l
This
(2.3.48)
to be o s c i l l a t o r y
i
1p
m=n Cm-1 i = mX - 1
immediately
are e q u i v a l e n t
is t h a t
from
since
P
n
~i
(2.3.52)
l
Theorem
2.3.2
because
> 0 .
2.3.3: Let
satisfy
the b a s i c
hypotheses
of T h e o r e m
2.3.1.
a)
If l i m o(t) t÷co
then
b)
If
(2.3.1)
o(t)
= ~
is o s c i l l a t o r y .
is n o n - d e c r e a s i n g
sufficient
(2.3.53)
condition
for
then
a necessary
(2.3.1)
and
to be o s c i l l a t o r y
is
that
oo
I tda (t) = ~ to Proof: P(t)
a) - ~
To p r o v e
follows for
all
b) we m u s t
immediately t .
from
(2.3.54)
Theorem
2.3.1
since
Hence
(2.3.9)
is i d e n t i c a l l y
that
(2.3.9)
is e q u i v a l e n t
show
satisfied. to
(2.3.54) . If
o
is n o n - d e c r e a s i n g
and
(2.3.9)
is
finite
then
co
It P ( s ) d s
< co
(2.3.55)
106
Consequently assume
o(t)
is zero.
must tend to a finite
Thus
P(t)
= -o(t)
limit,
which we can
and the latter
is non-
increasing. On the other hand if
It
P(t)
+
p2 dt
Hence
Corollary
(2.3.48) (a)
of
2.3.2(a)
are
oscillatory
Theorem
2.3.1.
[ m+l 0
implies where
f
that
all
solutions
is
any
function
of satisfying
109
Example
2:
If w e
let
b
(Un)
be as in E x a m p l e
1
-
and
6>0
(n + i) 1+6
n
1 above
then
so t h a t
(2.3.48)
I
i_i_
0
0 ci-i
b
< co m
has at l e a s t one n o n t r i v i a l
non-oscillatory
solution.
For m
I m=O
since
the s e r i e s
oo
i=0
b
~
~fo m ,=
i
_
terms
(m+l)
i+6
I m=i
b m
and
co
co
1 [ m=i
i
i=O C i _ l
have positive
oo
1 i+i
f
el_ 1
1 < i=l i + l
li (x + i) -1-6 dx -i
co
+
[ m=O
(m + i) -1-6
i.e. co
< 6-1
+ 0 (i) ~ i6 i i=l (i + i) oo
< 6-1
and
since
6 > 0
the
~ i=l
i---l--6+1 + 0(i) i
last t e r m is finite.
(2.3.62)
110
Example
3:
Let
cn
Pk-i
=
1
=
and
let
bn
=
(-l)n/n
,
n
= 1
.....
, 2
Then oo
and
thus
Pk-i
is odd.
By
the
verified
that
[ n=k
(-l)n n
=
(-i)
is p o s i t i v e
if
alternating
series
N lim ~ N+~ k=n
and
is
finite.
(Its
(b)
of
Theorem
2.3.2
k
can
IO
tk-I 1 +----td t
is e v e n
and
theorem
it
(2.3.63)
negative is
be
computed
via
2.3.63.)
is v e r i f i e d .
I
s k2
I
I
I
k=n-i
m=n
=
k=m-i
sk2
where
S k = m a x { P k , 0} Hence
m=n
A look
at
(2.3.63)
~ k=m-i
2 Sk
shows
=
that
k
readily
Now
m=n
if
exists
Pk-i
value
k
~ k=n-i
2 (k - n + 2) S k .
X S k2 < =
Moreover,
Thus
111
~[ k=n-i
ks k =
~[ k=n-i
k {I/
} tk +---i-~ d t
~
at
--
k=n-i Hence
2 [ Sk k=m - 1
[ m=n
and
so T h e o r e m
2.3.2
implies
A2
+
=
that
(-i) n n f (yn)
Yn-i
= 0
is o s c i l l a t o r y .
Example
4:
Let
c
n
= 1
,
b
n
=
( - l ) n /In+ 6 -
,
n = 1 , 2 , ....
Then oo
Pn-i
[ m=n
=
and
the
alternating
series
(-l)m m
+------~i
theorem
0
112
[ m=n
[ k=m-i
Pk
=
[ k=n-i
(k-n+2)P
and
k=n-i
k=n-I
m=k-i
m
co
X
1
k-
k=n-i
(k - i)
2+26
oo
k=n-1
(k - 1) 1+2~
max{flail c for all
(3.3.14)
, II~lIc}
t > c .
Then
i
t {¢(x)~(t) - ¢(x) C(t) }y(t) dv(t) c
[¢ (x)~2 (t)y (t)I dV (t) +
=
c
,
lx - ~0 IR2 < !4 from which
it w i l l
follow
that
ityEic =< 2R([~I + IBII for
all
y
T,2(v: (e oo))
•
t > c
It most
one
THEOREM
.
Thus and
follows
linearly
letting
t ÷ ~
hence
in
from
this
independent
is
that
L2(V; in t h e
solution
~
be
non-decreasing
and
I~ Iy(x ' I) 12d ~ ( x ) all
1
(3.3.9). solution
Proof: , ~
find
that
I) limit-point
can
be
in
case
at
L 2 ( V ; I)
3.3.2: Let
for
we
real If
of
two
is
complex
I ~ 0
(3.3.9)
This be
Im
or
the
then
in
there
that
for
> 0
b
> a
(3.3.17)
Y I 0
is
a solution
exists
at
least
of
one
L 2 ( V ; I)
standard
linearly
where
suppose
"nesting
independent
circles"
solutions
of
analysis.
Let
(3.3.9)
satisfying ~(a,
k)
= sin
~ ( a , k)
= cos
~
~ ' ( a , I)
= -cos
~ ' ( a , l)
=
sine
(3.3.18-19)
136
where
~ c
functions
[0 , z) of
1
Then for
, 4
fixed
x
, ~'
, 4'
(Appendix
are
III,
entire
section
i) .
If w e w r i t e
[}~](x)
then
it
follows
-
from
~(x)4'(x)
(3.3.18-19)
[~] The
solutions
Every of
~, 4
are
(x)
real
(3.3.20)
4(x)~'(x)
that
:
for
(3.3.21)
1
real
1
and
~(a,
l)cos
e + ~' (a , l ) s i n
e = 0
4(a,
l)sin
~ - 4' (a , l ) c o s
e = 0
solution
the
-
~
of
(3.3.9)
is,
up
to
satisfy
a constant
multiple,
form
8 = ~ + m4
where
m
a < b < ~ requiring
is
some
and
zeros must
8 c of
introduce
which
depends
a real
on
boundary
~
Now
condition
let at
b
by
that
g(1)
for
number
(3.3.22)
-: y ( b ,
[0 , z) the
The
entire
necessarily
be
l)cos
B + y' (b, l ) s i n
eigenvalues
function real
g(1)
(Appendix
of
8 = 0
(3.3.9) Since
III,
are
these
Theorem
(3.3.23)
then
the
eigenvalues
III.l.2),
137
(3.3.23) have
does
no
of
accumulation.
We
seek
m
such
now the
boundary
as
1
are
, b
functions in
meromorphic
If
we
let
B
varies
z =
image
D =
A
lies
on
of
=
~' ( b ,
m-plane.
cot
B
0
to
from the
and
Thus Cb
8 From
m
that
is
given
solution
8
A
the
zeros
above
simple
computa-
the by
circle
and
real
z
under
~'(b,
AD
- BC
z
0
satisfy
(3.3.24)
,
I
the
is
B+
Dm Cm
is
Theorem
, ~ ,
III.l.0),
the
moment, (-~,
then
~)
so
as
that
transformation
(3
C
=
~(b,
a circle
(3.3.23 have
which
, ~'
l
over
,
we
A+
%
'
I) ,
III,
for
Az + B Cz+D
=
Since
real
varies
=
B
for
b,
m
,
, b , 8) (Appendix
fix
axis
Cb
= m(l
,
will
, I) + %' (b , I) , l) + ~ ' ( b , I)
1
z =
so
consequently
(3.3.23).
of
and
real
~ ( b , I) I)
8~(b 8~(b
vary
entire
is
where
the
condition
cot cot
--
, 8
m
the
that
and
that
m
~'
identically
point
shows
Thus
vanish
finite
satisfies tion
not
if
I)
,
Cb and
3.24)
in
only
the if
m
(3.3.25)
the
image
of
Im
z =
0
,
138
( A + Cm) ( B + Dm)
Since
every
described
circle
see
mb
of
( A + Cm) ( B + Dm)
center
y
on
2
-
IYI
comparing
Cb
2
+ Y~
+ {~
coefficients
is g i v e n
radius
rb
- ~o
of
r
can
= 0
be
(3.3.27)
(3.3.26-27)
A5- B~ CD-
its
radius
that
center
by
mb -
and
and
(3.3.26)
= 0
by
r
we
with
-
is g i v e n
(3.3.28)
DC
by
I A D - BC 1 (3.3.29) r b = i ~ D _ De I
Substituting obtain
the
the
values
equivalent
for
A
same
way
we
find
1 =
Hence,
, C
, D
into
(3.3.26)
we
equation
[88] (b)
In t h e
, B
= 0
(3.3.30)
that
[~]
(b)
= AD
- BC
[~]
(b)
= DC
- CD
[~]
(b)
= AD
- BC
139 mb =
[#~] (b) [ ~ ] (b)
(3.3.31)
and _
rb
The c o e f f i c i e n t interior
of
of
Cb
mm
in
in the
1
(3.3.32)
(3.3.26)
m-plane
[80] (b)
is
[ ~ ] (b)
and so the
is g i v e n by
0 ,
Ifl The p o i n t s
m
are on
b la
then
Im(m) Im(1)
Ol2d
Cb
say,
if and o n l y
(3.3.38)
if
el2d.~ _ Ira(m)
(3.3.39)
Im(1)
The r a d i u s
rb
of
Cb
is also g i v e n b y
(3.3.40)
rb = 2 Im I ib I~ 12 d~) a
If,
say,
~
is c o n s t a n t
on some
rb = constant
and so the c i r c l e s
lies o u t s i d e
I
If
a < y < b < ~ ,
interval remain
£
f~
then for
the same u n t i l
then
f lel2d a
I
lel2d < Im(m) zm(x)
b e I b
141 and
so
C b ~ Cy
Thus
the
sequence
(C b)
if
b > y
of c i r c l e s
is
.
(3.3.41)
"nested"
in the
sense
(3.3.41) . Assuming sequence
(C b)
or a p o i n t then
its
~
shall
as
If the
which
is p o s i t i v e
is n o t
therefore
b ÷ ~
radius
(3.3.40)
that
eventually converge Cb
constant
to e i t h e r
converge
is n e c e s s a r i l y
given
the
a circle
to a c i r c l e
by
the
lim r b
C in
and hence
b
/a I.~12dv < ~o In this
case
within
Cb
,
if
m
is a n y p o i n t
(3.3.42)
on
C
then
m
lies
b > 0
Thus
ib
1¢ + ~ 1 2
dv
0
Since
(3.4.3)
y(x)
= a +
Ir(x)y(x)
includes (3.4.4)
interest of
=
is
(3.4.4). when
that
constancy,
treatment
of
can be
found
+
f
A
r(x)
e.g.
similar is
(3.4.3)
is
when
the W e y l
is e q u i v a l e n t
B(x-a)
(3.4.4)
the
indefinite also
r(x)
defined = 0
classification
in H e l l w i g to
treat-
a.e. for
[28].
integral
equation
x (x-s)y(s)d{o(s)
-Iv(s)}
(3.4.5)
a
then
using
include (t n)
the
methods
three-term
be
a given
outlined
recurrence
relations
1 we
in
can
then
(3.4.3).
For
let
sequence,
t 1 = a < tO < tI
0
.
by
the
introduction
is r i g h t - c o n t i n u o u s We
can
then
consider
of
and d e f i n e d
by
the m o r e
general
d~(x)
which
would The
case given
then
reduce
construction
(3.4.9). by
p(x)y' (x) -
to
in s e c t i o n
For example
(3.3.40)
thus
(3.4.9)
the
if we
ydo
without 3.3
radius
let
of
(3.4.11)
= ly
(3.4.10).
also the
applies
in the
circle
Cb
is
146 a = t_l
< t o < t I < '''
< tm_ 1 = b
then i~
(since
we
always
I 12d
assume
m-i
=
that
X0
a
~
is
n
I ~n12
(3.4.12)
continuous
at
a ) .
Consequently
1 m-i I) ~ 0
rb = 2(Im
For
the
latter
result
(5.4.6)].
The
nesting
recurrence
relations
Moreover the
space
weight
of
see
the
be
space
i.e.
(yn)
125-26
analysis
found
in
L 2 ( V ; I)
square-summable
an " '
a n l ~ n 12
[3, pp.
circle
can
(3.4.13)
~
sequences ~2(V ; I)
and
for
equation
three-term
[3, pp.
125-29].
is
equivalent
then
"with
respect
if a n d
only
to
to the
if
co
2 la n Y n I
< co
0
Thus
if
[3, p.
c
n
129,
> 0
,
Theorem
b
n
is
5.4.2]
any for
sequence any
1
and
a
> 0 n =
Im
I ~ 0
,
then
,
-CnYn+ 1 - Cn_lYn_ 1 + bnY n = lanY n
has
at
least
one
nontrivial
solution
~ =
(~n)
in
£ 2 ( V ; I) .
147
§3.5
LIMIT-POINT In
and
~
this
which
limit-circle
AND
LIMIT-CIRCLE
section
will
we
enable
0{
space
L 2 ( V ; I)
over
the
interval
This
space
was
defined
~ , 9
bounded
variation
10
such
~ ~
x > a
conditions the
on
limit-point
or
of
(x) -
in
some
establish
ydo
V(x)
[a , x]
,
be
=
is
and
(3.5.0)
ly
the
I =
total
variation
[a, ~)
,
of
a >-~
(3.3.3).
right-continuous
on
that
[a, ~)
o(x)
that
locally
there
of
exists
is n o n - d e c r e a s i n g
for
a solution
with
say.
(3.5.0),
with
Proof:
Let
initial
conditions
I = 10
y(x)
be
Theorem
3.2.0,
,
has
> 1
the
y(a)
by
functions
Suppose
- 10~(x)
y(x)
Then,
give
3.5.0: Let
Then
y
to
where
~(x)
LEMMA
us
classification
d~(x)
in t h e
shall
CRITERIA:
x
solution
= 1
,
y(x)
> a
of
y' (a)
is
y(x)
(3.5.1)
(3.5.0)
satisfying
= 0
a solution
the
(3.5.2)
of
the
integral
148
equation
y(x)
= 1 +
f
X
(x- s)y(s)d(o(s)
(3.5.3)
- Xo~(S))
a
for an
x ~ a interval
for and
Since
x e
y(a)
[a , a +
[a , a +
6]
6]
,
the
then
6 > 0
by
continuity
in w h i c h
integral
in
y(x)
(3.5.3)
there > 0
exists Then,
is n o n - n e g a t i v e
so
y(x)
Since in
= 1
y(a+
6)
~ 1
[ a + 6 , 61 ]
y(x)
> 1
x
there
6)
[a , a +
exists
Consequently
= y(a+
e
+
6]
61 > 0
for
x
in
(3.5.4)
such
that
such
an
(x- s)y(s)d(o(s)
y(x)
> 0
interval
- lO~)(s) )
+6 and
so
y(x)
Repeating of
real
this
> 1
process
numbers.
It
x e
we
obtain
is t h e n
lim
otherwise so w e
if
could
diction
lim
6
repeat
proves
that
n
=
the
6*
(3.5.5)
an
increasing
necessary
6
n
sequence
(3.5.5)
y(6*)
process holds
(6 n)
that
= ~
then
above
[a+ 6 , 6 1 ]
and
> 1 =
past thus
by 6"
continuity This
and
contra-
149
y(x)
THEOREM
3.5.1:
Let
3.5.0.
Suppose
further
> 1
~ , v
satisfy
(3.5.0)
Proof: solution
of
where
the
latter
there
exists
which
exists
hypotheses
of L e m m a
(3.5.6)
at
to s h o w
(3.5.0)
the
d~(t) I =
is l i m i t - p o i n t
It s u f f i c e s
all
that
la] Then
x > a
that,
is n o t
for in
by h y p o t h e s i s .
a solution
y(t)
of
some
I ,
L2(V;
I)
From
(3.5.0)
there Let
Lemma
such
that
is a I = 10
3.5.0 (3.5.1)
holds. Then
for
such
a solution,
ly(x) I2 Idv(t) I =>
hence
y
is not
COROLLARY
3.5.1:
Let
(a n )
in
L 2 ( V ; I)
be a s e q u e n c e
0 Let
(b) n
sequence.
be any
Ida(t) I =
given
such
that
and
(c n)
fan I =
sequence
another
positive
150 If
there
bn
cn
_
exists
a real
number
I
Cn_ 1 + 10 a n > 0
_
such
0
n=
that
0 , 1 , ...
(3.5.7)
then
CnYn+l
is
limit-point
solution
+ Cn-lYn-i
at
(yn)
co ,
such
i.e.
- bnYn
for
(3.5.8)
= lanYn
some
I
there
corresponds
a
that
co
X I anllYn 12
= ~
(3.5.9)
0
Proof:
We
equations
note of
continuous locally
(t n)
and
L(a,
of
form
(3.4.11)
bounded
co)
define
in p a s s i n g
The
that when
p(x)
variation
proof
Lemma
is
3.5.0 > 0
a step-function
~(t)
to
right-
satisfying
similar
extends
p(t)
-I
with
minor
changes.
with
jumps
at
the
by
require
n = 0, 1, has
the
and
We
here
....
be
constant
We y(t)
relation
(3.5.10) satisfied.
- v(t n-
that
solutions
recurrence
~ ( t n)
and
Moreover
define such
0)
on
o(t) that
= -a n
(3.5.10)
[tn_ 1 , t n)
as
in
Y(tn)
(3.4.7).
= Yn
, Then
satisfies
(3.5.0) the
(3.5.8). the
hypothesis
for
I = In ,
imply
that
(3.5.7)
(3.5.6)
implies
is
that
151
a - ~09
is n o n - d e c r e a s i n g .
Thus
Theorem
3.5..1 a p p l i e s
and
so
f
~Jy(t) 12 Ida(t) [ = a
which
implies
(3.5.9)
In this
form,
p.
Theorem
135,
THEOREM
3.5.1
5.8.2]
where
is a m i n o r the c a s e
a
extension n
> 0
of
is c o n s i d e r e d .
a , ~
be
variation
right-continuous
on
[0 , ~)
functions
locally
a necessary
limit-circle
and
of
and
(3.5.11)
I tJda(t) I < 0
Then
[3,
3.5.2: Let
bounded
Corollary
sufficient
condition
for
(3.5.0)
to be
is t h a t
(3.5.12) 0
Proof:
We
rewrite
y(x)
the
solution
= ~ + 8x +
f
of
(3.5.0)
as
x (x-s)y(s)d(o(s)-
(3.5.13)
l~(s))
0
Then only
(3.5.0)
L 2 ( V ; I)
Theorem of
is l i m i t - c i r c l e solutions.
12.5.2]
solutions
we
y , z
find such
if say Using
now
that (3.5.13) that
(3.5.13)
with
a result with
in
I = 0
I = 0
has
[3, p.
389,
has
a pair
152
The
solutions
limit
circle
Since
z(x)
are
÷ 1
x ÷ ~
(3.5.14)
z(x)
~ x
x + ~
(3.5.15)
then
then ~ x
y(x)
linearly
these for
independent.
solutions
large
x
we
IxlZ(sl12Id(s)l
Jx I
If
must
belong
will
have
(3.5.0)
to
is
L 2 ( V ; I)
zls112s21d (s)l s I
oo
> C IxS21dv(s)
Hence
(3.5.12)
is
since
y ÷ 1
then
over
satisfied. this
A
similar
forces
v
(3.5.16)
I
calculation
to be
of
shows
bounded
that
variation
[0, ~) Next,
bounded
if
(3.5.11-12)
variation
over
are
[0, ~)
I
°°lY(x
satisfied and
then
~
must
be
of
hence
2 idv(x) I
-~
As
, v
we
functions
variation
of
"formally
self-adjoint"
by
~[y] (x)
i
It
differential
following
right-continuous
is d e f i n e d
order
a property
function
are
is w e l l
of
of
§17 a n d
[15,
equivalence, p.
42]
we
shall
to g e n e r a l i z e d
expressions.
In the
I =
~ , ~
OPERATORS:
expression.
limit-point
the
case.
the n o t i o n
in t e r m s by t h i s
When
and
a second
DIFFERENTIAL
of a p a r t i c u l a r
equivalence
an a r g u m e n t
differential
defined of
in t h i s
interpreting
(3.5.0),
III.3).
is n o n - d e c r e a s i n g
now dealing
be
the c o n c e p t
above-mentioned
of
shall
"J-self-adjointness"
(3.5.0)
on
we
of an o p e r a t o r
out
(3.5.11-12)
OF G E N E R A L I Z E D
associated
the d o m a i n
will
the
interpreting
J-SELF-ADJOINTNESS In the
of
then
=
V
.
locally usual
d~(x)
of all
we
operator
generalized
as in s e c t i o n consists
The
assume
y' (x) -
3.2 of t h i s functions
that
9 , ~
are
of b o u n d e d
variation
denote
total
i
the
generated
differential
by the
expression
(3.6.0)
y(s)do(s)
chapter.
Thus
f E L 2 (V ; I)
the such
domain that
157
i)
f
ii)
f
is l o c a l l y has
absolutely
at e a c h
point
continuous
x e
[a, ~)
on
I
a right-derivative
!
f+(x) iii)
The
{ f'(x)
function
~(x)
E f'
(x)
-
fx
f(s)d~(s)
a
is iv) For
V-absolutely
~[f](x)
f ~ P
continuous
locally
on
I
.
e L 2 ( V ; I)
i
is d e f i n e d
by
if = i[f]
The
notions
expression when
~
of
"regularity"
(3.6.0)
are
if the
set
values
is b o u n d e d
growth
of
is n o t
of
o
in
in s o m e
then
In o u r
3.2
that
if
are due
definitions
and
below
[35,
p.
regularity"
of
249]
case
I = and
the
of
V(x)
if t h e
and,
in the
end
a
and
the
a
belongs
to K a c [a, ~)
a
is r e g u l a r the
set of
set of p o i n t s
in a d d i t i o n
to be
the b a s i c
the e n d
that
right-neighborhood
it is s a i d
regular
case
say
of g r o w t h "
from below
definitions
section
shall
"points
variation regular
latter
we
is b o u n d e d
is c o m p l e t e l y These
defined
"complete
is n o n - d e c r e a s i n g . In g e n e r a l
bounded
and
(3.6.1)
of
singular. to the
o a
its
of
is of .
The
If end
interval
a a
concerned.
[35]. It is t h e n assumptions
is c o m p l e t e l y
clear on
from
~, ~
regular.
of If
the
158
I =
[a, b]
in t h e
then
be
a, b
note
case
of
a finite
of
a solution
equal. by
that
This
setting
respectively
THEOREM
on
are
since
can
each some
both
completely
regular.
9 , ~
are
continuous
interval,
the
left
y(t)
also
equal
be to
interval
of
and
(3.5.0)
seen
by
~(a)
, 0(a)
at
,
right-
will
exist
extending
containing
a , b
~ , ~
and
~(b)
there past , ~(b)
[a , b]
3.6.1: Let
£[.]
ends
We
derivatives and
the
on
I =
I
[a, b]
Let
g
be be
a finite
interval
any
function
[y]
= g
in
and
consider
L 2 ( V , I)
The
equation
has
a solution
y(x)
satisfying
y(a)
y' (a)
if a n d
only
solutions
Proof:
We
if
of
the
the
note
function
homogeneous
that
(3.6.2)
f
is
= y(b)
= y' (b)
g(x)
0
(3.6.3)
= 0
(3.6.4)
=
is
equation
J-orthogonal i[y]
J-orthogonal
to
to
all
= 0
g
if a n d
only
if b f(x)g(x)dv(x) a
= 0
(3.6.5)
159 (The
J-orthogonality
L 2 ( V ; I)
,
see A p p e n d i x
This Lemma
I].
equation y(a)
theorem
(3.6.2)
= 0 ,
= 0
can be ~ r o v e d
has
y'(a)
3.2.0,
a unique
Theorem
conditions
solution
= I
,
z 2(b)
= 0
,
z 2(b)
y
and
=
I Ixl d
~[3.6.4)
p.
[46,
1.3.1
which
62,
the
satisfies
system
of
solutions
of
= 0
we
3
[ y' zj -yzj]b',
Ixl-
j = 1, 2 ,
reduces
and using
the b o u n d a r y
j: 1 (3.6.7)
(x) d9 (x)
z
(3.6.6)
to
[ -y' (b) g (x)
find
a "
= 0 ,
(3.6.6)
= 1
J
y (b)
Thus
in
Z[y] (x) zj (x) dg (x)
b a
z
J
£[zj]
above,
to
zj (x)d~(x)
ib y(x) a
I
in
!
3.2.1
g(x)
that
as
and Theorem
!
By n o t i n g
product
satisfy
z l(b)
:
exactly
be a f u n d a m e n t a l
z l(b)
Applying
J-inner
= 0
zi , z2
which
f r o m the
III.3.)
F o r by T h e o r e m
Let ~[z]
stems
is s a t i s f i e d
if and
only
if
j= 2
(3.6.7)
vanishes
for
160
j= 1 , 2 ,
i.e.
solutions
of
f
if
£[z]
J-orthogonal
= 0
and thus
N o w s i n c e the m e a s u r e absolutely V
continuous
the q u a n t i t y
with
(section
dV(x)
the c o n c l u s i o n
induced
respect
by
~ ,
s y s t e m of
follows. in
to the m e a s u r e
(3.6.0),
is
induced by
3.1)
d~ (x) dV(x)
Consequently,
to a f u n d a m e n t a l
exists
[V]
(3.6.8)
the e x p r e s s i o n
y' (x) -
yd~
= -dV(x----~" d~(x-----~ y' (x) -
yd~
a
(3.6.9)
V-almost
Thus
everywhere
if w e d e n o t e
(section
by
by
[24, p.
£%[y]
w e see t h a t
i, and T h e o r e m
the e x p r e s s i o n
to
£% £
dV(x)
is a n o t h e r by
(3.6.9),
£%[y] (x)
and b o t h of t h e s e (3.6.10)
Ex.
defined
by
A].
y ~ D
3.2),
£%[y] (x) -
related
135,
gives
rise
generalized i.e.
=
d~
~(x)
are d e f i n e d to
an
y' (x) -
for
yda
differential
expression
y c D
• Z[y] (x)
(3.6.11)
on the same d o m a i n
operator
(3.6.10)
Lt
on
D ,
D where
Thus D
is
161 the
domain
of
L
defined
earlier,
in t e r m s
of
the
Gram
that
for
y
e D
Ly = d___~_~. dV
LtY
or,
such
operator
,
(3.6.12)
J
defined
in A p p e n d i x
III.3,
L % = JL
If,
in T h e o r e m
then
we
can
conclusion usual
3.6.1,
replace will
then
orthogonality
we
assume
i[y]
in
follow in
(3.6.13)
that
v
(3.6.2)
with
is n o n - d e c r e a s i n g
by
£%[y]
and
J-orthogonality
L 2 ( V ; I)
,
since
the
being
v { V
the
in t h i s
case. !
We
now
define
a new
operator,
denoted
by
L
o
,
with
l
domain
DO
defined
by
[46,
p.
60]
!
P0 -- {f c P : f - 0
outside
a finite
interval
[e , 8] c (a , b) } (3.6.14)
!
The
restriction
of
the
operator
L
to
!
DO
defines
L0
!
Thus
for
y
e
DO
' L0Y
=
Ly
!
Similarly
we
can
define
(L~)
(3.6.15)
= Z [y]
%
(L o )
= Lfy
, %
by
y
~
:
Zt[y]
(D 0)
= DO
(3.6.16)
162
THEOREM
3.6.2: !
a)
If
y • 90
z c D
,
then
!
[/0y , z] =
where
[ , ]
side of
(3.6.17)
[y, Lz]
J - i n n e r p r o d u c t d e f i n e d by the left hand
is the
(3.6.5). !
Moreover,
the o p e r a t o r
L0
J-hermitian,
is
!
i.e.
!
!
[L0Y , z] =
y , z e 90
[y , [0 z]
(3.6.18)
!
b)
If
y
e
90
,
z
•
D
then
writing
LI
E
(/O)f
we
have (ily , z) = (y , Lfz)
where
( ,
)
is the inner p r o d u c t
Again,
the o p e r a t o r
(L1y, z) =
Proof:
iI
in
L 2 ( V ; I)
is hermitian,
(y, Llz)
i.e.
y, z E P
Both a), b) can be shown as in
(3.6.19)
[46, p. 61] m a k i n g use
of T h e o r e m 3.2.1 so we omit the details.
We now p r o c e e d as in operators
L0
Suppose £, £f
and
[46, §17]
in d e f i n i n g the
L%0
that the interval
are both r e g u l a r on
[a, b]
[a, b] .)
is finite.
(Then
163
We
define
DO =
and,
for
y
the
{y • D :
domain
DO
i.e.
similar.
any
y
operator
for
Proof:
L0
by
(3.6.20)
= ty
L% y
(3.6.21)
.
(3.6.22)
3.6.3:
For
the
operator
~ DO ,
% i0y =
and
the
y(a) = y(b) = y' (a) = y' (b) = 0}
toY
THEOREM
of
We
any
c DO ,
L0 y,
refer
z e D
[i0y , z]
=
[y , iz]
(3.6.23)
(L0%Y, z)
=
(y,
(3.6.24)
is
L+z)
J-hermitian
while
L %0
is h e r m i t i a n ,
z e DO ,
to
[L0Y , z]
=
[y , L0 z]
(3.6.25)
(L0%Y, z)
=
(y,
(3.6.26)
62,
I,
[46,
p.
L0%z)
II]
since
the
proofs
are
164
LEMMA
3.6.1: Rt 0
Let solutions
of
~ range
the
of
L0%
and = 0
~[z]
equation
M
let
be
the
set
of
all
.
Then H : Lg(V;
Proof:
Since
continuous so
M
all
solutions
functions
c H
It
is a l s o
readily
in
solution
(3.6.2),
with
in
DO
Hence
implies lies H
R% 0
in
is
if a n d
a Hilbert
THEOREM
i0
have
to
i%0
3.6.3,
the it
DO
then
solution
Z
if
the
they
all
that
replaced Thus
it
belong
the
H
y
,
and
y
of
y
states to
is
that
g
.
Since
M
(3.6.27)
is a
then
existence
is o r t h o g o n a l
decomposition
to
If i%
then
are
is a f i n i t e
2
by
3.6.1
equation
M
dimension
i~y
only
homogeneous
seen
Theorem
space
domain
Since
and
gonal
=
of
,
(3.6.27)
follows.
3.6.4: The
Proof:
Z%[y]
H
g c R %@
that
the
[a , b]
subspace
of
of
on
dimensional
= R %0 + M
I)
of
the
domain
D
suffices
to
is
zero.
(h, y) of
DO
~%[z]
= 0 = h
operator
is t h e
0
show
for
h
all For
y y
same
that
Letting
f0
be
~ DO
for
every such
E DO
is d e n s e
the
Let we
have,
h
element, z
be by
H
operators
element an
in
orthowe
any
Theorem
165
(z , L%0Y) =
and
so
z
previous
is o r t h o g o n a l lemma,
We is d e n s e theorem
expresses
the
THEOREM
3.6.5:
The symmetric
R %0
a set norm fact
is d e n s e
operators
0 ,
i.e.
is d e n s e
= 0
the
Thus
are
space
the
the
if
it
latter
90
space
% i0
and
by
h = 0
domain
Krein
z • M
in a K r e i n
topology.
in t h e
(h , y)
Consequently
that
i0
=
of
the
H
J-symmetric
and
respectively.
This
Note:
The
shown
to be
follows
rest
in
case
whose
Theorem
of
true
regular
operator
from
the
Theorems
results
this the
more
3.6.3-
in
[46,
general
operator
J-adjoint
ix 0
i0
4.
§17]
can
setting.
be
Thus,
is a c l o s e d
is e q u a l
to
i
similarly e.g.,
J-symmetric
[46,
p.
66,
i].
In a > -~
(Z%[z] , y)
£%[z] =
that
Hilbert
=
to
so
in t h e
i0
in t h e
and
recall
operator
Proof:
(L%z , y)
,
the we
singular
follow
the
case,
i.e.
approach
when
outlined
I = in
[a , ~) [46,
,
§17.4]
!
where
we
begin
with
the
operator
i0
defined
in
(3.6.14-15).
both
defined
on
the
!
We
recall
domain
~
that
L0
and
il
are
same
166
THEOREM
3.6.6: I
The
domain
of
definition
!
00
of
i 0
is
dense
in
!
and
L0
Proof:
is t h e r e f o r e
An argument
a
J-symmetric
similar
operator.
to t h a t of
[46, p.
68]
shows
that
in
.
I
00
,
when
viewed
as
the
domain
of
[1
'
is
dense
H
!
Thus is
LI
is a s y m m e t r i c
operator,
by T h e o r e m
3.6.2,
and
L0
J-symmetric.
W e n o w take space
topology
the c l o s u r e
of
L0
, L0
in the H i l b e r t
and d e f i n e
L0 = L0
it t h e n
follows
f r o m the p r e c e d i n g
closed
J-symmetric
i+
(and so
L)
that
i0
is a
operator.
We n o w p r o c e e d of
theorem
to find a p r o p e r t y
when
£%
of the d o m a i n
is in the l i m i t - p o i n t
0
case
in
L 2 (V ; I)
LEMMA
3.6.2:
[14].
F o r any set of six f u n c t i o n s {gq : 1 ~ q ~ 3} [0, ~) point
each being
and each having x • [0 , =)
,
locally
a finite
{fp : 1 ~ p ~ 3} absolutely
right-derivative
continuous at e a c h
on
167
d e t { [fp qq] (x) } = 0
x~
[0,~)
where !
[fg] (x)
Proof:
LEMMA
See
[14, p.
--- f(x)g+(x)
- g(x)f+(x)
374].
3.6.3: Let
be the
p
be a B o r e l m e a s u r e
s p a c e of s q u a r e
Suppose functions
that
which
integrable
f, g
X > 0
[0 , ~)
"functions"
and let
L2(~)
with respect
are c o m p l e x - v a l u e d
to
p-measurable
satisfy
f ~ L 2 0
the
Let
domain
the
iT = Ji
that
identity
a : 0}
domain
(3.6.35)
,
.
(3.6.36)
same
domain
D
and
f E D
= Lff
case,
note
Lagrange
is
, Lf
~- f~(0)sin
We
now
f Le
is
if
f, g c D
[fg] (0)
Next
D
by
and
Lff
or w h a t
and
L
operators
L
Similarly
3.6.2
earlier.
define
be
in L e m m a
(3.6.37)
proceed
to
show
that,
self-adjoint.
= 0
(Theorem
then
.
3.2.1)
(3.6.38)
shows
that,
for
171
I o { f ( x ),%9[g] (x) - ~[f] ( x ) g ( x ) } d V ( x )
= -[fg] (x)
Consequently X + ~
in
if
£%
(3.6.39)
is and
(L f , g]
and
so
i% e
it c o n t a i n s the
singular
to t h a t shown
in
symmetric
the
domain
case [46,
by
p.
the
3.6.7
of
the
L0
VII).
and
, we
(3.6.38)
In
in
The
proof
the
same
L0
of
find
(3.6.40)
L 2 ( V ; I)
operator
let
to
f ' g ~ De
is d e n 3 e
e
f , g c D
since
defined
this
fashion
is it
in
similar can
be
that
so t h a t
domain
ID
L0 =
[ief , g]
THEOREM
Theorem
DO
71,
and
t [f, L e g )
:
(3.6.39)
[fg] (0)
limit-point use
is
+
L
e
is
3.6.9: of
=
[f , Leg]
J-symmetric.
In t h e
limit-point
self-adjointness
following
i)
ii)
(3.6.41)
f , g e De
of
case,
L% e
the
if a n d
domain
only
if
D D
e
e
properties,
For
all
If
g c D
f e D
e
f, g e D
,
e
,
satisfies then
g e 0
e
[fg] (0)
= 0
[fg] (0)
= 0
,
for
all
is has
a
172
Proof: We theorem
note
that
of N a i m a r k
this
[46,
result
p.
73,
is a p a r t i c u l a r
Theorem
l] and
case
of a
c a n be p r o v e n
similarly. With shows and
that
D
defined
both
(i) a n d
consequently,
adjoint.
On the
the d e f i c i e n c y Consequently
(3.6.35)
are
[46,
if p.
a simple
satisfied
limit-point
hand,
indices the
in
(ii)
in the other
as
case
L% 26]
computation
in T h e o r e m L%
3.6.9
is s e l f -
is s e l f - a d j o i n t of the o p e r a t o r
then are
equation
(3.6.41)
(i~)* z = Iz
has
no n o n - t r i v i a l
self-adjoint
(0 , 0) .
solution
(3.6.41)
in
implies
L 2 ( V ; I)
that
Since
i%
is
the problem
L%z = Iz
z(0)cos
has
no
solutions
point.
Hence
THEOREM
3.6.10: The
L2(V;
I)
in
we have
equation
if a n d o n l y
self-adjoint.
e - z'(0)sin
L 2 ( V ; I)
Thus
e = 0
(3.6.28)
is l i m i t -
proved
(3.6.28)
is in the
if t h e o p e r a t o r
limit-point L%
, e c
case
[0, ~)
,
in is
173
In the
the
Hilbert
following
space
operators
L% e
self-adjoint. limit its
point
adjoint i
t
J-adjoint
by
and Lx e
Let
f
e D
and
Krein
Theorem
so
i
space
exists.
e
, g
[Lf
Let 3.6.10,
is
e
Lx
(i~)
respectively.
e
Then case
discussion
a
us
'
e
adjoint
of
suppose
that
(3.6.28)
J-symmetric
We d e n o t e
will
'
its
is
denote
the
in
i% e the
operator
domain
by
is
and Dx e
e Dx e
, g]
=
[f , Leg] x (3.6.43)
NOW
L % = JL
since
e
e
L% =
,
=
(L~)*
e
JL x = e
=
(JLe)*
L*J e
.
Moreover
L*J
(3.6.44)
e
"k
Substituting
(3.6.44)
into
(3.6.43)
we
find
g
that
E D
LeJ)
But
hence
g
e De
J-symmetric
Consequently D
c Dx e
e
.
Hence
Dx
c D
D
=
e
and
since
Dx e
and
so
be
J-self-adjoint
e
L i
e
e
1s
is
J-self-adjoint. On each
e
E
the
other
[0 , ~)
,
hand, so
that
let
i
we
have
L
=
Lx e
or
for
•
174
[f , Leg]
Using f,g
the E ~
Lagrange
=
[ief , g]
identity
in
(3.6.45)
we
find
that
for
e fx 0 = x+~lim 0 { f i - - ~ - g / e f } d v
Since
(3.6.45)
f , g E P
f, g ~ D
,
e
lim
g ( 0 ) f ' (0)
{f(x)g'
= x÷~lim [gf' _ ~,f]x0
- g' (0)f(0)
(x) - f' ( x ) g ( x ) }
= 0
=
"
so t h a t
(3.6.46)
0
X-~OO
for
all
f, g e
equality suffice
in to
fact show
We
e holds that
functions,
f e 0e
holds.
For
if
[fg] (x)
- f ( x ) g ' (x)
=
where
now
for
if
wish
all
show
f, g
f , g
' g e ~B
to
are
'
that
E ~
any
e , B c
the
For
two
latter
this
it w o u l d
real-valued
[0 , ~)
then
(3.6.46)
f , g c D
- f' (x)g(x)
[fRgR ] (x) + [fIgi ] (x) + i{ [figR ] (x) + [gifR ] (x) }
f = fR + ifI
'
g = gR
+ igI
Hence
for
given !
f c ~ are
,
fRc
real.
B ~ e
,
for
A similar B ~
Thus f E ~e
~e
, g
some result
[0, ~) let
e D8 ,
The
f , g ~ e
e e
[0 , ~)
holds
for
result
now
be
two
.
We
real will
since fI
fR(0)
e ~B
,
fR(O)
where
follows.
valued show
functions
that
under
with certain
175
hypotheses
on
a , ~
we
can
find
a
function
g*
E D
such
that
[fg] (x)
One
such
=
[fg*] (x)
condition
continuous
so
is
that
the
for
the
g*(x )
sense be
of
defined
the
some
a , b
absolutely
are
also
=
to
x
o
.
be
~-absolutely
function
~
,
~d~
measures
defined
to
(g(x) ~ ax+b
=
be
continuous
wish
Let
~-measurable
by
}
,
x
-> 1
[
,
0
< x
o
Let
and
have
(3.6.47)
determined. so
g*
We
< 1
need
g*(x)
to
be
that
g*(1)
We
large
by
g*(x)
where
all
following:
do
in
for
e D
= g(1)
(3.6.48)
where
~
is
as
above.
Consequently
g*(0)cos
(3.6.48-49) function
then
g*(x)
e
determine has
the
- g*' ( 0 ) s i n
a , b following
in
~
=
(3.6.49)
0
(3.6.47).
properties:
The
resulting
176
g*
{ A C I o c (0 , oo)
g +*'
and
(x)
-= g +*, (x)
G(x)
exists
Sx
-
for
each
x
> 0
g*(s)d~(s)
(3.6.50)
0
is
then
defined
for
G(x)
for
x
> 0
Since
~-absolutely more 0
< x
> 0
a
-
g*
1
,
G(x)
since
,
g
is e D6
Further-
9-absolutely
is
m-absolutely
continuous
continuous
L2(v; t0 ~)) (3.6.49)
there
large
and
From
> i
is
implies
exists
g*
therefore
[fg] (x)
=
lira
that
~ D
for
[fg*]
f
g*
~ D
such
that
E D
,
it
then
(x)
X-WCO
=
(3.6.46).
x
G(x)
X-~OO
by
fact
0
G(x)
~
in
g*
g
g e D8 ,
lim
x
that
Finally
x
and
if
Hence
Lg*
i =
continuous
(3.6.51)
if
=
x
the
0
preceding
discussion
that
[fg] (x)
:
0(i)
f , g
{ D
.
follows
and
177
Theorem case
3.6.7
now
and Theorem
self-adjoint. the
3.6.10
Hence
with
jumps o
on
with
difference
then
i%
.
same
forces
under
When
points
then
operator
is
J-self-adjoint
differential
since
hypothesis
on
~, ~
than
those
THEOREM
3.6.11:
other
to be implies
step-functions continuity
thus
the
result
in t h i s
in the o r d i n a r y
operators
i
in the K r e i n
continuous
general
of
and
A similar
operators
[0 , ~)
absolute
is s a t i s f i e d
ordinary
limit-point
above-mentioned
the
9
this
~ E
are b o t h
to
if it is s e l f - a d j o i n t .
weakening
the
respect
absolutely
is in the
i% ,
o , ~
and only
are
(3.6.28)
J-self-adjointness
of
o , ~
at the
that
the
self-adjointness
restrictions
of
implies
case
sense.
we may
space
if
holds
for
both
~,
In f a c t
include
previously
resulting
by
more
mentioned.
Suppose
(3.6.52)
So tido(t) I
CD
with tion
the
CD =>
313-14] case
converse SLP
false was
for o r d i n a r y
the p r o o f For
is n o t v e r y
suppose
that
[17,
to be v a l i d
differential
that i.e.
(3.7.7)
in g e n e r a l
shown
be
p.
313].
The
by E v e r i t t
expressions.
[17,
In the
implicapp. general
different. i
is
CD
at
infinity.
We
find
182
upon
integrating
IX
__
f(Ig)d~
by
=
parts
that,
(fg') (a)
-
for
all
(fg') (x)
f, g
IX
+
a
i
limit
as
for
,
(f'g' + fgd~)
a
Since
hand
e ~
is
CD
the
x ÷ ~
integral every
Moreover
tends
to
f , g ~ ~
~ lim
If p o s s i b l e ,
right-hand
tends
to a f i n i t e
f , [g c L 2 (V ; I)
since
a finite
limit
as
x + ~
the
left-
Consequently,
,
(fg')(x)
let
integral
exists
us
assume
that
lim
If(x)g'
(x) I =
and
is
e ~ 0
finite.
for
some
(3.7.8)
f,g
~ P
Then
I~I
> 0
X-WOO Thus
for
x > X
,
1 I f ( x ) g ' (~) I > ~
If
f
is u n i f o r m l y
inequality zero.
implies
bounded that
Consequently If
increasing
f
is n o t
sequence
g'
above
g'(x) { L2(X,
uniformly {Xn}
with
(3.7.9)
I~
on
[X, ~)
is u n i f o r m l y ~)
which
bounded xn ÷ ~
then
the
bounded
latter
away
from
is a c o n t r a d i c t i o n .
then
there
along
exists
which
an
183
f ( x n)
(3.7.9) and
then
implies
÷ co
that
,
n ÷
If(x) I > 0
for
x ~ X1 ,
say,
hence
1 If' (x)g' (x) I > ~
Integrating
the
latter
f' (x) f(x)
Isl
over
x => X 1
[X 1 , x n]
and
letting
n ÷ ~
we
find
If'(x)~'(x)
dx
=
X1 a contradiction,
by
f' , g'
( L 2 (a, ~)
Again,
in g e n e r a l ,
p.
313].
the
Hence this
now
interpret
relations, ordinary
§3.8
the
n
> 0
conclusion
implication
=>
these theory
differential
CD
for
(c n) all
having
is
since is
both
that
CD
irreversible
=>
SLP
[17,
SLP
=>
for
been
LP
.
(3.7.10)
three-term developed
recurrence in
the
case
of
expressions.
, (bn) n
=>
results
DIRICHLET CONDITIONS RELATIONS: Let
c
the
inequality,
Thus
DI
We
Schwarz
Let
FOR
be
THREE-TERM
real
(a n )
sequences be
RECURRENCE
and
a sequence
suppose of
real
that numbers
184
where
an
~ 0
sequence
where t
n
÷
both that
of
C_l ~
for
numbers
tn
tn_ 1 -
-
and
as
n ÷
Now
define be
these
o (t n)
Let defined
=
n=
a
an
increasing
by
1 Cn-i
t_l
be
(t n)
is
(3.8.1)
0 , 1 , ...
fixed.
We
also
assume
by
requiring
that
~ step-functions
constant
have
_
n
real
> 0
v , ~
all
on
v , o
[tn_ 1 , t n)
discontinuities
0 (t n
0)
_
cn
=
at
Cn_ 1
+
-
n=
l
the
b n
0,
(t n)
1,
...
only,
n=0,1,
,
that and
given
by
...
and
9 ( t n)
We that
also
-
suppose
neither Let
~(t n-
0)
that
~ , o
have ~
a
be
jump
=
at
-a n
are
,
n=
both
0 , 1 .....
continuous
(3.8.2-3)
at
a
and
infinity.
summable
and
y
y do
consider
the
differential
equation
£[y] (x)
=
d~(x)
(x) -
=
~(x)
x
e
[a , ~) (3.8.4)
where above
~ , o as
the
are
defined
solution
of
above.
Rewriting
a Volterra-Stieltjes
the
solution integral
of
the
185
equation
we
see that,
solution
y(t)
then
satisfies
Yn
-Cn Yn+l
using
is linear
the m e t h o d s
on
[tn_ 1 , tn)
the r e c u r r e n c e
- Cn-i Yn-i
of C h a p t e r
+ b n Yn =
and
if
i, the Yn
~ Y(tn)
relation
n=0,1,
an %n
... (3.8.5)
where Thus
~n H ~(tn) the d o m a i n
generalized gonal
Moreover space
of the o p e r a t o r
differential
curves,
continuous
D
i.e.
and
the
space
£2(Ia I)
,
expression
each
linear
function
on
generated
above
in
~
[tn_ 1 , tn)
L 2 ( V ; I) i.e.
i
f e i2(la I)
of p o l y -
is a b s o l u t e l y
for
becomes,
consists
by the
n= 0 , 1 , ....
in this
case,
the
if
oo
[. lanIIfn12
0
consequently
with
space
the
(3.8.24)
~ = 0
i
(This
setting.)
When
self-adjointness it
then
every
follows
of
the
that
L Moreover
generally
EXAMPLE
the
in
(3.7.10)
are
valid
and
~ Z2
and
3.8.1: c
n
= n
I1 Yn
b
a
n
= 1
and
if n = 2 m
let
some
m
> 0
if
n
2 TM
by
n
b
=
CnYn+l
n
Y0
,
= 1 n
Define
implications
irreversible.
Let
where
the
1)sin
= 0
+ Cn-I Yn-i
n = 1,
2,
...
that
Yn
Yn
say.
A computation
shows
191
if
zn
is a l i n e a r l y
Yn Zn+l
- Yn+l
by
the d i s c r e t e
z
c Z2
n
would
the
analog
LP
-
n
the
n
=
of the W r o n s k i a n
inequality
right
Cn Y n + l
is
-
a contradiction
while
solution
const
=
Zn
Schwarz
produce
finite
independent
side
+cn-I
applied since
the
diverges.
Yn-1
then
we m u s t
have
1 , 2 , ...
identity. to the left
Thus
latter
if
identity
side would
be
Thus
- bnYn
= 0
(3.8.25)
However
lim n Yn AYn
does
not
even
exist.
lim inf n +~
Thus
(3.8.25)
show
that,
irreversible next
result
THEOREM
In
fact,
n Y n A Y n = -i
is n o t
SLP.
in g e n e r a l , even
for
follows
,
lim sup n Y n n÷~
Other
the
implications
three-term
from
examples
remarks
AYn = 0
may
(3.7.10)
recurrence i,
be
2 of the
and
sufficient
condition
preceding
for
to
are
relations.
3.8.1:
A necessary
found
The section.
192
-Cn Yn+l
to be for
LP
all
in t h e
(here
a
all
n
lanl
is
that
(3.8.26)
(3.8.22)
should
hold
O)
> 6 > 0
for
(3.8.26)
independently
of
Proof:
let
If we
n
the
is l i m i t - p o i n t coefficient
f ~ D
,
and
that
0 < cn < M
fn ÷ 0
as
(3.8.22)
n ÷ ~ holds
[
for for
b
in the
£2(lal)-sense
n
then
]fn ]2 < 6 -I
bounded
all
.
Then
Thus
~
n
= la n y n
3.8.1: Let
for
+ bnYn
£2(lai)-sense
f, g c V
COROLLARY
- Cn-lYn-i
lanl [fn 12
0
of
a three-term
the
It
turns
recurrence
these
upper
of
non-real
case.
b
EQUATIONS
n
numbers
is
number
results
theorems
bound
obtained
(cf., [76],[77],[78]).
DIFFERENCE
we
the
illustrate
that
possible.
the
indefinite
Furthermore C_l
parts,
complement
for
in t h e
We
real
0 , 1 , ... , m - i
formally
negative
suppose
theorem
true.
examples also
we
(4.0.0-1)
analogous
sequence
..., m-i
be
another.
upper
STURM-LIOUVILLE WEIGHT-FUNCTION:
finite
real
In t h i s
of
the
the
separate
by means
n=
y(x,
X].
eigenfunction.
zeros
is
and
and
is a l s o
4.1
(4.0.0),
[0, i]
relation
therein
in
,
Let
eigenvalues out
Theorem
[0 , i]
,
by
302~
in
corresponding
[53]
p.
, and
second
-A(c n - lAY n-i ) + bnYn
= lanYn
a
let
fixed.
self-adjoint
n=
AN
INDEFINITE
0 , 1 , ... , m - i ~ 0
n c
n
We
order
WITH
any
,
> 0 shall
, be
difference
n=0
be
n=
-i , 0 , 1 ,
dealing
with
equation
' 1 ' .... m-i
(4.1.0)
200
where
A
is t h e
meter
and
m
forward
introduce
Y-I
(4.1.0-i)
(4.1.0-1)
define
is t h e n
y(l)
where
y_l(1) For
define
a
=
where
we
m-vectors
The also
following be
we
,
1
is a p a r a -
conditions
Ym = 0
an e i g e n v a l u e
(4.1.1)
problem.
complex)
, yl(1)
solution
m-vector
.....
y(1)
of ,
(4.1.2)
Ym_l(l)]
f =
(f0 ' fl . . . . .
f-1
their
[f , g]
=
m-i [ 0
summation
(4.1.3)
+ bnfn}
= fm = 0
define
of
we
fm-i )
by
= ani{-A(Cn-iAfn-1)
formula
A
= 0
i[f]
it t h a t
f ,g
= 0
m-vector
m-vector
take
boundary
(yo(1)
= ym(1)
£[f]n
the
(possibly
a given
the
operator,
> 2
If we
then
difference
by
definition.
"J-inner-product"
For by
(4.1.4)
fngnan
by
parts
given
[30,
p.
17]
should
useful, N
N+I
uk v k : [Uk_iVk] M
N
- [ vkAuk_ 1 M
M
(4.1.5)
201
We now define the c o l l e c t i o n
Q(f)
a quadratic
of all c o m p l e x
:
functional
m-vectors
f
Q(f)
with domain
by
(4.1.6)
Cf , ]~[f]] m-i
= c_11f012
+
I {CnJAfnl2+bnlfn12}
(4.1.7)
0
w h e r e we o b t a i n
(4.1.7)
from
(4.1.6)
upon
the a p p l i c a t i o n
of
(4.1.5).
THEOREM
4.1.i: Let
tion of 0
N
.
We
shall
proceed
208
to
show
that
orthogonal
the
to
Thus
e
n
in
(4.1.19)
must
e
in
M
(4.1.19)
must
n
satisfy
=
into
0 , ... , N - I
the
latter
equation
following
ej
~ n=0
fn(Ir)Yn(~j)
r =
0 , 1 , ... , N - I
always
has
a non-trivial
system
a set
of
property
(e i)
Since solution
The
Lemma
implies
}
M
that
z
is
.
we
find
that
of
N
equations
=
> N
the
last
0
equation
e 0 , e I , ... , e M _ 1
resulting
m-vector
z
then
Fix has
that
Q(z)
by
r=
0
the
where
the
so
unknowns,
[ j =0
such
chosen
have
m-i [ z fn(Ir) n=0 n
the
be
f0 ' fl ' "'" ' f N - i we
Substituting
can
4.1.1.
Moreover
> 0
Z_l
=
(4.1.21)
zm
=
0
that
Q(z)
=
[z , £ [ z ] ]
m-i =
[
n=0
z
£[z]
n
a
n
n
Hence
(4.1.7)
209
m-i
M-I
}rM[l
{r~0
n=0
erYn(Ur)
{s:0
es~sYn(Us)}an
M-I
Z
eres~s{~il
r,s=0
anYn(Ur)Yn(Us)}
M-I e r e s ~ s [ Y ( U r) , Y(U s) ]
r, s¼0
Now
since
(4.1.20)
[r ~ U s
for all
r, s ,
0 < r , s < M-1
,
implies
[y(u r) , y(u s) ] = 0
for all
r, s ,
0 < r , s < M-I
.
Hence
Q(z)
This, and
however,
the
EXAMPLE
is in c o n t r a d i c t i o n
theorem
is c o m p l e t e l y
(4.1.22)
• 0 .
with
M < N
(4.1.21).
Thus
=
Consider
proved.
4.1.1: Let
c
n
= 1 ,
b
n
= -2
and
a
n
(-i) n
the
problem
-A 2 Y n - i
- 2Yn = I (-i) n Yn
Y-I
= 0 = Ym
n= 0 ....
, m-i
(4.1.23a)
(4.1.23b)
210
where
m > 2
The c o r r e s p o n d i n g
2 -A Y n - i
with
the same b o u n d a r y
values
and c o n s e q u e n t l y If w e put
that,
if
m = 2k
the
zeros
negative
Y2k(1)
of
of
number,
are p r e c i s e l y
are all real. computation
shows
problem. 1
In fact
while
table
illustrates
k ,
of p o s i t i v e
and
the e i g e n v a l u e s
Y2k+l(1)
of The f o l l o w i n g
ym(1)
eigen-
= Y2k(-l)
has an e q u a l
definite
only even powers
m u s t h a v e o n l y real
then
zeros which
associated
(4.1.24)
a straightforward
Y2k(l)
and thus
problem
- 2Yn = lYn
conditions
Y0 = 1
"definite"
this:
Y2k(1)
of the
consists
of
has o n l y odd p o w e r s
211
TABLE
I
~0(~)
i
"'"
{&(x)
0
-1
~2(I)
-i
0
i
~3(I)
0
2
0
-i
~4(I)
1
0
-3
0
1
~5(i)
0
-3
0
4
0
-i
y6(l)
-i
0
6
0
-5
0
1
y7(1)
0
4
0
-i0
0
6
0
-i
~S(1)
1
0
-i0
0
15
0
-7
0
1
yg(1)
0
-5
0
17
0
-21
0
8
0
"'" "'" ''" "'" -'' "'" ''" .'' -i
---
o o .
The t a b l e power of
change
of the polynomial
(.4.1.22-23)
zeros must there and
table then
corresponding
coefficients
to
solutions
shows
even
be n o n - r e a l
of the
if
Z2k(1)
then
the
former
powers
and o c c u r
of
eigenvalues
indefinite we
simply signs.
is t h e p o l y n o has p o s i t i v e
I ; consequently
in c o n j u g a t e
of
the
t a b l e to p o s i t i v e
that
pairs of non-real
negative
To o b t a i n
sign change)
in t h e a b o v e
Y2k(1)
and o n l y
are M = k N = k
signs
of the corresponding
concerned.
(up to a p o s s i b l e
the negative
The r e s u l t i n g mial
the coefficients
I and o f t h e p o l y n o m i a l
coefficients problem
gives
pairs.
eigenvalues, (4.1.24-23).
its Thus
if m = 2k
,
212
§4.2
STURM-LIOUVILLE WEIGHT-FUNCTION The main
problems regular
DIFFERENTIAL
difference
for d i f f e r e n t i a l case,
is t h a t
number
of e i g e n v a l u e s
number
of e i g e n v a l u e s ,
the
"eigenfunction"
is m o r e is,
in b o t h
the
p(x)
cases,
> 0
latter
while
the
under
on
former
proper
number
and bounded shall ,
case it
eigenvalues constants.
following
~ L l o c ( a , b)
assumptions:
is s u c h
that
problem
-(p(x)y') ' + q(x)y
y(a)
admits point that
infinite
in the o t h e r
similar
the
an
a finite
In o n e
of n o n - r e a l
make
q(x)
has
while
by
in the
at m o s t
always
is t r i v i a l
finite
[a , b]
has
INDEFINITE
of e i g e n v a l u e
conditions.
the
AN
equations,
always
Still,
we
WITH
the h a n d l i n g
and difference
the
In t h i s s e c t i o n That
between
expansion
involved.
EQUATIONS
a denumerable
number
of a c c u m u l a t i o n . both
8.4.6].]
p
-i
set
in
Associated
case
(4.2.2)
= 0
(Conditions
the
(4.2.1)
of e i g e n v a l u e s
, q ~ L ( a , b)
In t h i s
orthonormal
: y(b)
: ly
,
see
having
which
guarantee
[3, p.
eigenfunctions
no
215, form
finite this
are
Theorem a complete
L 2 ( a , b) with
(4.2.1-2)
is the
"indefinite"
boundary
problem
-(p(x) z')'
+ q(x) z = Ir(x) z
(4.2.3)
213
z(a)
where such
r(x) that
measure. that p.
288]
is a r e a l - v a l u e d r(x)
takes
both
(4.2.4)
= 0
function signs
on
defined
some
on
subsets
"Indefinite ease" is c h a r a c t e r i z e d
The
both
= z(b)
q, r
have
and not being
a variable equal
sign
in
[a , b]
[a , b]
of p o s i t i v e by the (see
fact [53,
a.e.
4.2.1:
LEMMA
f
Let eigenvalue
be an e i g e n f u n c t i o n of
1
(4.2.3-4).
corresponding
to s o m e
Then
(plf'12+qlfl2)dx
= ~
fbrIfl 2 d x
(4.2.5)
a
Proof: over
We multiply
the
interval
fa
(pf')
Integrating boundary
LEMMA
and
(4.2.3) [a , b]
f dx + I
by to
F
f
and
first
conditions
rlf] 2 dx =
integral
(4.2.4)
both
sides
find
a
the
integrate
by p a r t s
the
result
la
qlf[ 2 dx .
and
applying
the
follows.
4.2.2:
Let eigenfunctions
I , H , f, g
I ~ ~
be
two
respectively.
non-real Then
eigenvalues
with
214
b r(x) f (x)g(x)dx
i.e.
f,g
l
are
J-orthogonal
= 0
(4.2.6)
in the K r e i n
space
L2
Irl)
and
f
b
g' {p(x) f'(x)
(x) + q ( x ) f ( x ) g ( x )
}dx = 0
(4.2.7)
a
Proof:
We have
along with
f(a)
(4.2.8)
g
by
we obtain,
upon
(l-i)
-(pf')'
+ qf = Irf
(4.2.8)
-(pg')
+ qg : urg
(4.2.9)
= f(b)
and
= g(a)
(4.2.7)
integration
the
latter
~ab
f
over
:
of the b o u n d a r y
Multiplylng
[a , b]
,
{(pg')
f-
integral
g}dx
= 0
and s u b t r a c t i n g
' {(pg')' f - ( p f ' )
because
by
r(x)f(x)g(x)dx
and i n t e g r a t i n g
= g(b)
by parts
(pf')
b [P(g'f- gf')]a
=
0
This
g}dx
we find
=
conditions.
the r e s u l t s
proves
(4.2.6)
since
215
Multiplying
(4.2.8)
by
g
and integrating
over
[a , b]
we obtain b b la {-(pf')' g + qfg}dx = X la rfg dx
Integrating
the first term in the left by parts we see that b -
la (pf,), ~
=
-
[pf,_g ] a b
+
dx
pf'g
= 'lab pf'q' dx
Thus
b{pf,~, + qfg}dx = ~
=
by
(4.2.6).
This completes
Associated A
defined
in
(A) = {y E L 2(a
where
for
with
L2(a , b)
,
~(A)
is the differential
with domain
'
c ACIo c(a
D(A)
,
,
Ay = -(py')'
If we let
0
the proof.
(4.2.1)
b) : y , py
y E ~(A)
rfgdx
be defined
by
+ qy
b)
defined
and
operator by
Ayc L 2 (a , b) }
216
D(A)
= {y • O(A)
: y(a)
= y(b)
(4.2.10)
= 0}
and let
Ay = Ay
then
A
is a r e s t r i c t i o n
a symmetric The regular
operator
[34,
following
y • D(A)
of
A
§4.11,
lemmas
Sturm-Liouville
to
(4.2.11)
D(A)
Theorem
is,
and i].
are p a r t of the t h e o r y
equation
in fact,
and can be f o u n d
of the
in
[34],
thus w e o m i t the proofs.
LEMMA
4.2.3: a)
above,
The r e g u l a r
is b o u n d e d
below,
Sturm-Liouville i.e.
there
operator
exists
A ,
defined y •
a constant
such t h a t
(Af , f) > y(f , f)
where
(
,
b) negative
Proof:
)
is the u s u a l
The o p e r a t o r
A
,
f • D(A)
inner product
has at m o s t
in
(4.2.12)
L 2 ( a , b)
a finite
number
of
eigenvalues.
For p a r t a)
Corollary].
For
see
P a r t b)
f • D(A)
[34,
§5.17,
is p r o v e d
,
in
Ex.
[34,
the e x p r e s s i o n
5.3 ° and §5.8,
§6.7,
Theorem
(Af, f)
2].
defines
a
217
quadratic
f u n c t i o n a l with v a l u e s
(Af, f) =
{plf'I2 + q l f I g } d x
This is i m m e d i a t e
LEMMA
f ~ 0(A)
if we f o l l o w the a r g u m e n t
leading
to
(4.2.5).
4.2.4:
We d e f i n e
D (Q)
by
oo
: {y ( L 2 (a , b)
D(Q)
: [ Iljl I (y , ~j)I 9 < oo} 0
oo
Q(y)
where
(lj)
(4.2.1-2)
= [ I ] (y , ~j) 12 0 J
, (~j)
Y { D(Q)
are the e i g e n v a l u e s
and e i g e n f u n c t i o n s
of
respectively.
Henceforth L 2 ( a , b)
( ,
)
will denote
Then the q u a d r a t i c
an e x t e n s i o n
of the f u n c t i o n a l
Q(y)
=
the inner p r o d u c t
functional (Ay , y)
Q(y) in
D(A)
in
D(Q)
,
i.e.
in is
y e N(A)
(Ay , Y)
or
~a~{PLY' for
f ~
Proof:
I2 + qlY
12}dx
: [ ljl (y , qbj)12 0
(4.2.13)
D(A) The proof of this t h e o r e m
is c o n t a i n e d
in
[34,
§6,
218
Theorem
i, pp.
6.1-6.5].
We now d e f i n e domain
D(Q')
D(Q')
given
another
quadratic
y(a)
and
for
Q' (y)
is d e f i n e d
The crux of the m a t t e r
LEMMA
~
and
L (a , b)
= y(b)
(4.2.14)
= O}
{plf'
+qlfl 2}
and e x t e n d s is the
(Ay , y)
following
[34, p.
6.6].
Q(y)
Q'(y)
lemma,
4.2.5: When
p(x)
of the q u a d r a t i c D(Q)
with
y e D(Q')
Q'(y) --
Then
Q'(y)
by
{y c A C ( a , b) : ply'l 2
=
functional
= D(Q')
> 0
a.e.
functional
Proof:
(Ay , y)
are
identical,
,
i.e.
and
{ply'f2 + q l y
where
the e x t e n s i o n s
}dx : [ ljl (y, %j) I2 0
y e D(Q')
This
important
result
is p r o v e d
in
[34, p.
6.8,
Theorem
3].
219
THEOREM 4.2.1: Let
p(x)
r(x)
as
value
problem
non-real
(cf.,
in the
[76])
> 0
hypotheses
(4.2.3-4)
N = the
following
If we
number
values
of
[a , b]
possesses
eigenvalues.
M = the
on
a.e.
,
q • L ( a , b)
(4.2.3-4).
at m o s t
The
a finite
and
eigen-
number
of
let
of p a i r s
of
distinct
non-real
eigen-
negative
eigenvalues
(4.2.3-4),
number
(4.2.1-2)
of
distinct
(which
we
know
is
finite
by
of
Lemma
4.2.3(b)),
then
M
Proof: of
We
let
(4.2.1-2)
Let
~0'
for
some
"'"'
< N
.
10 , 11 , ... , IN_ 1
arranged ~N-I
be
the
in an
increasing
the
corresponding
be
f • D(Q')
(4.2.15)
we
have
(f , ~j)
negative
order
eigenvalues
of m a g n i t u d e .
eigenfunctions.
= 0 ,
j = 0 .....
If N-I
,
then
Q' (f)
> 0
since co
Q' (f)
= [ ljJ (f , ~j) I 2 N
(4.2.16)
220
and
the
I. > 0 3 We
non-real
now
let
Z0 ' ~1 ' "'" ' ~ M - I
eigenvalues
[i
We
write
the
of
z.(x) 1
replacing of
M
0
~ Zj
corresponding
z 0(x)
Then
(4.2.3-4
g
by
mutually
arranged
i
.....
i= 0 ..... f
,
j
N .
that
M
Then is o r t h o g o n a l For
it
it
shall
so
be
is p o s s i b l e
(in t h e
is n e c e s s a r y
(f , }j) and
=
[ 0
e.z. (x) 3 3
chosen
(4.2.18)
Assume,
later.
to c h o o s e
L 2 (a , b ) - s e n s e )
the to
(ej)
so t h a t
~0 . . . . .
that
: 0
if p o s s i b l e
j = 0 , 1 , ... , N - I
~N-I
f
221
M-I ei(z i , ~j)
j = 0 , 1 , ... , N-1
= 0 ,
i=0
The l a t t e r
constitutes M > N
a set of .
Thus
N
unknowns
where
this
solution
(ej)
not all
necessary
that,
for such a c h o i c e
zero w h i c h
linear system
equations
(ej)
It is then ,
Q' (f) > 0
because
of a p r e c e d i n g
Q'(f)
remark.
M
has a n o n - t r i v i a l
we fix. of
in
(4.2.19)
Moreover,
= Q'{Zejzj}
=
i
n
i
_
-!
_
_
a p(Tejzj) (7eiz ~) + a(Zejzj)~ (Zeiz i) M-I
I~ {P z'i' j i + q zj~i}dx
ejei i,j=0
But since applies
Pi ~ ~j
for all
Lemma
4.2.2
and so ~ab {p
for all
0 =< i , j =< M-I
i , j ,
zI jzi + q z j ~ i } d x
0 < i , j < M-I
.
= 0
Consequently
Q' (f) = 0
which
contradicts
completely
proved.
(4.2.19).
Thus
M < N =
and the t h e o r e m
is
222
The Richardson make
the
preceding [53]
theorem
mentioned
following
where ous
p(x)
and
at
L2(Ir[
and
is n e g a t i v e
continuous sign
> 0
on
least
once
functions
f
[a , b]
space
,
III
discussion
these
Let Appendix
1.4,
those
the
chapter
equation
,
q(x)
that
is
and r(x)
case
the
continur(x)
f]
and
-
ifl 2 Irldx
the
indefinite
space
(equivalence
classes
-
If I2
the
references
r
dx
A
= y(b)
= 0
eigenvalues,
in
some
cases
has o n l y real e i g e n v a l u e s ,
of a c c u m u l a t i o n ,
clustering
at m i n u s
with
infinity
infinity.
The s e c o n d [53, p.
301,
Theorem
4.2.2
boundary
problem
J-self-adjoint
p a r t of this
Theorem
VII].
remains
conditions
true
time.
changes
in the a r g u m e n t ,
self-adjoint
It w o u l d
Theorem
4.2.1
differential
is due to R i c h a r d s o n
seem plausible
for a r b i t r a r y
t h o u g h we
present
theorem
shall
"J-self-adjoint"
not go into
extends,
with
to the g e n e r a l
that
this
at the
appropriate
even order
formally
equation
(_i) n[poy(n) ) (n) + (-i) n - l ( p l y ( n - l ) ] (n-l) + ... + Pn y = fry
y(J) (a) = y(J) (b) = 0
where [a , b] that
P0 > 0 , Pk
and
where e C
(n-k)
i
Pi(X)
changes
is in the r a n g e
(a, b)
j = 0 .....
n-i
s i g n at l e a s t 0 < i < n
(see A p p e n d i x
III.4).
once
in
and w e a s s u m e
CHAPTER
5
INTRODUCTION: In t h i s differential
operators
i[y] (x)
where
I =
variation Chapter
=
d~(x)
[a , ~)
,
3.
We
we
y(s)do(s)
satisfying
x
o
a lemma
[23]
non-oscillatory
and
~
I
(5.0.0)
of b o u n d e d -
the b a s i c
by proving
of G l a z m a n
of g e n e r a l i z e d
expression
is n o n - d e c r e a s i n g ,
begin
the
spectrum
y' (x) -
to a t h e o r e m
is to r e l a t e
the
by the
~ , o
shall
study
generated
~
locally,
in c o n t e n t which
chapter
assumptions which
is s i m i l a r
the purpose
behaviour
of
of
of
solutions
of
Z[y] (x) = ly(x)
to t h e then
finiteness
use
some
for the
spectrum
of a n y
which
shall
and
then
the
spectrum
non-oscillation
criteria
we
of
obtain
discreteness self-adjoint
define
later.
an e x t e n s i o n
to t h e
results of
the
left
of
I .
from Chapter negative
extension We
(5.0.1)
x e I
shall
part
of a t h e o r e m
the
of M.
of
the operator,
latter Sh.
can
2 to o b t a i n
of t h e m i n i m a l apply
We
result
Birman
226
[23,
p.
93]
gives
discreteness operators ous
of the
in the
spectrum
generalized bounded show
spectrum
polar
(also
the
We
note
that
always
the p r o o f s , will
be o m i t t e d .
operators in
[35],
are
are
generality
§5.1
in a s s u m i n g
shall
[y] (x) =
I =
general we
SPECTRUM
be d e a l i n g
d dr(x)
[a, ~)
,
can
case
o
is of
we
shall
is b o u n d e d
extends
a result
setting,
shall
property
the
we
found
shall in
adaptable
that
the
this
theory
[23] and
not or so
resulting
c a n be
found
operators
thus
defined
is no g r e a t
loss
of
in g e n e r a l .
OF G E N E R A L I Z E D
with
be
for
space
general
though
are
assume
the
the
the
either
so t h e r e
y'+ (x) -
v
which
continu-
when
extension
Conditions
and this
the
of t h e s e
latter
be u s e d
applications
single-valued
THE DISCRETE OPERATORS: We
where
In the
case
consequently
These
Moreover
study
is n o n - d e c r e a s i n g
shall
single-valued.
[36].
indeed
~
spaces
in the m o r e
bound
the
p = 1
space,
the d e t a i l s :
in the
for
differential
spectrum)
In the
lower
since
in t h e s e
then
essential
I
case
is a H i l b e r t
give
shall
self-adjoint
an e x p l i c i t in the
of o p e r a t o r s
of
condition
second-order
We
the
on all
[16]
I)
sufficient
particularly
of E v e r i t t
L2(v,
of
corresponding
and give
and
case.
called
operators
variation
that
below
a necessary
DIFFERENTIAL
expression
y(s)do(s)
is a r i g h t - c o n t i n u o u s
x • I
(5.1.0)
non-decreasing
227
function locally as
and
~
on
usual,
I
In c a s e
that
both
In t h e for
the
compact we
space
of
mean
finite
the
v , ~
in
[a , b])
a function
of
variation
is
finite
continuous
at
the
, b]
on
functions
[a , b]
By
vanishes
we
assume,
end-points.
( A C 0 [ a , b])
continuous
which
bounded
I
AC[a
absolutely
and
interval
are
following,
support
shall
some
is r i g h t - c o n t i n u o u s
will
stand
(having
a finite
identically
function
outside
interval.
We
define
the
spaces
S(~ , 8)
and
S ( ~ , 8) =
{z(x) e A C [ e , 8] : z' e T, 2 (~ , 8)
T(e,
{y(x) e A C [ e ,
T ( ~ , 8)
by
z(~) = z(8) = 0}
and
!
8) =
8] : y + ( x )
exists
everywhere
on
I
[a, 8]
,
y+(x)
is
[~ , B]
,
dy ~./ d r
c L 2 (~ ; [e , B])
y(~)
Note
= Y(8)
v-absolutely
= y' (~)
= y' (8)
continuous
on
and
= 0}
that
T ( ~ , 8) c s ( ~ , 8)
The
first
(see
the
LEMMA
result book
similar
Glazman
to
[23,
an p.
important 35,
Lemma
result
of
Krein
5]).
5.1.0:
Let [~ , 8]
of
is
z(x)
Then
for
e S(e, any
B) e > 0
be ,
a finite there
function
exists
with
a finite
support function
228
y(x)
having
e T(~ , B)
the
same
support
as
z(x)
and
such
that
(5.1.1)
IQ[ z] - Q[Y] I < s where
Q[z]
Proof:
We
This
said,
case
where
Q'[z]
shall
{Iz'12dx+ IzI2d (x)} .
adapt
it w i l l
be
the
sufficient
g { constant,
defined
is b e c a u s e
or
in
for the
the
-
the
situation.
lemma
quadratic
in the
functional
5.1.3)
IZ' 12 dx .
fundamental
nature
c a n be r e a l i z e d
by
i.e.
{yj(x) }
is any
sequence
for all
j ,
y- (~) = 0 ]
to t h i s
to p r o v e
"Q-metric" If
[23]
by
Q' [z]
This
ideas
(5.1.2)
of any
its n a t u r e
[yj (x) - Yk(X) I
0
there
exists
This
function
c
,
Q[z] - Q[Y] I
fooIzl2dv"
~0
(5.i.ii)
a
Proof:
We
can
assume
non-oscillatory,
then
y'(x)
=
c
that
+
f
0
=
0
Since
[y] (x)
=
0
is
x y(s)do(s)
x
c
[a , ~)
(5.1.12)
a
and
so
exists
the a
latter
solution
equation y(x)
is of
non-oscillatory.
(5.1.12)
such
that
Thus
there
y(x)
~
0
for
235
all
x ~ x0
X => x 0 , u'
if
there
(x 0 , X) u
can
by
.] be
for
COROLLARY
is n o
function
,
u ( x 0)
were
one
Thus
all
u
= 0
then
such
every
by
then
the minimal
that
of
Q[u]
would Q[u]
, ]
< 0
> 0
a zero
B
all
,
.
Since e,
for
~ 0
any
e A C [ x 0 , X]
have
with
,
Q[u]
u
for
in
such
replaced u
~ J
and
(5.1.9).
the
(5.1.11)
spectrum
operator
Proof:
This
THEOREM
5.1.0:
L0
follows
A necessary
is
from
and
to b e o s c i l l a t o r y spectrum
of
lying
the
The
any
for
of
argument
any
finite
[23,
pp.
sufficient
10
is
be
for
all
z e G
self-adjoint to
the
34-35,
left
is t h a t extension an
similar
of
that
of
I = 0
28].
for
(5.1.13)
the of
infinite
to
for
extension
Theorem
condition
and
= ly(x)
I = l0
self-adjoint
left
holds
of
[y] (x)
Proof:
and
,
u
then
that
y(x)
such
that,
5.1.2:
Then
to
such
= 0
as e l e m e n t s
E G
implies
u
= u(X)
respectively,
Suppose 10
for
regarded
x0 , X
thus
I.i.0
there
c B V ( x 0 , X)
[For
Theorem
part
of
the
the minimal set.
in
[28,
p.
40,
operator,
236
Theorem
31].
oscillatory
We for
solution
assume
10 = 0
yl(x)
B > ~ > tI an
can
of
Then
eigenfunction
that
then
(5.1.12) on
[~,
for
8]
If
x = tI
which
to
at
can
I = 0
(5.1.13)
there
vanishes
yi(x)
corresponding
£ [y]
I0 = 0
be of
is
exists e,
B
a
where
regarded the
as
problem
= 0 (5.1.14)
y(~)
If we
let
el
= e
and
principles
it
follows
81
the
have
a negative
corresponding
(
,
)9
Thus
applying
such
that
> B ,
then
(5.1.14)
the
Lemma
inner we
(L0~ I , ~i)
Choosing eventually
t 2 > 81 obtain
we an
11 we
can
with
product can
infinite
find
= Q[~I]
iterate
Writing
see
= l l ( f I , fl)
5.1.0
variational
= y ( B 1) = 0
eigenfunction
is
from
along
eigenvalue
Q[fl]
where
= 0
that
y(el)
should
= y(B)
the
sequence
for
fl(x)
that
< 0
in the
space
a function
L2(v) ¢i
in
< 0 .
construction of
finite
and
we
functions
237
~k ~ D(Lo)
with
disjoint
(L0} k , ~k )
Applying tive
now Theorem
part
of the
Conversely for imply This
that
completes
As
just
spectrum the
32 of
of C h a p t e r
proved,
that
it is o s c i l l a t o r y must
p.
that
2 e ....
15]
we
is an i n f i n i t e
that
(5.1.13)
5.1.1
along left
find
that
the n e g a -
set.
is n o n - o s c i l l a t o r y
with
of
Corollary
0
must
be
5.1.2 finite.
proof.
[23, 3.) if
we o b t a i n
§2.14].
for all
Theorem
(For t h i s
It a l s o
(5.1.13)
follows,
31 of
reduction from
is o s c i l l a t o r y
I > 10
[23,
Hence
the for
one
§2.12]
see
the
theorem ~ = l0
of t h r e e
then cases
occur:
i)
It is n o n - o s c i l l a t o r y
for all
2)
It is o s c i l l a t o r y
all
3)
There
is some
oscillatory
This
[23,
to the
an a p p l i c a t i o n
and Theorem methods
Lemma
such
k=lt
,
13 of
suppose
Then the
< 0
spectrum
let us
10 = 0
supports
also
THEOREM
applies
and
for
~0
such
for
to t h r e e - t e r m
l
1
that < l0
recurrence
for
~ > 10
it
is
it is n o n - o s c i l l a t o r y .
relations.
5.1.2: Let
~
satisfy
the u s u a l
hypotheses
and
suppose
that
238
o(t)
tends
is zero.
to a f i n i t e
Suppose
limit
furthermore
lim t÷~
Let
~
for
the
when
~(~)
Proof:
The
can
assume
of
t(~(~)
(5.1.15)
a necessary
(5.1.13)
- v(t))
and
sufficient
to be d i s c r e t e
= 0
is t h a t
(5.1.16)
0
is there
that
n If(Xk+h
(I.i.3)
k) - f ( x k) I < e
k=l
for such
every that
n
disjoint
subintervals
(x k , x k + h k)
of
[a , b]
258
n
hk
(I.i.4)
0
(I.3.2).
~
for
a ~ x
p(x)
=< c O + c i
p(x)
to
and
, cI > 0
Assume
and
if
~ b
continuous
is
and
~
non-decreasing
and
is
non-
right-
,
p(t)da(t)
(I.3.11)
then
< c O exp
possible
(I.3.12)
{c I [o(x) - o(a) ] }
that
y(t)
, z(t)
are
two
solutions
Then
Jy(t) - z(t) J _
0
for
all
f • F
L p ( - ~ , ~)
in
,
p
> 1
,
if
> 0
such
P
For
137]
functions
lif[l
b)
invariant,
space
25].
A
a)
a convex
there ,
that,
for
all
f • F
(!I.l.l)
< M
is a
,
6(e)
> 0
such
that,
281
II f (x + h) - f ( x ) I I
whenever
c)
If
lhl
EA =
all
< 6
{x e I R :
f • F
I x - x01
> A
The Ascoli
for
is
the
theorem
is
0
(II.l.3)
induced
the
LP-norm
" L P - a n a l o g '' o f
on
EA
the Arzel~-
II.1.2: Let
> 1
norm
above
then,
theorem.
COROLLARY
p
the
fixed}
, x0
,
l i m IIfllE A A+oo where
(II.l.2)
< E P
,
F
a > -~
There
be
a family
,
satisfying
is a n
M
of
> 0
the
such
IIfllp If
EA =
and
for
{x : all
A < x f c F
functions
in
following
that,
for
L P [ a , ~) conditions:
all
f • F
then,
for
given
E>0,
,
IIf II~A --< ~ if
A
For f e F
is
£ > 0 ,
sufficiently
there
,
(II.l.4)
=< M
< ~}
,
is
(II .1.5)
large.
a
6(e)
> 0
such
that,
for
all
282
] f (x)
dx
< £
(II.l.6)
~a
whenever
d)
For
lhl
e > 0
f c F
< 6
there
whenever
Proof: a.e.
Set on
LP(~)
i)
ii)
Ihl
F
F =
to w h i c h
0
such
0
,
there
is a n
(II.l.4)
A 0 ~)
that
< e
A => A 0
clearly
implies
then
(-~ , a)
I-A If(x)[P dx
This
such
< e
P
(in t h e
IA[f(x) IPdx
and
> 0
•
Then
we
(II.l.l)
For
n(e)
h) - f ( x ) [ ]
is c o m p a c t
( - ~ , a)
and
an
,
[If(x+
Then
is
that
= 0 < e
(II.l.3)
A => A 0
is
satisfied.
> 0
283
Let
iii)
e > 0
If
0
0
a rational
r > t
such
I°(t) - ° ( r )
where further
t < r < t + 6(g0 ' t) so
an
there
for which
(II .i .i0)
IZm(t) - z(t) I -> gO
Choose
is
that
g0 I < 12a
If n e c e s s a r y
(II.l.ll)
restrict
r
that
go t < r < t + l ~ .
(II.l.12)
287 Since
z
m
(r) ÷
z(r)
aS
m
,
÷
there
is
N ( S , r)
an
for
which
IZm(r) for
all
m
gO
I
- z(r)
0 ,
IZm(S n) - z(t) I
N e)
that
n -> NI(s)
z
of
m
on the r a t i o n a l s
implies
that
IZm(S n) - z(s n) I < s_ 2
where
N2
is i n d e p e n d e n t
inequalities
there
Finally,
t
if
follows
before
n
.
From
these
last
two
(II.l.14).
is rational,
z(t)
by d e f i n i t i o n
of
m -> N 2 (£)
while
if
t
= lim Zm(t) m+~ is i r r a t i o n a l ,
so that
z(t)
= lim n÷~
z(s n)
= lim Ilim
Zm(Sn) }
we
let
s
n
be as
289
= lim ( l i m m ÷ ~ -n÷~
on a c c o u n t
of
Zm(Sn) }
(II.l.14),
= l i m Zm(t) m+~
N o w an a r g u m e n t convergence on
similar
is u n i f o r m
to a p r e v i o u s
one
shows
on the i r r a t i o n a l s
t h a t the
and so e v e r y w h e r e
K . !
Thus
for
xk c B n
of a s u b s e q u e n c e of
,
x k H zk
which
we have
converges
shown
uniformly
the e x i s t e n c e on c o m p a c t
subsets
[T , T + n) !
Moreover that of the
the u n i f o r m
(xk)
and
Ilxk -xll~ =
thus
sup t~ [T, ~)
convergence letting
x
of the
(x k)
be t h e i r
implies
limit,
Ixk(t) - x(t) 1
max I sup
,
T+n
,
we
must
then
have k
÷
Hence
sup te[T+n,~)
IXk(t ) - x(t) I =
Ic k
-el
÷0
as
k
÷~
sup te[T,T+n)
as
k ÷ ~
Thus
from
previous
llxk -xiI ~ ÷ 0
Similarly
IXk(t) - x ( t )
since
as
I ÷ 0
considerations.
k
÷
O(t)
is
bounded
[T + n
, ~)
away
from
zero
on
[T , T + n)
!
and
xk
~ 0
on
,
!
!
xk(t) sup te [T,~)
oo
- z(t)
Q(t) !
xk(t) sup tc IT, T + n )
- z(t) I
(II.l.15)
Q(t)
!
since Then
both
xk
uniform
(II.l.15)
z
are
convergence
of
tends
and
to
zero
as
identically the k ÷ ~
zk
to
zero z
(Also
on
[T + n
, ~)
implies
that
z = x'
follows
291
from
this.)
Hence
~~ !
llxk
and
so
B
n
is
xlJB = IExk
xll~ ÷
÷0
k÷~
compact.
,
!
APPENDIX
§l.
EIGENVALUES
related
OF
GENERALIZED
We
shall
mainly
to
eigenvalue
be
£[y]
THEOREM
was
defined
initial
basic
results
= ly
(III.l.l)
in C h a p t e r
value
Y(a)
where
e,
unique
solution
B ~ •
functions
,
3.
problem
fixed
follows
= ~
and y(x,
y ( x , I)
For
solutions I)
some
problems
~[y]
y(x,
with
EQUATIONS
III.1.0: The
Proof:
DIFFERENTIAL
concerned
i[y]
where
III
1 I)
,
from
is a n e n t i r e
the
~y
(III.l.2)
y' (a)
=
and moreover are
function
for
and
3.2.0 of
parameter
entire
existence
Theorem
(III.l.3)
8
is a c o m p l e x
, y ' ( x , I)
1
=
1
has
fixed
x
functions
uniqueness
and can
[3, p. be
,
the
of
1
of
341].
found
the
in
That [3,
293
p.
355].
p.
216,
[35]
The
complete
Theorem
also
2].
applies
result
is f o u n d
We o n l y
when
~
need
is,
in
[35,
p.
to n o t e
that
the p r o o f
more
generally,
250
and in
of b o u n d e d
variation.
THEOREM
III.l.l: Let
such
that
o
for
be some
a non-decreasing set
E c
[a, b]
~d0(t)
Let
V
be a r i g h t - c o n t i n u o u s
such
that
for
some
set
Then
the
function
,
(III.l.4)
e
function
F c
~
> 0
right-continuous
of b o u n d e d
variation
[a, b]
Id~(t) l > 0
(III.l.5)
problem
;x du(x)
y'
(x) -
}
y(s)de(s)
=
Xy(x)
(III.l.6)
a
with
Proof:
e,
6 c
y(a)cos
~ - y' (a)sin
~ : 0
y(b)cos
8 + y' (b)sin
8 = 0
[0, ~/2)
If p o s s i b l e
the p r o b l e m
and
has
let
y(x,
I)
only
I , the
real
(III.i.7"8)
eigenvalues.
Im I ~ 0 , corresponding
be an e i g e n v a l u e eigenfunction.
of
294 We m u l t i p l y
(III.l.6)
and i n t e g r a t e
by
with
respect
to
to o b t a i n
la
{
y(x,
l)d y' (x, X) -
Integrating
yd~ a
the left h a n d
}
Using
-tan
the b o u n d a r y
81Y' (b
Combining parts
'
, x) 12 a x +
fa~ ly(x ,
conditions
I) 12 - tan elY' (a
the latter
ly(x
l)
d~(x)
side by parts,
[..y ( x , t ) y ' (x , t ) ] b = I b { l y ' ( x a a -I
= -I
into the
I)
ly(x,X)
I~ dr(x)
(III.i.7-8)
'
I) 12 =
former
J2,~o(x)}
(III.l.9)
we
[y(x
find,
'
l)y' (x
and t a k i n g
'
I)] ba
imaginary
we o b t a i n
(Im I)
and thus,
since
= 0
(III.l.lO)
Im I ~ 0 ,
i Inserting
I~ly(x 'I) I~ d~(x)
the latter
b lY(x , 11 12 d~(x) a
into
(III.l.9)
(III.l.ll)
= 0 .
we m u s t
therefore
have
295
I~{ly
'(x,
= -tan
But
since
negative Hence
we must
i From
the
[a, b] zero.
to
the
,
{lY' (x
latter
11
it
12
dx
side
is
hand
side
is
necessarily
strictly
non-negative.
+ ly(x , I)I 2 do(x) } = 0
follows
(III.l.4)
Hence
that
finally
y ( x , I)
therefore
implies
.
H 0 y
y(x,
implies on
is a c o n s t a n t
that
[a , b]
is an Im
I)
this
constant
which
,
hence
must
is c o n t r a r y
eigenfunction.
I = 0
on
This
all
contra-
eigenvalues
real.
III.l.2: Let
variation
~
and
be
some only
Proof:
set
We
assuming
and, F c
real
a right-continuous
suppose
non-decreasing
by
right
have
that
THEOREM
has
hand
requirement
diction
on
left
the
(III.i.12)
a
and
be
are
b
[0 , ~/2)
the
~)12d~(x)}
l) J2 - tan(~ ly'(a, 1) J2
Bly'(b,
e , 8 e while
1) 12 dx + l y ( x ,
that
eigenvalues
the
as
Then if
in
the
existence
of
of b o u n d e d
is r i g h t - c o n t i n u o u s
additionally,
[a, b]
proceed
~(x)
function
that the
satisfy
problem
e , B E
proof
it
and
(III.l.5)
(III.i.6-7-8)
[0 , ~)
of
a non-real
the
previous
eigenvalue
theorem 1
and
296
arguing,
as
, B ~ 7/2
in t h a t ,
(III.l.10).
we
proof,
take
Since
until
imaginary
Im
I z 0
we
reach
parts
in
(III.l.9). (III.l.9)
If
obtaining
,
lab]y(x , I) 12 dr(x) = 0
But
since
y ~ 0
impossible
and
v
satisfies
y
F
is
has
positive
a solution
of
solution.
This
values
exist.
is
this
is
{ 0
on
F
be
the
unless
IF l y ( x ,
Since
(III.l.5)
can
treated
l)12d~(x)
~-measure
(III.l.6)
contradiction The
separately
case with
:
0
y ( x , l)
then
y
shows
that
when
.
must
either
a similar
trivial
no n o n - r e a l or b o t h
argument
eigen-
of
to
Since
e,
the
B = 7/2
one
above.
We bounded
note
variation
(III.i.6-7-8) The
latter
singular
§2.
need
case
LINEAR
refer
that
on
see
the
if
[a , b]
not
situation
For may
here
be
was
all
~
is a n
then
the
real
even
illustrated
papers
[i0],
OPERATORS
IN A H I L B E R T
the
notions
to a n y
basic book
on
function
eigenvalues if
~ , 6 E
in C h a p t e r
4.
[0 , ~/2] For
the
SPACE: Hilbert
analysis.
spaces The
of
of
[ii].
regarding
functional
arbitrary
one
space
.
297 L 2 (V ; I) such
defined
by
those
(equivalence
classes
of)
functions
that
llfll---{ I
If(x)]2 d V ( x ) } ½
< oo
(III.2.0)
I
is
a Hilbert
space
whenever
V(x)
is
a
non-decreasing
function. If
HI
, H2
consists
of A
to
x
Hilbert
pairs
if
given x
spaces
{x I , x 2 }
operator
xn ÷
that
A
any
, Ax n +
operator
y
xI
e HI
a Hilbert
sequence
Xn
then
admits
A
where
in
direct
their
x
sum, ,
space
c DA
c DA
x2 c H2 H
is
( domain and
a closure
HI ~
A
y
if
said
of
A)
= Ax
and
only
if
relations
6 DA
n
imply
, x ' e DA n
that
y
The of
two
linear
An the
all
closed
be
such
are
those
x A
fact
a
of
vectors
for
x
subspace
S
the for
÷ x
x
, x ' ÷ n
x
closure which
{AXn}
~ D(A)
set
÷
n
, Ax
÷
n
y
, Ax . ÷ y ' n
y'
domain
xn
Then
=
, x
,
Ax
a H S
is is
D(A)
there
is
converges
H
lim
dense
dense
in
Ax if
precisely
xn
satisfying
as
c DA
n
÷
n its
H
consists
if
closure and
only
S = H if
there
In is
H2
298
no
nonzero
vector
Let of
A
definition The
in
be in
set
H
any H
of
all
which
D*
for
of
all
the
defined
x
c D
operator
Denote
the
vectors
and
is
simply An
x,
y
z e H
e D
called
E D*
the
operator
A
A*y
adjoint is
said
A
to
said
be
by
defines
denoted
D
the
by
A*
hermitian
if,
.
domain ,
and
hermitian
operator
z
of to
A be
. for
all
be
An
operator
A
=
(x , Ay)
with
a dense
domain
of
definition
symmetric.
to
with
self-adjoint complex if
A
a bounded
operator points
if
A
a dense
the
1
1
is
inverse defined are
domain
of
definition
is
said
= A*
number
operator
regular
=
,
(Ax , y)
is
,
,
A
domain
by
y
A*
that
some
S
a dense
such
(x , z)
A
with of
y
to
to
domain
=
adjoint
operator
orthogonal
linear
(Ax , y)
holds
is
a
regular point
( A - II) on
called
the
-i
whole
of
exists
and
space
H
the represents
.
points of the spectrum
All of
nonA
299
The
set
of all
discrete spectrum of All
other
eigenvalues A
of
A
constitutes
.
points
of the
spectrum
(if any)
points of the continuous spectrum
(essential
The
constitutes
collection
of all
such
.
spectrum of
spectrum of
A
discrete
continuous
and The
Hilbert
spectrum
space For
operators
§3.
we
general
refer
to
general
product
spaces
[7] and
[40].
and,
A linear [ , ]
spectra
of a n y
OPERATORS
For
points
A
of
are
spectrum) the
called of
A.
continuous
is the u n i o n
of the
A
self-adjoint
operator
A
in a
is real.
the
LINEAR
product
The
the
theory [46,
of e x t e n s i o n s
information
SPACE: concerning
in p a r t i c u l a r ,
K
is c a l l e d
symmetric
~14].
IN A K R E I N
space
of
with
Krein
indefinite spaces
a generally
we
inner
refer
indefinite
to
inner
a Krein space if
K = K+[+]K_
where
K+(K_)
product
[
denotes
a direct
inner
,
is a H i l b e r t
product
] (-[
,
])
,
] ,
with
respectively.
sum which [
space
is o r t h o g o n a l
i.e.
respect The with
to the
symbol respect
inner [~] to the
300
~+ ~ K_ = {o}
whenever
f+
e K+
, f_
A positive defined
on
K
definite
is
orthogonal
inner
here
then
,
g = g+
of
K
that
different are
all
space
- P_
different
inner
to be
induced
Hilbert
Example:
Let
of b o u n d e d
[ H V]
by
the
function
V
is c a l l e d
(
,
)
can
be
=
g_]
and
K±
e K±
if
P±
denotes
The
important
of
K
generate
the
corresponding
the
then
(Jf , g)
+ P_
= I
,
)
but
Topological
interpreted
in
the
concepts norm
fact
norms
in a K r e i n
topology
of
the
space.
be
variation
(Im
is e q u i p p e d
(
f± , g±
decompositions
[7].
~(x)
m
,
space
on
P+
products
equivalent
are
and
- If,
+ g_
a Hilbert
projector
J = P+ is
= o
product
If+, g+]
[f , g]
where
f]
by
f = f+ + f_
(K , ( , )]
[%,
K_
e
(f, g) =
where
,
on
denote
the with
~
a right-continuous [a , ~)
the
signed
respectively
total such
variation a measure
,
function
a > -~
measure
measure and
the
Let (measure)
(Chapter
locally
3, of
norm
induced
section ~
and
i) . when
is d e f i n e d
L2 by
301
Ilfll
then
L 2Iv ; (a, =))
=-
f2
= K
f (x) 12 dV(x)
is a H i l b e r t
0
on
(n-k)
boundary
we
shall
(a, b)
[a , b] problem
assume
k=
that
0 , ... , n
EQUATIONS
304
(n-k)
n
(-i) n-k (pkf (n-k) ]
=
If
(III.4.1)
k=0
f(J) (a) = f(J) (b) : 0
is s e l f - a d j o i n t values
[9, p. 201, Ex.
are real,
accumulation. eigenvalues
Also
3].
in
(III.4.1-2),
L 2 ( a , b)
point
f
of
to d i s t i n c t
Moreover,
then
series
(III.4.2)
the e i g e n -
no f i n i t e
corresponding
(III.4.2)
convergent
Consequently
and h a v e
eigenfunctions
satisfies
into a u n i f o r m l y of
below,
are o r t h o g o n a l
f 6 c n ( a , b)
Yk(X)
bounded
j = 0 , .... n-i
if
can be e x p a n d e d
of the e i g e n f u n c t i o n s
i.e.,
co
f(x)
:
[
(III.4.2)
(f , yk)Yk(X)
k=O
where we assume result,
see
that
[9, p.
llyklI : 1
197,
Theorem
As u s u a l w e d e f i n e
D(A)
for all
k
latter
4.1].)
an o p e r a t o r
= {f ~
(For the
C n
A
with
domain
(a, b) }
such t h a t
Af :
If we
D(A)
n [ k=0
(n-k) (-i) n-k (pk f (n-k) ]
(III.4.3)
let
=
{f ~ D(A) : f(i) (a) = f(i) (b) = 0
for
i= 0 , 1 .....
n-l}
305
and
Af = Af
then
integrating
f E D(A)
(Af , f)
by p a r t s
we
find
fb n (Af, f) = I~ [ P n _ j l f ( J ) 12 dx a j=0
f • D(A) (III.4.4)
Now
from
Yk(X)
(III°4.2)
in
and
L 2 ( a , b)
the
we
completeness
also
have,
of the
for
eigenfunctions
f • D(A)
co
(Af , f) :
[
(III.4.5)
lkl (f , Y k ) I2
k=O
Thus
if w e
negative have
let
I0' 11'
eigenvalues
of
(f, yk ) = 0 ,
" ' " ' IN-I (III.4.1-2)
k= 0, 1,
rest
For,
of
the
argument
an a d a p t a t i o n
non-real
now
of L e m m a
eigenvalues
and
N
distinct
if for ,
some
f
we
4,
§2.
then
> 0
follows 4.2.2
that
shows
in C h a p t e r
that,
if
I , ~
are
of
f(i) (a) = f(i)(b)
Af = Irf
the
. . . , N-I
(Af , f)
The
be
= 0
i = 0 , ... , n-i
,
(III.4.6)
where
r(x)
is,
say,
continuous
on
[a , b]
and
changes
sign
306
at l e a s t
once
there,
and
I ~ [ ,
then
~b J|a f ( x ) g ( x ) r ( x ) d x
i
b
n
where
f , g
are
(j)~(j) P
a
j=O
the
(III.4.7)
= 0
.f n-]
(III.4.8)
dx = 0
eigenfunctions
corresponding
to
I ,
respectively.
Thus
we
let
(III.4.6)
H 0 , ..., HM_ 1
such
that
eigenfunctions
be the
H i ~ [j
%0 ' ~1 . . . . .
,
non-real
eigenvalues
0 =< i , j =< M -
%M-I
Since
1 ,
%i(x)
of
with ~ D(A)
we
have
for
i = 0,
fb
n
a
j=0
P n-j I ~i(j) 12 dx
..., M-
1
Thus
as in C h a p t e r
possible
to c h o o s e
(f , yk ) = 0 ,
4.2, the
we
ej%j
we
see
that
substituting
(x)
coefficients N- 1
(Af , f)
But by
let
M-I [ j=0
k= 0 .....
(III.4.9)
(~i' Y k )12
0~ Ik
=
f(x)
and,
=
(III.4
if e
This
M > N ]
such would
then
it is
that then
imply
> 0
in the
latter
relation
and
that
9)
307
expanding
the
form,
we
shall
find
(Af, f
on
account
the
of
(III.4.7-8).
that
= 0
This
contradiction
then
proves
result.
Note: idea
The
is
to
problem
here
approximate
is
the
the
following:
eigenvalues
of
Richardson's the
continuous
problem, !
(py')
+
y(0)
by
the
eigenvalues
of
the
(q+ Ik)y
= y(1)
= 0
= 0
discrete
problem,
2 m
A ( P i A Y i _ I)
i = 0 , 1 , 2 .... points The
i/m
claim
eigenvalues
,m
are
where
by
to be
that
the
discrete
Y0
are
approximations
problem.
However
the
denoted
appears of
+ qiy i + IkiY i = 0
to it
the
values Yi
' Pi
for
of
problem
y , p , q , k
' qi
large
,
' ki
values
at
the
respectively. of
m
the
with
= Ym = 0
eigenvalues
is n o t
at
all
of
clear
the that
above if
continuous
the
discrete
308
problem
has
non-real
eigenvalues
then
approximate
non-real
eigenvalues
in t h e
it is c o n c e i v a b l e
that
seem
information
the
as
if e n o u g h
latter
eigenvalues criteria
possibility. may
on the
existence.
these
exist
and
limits
so o n e
may
be real.
in s o m e needs
which
will
necessarily
continuous
is p r o v i d e d
In f a c t
coefficients
these must
in
[53]
cases
case.
For
It d o e s
not
to e x c l u d e
no n o n - r e a l
to e s t a b l i s h guarantee
some
their
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~,
Scotland• Verlag,
Equations
three-
Rep. Acad.
Sturm-Liouville
in the Proceedings
Differential
a
171-175.
, Indefinite
To appear
for
of the C o n f e r e n c e
1982,
Lecture Notes
problems,
University
on
of Dundee•
in Mathematics, Springer-
N.Y.
[77]
, function
Some
remarks
associated
with
a
on
the
second
order order
of
an
entire
differential
%
equation,
Proceedings Operators• Notes
appear
TO
I
in Tribute
of the S y m p o s i u m on Differential University
of Dundee,
in Mathematics•
[78] of
Scotland,
Springer-Verlag,
, Asymptotic values
to F.V. Atkinson,
non-definite
To appear
Sturm-Liouville
Scotland•
Springer-Verlag W.T.
Reid•
Journal
of Math.
and
related
J. Math.•
H. Weyl,
Uber
Notes
of
University
in Mathematics,
8,
die
Functionen,
(1959)
matrix
i_00, (1966),
und
differential
linear
Riccati
gewohnlicher
singularitaten
willkurlioher
Proceedings
Operators,
linear
and Mech.,
Illinois
mit
eigen-
N.Y.
• Generalized systems
[81]
Lecture
Generalized
[80]
the
problems,
to F.V. Atkinson,
the S y m p o s i u m on Differential
[79]
of
I
in Tribute
of Dundee,
N.Y.
distribution
&
Lecture
systems,
705-726.
differential integral
equations,
701-722
Differentialgleichungen zugehorigen
Mat.
Ann.,
Entwicklungen
68,
(1910)•
220-269.
Subject
Conditional
Dirichlet,
Difference Equation,
Index
180 15
Dirichl et property,
180
Generalized derivatives,
120 ff.
Generalized ordinary differential
expressions,
Generalized ordinary differential
operators,
Green's
function,
25-27,
273 ff.
Indefinite weight-function, J-sel f-adjointness,
197 ff.
156 ff°
Limit-circle,
132, 147 ff.
Limit-point,
132, 147 ff.
Non-oscillatory
equation,
30
Non-oscillatory
solution,
30
Oscillatory
equation,
30
Oscillatory
solution,
30
Picone 's identity,
3
Strong Limit-point, Sturm comparison
180-181
theorem,
i0 ff.
Sturm separation theorem,
4, 22 ff.
Three-term recurrence relation, Volterra-Stieltjes
integral
Weyl classification,
16
equation,
129 ff.
29
123 ff.
156 ff., 225