Volterra Stieltjes Integral Equations and Generalized Ordinary Differential Expressions Preprint, 1983
Errata •
p.90: Proof of Theorem 2.3.1; delete the remark in parentheses “since -y(t) is a solution” (comment: clearly this is not assumed; if y(t) < 0 in (2.3.11) define g(t) by the negative of the right-side of (2.3.12)) • p.244, Statement of Theorem 5.2.2: Replace the word “precisely” by “contained in” (comment: as this is what is proved there).
Quest' opera
e
umilmente dedicata
ai miei cari genitori Giosafat Oliviana e al mio fratello Marco A.M.D.G.
e
PREFACE The aim of these notes is to pursue a line of research adopted by many authors (We Feller, M.G. Krein, 1.5.
Kac,
F.V. Atkinson, W.T. Reid, among others) in order to develop a qualitative and spectral theory of Volterra-Stieltjes integral equations with specific applications to real ordinary differential and difference equations of the second order. We begin by an extension of the classical results of Sturm (comparison theorem, separation theorem) to this more general setting.
In chapter 2 we study the oscillation theory
of such equations and, in Chapters 3,4,5, apply some aspects of it to the study of the spectrum of the operators generated by certain generalized ordinary differential expressions associated with the above-mentioned integral equations. In order to make these notes self-contained some appendices have been added which include results fundamental to the main text.
Care has been taken to give due credit to those
researchers who have contributed to the development of the theory presented herein - any omissions or errors are the author's sole responsibility. I am greatly indebted to Professor F.V. Atkinson at whose hands I learned the subject and I also take this opportunity to acknowledge with thanks the assistance of the Natural Sciences and Engineering Research Council of Canada for continued financial support.
My sincere thanks go to Mrs. Frances Mitchell
VI
for her expert typing of the manuscript. Finally, I am deeply grateful to my wife Leslie Jean for her constant encouragement and patience and I also wish to thank Professor A. Dold for the possibility to publish the manuscript in the Lecture Note series.
Angelo B. Mingarelli Ottawa, April 1980.
TABLE OF CONTENTS
x
INTRODUCTION CHAPTER 1 Introduction
1
1.1.
Comparison Theorems for Stieltjes IntegroDifferential Equations .. . . . . . . . . . . . . . . . .
1.2.
Separation Theorems
20
1. 3.
The Green's Function
25
4
CHAPTER 2 Introduction 2.1.
Non-Oscillation Criteria for Linear Volterra-Stieltjes Integral Equations
28 29
2.1A. Applications to Differential Equations
52
2.1B. Applications to Difference Equations
60
2.2.
74
Oscillation Criteria
2.2A. Applications to Differential Equations
80
2.2B. Applications to Difference Equations
82
2.3.
An Oscillation Theorem in the Nonlinear Case . . . . . . . .. . . . . .. . .. . .. . . . . . . . . .. . . . . .
Addenda
87 113
CHAPTER 3 Introduction
118
3.1.
120
Generalized Derivatives
VIII
CHAPTER 3 (continued) 3.2.
Generalized Differential Expressions of the Second Order
123
3.3.
The Weyl Classification
129
3.4.
Applications
143
3.5.
Limit-Point and Limit-Circle Criteria
147
3.6.
J-Self-Adjointness of Generalized Differential Operators . . . . . . . . . . . . . . . . • .
156
3.7.
Dirichlet Integrals Associated with Generalized Differential Expressions
180
3.8.
Dirichlet Conditions for Three-Term Recurrence Relations ....•..•...•....•.•.
183
CHAPTER 4 Introduction
197
4.1.
Sturm-Liouville Difference Equations with an Indefinite Weight-Function .
199
4.2.
Sturm-Liouville Differential Equations with an Indefinite Weight-Function
212
CHAPTER 5 Introduction
225
5.1.
The Discrete Spectrum of Generalized Differential Operators . . . . . . . . . . . . . . . . . .
226
5.2.
The Continuous Spectrum of Generalized Differential Operators .. . . . . . . . . . . . . . . . .
242
1.1.
Functions of Bounded Variation
256
1.2.
The Riemann-Stieltjes Integral
258
1.3.
General Theory of Volterra-Stieltjes Integral Equations
264
1.4.
Construction of the Green's Function
273
APPENDIX I
IX
APPENDIX II ILL
Compactness in L
P
280
and Other Spaces
APPENDIX III III.l.
Eigenvalues of Generalized Differential Equations .
292
III. 2.
Linear Operators in a Hilbert Space
296
III. 3.
Linear Operators in a Krein Space
299
III. 4.
Formally Self-Adjoint Even Order Differential Equations with an Indefinite Weight-Function
.
303
BIBLIOGRAPHY
309
Subject Index
318
INTRODUCTION Let p,q: 1-+ IR, p(t) > 0
a.e.
Lebesgue measure) and lip, q E L(I)
(in the sense of [a,b]
where I
c
IR
Consider the formally symmetric differential equation
= 0,
(p(t)y')' - q(t)y
t
E
(1)
1.
By a solution of (1) we will mean a function y: I -+ C , Y E AC(I),
(i.e., absolutely continuous on I) such that
py'E AC(I) and y(t) satisfies (1) a.e. on I.
Let y E I.
Then a quadrature gives, for t E l , p(t)y' (t)
=
13 +
t
J
y(s)q(s)ds
Y
(py') (y).
where 13 a(t)
Jt a
Since q E L(I)
its indefinite integral
q(s)ds exists for t E l and a E AC(I).
Hence y
will be a solution of (1) if and only if y(t) satisfies a Stieltjes integro-differential equation of the form p(t)y' (t.)
=
13 +
Jt
y(s)da(s),
tEl,
(2)
y
where the integral may be interpreted, say, in the RiemannStieltjes sense. whenever
On the other hand (2) also has a meaning
a E BV(I)
is continuous on I.
(i.e., bounded variation on I) and y Hence equations of the form (2) may be
used to deal with differential equations (1). need not be continuous on I
Moreover a
(as long as we require a solution
of (2) to be continuous on I) and so (2) can be used to treat discrete problems, e.g., difference equations (or threeterm recurrence relations) as well as continuous problems
XI
as we have seen. 0 corresponds to
The
vet) non-decreasing and the
case of unrestricted ret) corresponds to V(t)E BV(I). In the former case the operator defined by the differential expression is formally symmetric (under suitable domain 2
restrictions) in the weighted Hilbert space L (I,dv).
In
the latter case the operator is J-symmetric in a Krein (Pontrjagin) space, since the measure induced by vet) is a signed measure.
XIV
Expressions of the form (5) were first considered by
w.
Feller [68J,[69J,[70J,[71J,[72J,[73J in the case when
aCt) :: constant on I, p(t) function on I, a
E
:: I, and
v a given non-decreasing
(cf., also Langer [41J).
The more general case
BV(I) was treated by 1.5. Kac [35J,[36J,[37J when v is
monotone, cf., [46,p.49J.
CHAPTER 1 INTRODUCTION: In this chapter we shall study the Sturmian theory of Stieltjes integro-differential equations; that is, equations of the form
p(t)y' (t)
c + It y(s)do(s) a
defined on a finite interval
I = [a, b)
and
(1. 0.0)
p,
0
are real
valued right-continuous functions of bounded variation on and
p(t) > 0
I
there.
Historical Background: The comparison and separation theorems of Sturm comprise what we call the Sturmian theory.
Comparison theorems
for the scalar equation
(p(t)y' (t))' - q(t)y(t)
o
(1.0.1)
were first obtained by Sturm [58, p. 135] in his famous memoir of 1836.
In that paper Sturm considered the equations
o
(1. 0.2)
2
a on a finite interval and showed that if G < G 1, 2
(1.
a
< K
2
0.3)
K1 '
equality not holding everywhere on the interval,
then between any two zeros of some solution of (1.0.2) there is at least one zero of any solution of (1.0.3).
This is the
result usually known as the sturm-Comparison Theorem.
Sturm's
proof depended upon the introduction of a parameter in the coefficients which allowed him to pass continuously from to
K 2
and from
G 1
to
G 2,
K1
as the parameter was increased,
and then he studied the location of the zeros of the solutions as the parameter varied. valid for all
t
1,
t
2
E
It also depended upon the identity I ,
which can be obtained by an application of Green's theorem [ 13, p. 291].
It seems [58, p. 186] that Sturm carne to the conclusion of the comparison theorem by first having shown it true for the case of a three-term recurrence relation or second order difference equation though the latter result was not published. A discrete analog of the comparison theorem was published by Fort [21, p.
whose method of proof was, in essence, that
of Sturm applied to difference equations instead of differential equations.
3
In 1909 Picone [48, p. 18] gave by far the simplest proof of the comparison theorem in the continuous case.
He
made use of the formula t
[:l ( K yz' - K y'
z
2
1
Z)] t
2
1
(1. 0.5)
commonly known as the Picone Identity.
The use of (1.0.5)
allows an immediate proof of the Sturm Comparison Theorem [33, p. 226]. (cf., also
[74]).
One important extension of the comparison theorem was that of Leighton [42, p. 604] who interpreted the theorem in a variational setting: Q[y] y
associated with (1.0.2-3) acting on functions 1
C (a, b)
E
He made use of a "quadratic functional"
and
= y (b) = 0
y (a)
termed 'admissible').
For such
(such functions were
y,
(1. 0.6)
The main result was that if some non-trivial admissible function y
had the property that
Q[y] < 0
then every real solution
of (1.0.3) would have to vanish at some point in
(a, b)
Swanson [59, p. 3] weakened Leighton's condition
Q[y] < 0
Q[y]
0
for
y
t
0
reaching the same conclusion provided
the solutions were not constant multiples of
y.
to
4
The Sturm-Separation theorem states that the zeros of linearly independent solutions of, say, separate one another.
(1.0.2) interlace or
A similar result holds for three-term
recurrence relations and in fact a more general result is known in the latter case.
(See section 2) .
In section 1 we shall give an extension of the aforementioned "Leighton-Swanson Theorem" to the class of integral equations (1.0.0) and give, as corollaries, the corresponding continuous and discrete versions of the comparison theorem. In section 2 we give a proof of the Sturm Separation Theorem for (1.0.0) and give some applications to both differential and difference equations.
We conclude this
chapter with a study of the Green's function for boundary problems associated with
(1.0.0)
and its application to the
problem of finding an explicit representation for the solution of the non-homogeneous problem.
§l.l
(See section 3).
COMPARISON THEOREMS FOR STIELTJES INTEGRO-DIFFERENTIAL EQUATIONS: Let
p. (t) 1
(J.
1
(t)
i
=
1 , 2,
functions of bounded variation over p. (t) 1
> 0,
t
E
[a, b]
We assume that
[a , b]
i = 1 , 2,
functions are right-continuous on
be real valued
and that all four
[a, b]
with each possess-
ing a finite number of discontinuities there. simplicity only.
(This is for
In the following chapters this hypothesis
can be omitted, in most theorems, without affecting the conclusions.)
We will, in general, assume that all these
5
functions are continous at lim a(t)
exists as
t
+
a, b ,
and if
b
00
then
00
Consider the equations
P1 (t)u' (t)
c + It u(s)da (s) 1 a
P2 (t)v' (t)
c '
+
(1.1.0)
C
(1.1.1)
v (s) do 2 (s)
where by a solution of (1.1.0), say, we mean a function u(t)
E
AC[a, b]
each point
t
E
with
P1(t)u'(t)
E
BV(a,b)
satisfying (1.1.0) at
[a, b]
Associated with the pair (1.1.0-1) is the quadratic
Q[u]
functional
{u
U
E
with domain
AC [a , b]
, P 2 U'
E
BV(a , b)
, u (a)
u(b)
= a} (1.1.2)
and where, for
u
E
D ' Q (1.1.3)
We can now state and prove an extension of the LeightonSwanson result.
THEOREM 1.1.0: Let
i
=1
, 2 ,
be defined as above and let
6 U
E
D , Q
u pO,
be such that
QIu l
(1.1.4)
< 0 •
Then every solution of (1.1.1) which is not a constant multiple of
u(t)
vanishes at least once in
Proof:
Assume, on the contrary, that
(a, b)
and let
a < s < t < b.
Then
v
(a, b)
•
does not vanish in P2v'/v
E
BV b) 10 c(a,
and so
(1.1.5)
exists.
Case 1:
v(a)
0,
v(b)
For
t
Js
0 •
satisfying (1.1.4),
u
dl-VI)
2 (P2 v-
(1.1.6)
(1.1.7)
where in passing from (1.1.6) to (1.1.7) we used the equation (1.1.1).
Integrating (1.1.5) by parts we find that
7
v'u'u
(1.1.8)
v
Combining (1.1. 7),
(1.1. 8) and adding
to both sides we obtain,
[
P2
u
V'
v
2] t +
Jt
s
s
P u' 2 dt - 2 2
[p
+
v' 2 t
Jt s
P v'u' 2
J: p,{U' _ t t
[P2 V' Js
+
v
'2
(1.1.9) for
a < s < t < b . Hence if we let
obtain, since
v(a)
,
s
a +0
-+
t
-+
b - 0
in (1. 1. 9) we
a ,
v(b)
(1.1.10)
Q[u]
The hypothesis on we must have [a , b l
u
implies
a
or that
which we excluded.
Q[u] u
=
0
but since
is a multiple of
v
1- a , v
This contradiction shows that
on v
8
must vanish at least once in
Case 2:
= v(b) =
v(a)
(a, b)
0 •
To settle this case it suffices to
that in
(1.1.9) , 2
lim t+b-O
u (t)P2 (t)v' (t) v(t)
o
(1.1.11)
o .
(1.1.12)
and 2
lim s+a+O
u (s)P2 (s)v' (s) v(s)
It is possible to show that solutions to the initial value problem (1.1.1),
v(a)
=
c
See Appendix I and [3, p. 341]. v' (a)
0 •
=
P2(a)v' (a)
1,
Thus since
c
are unique:
2
v(a)
0,
(The prime here usually represents a right-
derivative which is an ordinary (two-sided) derivative if is continuous at the point in question.) since
o
v(b)
P2 (b ) v ' (b)
O.
Similarly [3, p. 348],
Hence
2
lim s+a+O
u (s)P2 (s)v' (s) v(s)
P2 (a) v
provided the latter limit exists.
02
I
(a)
u
2
(s)
lim V""T""S) s+a+O
The hypothesis on
(1.1.13)
02
implies that it is continuous in some right-neighborhood of Thus
P2(t)v' (t)
Similarly
P2(t)
is continuous in such a neighborhood. is continuous in some, possibly different,
right-neighborhood of
a.
Hence
v' (t)
is continuous (i.e.
a.
9
is an ordinary derivative) in some right-neighborhood (a,a+o)
o
> 0 •
In the same way it can be shown that ordinary derivative in (a, a + n) 2(t))' (a, a+n) , (u = 2u(t)u'(t)
n > O. Since
u' (t)
is an
Thus in u, v
E
AC [a , b] ,
we can apply L'H;pital's theorem to the limit in the right of (1.1.13) to obtain
u
2
(s)
lim s-+a+O
lim V"TS) s-+a+O
2u(s)u ' (s) v' (s)
o since, as we saw above,
v' (a)
7
O.
Hence the limit (1.1.12)
exists and is zero. Similarly it can be shown that (1.1.11) holds. Combining (1.1.11),
(1.1.12) and letting
s-+a+O,
t-+b-O
in (1.1.9) we obtain (1.1.10) again and thus derive a contradiction.
Case 3:
v (a) = 0,
v (b)
7
0
or
v (a)
7
0,
v (b) = 0 .
This case is easily disposed of as it is simply a combination of Cases 1 and 2 leading to (1.1.10) via (1.1.9) and (1.1.11-12).
This proves the theorem.
Associated with (1.1.0) is the quadratic functional Q' [u]
with domain
10
{u:
u
AC [a , b]
E
, P 1u
I
E
BV(a , b)
, u (a )
u(b)
= o}
(1.1.14) and (1.1.15)
Q' [u]
COROLLARY 1.1.0:
Let u(a)
u(b)
u
=
(Swanson [59, p. 4], Leighton [42, p. 605, Cor. 1]). be a non-trivial solution of (1.1.0) with
0
Then every solution constant multiple of
u
v(t)
of (1.1.1) which is not a
must vanish at least once in
(a, b)
provided
o .
Proof:
Let
u
be a solution of (1.1.0),
u(a)
(1.1.16)
u(b)
o .
Then
[up 1u
I ]
ba _
I
b
a
p 1u I 2 d t
(1.1.17)
Using the equation (1.1.0) in the left-side of (1.1.17) we find that Q' [u]
[uPl u' ]
o .
b
a
(1.1.18)
11
(1.1.16) now says that
Q' [u] - Q[u] > 0
or, because of
(1.1.18) , Q [u]
Since
u
(1.1.19)
< 0 •
is not a constant multiple of
applies and hence
v(t)
v,
Theorem 1.1.0
vanishes at least once in
(a, b)
.
Swanson's extension [59, p. 4] of Leighton's Theorem [42] is obtained by setting
t
0. (t) 1
[a, b)
E
,
i
=1
, 2 ,
(1.1.20)
in (1.1.0-1) and in (1.1.16).
COROLLARY 1.1.1:
(Sturm Comparison Theorem)
Let
q.EC[a,b],
i
1
=1
, 2
and suppose that
If
and
uta)
for which
=
=
u(b)
v(c)
=
0
0,
o
(1.1.21)
o
(1.1.22)
then there is at least one
whenever
v
c E (a, b)
is a solution of (1.1.22)
which is not a constant multiple of
u
12
Proof:
Let
0. (t)
be defined as in (1.1.20).
l
follows from Corollary 1.1.0 on account that is non-decreasing on
[a, b]
The result now
°1 (t)
- 02(t)
by the above hypothesis.
We now interpret these results for a three-term recurrence relation.
Let
(1.1.23)
b
be a fixed partition of the interval
[a , b]
and let
c
, c m- 1 be a given positive real sequence. -1 ' Co ' c 1 ' Let b ' b ' , b m- 1 be an arbitrary real sequence and o 1 define a function p(t) by setting on [a , b l
p(t)
for
c n- 1 (nt - tn- 1)
n= 0,1,2, ... , m.
if
Then
(1.1.24)
t E [ t n- 1 , tn)
p(t)
is a positive right-
continuous function of bounded variation with jumps, if any, at the
{t.} . l Now define
o(t)
on
[a, b]
by requiring that it be
a right-continuous step-function with jumps at the
{t
i}
of
magnitude o(t ) - o(t n n where With
0)
-b
n
(1.1.25)
n = 0 , 1, ..• , m-l . p(t)
, o(t)
as defined above consider (1.0.0).
On
13
[a, to)'
=
o(t)
It
constant, hence,
Ydo _ 0
(1.1.26)
a
and so (1.1.0) implies that
p(t)y' (t) = c
But
p(t)
=
p(a)
on
(1.1.27)
p(a)y' (a)
[a, to)
because
p(t)
is also a step-
function, hence (1.1.27) implies that
y' (t)
In fact, letting
y(t
(1.1.28)
y' (a)
n)
yn'
n = 1 , 0 , 1 , ... ,m,
then
(1.1. 27) gives
y' (t)
(1.1.29) Hence p(t)y' (t)
p(a)y' (a)
(1.1.30)
14
Now let
t
[t
E
n_ 1,
from (1.1.30) that
t
,
n)
1
p(a)y' (a) +
p(a)y' (a) +
p(a)y' (a) +
p(a)y' (a) +
since
a
is constant on
constant on there,
[t
y' (t)
m.
When
p(t)y' (t) = c_ (Yo - y-1) 1
Thus
p(t)y'(t)
n
r a
n = 0, for
t
we know [a, to) .
E
yda
n-1
t.
L f
i=O
n-1
L
i=O n-1
L
i=O
yda +
l
t i- 1 t.+O
r. l
rt
Jt
yda n-1
yda + ft yda t n_ 1+O
y(t.) (a(t.) - a(t. - 0)) + 0 l
l
l
[t
is n- l ' t n ) . Hence p(t)y' (t) and since p(t) satisfies (1.1.24)
n- l ' tn) is also constant so that
y' (t)
Yn - Yn-1 t - t n n-1
t
[t n- l ' t n ) .
E
Consequently, p(t)y'
(t)
t
E
[t
n- l ' t n ) . (1.1.31)
This is true for each
n
in the range considered.
(1.1.0) gives
If
15
p(t -O)y'(t -0) + It ydo n n t -0
p(t)y' (t)
n
+0
t
p(t
n
-O)y'(t -0) + n
It
ydo
n n
-0
c n- 1 (Yn - Yn-1) + y(t n) (o(t n) - o(t n -
0») (1.1.32) (1.1.33)
where we have used (1.1.31) and (1.1.25) in obtaining (1.1.32), (1.1.33) respectively. By (1.1.31) we find that t
E
[tn' t
n+ 1)
•
p(t)y' (t) = c n(Yn+1 - Yn)
if
Combining this with (1.1.33) we obtain
(1.1.34) or c y + c y n n+1 n-l n-1
(c + n
C
n-l
-
b )Y n n
o
(1.1.35)
which is equivalent to
o where
(1.1.36)
represents the forward difference operator,
n = y n+1 - y n· Summarizing then, we see that when
p(t)
, o(t)
are
defined as in (1.1.24-5) respectively, the Stie1tjes integro-
16
differential equation (1.1.0) has solutions which are polygonal curves whose "vertices" are the points the
(tn' Yn)
and
satisfy the three-term recurrence relation (1.1.35)
or the second-order difference equation (1.1.36) for n
=0
, 1 , 2 , ••• , m-l .
An argument similar to the one above shows that if, instead of (1.1.25), we require
a(t)-a(t-O) n n then (1.1.0), with the same
(1.1.37)
p
as in (1.1.24), will give rise
to the recurrence relation
c n y n+l + where
C
n-l Yn-l - b n Yn
o
(1.1.38)
n = 0 , 1 , 2 , ••• , m-l .
The initial conditions
y(a)
=
a
p(a)y' (a)
(1.1.39) =
S
(1.1.40)
associated with (1.1.0) become, in the case of a recurrence relation, =
(1.1.41)
a
s
(1.1.42)
17
on account of (1.1.30). A fundamental solution, i.e. one in which
a
=
0,
B
1,
will then become
o
(1.1.43)
(1.1.44)
in the case of a recurrence relation (1.1.38).
(In this
respect see [3, p. 97]). With the
t
n
defined as in (1.1.23) we suppose given
two arbitrary real finite sequences and two positive sequences Let
C5. 1
(t)
,
i = 1 , 2,
c
b
n
qn' n = 0, 1, ... , m-l n, n = -1 , 0 , 1 , ... , m-l
be step-functions on
[a, b]
with
saltus b
n
(1.1.45)
(1.1.46) n
=0
Let
, 1 , •.• , m-l .
p. (t) 1
,
i
=1
, 2,
be defined by
c n- l(t n - t n- 1)
t
E
[t n- l ' t n )
(1.1.47)
r n- l(t n - t n- 1)
t
E
[t n- l ' t n )
(1.1.48)
18
where
o,
n
i=1,2,
1
...
,
, m-l .
Then the
and, along with the
0. (t) J.
for ,
continuous and of bounded variation on
i=1,2,
are right
[a, b]
Consider now (1.1.0-1) with the above choice of 0i' points
i=1,2. t.
J.
Pi'
The solutions of (1.1.0-1) evaluated at the
will then satisfy the recurrence relations
c n u n +1 + c n- 1 u n- 1 -
(c n +
n- 1 + b n ) u n
0
r n v n +1 + r n- 1 v n- 1 -
(r n + r n- 1 + q n ) v n
0
C
and
where
n = 0 , 1 , '"
, m-l
respectively.
The latter are
equivalent to
i'l (c n- li'lu n- 1) - b n u n
o
(1.1.49)
i'l (r n- li'lv n- 1) - q n v n
o
(1.1.50)
n = 0 , 1, ... , m-l.
We can now state a discrete analog of
the Sturm comparison theorem one form of which was proven by Fort [21, p.
] .
COROLLARY 1.1.2: > 0 and b n n equality not holding for every n
Let
c
n
> r
i'l(c n- li'lu n- l ) - b n u n
for
n=O,l,
u
o
m
=
0
. .. , m-l , and
(1.1.51)
19
then there is at least one node of
o
lI(r n- lllv n- 1) - q n v n in
(a,
(1.1.52)
b)
REMARK: We note that the condition lent to
u(a)
u(b)
=
0
when
u -1
u
=
um
=
0
is equiva-
is considered a solution
of (1.1.0). By a node we mean a point on the abscissa where the "polygonal curve" defined by the finite sequence
v
crosses
n
the axis.
Proof:
The condition
implies that
c
along with (1.1.47-8)
> r > 0 n = n
P1 (t) > P2 (t)
> 0
.
Moreover, since
b
n
> q = n
we find from (1.1.45-6) that
(J
Since
1
(t
n - 0) -
°2 (t n -
(1.1.53) implies that
for
t
E
[a, b]
°1 (t)
-
O
(1.1.53)
for each
are step-functions on
n,
0) •
2
(t )
is non-decreasing
This, along with the above Remark, shows
that Corollary 1.1.0 is applicable and hence the equation (1.1.1) has at least one zero in (a, b) to the required conclusion.
which is equivalent
20
Note:
In general, a comparison theorem for equations of the
form
c n Yn+l +c n-l Yn-l -b n Yn
under the assumptions For example let r
n
that
q
= c/2
n c
n
> r
c
n
c
= c > 0
for each b
n
in this case,
n
> r
n
> qn
(1.1.54)
o
(1.1.55)
for all n=O,l
n
,
,
and
...
, m-l
b
n
,
= 3c
We see then
but a simple computation shows that,
has no nodes eventually while (1.1.55)
will have nodes for large
§1.2
(1.1.54)
is not available.
n
,
n
o
n.
SEPARATION THEOREMS: In this section we prove the classical Sturm separation
theorem, on the separation of zeros of linearly independent solutions, as a consequence of the results in section 1.
In
the case of finite differences this result was also probably known to Sturm [58, p. 186] as one can gather from the remarks at the end of his memoir. If
c
n
> 0
for all
n,
then the nodes of solutions
of
c n Yn+l +c n-l Yn-l -b n Yn
o
(1. 2.0)
21
separate one another if these are linearly independent.
(The
proof of this result will follow below.) The Sturm separation theorem is not valid in general in the case of a general three-term recurrence relation
o .
(1. 2 .1)
Bacher [6, p. 176] points out that the separation property for solutions of (1.2.1) holds if
P
for all
n
n
R
n
> 0
(1. 2.2)
in the range considered.
false, in general, if (1.2.2) fails. an example the case where
P
n
= 1 ,
The result is however He gives [6, p. 177] as
Q = R =-1 n n
for all n.
The nodes of the linearly independent solutions corresponding to the initial values Yo
=
6
y-l = 0,
Yo = 1
and
y-l = -10 ,
do not separate one another. One proof of the separation property of (1.2.1) under
the hypothesis (1.2.2) was given by Moulton [45, p. 137].
We
note that the condition (1.2.2) is the analog of the condition p(t) > 0
for the equation
p(t)y" + q(t)y' + r(t)y
If
p(t) > 0
o .
(1.2.3)
then the zeros of linearly independent solutions
22
of (1.2.3) separate one another.
(One way of seeing this is
that (1.2.3) can then be transformed into an equation of the form (P(t)y')' + Q(t)y
where
P(t) > 0
o
(1. 2.4)
and the result follows from the separation
property of the zeros of (1.2.4).)
THEOREM 1. 2 . 0 : The zeros of linearly independent solutions of
c + It y(s)dcr(s) a
p(t)y' (t)
(1.2.5)
separate one another.
Proof:
(1.2.5) has two linearly independent solutions
which generate the solution space [3, p. 348]. and to, say,
u
find that
If we now set
in (1.1.16) we can apply Corollary 1.1.0 when
v
u, v
u
vanishes at two consecutive points to
must vanish in between since
stant multiple of
v
is not a con-
u.
In particular if
o
E
C' (a , b)
the classical Sturm separation theorem.
we immediately obtain
23 COROLLARY 1.2.0: If
C5
C I (a , b)
E
and
C5 '
(t)
= q (t)
t
E
[a, b)
then the zeros of linearly independent solutions of
(p(t)y')
I
-
q(t)y
o
(1.2.6)
separate each other.
Porter [49, p. 55] showed that two linearly independent solutions of (1.2.0) generate the solution space and considered the limiting process which takes a difference equation to a differential equation. Defining
C5,
P
as in (1.1.24-25) we obtain the discrete
analog
COROLLARY 1. 2 . 1 : If n
c
n
> 0,
= 0 , 1, ••. , m-l
n
= -1
, 0 , ... , m-l
and
b
n
is any sequence and
n-
n- 1)
- b n Yn
o
(1. 2.7)
then the nodes of linearly independent solutions separate one another.
As an application of Corollary 1.2.1 to the recurrence relation (1.2.1) we state the following [45, p. 137].
24 COROLLARY 1.2.2: Let
P
Q ,R n n
n,
be real finite sequences and
o for
n
=0
(1.2.8)
, 1, ... ,m-l .
If
P
> 0
R
n n
n
=0
, 1 , ••• , m-l
(1.2.9)
then the nodes of linearly independent solutions of (1.2.8) separate each other.
Proof:
The idea is to show that (1.2.8) under the hypothesis
(1.2.9) can be brought into the form (1.2.7) after which we simply apply Corollary 1.2.1. Let
c_
l
> 0
P
and consider the recurrence relation
n
n
R c n- l n
(1.2.9) implies that
c
n
> 0
=0
for
, 1 , ••• ,m-l .
n
=
0 , 1 , ••• , m-l
(1.2.10)
since
If we now set
(1.2.11)
for
n
= 0 , 1, ... ,m-l,
then a simple computation shows that
25
with the substitutions (1.2.10-11),
(1.2.7) reduces to the
three-term recurrence relation (1.2.8).
Hence the result
follows:
§1.3.
The GREEN'S FUNCTION: In Appendix I to this
work
we have shown the exis-
tence of a Green's function for the inhomogeneous problem
lji(t)
CI.
+ 13
Ja t
1 p
+
Jt
IS
1
P (s )
a
ljido ds + Jt f p
a
a
o
(1.3.0)
(1.3.1)
where
{M .. lji ( j -1)
j=l
1J
(a)
+ N .. P (b) lji ( j 1J
-1) ( b) }
i
=
1 , 2 ,
(1.3.2)
and the
M.. , N .. 1J
1J
are real constants, under the hypothesis
that the homogeneous problem (with
f
conditions (1.3.1) is incompatible.
=0
)
a.nd the boundary
(By this we mean that the
homogeneous equation with homogeneous boundary conditions has only the zero solution.)
If
f
=
0
in (1.3.0) then the
resulting integral equation is of the form (1.0.0). If ous on with
0
[a, b) o'
=
q.
E
C' (a, b)
and
p(t)
then (1.3.0) with
is positive and continuf
=
0
reduces to (1.2.6)
In this case the "derivative" appearing in
(1.0.0) is continuous and the Green's function reduces to the
26
usual one.
(See Appendix I, p. 278 .)
On the other hand if then (1.3.0) with
o
f
recurrence relations.
p(t)
,
a(t)
are step-functions
can be made to include three-term
In this case and, more generally, for
difference equations of higher order the Green's function seems to have been first constructed by Bacher [5, p. 83]. Another treatment was given by Atkinson [3, p. 148]. We showed in Appendix I that if (1.3.0-1) with is incompatible then the unique solution of (1.3.0-1)
f
0
is
given by
e
for t
E
x
E
[a, b]
[a, b]
and
same points where
In the particular case when f(t)
p(t) = 1 ,
is a step-function with jumps at the
a(t)
has its jumps and if we denote by
f(t.)
f.
- f(t. - 0)
l
l
where, as usual, the
(1. 3.3)
G(x, t)df(t)
1j!(x)
t. l
(1.3.4)
l
represent the jump points of
f
,
then a simple computation shows that
t a
m-l
L
i=O
G(t
n
G(t
, t)df(t)
n,
til • (f(t
i)
- f(t
i
- 0)) (1.3.5)
27
and if we write
o
G . _ G(t , t.) nl. n l.
< n,
i < m- 1
we find
that m.... l ljin
This
ljin
I
i=O
G .f. nl. l.
(1.3.6)
then represents the solution to the corresponding
inhomogeneous difference boundary problem.
Usually (1.3.6) is
derived directly using methods of finite differences. for example [3, p. 149] and [5, p. 84].)
(See
For further details
see Appendix I, section I.4. We note that when
p(t)
, a(t)
functions of bounded variation on
are continuous
[a, b]
then the derivative
appearing in (1.0.0) is continuous everywhere and so, from Appendix I, the discontinuity in the first derivative of the Green's function is given by
G (t + 0 , t) - G (t - 0 , t) x
x
1 p(t)
(1.3.7)
which is the usual measure of discontinuity of the Green's function associated with a second-order linear differential equation of the form (1.2.6).
CHAPTER 2 INTRODUCTION: There is a very extensive literature dealing with the subject of oscillation and non-oscillation of real second order differential equations on a half-axis (see, for example, [59]).
On the other hand there is little known about
establishing criteria for the oscillatory and non-oscillatory behaviour of solutions of difference equations.
In the
particular case of three-term recurrence relations some results can be found in [23, pp. 126-128] and more recently in [32, p. 425]. [12],
Other results are more or less scattered:
[21],
[20]. In this chapter we shall be concerned with obtaining
some oscillation and non-oscillation criteria for linear and non-linear Stieltjes integral equations on a half-axis.
It
will be noted that if one makes an hypothesis on the integral of the potential
q
in
y" - q(t)y
o
t
E
[a, (0)
(2.0.0)
which will guarantee the existence of oscillatory or nonoscillatory solutions, then a certain discrete analog will
29
exist for a three-term recurrence relation. In section 1 we give some non-oscillation criteria for Stieltjes integral equations and their applications to second order difference equations.
In section 2 we give some results
on the oscillatory behaviour of solutions and in section 3 we extend a result of Butler [8, p. 75] and state a necessary and sufficient condition which guarantees that all continuable solutions of a non-linear equation are oscillatory.
As a
corollary we shall obtain the discrete analog of Atkinson's theorem [2, p. 643].
Various examples are included which
should help visualize the theorems stated.
§2.l
NON-OSCILLATION CRITERIA FOR LINEAR VOLTERRA-STIELTJES INTEGRAL EQUATIONS: In the following, we shall usually be considering
equations of the form
y' (t)
where
0
c + It y(s)do(s) a
t
E
[a, 00)
(2.1. 0)
is a right-continuous function locally of bounded
variation on
[a, 00)
Because of the applications we shall
assume, in addition, that the number of discontinuities of remains finite in finite intervals.
0
The theorems proved here
can also be extended to equations of the form
p(t)y' (t)
c +
r a
y(s)do(s)
(2.1.1)
30
in the case when
p(t) > 0 ,
(2.1.2)
00
p
satisfying the usual conditions stated in Chapter 1.
every equation of the form (2.1.1), where
p
For
satisfies
(2.1.2), can be transformed into an equation of the form (2.1.0) by the change of independent variable
t t---»
which will take
[a ,00)
T
(t)
into
=
It 1. a
(2.1. 3)
p
(See Appendix I,
[0,00).
equation (1.3.14).)
DEFINITION 2.1.1: A solution of (2.1.0) is said to be oscillatory if it has, to the right of
a,
an infinite number of zeros and is
non-oscillatory if there is some
zeros when
t
to
E
m
such that it has no
to
From the Sturm separation theorem, Theorem 1.2.0, we see that if one solution is oscillatory (non-oscillatory) then all solutions are oscillatory (non-oscillatory). Equation (2.1.0) is said to be oscillatory (nonoscillatory) if all of its solutions are oscillatory (nonoscillatory) .
Unless otherwise stated we shall, in the following,
31
assume that
o(t)
,
appearing in (2.1.0), has a limit at
00
Le. lim
(2.1.4)
0 (t)
t-+ oo
exists and is finite.
0(00)
Denoting this limit by
assume it is zero (for if we let has the same properties as
,(t)
(2.1.0) remains unchanged if
o(t) - 0(00)
=
,(00)
and
0
=
0
then
,
Moreover,
is replaced by
0
we can
,).
The first result is an extension of a well-known theorem of Hille [31, p. 243] which relates the non-oscillatory behaviour of (2.1.0)
to the existence of solutions of a
certain non-linear integral equation. THEOREH 2.1.1: Let
0
be right-continuous and locally of bounded
0(00)
variation satisfying (2.1.4) with
and sufficient condition for (2.1.0)
= o.
Then a necessary
to be non-oscillatory is
that the integral equation
v(t)
=
o(t) +
C
v 2(s)ds
have a solution, for sufficiently large integrable at infinity
Proof:
(cf.,
(2.1.5a)
t,
which is square
[80]).
To show that the condition is sufficient assume that
(2.1.4) has a solution then implies that
v(t)
V E
some
(2.1.4)
is right-continuous, locally of
32
bounded variation and
v(oo)
O.
Put y(t)
exp Jt v(s)ds .
(2.1.5b)
a
Then
y(t)
is locally absolutely continuous and so
=
y' (t)
v(t)exp Jt to
v(s)ds
(2.1. 6)
everywhere, as a two-sided derivative, except possibly the jump points of Letting
v(t)
h > 0,
t
which are the same as those of
o(t)
arbitrary,
e x p [(t+h Y (t+h) - Y (t)
y(t) .
h
v (s)
dS) (2.1. 7)
{
h
Now
t +h Jt
for each
h
>
0,
v-I
)
fixed
use Theorem H of Appendix
(2.1. 8)
t I
Hence we can let
+
0+
and
to find that
I It+h lim 11 t v(s) ds
h+O+
h
v (t)
(2.1. 9)
33
while the other terms are zero by virtue of (2.1.9) and the continuity of the integrals. Hence letting
h
+
0+
in
we obtain, from above,
y' (t)
(2.1.10)
y(t)v(t)
where the derivative is in general understood as a rightderivative which is locally of bounded variation.
r r
Thus if
dy' (s)
to
d (y (s) v (s) )
to
I
t
v dy +
Y dv
to
to
where we have
I
t
by equation (2.1. 10) ,
(2.1.11)
Theorem K of Appendix I now implies that the first integral in (2.1.11) vanishes for all
t
and hence
34
y' (t)
so that
y(t)
equation. for
t
> =
is a positive solution of the above integral
This implies that (2.1.0) has a positive solution t
0
and hence is non-oscillatory.
To prove the necessity we suppose that (2.1.0) has a non-oscillatory solution positive for For
y(t)
t > to . t > to
we set
v(t)
Then
v(t)
which we can suppose is
z.'.J!.L
(2.1.12)
y(t)
is locally of bounded variation on
[to' 00)
is right-continuous.
Hence, for
t > to '
r
to
y(lS) dy' (s) -
r [?) 0
2
ds
and
35
Hence
v(t)
for
t
to'
(2.1.13)
Since
o(t)
has a limit at
that the same must be true of Suppose, if possible, that square-integrable at
v(t)
v(oo)
y' (t) < 0
there is a
t
for
(2.1.13)
shows
. a
O.
Then
v
cannot be
and so (2.1.13) implies that
00
lim v(t) t-reo
Hence
=
00
t
> =
t
1
(2.1.14)
_00
because of (2.1.12).
Moreover
such that when
2
(2.1.15)
If we let
t3
=
(2.1.13) with
max{t o ' t 1 ' t 2 }
t , to
then using (2.1.15) in
replaced by
T, t
3
respectively, we
obtain
v(T) < -1 +
JTt 3
whenever
T > t
=
3
.
y(s)
(2.1.16)
We now use Gronwall's inequality [9, p. 37,
Exercise 1] in (2.1.16) to obtain
36
v(T)
« -1 -
I:, IY;(;:/ I {JJy; I exp
-exp JT t
}dS
y(s) 3
Thus, by (2.1.12),
(2.1.17)
for all
T
t
(2.1.17) implies that
•
3
positive which is a contradiction.
y(t)
cannot remain
v(oo) = O.
Hence
We can
now rewrite (2.1.13) as
v(T) - v(t)
where
T
that
v(t)
t
t
o (T) -
Now letting
3.
satisfies (2.1.4) for
0 (t)
T
-
J
v
2
(2.1.18)
in (2.1.18) we find
00
t
T
t
t
3.
This completes
the proof.
THEOREM 2.1.2: Let
0
1 '
O
2
bounded variation on 0;(00)=0,
i=1,2
be right-continuous functions locally of [a, 00)
satisfying (2.1.4) with
37
Assume that
(2.1.19) If
(2.1.20)
has a solution for
t > to
then
(2.1.21)
v(t)
has a solution for
Proof:
t > to .
We shall make use of the Schauder fixed point theorem
(Appendix II, Theorem 2.1.1).
With the Banach space
and the usual norm we consider the subset
x where
{v
E
vi (t)
For
L2
is as in (2.1.20). v
E
X
we define an operator
a E [0, 1]
and
T
on
X
by
(2.1.23)
(Tv) (t)
If
(2.1.22)
(to ' co)
x , Y EX,
38
I ax
+
(1 - a) y
I
< a Ix
I
+
(1 - a)
Iy I
(2.1.24) and hence
X
For
is convex. v EX,
I (Tv)
(t)
I
to.
- (Tx) (t) [2
t
Tx
n
- Tx I 2
Hence
0
To show that of Appendix II. ITxl
vi
T
-+
0
as
n
-+
(2.1.31)
00
is continuous.
TX
is compact we use Corollary 11.1.2
(11.1.4) is satisfied since if
x EX,
and so
(2.1.32)
= {t: to
E A
choose
sufficiently large so that
A
A
t < oo}
If we let
then given
>
0,
we
(2.1.33)
This will then imply that
(2.1.34)
for all
x E X
by virtue of (2.1.32).
This proves (11.1.5).
TO prove (11.1.6-7) we need some additional information. since
is a solution of (2.1.20),
2
vi
(t)
>
1° 1 (t) I 2
(2.1.35)
41
and so
01
L
E
2
(to' (0)
•
By the same argument,
{C
(t)
and so
(2.1.36)
The following theorem [24, p. If
f
E
LP[t
o'
(0)
,
P > 1,
Ilf(x+h) -f(x)ll
Since
01
L 2 [t
E
o
L
2
, (0)
[to ' (0)
] will also be useful. then
p
as
0
-+
h
0 .
-+
we have from (2.1.19)
(2.1.37)
that
and thus
110 2(t+h) -0 2(t)11
-+
0
h
as
0
-+
(2.1.38)
on account of (2.1.37) . Similarly if we set
V(t)
C
2
Vi
then I!V(t+h) -V(t)11
-+
0
as
h
-+
0
(2.1.39)
42
because of (2.1.36). Thus if
x EX,
>
0
(2.1.40)
if
Ihl < 8 ,
by the continuity of the integral.
This proves
(11.1.6) •
For
x EX,
> 0
II (Tx) (t+h) - (Tx) (t)11
Ilo 2(t+h) -02(t) +
From (2.1.38) we can choose
h
+
II
Jt+h
x
2
ds II
2
t
< Ilo 2(t+h) -o2(t)11
t
Jt+h x
so that if
(2.1.41)
II • jhl < 8 1
then
(2.1.42)
Similarly there is a
8
such that whenever
> 0
2
Ihl < 8
2
(2.1.43)
Ilv(t+h) -V(t)11 < Thus
II
Jt
t+h
x
2
11
to
and
(2.1.45) If v
has a solution for
1
(t)
t > to
v (t)
has a solution for
Proof:
01 (t)
+
"" v 12 ds ft
then
= ± o 2 (t)
+
f:
v
2
ds
t > to .
This follows immediately from the theorem.
THEOREM 2.1.3: With
(2.1.46)
as above and
(2.1.47)
44 (2.1.48)
suppose that y' (t)
is non-oscillatory.
z'
(2.1.49)
Then
(2.1.50)
(t)
is non-oscillatory.
Proof:
This is immediate from Corollary 2.1.2 and Theorem
2.1.1.
THEOREM 2.1. 4 :
Let
0(t)
satisfy the conditions of Theorem 2.1.1.
If
tl0 (t) I
to > 0
(2.1.51)
then (2.1.0) is non-oscillatory.
Proof:
Let
01 (t)
Theorem 2.1.3.
=
1/4 t
and
02 (t)
==
0(t)
and apply
This is permissible since (2.1.49) is then
equivalent to y" +
1 4t
2
Y
o
(2.1.52)
45
which is a non-oscillatory Euler equation [59, p. 45].
The
result now follows.
COROLLARY 2.1.3: (2.1.0) is non-oscillatory if
lim sup tlo(t) t++ oo Proof: t
I
0
or
y(t) < 0
for
t;; t
1
•
If
y(t)
> 0 ,
46
Theorem 2.1.1 implies that (2.1.46) has a solution t ;;: t* for
maxi to ' t i}
vi (t)
Hence (2.1.47) has a solution
for
v(t)
t > t* (because of Corollary 2.1.2) which corresponds to
some non-oscillatory solution suppose
z(t) > 0
for
z(t)
t > t*
of (2.1.50).
We can
Since the proof of Theorem
2.1.2 guarantees that
Iv(t)
I
t > t*
< vi (t)
we can recover the non-oscillatory solutions
(2.1.56)
y, z
to find
that z (t) < Y (t)
If
z(t) < 0
for
other hand if
t;;: t*
y(t) < 0
t > t* .
(2.1.57)
the last line is clear. for
t;;: t
i
then
On the
-y(t) > 0
the above argument shows that there is some solution
and z(t)
such that z(t) < -y(t)
t > t* .
(2.1.58)
This completes the proof.
THEOREM 2.1.6: Let
o(t)
satisfy the hypotheses of Theorem 2.1.1.
If (2.1.59)
47
then (2.1.0) is non-oscillatory.
Proof:
By Theorem 2.1.1 i t suffices to show that (2.1.4) has
a solution for sufficiently large
We shall again make
t
use of the Schauder fixed point theorem. Let
X
be a subset of
defined by
Iv (t)
X
-
0 (t) I
10 (t) I ,
to}
. (2.1.60)
For
v
X
E
we define a map
(Tv) (t)
If
a E [0, 1]
1
au
and
T
by
I:
o (t) +
Ia (u -
01
0)
I
< a u - 01
X
2
(2.1.61)
ds .
u, v EX,
+ (1 - a) v -
This shows that
v
0
in Theorem 2.1.7 and then state the
converse.
2.1. 8 :
THEOREM
Let With
O
o(t)
satisfy the hypotheses of Theorem 2.1.1.
defined as in (2.1.65) suppose that (2.1.64) is non-
2
oscillatory. Then (2.1.0) is nonoscillatory and for each nontrivial solution
z
of (2.1.64)
there is a solution
such that
o for
t > t
Proof: space
X
where
< Y (t)
to .
is no longer required to be
Equality in (2.1.84) is attained in the case of
the Euler equation (2.1.52).
THEOREM 2.1. 7A: Let
a(t)
satisfy the hypotheses of Theorem 2.1.6A
along with (2.1.84).
If
A(t) > 0
y" + 4A 2 (t)y
for large
t
then
o
(2.1.85)
is non-oscillatory.
Proof:
Refer to Theorem 2.1.7 with
o(t)
- A(t)
•
Whether (2.1.68) being non-oscillatory implies that (2.1.85) is, appears to be an open question [59, p. 93] which we shall discuss in section 2.2.
COROLLARY 2.1.4A: Let
A(t)
> 0
and suppose that (2.1.84) is satisfied.
Then (2.1.68) has a non-oscillatory solution
Iy (t.) I
< exp {2
r t"
A (s) ds } .
y(t)
such that,
(2.1.86)
59
Proof:
This follows from Corollary 2.1.4 with
o(t) - A(t)
.
THEOREM 2. 1 . 8A : Let
A(t)
be defined as in (2.1.68) and suppose that
z" + is non-oscillatory.
4A
2
(t)
o
z
(2.1.87)
Then
o
y" + a(t)y
(2.1.88)
is non-oscillatory and for each non-trivial solution (2.1.87) there is a solution
o for
t
Proof:
< y(t)
y(t)
t
sufficiently large, say,
of
of (2.1.88) such that
{r
< [z Lt.)
z(t)
t
IA(S) IdS}
(2.1.89)
1
t
1
.
This is an application of Theorem 2.1.8. The first part of the theorem is identical with a
theorem of Hartman and Wintner [27, p. 216] though the estimate (2.1.89) is stronger than the corresponding estimate in [27] where the absolute value sign about not appear.
A(t)
in (2.1.89) does
Thus the first part of Theorem 2.1.8 extends the
Hartman-Wintner result cited
above
to equations of the type
(2.1.0), while the second part extends the corresponding result only when
o(t) > 0
in Theorem 2.1.8.
60
2.1B
APPLICATIONS TO DIFFERENCE EQUATIONS: In this subsection we apply the theorems of section
2.1 to recurrence relations of the form
o
c y + c y + b y n-l n-l n n n n+l where
n
=
-1 , 0 , 1 , ... ,
(2.1.90)
is any given real
(b )
n
n = 0 , 1 , ••• •
sequence,
We shall assume, unless otherwise specified, that
(2.1.91)
00
be satisfied as an extra condition upon the We saw in Chapter 1 that if
o(t)
c
n
is a step-function
with jumps, at a fixed increasing sequence of points where
t -1
=
a
(t )
n
and (2.1.92)
n=O,l, ... , of magnitude
o(t) - o(t -0) n
for
n
=0
, 1 , 2 , •.. ,
n
-b
n
- c
n
- c
(2.1.93)
n-l
then (2.1.0) gives rise to solutions
of some "extended" recurrence relation in the sense that the resulting solution is a polygonal curve [a , 00)
y(t)
which has the property that if we write
defined on y
n
:: y(t ) n
61
then the sequence
(Yn)
is a solution to the three-term re-
currence relation (2.1.90) for
n
We note that, whenever for given sequences
m
> 0
b
,1, . . . •
o(t)
,
n
=0
is defined by (2.1.93)
then for
, m
o (t)
o (a)
-
L o
(b
n
+c
n
+c
n- 1)
This follows from (2.1.93) and the relation
(2.1.94)
•
o
(t
n
- 0)
= o(tn- 1) . THEOREM 2.l.1B: Let
o(t)
be defined as in (2.1.94) and assume that
exists and is zero.
0(00)
Then a necessary and sufficient
condition for (2.1.90) to be non-oscillatory is that
v(t)
where L2
o(t)
o (t) +
I:
v
2
ds
(2.1.95)
is given by (2.1.94), have a solution which is in
at infinity.
Proof:
This follows immediately from Theorem 2.1.1 and the
results of Chapter 1.
REMARK: A solution
(Yn)
of (2.1.90) is said to be oscillatory
62
if the sequence exhibits an infinite number of sign changes and non-oscillatory if, for constant sign.
n
N,
the sequence retains a
The discrete version of the Sturm separation
theorem shows that if a solution is oscillatory (nonoscillatory) then all solutions inherit the same property. Moreover the transition from (2.1.0) when
°
to (2.1.90), in the case
is given by (2.1.94), shows that a given solution of
(2.1.0) is oscillatory (non-oscillatory) if and only if the corresponding solution of (2.1.90) is oscillatory (nonoscillatory) . Thus, with
°
defined as in (2.1.94),
c + It y(s)do(s) a
y' (t.)
t
E
[a, (0)
(2.1.96)
is oscillatory (non-oscillatory) if and only if
o
c n y n+l + C n-l Yn-l + b n Yn
n=O,l, ..•
(2.1.97)
is oscillatory (non-oscillatory). The latter theorem thus gives the discrete version of Hille's theorem [31]. For given sequences functions
01
'
02
on
c
n
[a, (0)
> 0 ,
n
we define step
by setting
m I' L.
b
(b
n + c n + c n- 1)
(2.1.98)
63
m > 0,
and m
Io
02 (a)
(g
n
+
c
n
+c
n-
We recall that the (2.1.92) a fortiori, so that With
°1
'
°2
t
n
-->-
00
as
c
(2.1.99)
1)
also satisfy
n
n -->-
00
so defined we obtain the discrete
analogs of Theorem 2.1.2 and Corollary 2.1.2 denoted by Theorem 2.1.2B and Corollary 2.1.2B respectively.
Since the
latter two results can be stated in the same way as the former two, we shall omit them and it shall be understood that when we refer to either of Theorem 2.1.2B or its corollary we shall mean Theorem 2.1.2 or its corollary with 01
'
02
THEOREM
given by (2.1.98-99).
2.1. 3B : Let
c
n
>
0
and satisfy (2.1.91).
Suppose that
m lim I (b + c + c 1) n n nm-->-oo 0
(2.1.100)
m lim I (g +c +c 1) n n nm-->-oo 0
(2.1.101)
both exist and are finite
(so that the series need only be
conditionally convergent) . Suppose further that
64 00
L
m for
(c
m > mo'
n
+C
n-
1 +b )
>
n
I
I
m
(c n
+ C n- 1 + g n )
I
(2.1.102)
If
c n y n+l + c n-l y n-l + b n Yn
o
(2.1.103)
o
(2.1.104)
is non-oscillatory then
is non-oscillatory.
Proof:
Define
01
'
02
by (2.1.98),
(2.1.99) respectively.
(2.1.100-101) are then equivalent to requiring that both 01 (00)
,
exist and be finite.
02(00)
Since we can alter these o (00)
by an additive factor, we can assume that
1
= o 2 (00) = 0
•
This then implies that 00
01 (a)
L
(c n + C n- 1 + b n )
L
(c n + C n- 1 + g n )
0
o 2 (a)
Hence, for
t
E
0
(2.1.105)
.
(2.1.106)
[tm- l ' t m) , 00
o 1 (t)
L
m
(c n + C n- 1 + b n )
(2.1.107)
(c n + C n- 1 + g n )
(2.1.108)
00
o 2 (t)
L
m
65
Thus the requirement that (2.1.48) be satisfied for large
t
is equivalent to the requirement that (2.1.102) hold for large m.
From the remark we see that (2.1.49) must be non-
oscillatory.
Hence Theorem 2.1.3 applies and hence (2.1.50)
is non-oscillatory.
Consequently (2.1.104) is also non-
oscillatory and this completes the proof. The latter theorem is therefore the discrete analog of the Taam result [60].
Simultaneously i t provides an
extension of the discrete version of the theorem of wintner [63]
and Hille [31]
(see Theorem 2.1.3A).
Thus for example,
if > 0
n=O , 1 ,
...
(2.1.109)
gn > 0
n=O , 1 ,
...
(2.1.110)
b
n
and co
L
m
co
b
n
>
L
m
gn
o ,
m > m
(2.1.111)
then (2.1.104) is non-oscillatory if (2.1.103) is.
This would
be the formulation of the discrete analog of Hille's theorem [31] •
THEOREM 2.1. 4B:
c
Let sequence
(b )
n
n
> 0
and satisfy (2.1.91).
For a given
assume that (2.1.100) exists.
If
66
(2.1.112)
then (2.1.90) is non-oscillatory.
Proof:
We define
by (2.1.93).
G
Then, for
t
E
[tm- l ' t m) ,
we shall have
G
L
(t)
m
(c
(2.1.113)
n + C n- 1 + b n ) •
For (2.1.51) to hold for large
t
it is necessary that
(2.1.114)
for all
t t
we let
-+
m
t_
l
m
=
a
mo
=
0
.
m
is sufficiently large.
Thus
in (2.1.114) and use (2.1.92) to obtain
tm
{a +
for
when
[tm- l ' t m)
E
I
0
_1 c n- l
}IIm (cn +
C
n- 1 + b n )
I
0
t
E
(2.1.131) then the differential equation
z"
2
o
+ 40 (t)z
(2.1.132)
is non-oscillatory.
Proof:
This follows immediately from Theorem 2.1.7.
COROLLARY 2.1.4B: Let
c
n
, b
n
satisfy the hypotheses of Theorem 2.1.7B
and suppose that (2.1.131) holds. oscillatory solution
(Yn)
Then (2.1.90) has a non-
such that for
2t
t
Iy n I
< ex p {
n
0 (s)
dS}
n
N , (2.1.133)
73
where
Proof:
a
is as in (2.1.131).
Follows from Corollary 2.1.4.
THEOREM 2.1. 8B : Let
a(t)
be as in Theorem 2.1.7B except that
need not be non-negative.
Assume that
2
o
z"+4a(t)z is non-oscillatory.
a
(2.1.134)
Then
c n y n+l +
C
n-l Yn-l + b n Yn
o
is non-oscillatory and for each non-trivial solution (2.1.134) there is a solution
(Yn)
(2.1.135)
z
of
of (2.1.135) such that
t
where
Proof:
o
< yn
0
is fixed and
(2.2.26)
tA(t)
then (2.1.70) is oscillatory.
Proof:
This is a consequence of Theorem 2.2.1 where
a(t)
:: A(t) The first part of this theorem is due to Wintner [63, p. 260] and the second part follows almost immediately from this result (see [44, p. 131],
[63, p. 259]).
81
THEOREM 2.2.2A: Let
a(t)
and suppose that
satisfy the conditions of Theorem 2.1.6A A(t)
> 0 .
If
(
2 A (s)ds
0 ,
(2.2.28)
then (2.1.70) is oscillatory.
Proof:
This follows from Theorem 2.2.2. The above theorem is due to Opial [47, p. 309].
THEOREM 2. 2 . 3A : Let
a(t)
be continuous on
[a, 00)
and suppose that
( 2 . 1. 6 8 ) exi s ts . If
JOO a(s)ds a
then (2.1.70) is oscillatory.
00
(2.2.29)
82
Proof:
We let
= -
o(t)
Jt a(s)ds a
in Theorem 2.2.3.
The latter theorem was proven by Fite [19, p. 347] in the case when
a(t)
0
p. 115] for general
§2.2B
and was extended by wintner [61,
a(t)
APPLICATIONS TO DIFFERENCE EQUATIONS:
THEOREM 2.2.1B: Let the
c
n
, b
n
satisfy the hypotheses of Theorem
2.1.4B and assume further that
defined in the proof
G
m
m > m
of Theorem 2.1.6B, is non-negative for
= a
.
If
m
{ Ia
1
} mI 00
c n-l
(c n + C n- 1 + b n )
m a (2.2.31)
where
Proof:
> 0
is fixed, then (2.1.90) is oscillatory.
The first part is a consequence of Theorem 2.1.4B.
An argument similar to the one used in the proof of the latter theorem shows that (2.2.3) is equivalent to (2.2.31).
83
As a consequence of this we obtain in particular,
COROLLARY 2.2.1B: Let
(b)
be any sequence whose series is conditionally
n
convergen t and 00
(2.2.32) If 00
m
I
m
bn
my
>
>
for some
E >
0
if
m
C+l
m
-2
dx
y
m+l 1 4" +
x
(2.2.36)
E
is sufficiently large since
y >
1:-4
Consequently the above corollary implies that (2.2.35) is oscillatory. Using the discrete Euler equation (2.2.35) as a comparison equation we can deduce the following result. (b) n
is a positive sequence such that, for fixed
E
If
> 0 ,
(2.2.37)
then (2.1.116) is oscillatory.
[4"1+ ) E
then
bn
gn
0
For if we let
(n
+
1) -2
(2.2.38)
and Theorem 2.1.3B would lead immediately
to a contradiction if (2.1.116) was assumed non-oscillatory. Similarly it can be shown that if
0
(2.3.4)
n=1,2, ... ,
and continuous, in terms of an integral condi-
tion on the coefficient.
This has recently been generalized
[8 ] to equa tions y" + p(t)f(y)
where
p(t)
o
is unrestricted to sign and
into a "superlinear" equation.
(2.3.5)
f
turns (2.3.5)
The result which we shall
prove later on will give, in particular, a necessary and sufficient condition for the difference equation
2
6. Y n 1 + b n f (y n )
to be oscillatory.
o
As a corollary we shall obtain the dis-
crete analog of Atkinson's theorem [2], i.e. positive then, for
(2.3.6)
n=O,l, ...
If
k > 1 ,
o
n=O,l, ...
(b
n)
is
89
has a non-oscillatory solution if and only if 00
I
o
nb
n
0
for all
y
7-
o.
f'(y) > 0
and
oo
dt
1
f (t)
J b)
T
J
T
J
t
dt
< 00
f (t)
- 00
0
Y (t)
We let
g (t)
=
f
f
(t)
Y (t)
J
(2.3.12)
where the prime represents in general a right-derivative. Then
g{t)
variation on
shall be right-continuous and locally of bounded (to' 00)
An integration by parts shows that, for
to < t < T ,
91
I
T 1 t f(y(s») dy' (s)
T f'(y)(y,)2 t ds f(y)2
J
g(T) - g(t) +
(2.3.13)
where we have omitted the variables in the integrand for simplicity.
Moreover, an application of the integral equation
(2.3.1) shows that
T 1 It f (y)
dy I = -
ITt
(2.3.14)
de .
Hence combining (2.3.13-14) we find
=
g(t)
whenever
g(T) + er(T) - er(t) +
to < t < T.
I:
f'
ds
(2.3.15)
Our basic assumption leads us to two
cases: I)
lim sup T-+oo
II)
Case I: t
=>
t
0
(
lim sup ( T-+oo
P(s)ds
some
+00
P (s) ds
to
t > to.
(2.3.16)
(2.3.17)
(2.3.16) implies that the relation is valid for all .
If there is a sequence
T
n
t
00
such that
(2.3.18)
then for
n
sufficiently large we shall have, for
92 T
n
Tn
g(t)
g (T ) + 0 (T ) -
n
G (t)
n
+
f
f
I
(y) g
2 ds.
(2.3.19)
t
Hence
It
T
g(t)
so that letting lim sup
>
n
Tn
It
n do +
00
we obtain
f' (y)g
2
g(t)
ds
(2.3.20)
P(t)
Taking the
of both sides of the latter we obtain a contradiction
on account of
(a).
If no such sequence exists then we must have
g (t)
< 0
(2.3.21)
y' (t)
< 0
(2.3.22)
and so
(2.3.15) now implies
g(T) < g(t) -
Moreover (2.3.16) implies that large of some
t
I:
(2.3.23)
do .
P(t) > 0
for arbitrarily
(not necessarily all such) which shows the existence t2
t
1
such that
(2.3.24) if we assume that
P(t)
t
'
2
g(T) < g(t ) - -K < 0 •
(2.3.25)
2
Replacing
t
by
t
in (2.3.15) and using (2.3.24 - 25) we
2
obtain
+
g(T)
If we write t > t
=
2
r t
f' (y (t) )
¢(t)
fl (y) Iy' f (y)
g(T)
-K +
g ds .
(2.3.26)
2
Iy
1
fly (t)}
and so
I
(t)
r t
I
then
¢(t) > 0
for
(2.3.27)
¢(s)g(s)ds .
2
An application of the Gronwall inequality to (2.3.27) then gives
g(T) < -K
f (y (t
2)
)
f(y(T))
T > t2
(2.3.28)
T > t
(2.3.29)
Le. y' (T)
0 t
P (s) ds
I
such that
T > t
< Mt
We now proceed as in Case I.
(2.3.31)
.
If there is a sequence
such that (2.3.18) holds for large
n
T
n
t
00
we find from (2.3.20)
that g2 (t) >
y (t)
i)
Now either
ii)
,;
i)
Let
y(t)
>
s > 0
there is Y (t ) n cS
> 0,
(2.3.32)
(t)
t > t3 '
(t)
such that
n
or t
n
too,
4- 0
t > t
c = inf {r : (u)
•
3
cS
Since
u < oo} > 0
(2.3.33)
(2.3.20) then implies that
g(t) > P(t) + c (g2 d S
(2.3.34)
95
g ( 1:) > P (t) + c (
Integrating both sides over
T +
[t, T)
we get a contradiction to
ro
(2.3.35)
P (r) dr
and taking
(a)
lim sups
as
since by hypothesis the
integral of the right side of (2.3.35) is divergent. ii)
Let
For large
t
n n
t
ro
and y(t )
0 >
t(t)n
fixed,
t
ds f(s)
+
y(t ) n
be such that
t > t
3
t
0
.
, ro
> It n (p (s) + I
s
f'
because of (2.3.20) , in the limit, and (2.3.32) . Thus Y (t )
••
ds f(s)
t (2.3.36)
> It n P(s)ds
so that (
lim inf n+ ro l
y(t )
••
ds f(s)
t
1
> lim inf It n P(s)ds n+ ro
and so
o because of But
(b).
I
y (t )
o
ds f(s)
T+n.
f(y)dcrds
[T, T + n )
,
[T,T+n].
E
AnY
(2.3.41)
(2.3.42)
has a right-derivative at
given by
(A y) n
1
(t) =
Joo t
f(y(s))dcr(s)
•
then integration by parts shows that
(
f(y(s))dcr(s)
f(y(t))
r r{r t
de +
s
t
do } f' (y)y'ds •
Hence
f(y)dcr = f(y(t))P(t)
(
Thus for
T::; t
+ (
P(s)f ' (y(s))y' (s)ds •
::; T + n ,
and proceed to show as in [8] that
I (AnY)' if
T
(t)
I
is so large that
< Q(t)
t
E
[T, T
+ n)
(2.3.43)
100
2b
If
t
T
+n ,
(A
(I n
y)
I
P ( s) ds
I
(t)
=
hence
0
> T .
t
A
n
(B )
B
c
n
•
T so large that
If, in addition, we require
(
< 1
Q(s)ds < 1
> T
t
(2.3.44)
then o < (An y) (t)
< 2
t
> T ,
(2.3.45)
since we can estimate the inner integrals in (2.3.41-42) by (2.3.43) and (2.3.44)
then gives
also follows from (2.3.42)
(A
If
T < t
1
< t
n
2
y) (t)
(2.3.45).
For
y
E:
B
n
it
that
t>T+n.
constant
< T +n ,
t 2
I ft 1
f
(y (s)
) do (s)
t
f' (y)y'
If necessary we can restrict
T
J s
I
2
do ds
further by requiring that
101
t
b
IJ
Q (s)
2
do
s
I
< 1
Substituting this in the former equation we obtain
Thus
A
n
(B
n
)
B
c
•
n
There remains to show that
A
n
is continuous:
This
can be done as in [8, p. 82] with the appropriate modifications in the definitions of
a(o)
, b(o)
there.
Le. a(o)
sup{lf(y) -f(x)!
0 < x,y < 2,
b(o)
sup{lf'(y) -f'(x)!
0 < x,y < 2, Iy-xl < o} .
From the above definitions we see that as b(o)
+
0
since If
given
E;,
f
E
C'
c (0)
0
+
0
both
.
0
is chosen sufficiently small so that, for
II x
- yll < 0 ,
2
where
Iy-xl < o}
c (0)
n=m c n n
N+oo
.
-00
Q.J- 1 - max{-P j_ l
I
o}
and 00
R.J- 1 00
Then
1 c. R.J
0
for
n
N.
Then a necessary and
105
sufficient condition for (2.3.48) to be oscillatory is that 00
00
00
+
Proof:
00
•
(2.3.52)
This follows immediately from Theorem 2.3.2 because
(2.3.51- 52) are equivalent since
P
> 0 .
n
THEOREM 2.3.3: Let
0
satisfy the basic hypotheses of Theorem
2.3.1. a)
If
lim
(2.3.53)
0 (t)
t-+oo
then (2.3.1) is oscillatory. b)
If
o(t)
is non-decreasing then a necessary and
sufficient condition for (2.3.1) to be oscillatory is that
r
tdo (t) =
(2.3.54)
00
to
Proof: P(t)
=
a)
follows immediately from Theorem 2.3.1 since for all
00
t.
Hence (2.3.9) is identically satisfied.
To prove b) we must show that (2.3.9) is equivalent to (2.3.54) . If
0
is non-decreasing and (2.3.9) is finite then (
P(s)ds
0 ,
108
so that (2.3.58) is equivalent to (2.3.54). In particular we can choose all
n
when
-1,0,1, '" b
n
.
a = -1
and
c
n
1
for
We then obtain from (2.3.48) that,
0 ,
>
/:,.2 Y
+ b f n
n-l
o
(y )
n
(2.3.60)
is oscillatory if and only if 00
mb
(2.3.61)
m
This is the discrete analog of Atkinson's theorem [2] which follows from the previous corollary.
Example 1: b
n
Let
=l/(n+l)
and if we choose m
L{ Lc
00
0
0
c
= n + 2, n n=O,l, a = 0,
n = -1 , 0 , 1 , ...
and let
Then (2.1.91) is satisfied
then 00
L {I +1.+ 2
1 }b m i- l
0
1 ••• +_l_} m + 1 m+l
00
>
1 L m+l
00
0
Hence Corollary 2.3.2(a) implies that all solutions of (2.3.48) are oscillatory where (a) of Theorem 2.3.1.
f
is any function satisfying
109
Example 2:
If we let
b
be as in Example 1 above and
n
1 (n + 1) 1+8
n
o
> 0
then
_l_} bm
Yo { I o ci - l
t.
But (2.3.74) implies that
t
f(s)ds
Hence the latter holds for all
t
E
(2.3.76)
[tm- l ' t m )
and thus applying Corollary
2.1.2 again we find that (2.3.73) has a solution and thus (2.3.64) has a non-oscillatory solution which implies that it is non-oscillatory.
This completes the proof.
As a consequence we immediately obtain that the discrete Euler equation 2
/:; Yn-l +
is oscillatory when
Y >
41
Y
+ 1)
(n
2 Yn
o
(2.3.77)
and non-oscillatory when
y
< ! = 4
(see example I, section 2.1, and Example 1 of section 2B). Furthermore we shall have [30, p. 30],
o non-oscillatory when
A S 0
n=O,l, ...
and oscillatory when
A > 0
117
because of the analogous property for
y" +"Ay
o.
CHAPTER 3 INTRODUCTION: The purpose of this chapter is to provide a basic framework for the theory of operators generated by the Volterra-Stieltjes integral equations encountered in the preceding chapters.
The method used here will show that
these integral equations can be thought of as defining generalized differential operators. undertaken by I.S. Kac
[35]
though the application there was
only to differential equations. been used by H. Langer
Such a formalism was
A different formalism has
[41] to deal with the notion of an
operator defined by a Stieltjes integral equation of the form (2.1.0).
The method which we shall use here is a natural
extension of that used by Kac
[35] and its applications will
include differential equations and in particular, SturmLiouville problems with indefinite weight functions and difference equations. In section 2 we shall proceed to define the generalized differential expression
.2 [f]
l
[f (x) -
r a
f(s)do(s)
J
(3.0.0)
119
where
v, a
are real right-continuous functions of bounded
variation, after having given the background material in section 1. In section 3 we shall study the Weyl classification (limit-point, limit-circle) of singular generalized differential operators with an application to the particular case
_y" + q(t)y
h(t)y
where the weight-function interval.
r(t)
t
E
(3.0.1)
[a, 00)
vanishes identically on some
Other applications will include the three-term
recurrence relation
-c n y n+l - c n-l y n-l + b n Yn where
c
n
>
o.
Aa y
(3.0.2)
n n
These will be discussed in section 4.
In section 5 we give some criteria which can be used to determine whether a certain equation is in the limit-point or limit-circle case.
In section 6 we shall be considering
the self-adjointness and, more generally, the
J-self-
adjointness of such generalized operators. In section 7 we discuss the finiteness of Dirichlet integrals associated with (3.0.0) and consider the chain of implications [39]
DI => CD => SLP => LP
120
where these abbreviations stand for Dirichlet, Conditionally Dirichlet, Strong Limit-Point, Limit-Point respectively. Finally, in section 8 we define these notions for a three-term recurrence relation and give some examples.
§3.1
GENERALIZED DERIVATIVES:
v
Let
be two real right-continuous functions
locally of bounded variation on at each interior point
y
lim
E R
a > _00.
Then
,
u (x )
x->-y±O
[a, 00)
lim
x->-y±O
v (x)
both exist and are finite. Associated wi th u (or v ) defined on intervals
(a, (3]
m (a, (3]
u
is a set function
and
-
[a , (3]
in
m
u
[a, 00)
(3.1.0)
u to)
(3.1.1)
[a , (3]
When
is non-decreasing then
a-fini te Borel measure on function
by
[a, 00)
induces a
[55, p. 262].
Since every
of bounded variation is a difference of two non-
decreasing functions such a function will induce a signed Borel measure on
[a, 00)
a-finite
[55, p. 264, ex. 11] which is
fini te if the original function is bounded on
[a, 00)
We
121
will denote such a measure by set
Then, for every Borel
m
u
E ,
(3.1.2)
m (E)
u
where jl
+ , mjl mjl
are the positive and negative variations of
obtained by its Jordan decomposition [24, p. 123].
each function
jl
right-continuous and locally of bounded
variation induces a
a-finite signed Borel measure on
satisfying (3.1.2).
The measure
[m
where
+
mjl , mjl
u
1m I u
defined by
(3.1.3)
are as in (3.1.2) is called the total varia-
jl , v
jl.
are signed measures we say that
absolutely continuous with respect to every measurable set If
jl(x)
is
[a, 00)
I (E)
tion (or total variation measure) of If
Thus
E
for which
[m v
v
if
I (E)
u
is
[m u I (E) = 0 0
for
[24, p. 125] .
v-absolutely continuous there exists a finite-
valued measurable function
m (E) u for every Borel set
E
¢
such that
J
E
¢dm
(3.1.4)
v
[24, p. 131, ex. 4].
The function
¢
appearing in (3.1.4) is called the Radon-Nikodym derivative of jl
with respect to
v
It is unique in the sense that if
is another measurable function with this property then
¢ =
122
v-almost everywhere (that is they are equal everywhere except possibly on a set When and
¢
E
with
[rn v
I (E) = 0) .
is non-decreasing and right-continuous
is a non-negative Borel measurable function the
Lebesgue-Stieltjes integral of
¢
with respect to
is
defined by
J ¢ (x) If
¢
(x) -
J
(3.1.5)
.
is both positive and negative it is integrable with
respect to
When
if it is integrable with respect to
is of bounded variation
t a
¢ (x) dlJ (x)
agrees with the ordinary Riemann-Stieltjes integral whenever the latter is defined [55, p. 261]. When
f1, v
are of bounded variation and
is
v-absolutely continuous,
dv exists
(x)
lim
- MO
v (x
+ h) -
mv-almost everywhere, in particular when
point of discontinuity of
v
it exists and
v(x+O) -v(x-O) ¢ (x)
(3.1.6)
v (x - h)
x
is a
123
where to
v
¢
is the Radon-Nikodym derivative of
defined in (3.1.4).
with respect
(For general information on these
derivatives, see [24, p. 132].) From (3.1.4-5), for
a,S
E
S±O
I IS±O
-
a±O
[a, (0)
,
[18, p. 134],
¢ (x ) dv (x)
(x) • dv (x)
a±O d v (x)
When
v
are of bounded variation and have no common points
of discontinuity then
I
S±O
13+0
a±O
-
IS±o a±O
.
This is the general formula for integration by parts. is a single point i ts (or
§3.2
When
x
u (or v ) measure shall be denoted by
It is defined by (3.1.1).
v{x}).
GENERALIZED DIFFERENTIAL EXPRESSIONS OF THE SECOND ORDER: In this section we shall essentially pursue the
approach of Kac [35] in the definition of a generalized differential expression of the second order on some interval I,
i.e. for
x
E
I,
a
R,[y] (x)
==
d {y+(x) , -dv(x) -
E
I
fixed,
Ja+O+O y(s)do(s) } X
(3.2.0)
124
where and
was assumed to be locally of bounded variation on
G
I
was non-decreasing.
v
It shall be convenient to assume that addition, right-continuous and bounded variation.
v
G
is, in
an arbitrary function of
We shall see below that it is still
possible, in the latter case, to define (3.2.0).
Basic assumptions: Throughout the remainder of this chapter we shall assume that:
v,
a)
both have at most a finite number of discontinuities
G
in finite intervals (see the hypotheses of §l.l). b)
If
I
is a finite interval then both
continuous at the end-points of infini te interval then both
v,
If
I G
v,
G
shall be
I
is a semi-
shall be continuous
at the finite end, and that neither has a discontinuity at infinity, i.e.
lim v(x)
lim
X-HO
X-+oo
G
(x)
both should exist (may be infinite) . Thus both a), b) shall be assumed in addition to the usual hypotheses of right-continuity and bounded variation for
v ,
G
•
125
Let
v
be two right-continuous functions of bounded
variation. induces a
As we saw in the previous section each of these a-finite signed Borel measure on
addition, we assume that respect to
m
m
u
I.
If, in
is absolutely continuous with
then the Radon-Nikodym derivative
v
dm u
(3.2.1)
¢ - dm
v
exists
v-almost everywhere and we have relation (3.1.4).
Moreover (3.2.1) agreees with (3.1.6) Let
v, a
v-almost everywhere.
be two right-continuous functions of
bounded variation on A function
[a, b] . f
is said to belong to the class
i)
f
is absolutely continuous on
ii)
f
has at each point
derivative
,
f+(x)
x
E:
[a, b]
[a, b]
,
a right-
and the function
_ f:(X) - JX f(s)da(s) a
is
v-absolutely continuous on
We note that (ii) necessitates that variation on
[a, b]
an "associated number"
The quantity [35, p. 212].
[a, b] be of bounded can be termed
126
Thus the preceding discussion shows that if
f
E
Vv
the
quantity
.Q,
exists
{< (x) -
[f] (x) - - dv
v-almost everywhere on
r a
(3.2.2)
f (s) do (s) }
[a, b]
(i.e., it has meaning
everywhere except possibly on a set on which the total variation of
v
is zero).
A particular case of a generalized differential expression is (2.1.0). m v
To see this we let
is Lebesgue measure and let
(2.1.0).
y(x)
v(x)
=x
so that
be any solution of
It is clear that
(x) - JX y(S)dO(S)}
0
a
so that equations of the form (2.1.0) can be brought into the form (3.2.2). By a soZution of the generalized differential equation
(x) -
is meant a function
f
E
V
r a
v
f(S)dO(S)}
, l/J
(3.3.20)
- l/J (x) cj>' (x)
(3.3.21)
1 .
are real for real
A
and satisfy
cj>(a, A)COS a + cj>' (a, A)sin a
0
l/J(a, A)sin a - l/J'(a, A)COS a
= 0
Every solution
cj>
of (3.3.9)
is, up to a constant multiple,
of the form
e where
m
a < b
+ ml/J
(3.3.22)
is some number which depends on
A.
Now let
and introduce a real boundary condition at
b
by
requiring that g(A)
for
S
E
[0,
TI)
- y(b, A)COS S + y'(b, A)sin S
•
o
(3.3.23)
The eigenvalues of (3.3.9) are then the
zeros of the entire function
g(A)
Since these eigenvalues
must necessarily be real (Appendix III, Theorem 111.1.2),
137
(3.3.23) does not vanish identically and consequently the zeros have no finite point of accumulation. We now seek
m
such that the solution
satisfies the boundary condition (3.3.23).
e
above
A simple computa-
tion shows that cot S ¢ (b , A) + ¢ , (b , A) cot S 1jJ (b , A) + 1jJ' (b , A)
m
Thus as
m = m (A
vary
A
are entire functions of
1jJ'
m
S
A, b ,
z
= cot S and fix
S varies from
0
to
TI,
b, A
I
,
1jJ ,
A.
for the moment, then as
varies over
Z
¢, ¢
(Appendix III, Theorem 111.1.0),
A and real for real
is meromorphic in
If we let
Since
b , S)
,
(_00, 00)
so that
the image of the real axis under the transformation Az + B +D
m
A = ¢ (b , A)
where D
=
1jJ'
(b, A)
m-plane. lies on
Thus C b•
and
e
- Cz
B=¢'(b,A)
AD - BC
C
=
1jJ (b , A)
C in the b will satisfy (3.3.23) if and only if m 7
0,
(3.3.24)
'
is a circle
From (3.3.24) we have
z = _B+Dm
(3.3.25)
A+ Cm
so that the circle is given by
C b,
which is the image of
1m z
0,
138
(A + Cm)
(B
+
Dm)
(A
Since every circle with center
+
Cm)
o .
(B + Dm)
and radius
y
r
(3.3.26)
can be
described by
o
(3.3.27)
we see on comparing coefficients of (3.3.26-27) that center of
is given by
AD - BC CD - DC and its radius
(3.3.28)
is given by lAD - BC I
(3.3.29)
ICD - DC I A , B , C , D
Substituting the values for
into (3.3.26) we
obtain the equivalent equation [ee] (b)
0
.
(3.3.30)
In the same way we find that
1
Hence,
[1/1] (b)
AD
[1/Il/J] (b)
DC - CD
(b)
AD
BC
BC
139
mb
[ 0,
if and only if
b
is any sequence and an 0 n [3, p. 129, Theorem 5.4.2] for any A 1m A 0 ,
Aa y
-c n y n+l - c n-l y n-l + b n Yn has at least one nontrivial solution
then
n n
w
=
(w) n
in
£2(V
I).
147
§3.5
LIMIT-POINT AND LIMIT-CIRCLE CRITERIA: In this section we shall give some conditions on
and
v
which will enable us to establish the limit-point or
a
limit-circle classification of
{Y'(X)-r Yd a}
in the space
v
L
2(V;
I)
where
over the interval
(x )
V(x)
[a, x ]
and
(3.5.0)
AY
a
is the total variation of I = [a, (0)
,
a >
-00
•
This space was defined in (3.3.3).
LEMMA 3.5.0: Let
a, v
be right-continuous functions locally of
bounded variation on AO x
Em a,
such that
[a, (0)
Suppose that there exists
a(x) - AOV(X)
is non-decreasing for
say.
Then (3.5.0), with
A
AO'
y(x) > 1
Proof:
Let
y(x)
has a solution x > a •
y(x)
with (3.5.1)
be the solution of (3.5.0) satisfying the
initial conditions y(a)
Then, by Theorem 3.2.0,
o •
1
s ' (a)
y(x)
is a solution of the integral
(3.5.2)
148
equation y(x)
for
x
a.
an interval for
x
E
+
1
Since
r(Xa
8 > 0
[a, a + 8]
(3.5.3)
AO\)(S))
then by continuity there exists
1
y(a)
[a, a+ 8]
s)y(s)d(a(s) -
y(x) > O.
in which
Then,
the integral in (3.5.3) is non-negative
and so y (x )
Since in
y (a + 8) > 1
[a + 8 , 8
y(x)
> 1
X
E
[a, a + 8]
there exists
8
Consequently for
1]
y(a+ 8) +
JX
a+8
such that
> 0
1
x
(3.5.4)
y(x) > 0
in such an interval
(x- s)y(s)d(a(s) -
AO\)(S))
and so y (x)
> 1
Repeating this process we obtain an increasing sequence of real numbers.
n
)
It is then necessary that
lim 8 n+ oo
otherwise if
(8
lim 8
n
8*
n
then
(3.5.5)
00
y(8*)
1
so we could repeat the above process past diction proves that (3.5.5) holds and thus
by continuity and 8*.
This contra-
149
y(x) > 1
THEOREM 3.5.1: 3.5.0.
Let
a, v
x
> a
•
satisfy all the hypotheses of Lemma
Suppose further that
('ldV(t) a
I
(3.5.6)
00
Then (3.5.0) is limit-point at
Proof:
It suffices to show that, for some
solution of (3.5.0) which is not in
L
2(V;
where the latter exists by hypothesis. there exists a solution
y(t)
A,
there is a
I)
From Lemma 3.5.0
of (3.5.0) such that (3.5.1)
holds. Then for such a solution,
CldV(t) a
hence
y
is not in
L
2
(V; I)
I
00
•
COROLLARY 3.5.1 : Let
(a ) n
be a sequence such that 00
L 0
Let
(b ) n
sequence.
la n I
00
be any given sequence and
(c ) n
another positive
150
AO such that
If there exists a real number
b
n
- c
- c
n
n-
1 + A a 0
>
n
n=O, 1, ...
0
(3.5.7)
then cy +c Y -by n n+l n-l n-l n n is limit-point at solution
(Yn)
Aa
y
(3.5.8)
n
A there corresponds a
i.e. for some
00
n
such that
(3.5.9)
00
Proof:
We note here in passing that Lemma 3.5.0 extends to
equations of the form (3.4.11) when
p(x) > 0
right-
continuous and of bounded variation satisfying locally
L (a ,
The proof is similar with minor changes.
00)
We define a step-function (t)
n
p(t)-1
v(t)
with jumps at the
by v (t ) n
and require that n=O,l, ... has solutions
v
v (t
-a
- 0)
be constant on
We define y(t)
n
[t n- l ' t
n) as in (3.4.7).
o(t)
such that
(3.5.10)
n
y(t
n)
= Yn
Then (3.5.0)
satisfies the
recurrence relation (3.5.8). (3.5.10) and the hypothesis imply that (3.5.6) is satisfied.
Moreover for
A = An '
(3.5.7) implies that
151
o - 1..
0
is non-decreasing.
V
Thus Theorem 3.5,.1 applies and so
j""ly(t)1 a
2!dV(t)[ 00
which implies (3.5.9) In this form, Corollary 3.5.1 is a minor extension of [3, p. 135, Theorem 5.8.2] where the case
a
n
> 0
is considered.
THEOREM 3.5.2: Let
0, V
be right-continuous functions locally of
bounded variation on
[0,
r t
and
00)
I do (t) I
-00 by
V.
As usual we denote the total The operator
L
generated by the
"formally self-adjoint" generalized differential expression
[y] (x)
-
y' (x) -
r
y(S)da(S)}
is defined as in section 3.2 of this chapter.
V of
L consists of all functions
f
E
(3.6.0)
Thus the domain
L 2(V; I)
such that
157
i)
f
is locally absolutely continuous on
ii)
f
has at each point
iii)
The function ].1
is iv) For
x
L
V
E
r
(x) :: fl (x) -
00)
a right-derivative
f(s)da(s)
a
V-absolutely continuous locally on
t[f] (x)
f
[a,
E
I .
E
L
2(V;
I .
I)
is defined by
Lf
(3.6.1)
t[f]
The notions of "regularity" and "complete regularity" of the expression (3.6.0) are defined in [35, p. 249] in the case when
v
is non-decreasing. In general we shall say that the end
if the set of "points of growth" of
V(x)
a
is regular
and the set of its
values is bounded from below and if the set of points of growth of
is bounded below and, in addition
a
bounded variation in some right-neighborhood of
a
a
is not regular then it is said to be singular. is completely regular if
a
I
= [a, a
The end
a a
00)
[35]. It is then clear from the
latter definitions and the basic assumptions on section 3.2 that the end
If
belongs to the interval concerned.
These definitions are due to Kac In our case
is of
v, a
is completely regular.
of If
158
I
[a , b]
then the ends are both completely regular.
We note that since
v,
are continuous at
0
a, b ,
in the case of a finite interval, the left and rightderivatives of a solution and be equal. a, b
y(t)
of (3.5.0) will exist there
This can also be seen by extending
by setting each equal to
v(a)
, o(a)
respectively on some interval containing
and
v, v(b)
0
past , o(b)
[a, b]
THEOREM 3.6.1: Let
Q, ['J
on
I
I
=
[a, b]
Let
g
be a finite interval and consider 2(V,I) be any function in L The
equation Q,
has a solution
y(x)
[y]
(3.6.2)
g
satisfying y(a) y' (a)
if and only if the function
y(b)
0
y' (b)
g(x)
We note that
f
is
(3.6.4)
0
is
solutions of the homogeneous equation
Proof:
(3.6.3)
J-orthogonal to all
Q,[y]
J-orthogonal to
=
0 .
g
if and only
if b
Ja
f(x)g(x)dv(x)
o .
(3.6.5)
159
(The
J-orthogonality stems from the
2
L (Vi I)
,
J-inner product in
see Appendix 111.3.)
This theorem can be eroved exactly as in [46, p. 62, Lemma 1].
For by Theorem 3.2.0, and Theorem 1.3.1 the
equation (3.6.2) has a unique solution which satisfies y (a)
=
0,
0
z1' z2
Let
0
=
Y I (a)
be a fundamental system of solutions of
which satisfy
1
o
o
1 .
Applying Theorem 3.2.1 to b
Ja
g (x)
Ja
]
]
conditions above,
b
[z . ] (x)
=
]
Jab
(x)
d v (x) -
[y I
]
b
z.
and
z. (x) dv (x)
Ja y (x) By noting that
y
0
j
=
we find
z.] (x) dv (x) -
Z • -
1,2,
]
-, b y z .] • ] a
(3.6.6)
and using the boundary
(3.6.6) reduces to
g (x) Z . (x) d v (x) ]
{ -y' (bl
j = 1
(3.6.7) y(b)
j
=
2
Thus '3.6.4) is satisfied if and only if (3.6.7) vanishes for
160
j
=1
, 2,
solutions
i. e.
Of
f
if
t[z]
=
J-orthogonal to a fundamental system of
0
and thus the conclusion follows.
v,
Now since the measure induced by
in (3.6.0), is
absolutely continuous with respect to the measure induced by V
the quantity (section 3.1)
dv(x) dV(x)
exists
[V]
(3.6.8)
.
Consequently, the expression
{y'(X) -
J:
Yda}
•
{Y' (x) -
r a
Yda} (3.6.9)
V-almost everywhere by [24, p. 135, Ex. 1, and Theorem A] . Thus if we denote by
t t [Y]
the expression defined by
YE V
v
(section 3.2),
tt[y] (x)
we see that related to
tt t
-
{Y' (x ) -
r a
(3.6.10)
Yda}
is another generalized differential expression by (3.6.9), i.e.
for
Y
E
Vv
dv dV (x) • t [Y] (x )
(3.6.11)
and both of these are defined on the same domain (3.6.10) gives rise to an operator
Lt
on
V,
v
\)
where
Thus D
is
161
the domain of
L
y ED,
defined earlier, such that for
dv dV· Ly
or, in terms of the Gram operator
(3.6.12)
J
defined in Appendix
III. 3,
JL .
(3.6.13)
v
If, in Theorem 3.6.1, we assume that then we can replace
£[y]
in (3.6.2) by
conclusion will then follow with usual orthogonality in
L
is non-decreasing
2(V
i
£t[y]
and the
J-orthogonality being the
I)
since
v
=V
in this
case. We now define a new operator, denoted by
, Vo
domain
= [f
E
V
defined by
f
=0
with
[46, p. 60]
outside a finite interval
[a, S]
c
(a, b l ] (3.6.14)
The restriction of the operator Thus for
y
L
to
v'o
defines
L' o
E
Ly
Similarly we can define
,
(L n)
(3.6.15)
,Q,(y]
v'o (3.6.16)
162
THEOREM 3.6.2: a)
If
Y
,
E
VO '
Z
V
E
then (3.6.17)
[y, Lz]
where
[,]
is the
J-inner product defined by the left hand
side of (3.6.5). Moreover, the operator
L'
o is
J-hermitian, i.e.
y,ZEV
b)
If
v' YE:o
Z
E
,
(3.6.18)
O'
V then writing
we
have
where
is the inner product in Again, the operator
L
1
2
(V; I)
•
is hermitian, i.e.
y,
Proof:
L
Z
E
V •
(3.6.19)
Both a), b) can be shown as in [46, p. 61] making use
of Theorem 3.2.1 so we omit the details.
We now proceed as in [46, §17] in defining the operators Suppose that the interval i , R- t
are both regular on
[a, b]
[a, b] .)
is finite.
(Then
163
V
We define the domain
{y
and, for
y
E
y(a)=y(b)=y'(a)=y'(b)=O}
V:
E
L
o of the operator
o by (3.6.20)
Vo ' (3.6.21)
Ly
(3.6.22)
THEOREM 3.6.3: For any
y
Vo '
E
Z
E
[LoY, z] ==
t
( L oY , z)
and the operator
i. e.
for any
L
y, z
o E
is
V0
V (3.6.23)
[y, Lz]
t
(3.6.24)
(y , L z)
J-hermitian while
Lto
is hermitian,
'
(3.6.25)
(3.6.26)
Proof: similar.
We refer to [46, p. 62, I, II] since the proofs are
164
LEMMA 3.6.1: t Ro =
Let
range
0
Lto
f
and let
be the set of all
M
Q,t[z] = 0 .
solutions of the equation Then
t
H
Proof:
Ro
Since all solutions of the homogeneous equation are
continuous functions on so
M
c
H.
[a, b]
H
Hence
lies in H
Hand
is a finite
2.
replaced by
Q,t[y]
implies that
M
of dimension
solution of (3.6.2), with
Vo •
they all belong to
It is also readily seen that
dimensional subspace in
in
(3.6.27)
+ M .
Y
If
Q,t
is a
then
y
Thus the existence of
y
Theorem 3.6.1 then states that
M.
if and only if it is orthogonal to
is
g
Since
is a Hilbert space the decomposition (3.6.27) follows.
THEOREM 3.6.4: The domain
Proof:
LO
o of the operator
Since the domain
and
gonal to have then solution of 3.6.3,
V
Vo
L
o
is the same for the operators
it suffices to show that every element is zero. (h , y)
=
0
Q, t [z] = h
Letting for all For
H.
is dense in
h
h
ortho-
be such an element, we Let
z
be any
we have, by Theorem
165
(L
and so
z
t z, y)
is orthogonal to
previous lemma, and so
o
(h , y)
Consequently
£ t [z]
-=
E
M
by the
h = 0
i.e.
0,
z
We recall that a set is dense in a Krein space if it is dense in the Hilbert norm topology.
Thus the latter
Va
theorem expresses the fact that the domain
La
operator
of the
H.
is dense in the Krein space
THEOREM 3.6.5: The operators
are
and
J-symmetric and
symmetric respectively.
Proof:
Note:
This follows from Theorems 3.6.3 - 4.
The rest of the results in [46, §17]
can be similarly
shown to be true in this more general setting. in the regular case the operator operator whose
LX
J-adjoint
a
La
Thus, e.g.,
is a closed
is equal to
L
J-symmetric
[46, p. 66,
Theorem 1].
In the singular case, i.e. a > -00,
I = [a, 00)
we follow the approach outlined in [46, §17.4]
where we begin with the operator We recall that domain
when
,
Vf") •
I
La
and
L1
L'
a
defined in (3.6.14-15).
are both defined on the same
166
THEOREM 3.6.6: The domain of definition
L'
and
is therefore a
o
Proof:
is dense in
of
o
H
J-symmetric operator.
An argument similar to that of [46, p. 68] shows that
Vo ,
when viewed as the domain of
Thus
L
I
V'
L
1
H
is dense in
L'
is a symmetric operator, by Theorem 3.6.2, and
1
0
J-symmetric.
is
We now take the closure of
L
, o
-L ,
in the Hilbert
o
space topology and define
Lo
it then follows from the preceding theorem that closed
is a
J-symmetric operator. We now proceed to find a property of the domain
Lt
of L
(and so
L)
when
£t
V
is in the limit-point case in
2(V;I)
LEMMA 3. 6 • 2 :
[ 14] •
For any set of six functions {g
q
:
1
q
[0 , 00) point
3}
{f p:
1
P
3}
each being locally absolutely continuous on
and each having a finite right-derivative at each x
E
[0, 00)
167
det {[f
p
q ] (x) } q
o
X
E
[0, 00)
where [fg] (x)
Proof:
See [14, p. 374].
LEMMA 3.6.3: Let
be a Borel measure on
II
and let
[0, 00)
2
L (u )
be the space of square integrable "functions" with respect to ll.
Suppose that
f, g
are complex-valued
u-rnea surab Le
functions which satisfy
2 gEL ( ll i [ 0 , X » )
for all
X > 0
and that
Then
o .
lim
Proof:
This result can be proven in exactly the same way as
in [15, p. 42] with the necessary modifications.
Let
L
t
be the operator defined by (3.6.12),
Then by Theorem 3.3.2, for
1m A
0,
the problem
(3.6.10).
168
"Ay
on
[0,
has at least one nontrivial solution in I
=
[0, (0)
(3.6.28)
(0)
L 2(V; I)
where
Using this result and Theorem 3.3.1 along with
Lemmas 3.6.1-2 we can show, by adapting the argument of Everitt [15, pp. 42 - 45] to our situation, that whenever (3.6.28) is limit-point there follows
lim
x+ oo
{f (x) g
(x) -
f: (x) g (x)
}
o
(3.6.29)
Conversely if (3.6.28) is limit-circle, then it must be so for "A
=
o.
In this case it is possible to find two real linearly
independent solutions
¢,
of
o which satisfy (3.3.18-19) say. L
2(V;
I)
and consequently in
(3.6.30)
By hypothesis these are in D.
Moreover, by (3.3.18-19)
hence lim
(x)
0 .
x+ oo
Summarizing, we obtain
THEOREM 3.6.7: A necessary and sufficient condition for (3.6.28) to
169
be limit-point is that for all
f, g
E
V
(3.6.29) be
satisfied.
THEOREM 3.6.8: Let
a
satisfy (3.5.11).
Then (3.6.28) is limit-
point if and only if
try]
is limit-point (in the
(3.6.31)
"Ay
2
L (V ; I)
sense)
where £
and
are
related by (3.6.11).
Proof:
For Theorem 3.5.2 implies that (3.6.28) is limit-
circle if and only if
(3.6.32)
where
V(t)
However the latter is equivalent to
00
(3.6.33)
and Theorem 3.5.2 again implies that (3.6.31) is limit circle if and only if (3.6.33) and so (3.6.32) is satisfied.
The
result now follows.
COROLLARY 3.6.1: In order that (3.6.31) be in the limit-point case at infinity (in the space
2 L (V; I))
it is necessary and
170
sufficient that
o
lim [fg] (x) x-+ oo
where of
(3.6.34)
is defined in Lemma 3.6.2 and
[
L , Lt
a
La
of
V
where
a
is the domain
defined earlier.
La , Lat
We now define the operators
V
V
Let the domain
be defined by
{f
a
[0, 'IT)
E
V:
E
,
f(O)cos
and for
La t
Similarly
L a
0.-
f
,
.
f+(O)sln a
Lf .
f
f
Lat
in the limit-point case,
= JL a t La
First of all we note that if
(3.6.35)
Va
E
(3.6.36)
is defined on the same domain
or what is the same,
O}
E
Va
V a
and
(3.6.37)
We now proceed to show that, is self-adjoint. f, g
[fg] (0)
E
o .
Va
then (3.6.38)
Next the Lagrange identity (Theorem 3.2.1) shows that, for f,gEV
a
,
X>O
171
-[fg] (x) + [fg] (0)
Consequently if
x
+
00
is limit-point and
f , g
V
E
ex
, we let
in (3.6.39) and use Theorem 3.6.7 and (3.6.38) to find
(f,Ltg) ex
and so
(3.6.39)
•
L
t ex
(V
is symmetric
it contains the domain the singular case by
f , g
E
V
is dense in
ex
(3.6.40)
ex
L2
(V ; I)
since
V
o of the operator L o defined in L = The proof of this is similar o
to that in [46, p. 71, VI])!.
In the S3-me fashion it can be
shown that
[L
so that
Lex
ex
is
[f , L g]
f, g]
f , g
ex
E
V
(3.6.41)
ex
J-sYmmetric.
THEOREM 3.6.9:
V
In the limi t-point case , the domain
domain of self-adjointness of
if and only if
V
ex
ex
the following properties,
i) ii)
For all g
If
f
E
E
Vex ,
f, g
V
E
V
ex
satisfies then
g
E
Vex
[fg] (0)
o ,
[fg] (0)
o
for all
is a has
172
Proof: We note that this result is a particular case of a theorem of Naimark [46, p. 73, Theorem 1] and can be proven similarly. With
Va
defined as in (3.6.35) a simple computation
shows that both (i) and (ii) are satisfied in Theorem 3.6.9 and consequently, in the limit-point case, adjoint.
Lt
On the other hand, if
a
L
t a
is self-
is self-adjoint then
the deficiency indices [46, p. 26] of the operator are
(0, 0) .
Consequently the equation
(3.6.41)
AZ
has no non-trivial solution in
L
2(V;
Since
I)
Lat
is
self-adjoint (3.6.41) implies that the problem
Ltz
AZ
Z(O)cos a - z' (O)sin a
has no solutions in point.
L
2(V;
I)
0
Thus (3.6.28) is limit-
Hence we have proved
THEOREM 3.6.10: The equation (3.6.28) is in the limit-point case in if and only if the operator self-ad-joint.
Lt , a a
E
[0, 'IT)
is
173
In the following discussion
(L t)
a
*
,
will denote
LX, a
the Hilbert space adjoint and Krein space adjoint of the
Lat , La
operators
self-adjoint.
respectively.
Then by Theorem 3.6.10,
L
limit point case and so its
Let us suppose that
Let
exists.
LX a
J-adjoint f
0
E
a
a , g
[L
ct
E
is a
(3.6.28) is in the
J-symmetric operator and
We denote its domain by
oax
f , g]
[f , L aXg ]
(3.6.43)
Now since
L
t a
JL ct
Moreover
(3.6.44)
Substituting (3.6.44) into (3.6.43) we find that
g
E
* O(LaJ)
But
hence
g
0
E
J-symmetric
.
a
0
a
Consequently c
OX a
Hence
OX
c
ct
0
ct
0
and since
ct
L
a
is
vX a
and so
be
J-self-adjoint for
L
a
is
J-self-adjoint. On the other hand, let each
a
E
[0,
TI)
L
ct
so that we have
L
a
174 [f , L g]
f , g
[L f , g]
a
a
E
V
(3.6.45)
a
Using the Lagrange identity in (3.6.45) we find that for
o Since
lim JX{fIg - gL f}dV x..... oo 0 a a
f, g
V
E
g (0) f'
a
lim [gf' x..... oo
=
f, g
Va .
E
suffice to show that if f
EVa
'
holds.
For if
[fg] (x)
_ f (x) g' (x )
f
E
V
f
:l'
,
a
A
S
V
E
R
are real.
S
f, g
f = f + if R r
where
E
a
Thus let f
E
V
a
, g
g
E
E
V
f, g
E
E V
s ,
VS'
a,
S E [0, TI)
then (3.6.46)
f' (x)g(x)
,
g = gR + ig r
.
a
E
[0 ,
Hence for given
:l'
f
since
TI)
f
r
E V
s
R
(0)
,
where
The result now follows.
f , g
S
For this it would
are any two real-valued
for some
TI)
V.
f, g
similar result holds for [0 ,
(3.6.46)
We now wish to show that the latter
equality in fact holds for all
functions,
so that
o
lim {f(x)g' (x) - f' (x)g(x)} x ..... oo for all
o
g' (0) f (0)
(0)
.
be two real valued functions with a
We will show that under certain
(0)
175
hypotheses on
o , v
g*
we can find a function
Va
E
such
that [fg] (x )
[fg*] (x )
for all large
One such condition is the following: continuous so that for some
Let
v-absolutely ¢ ,
¢dv
in the sense of the measures defined by
0
and
v.
Let
be defined by
g*(x)
where
be
0
v-measurable function
do
g*(x)
x.
a, b
1
x
q Ix )
o
ax + b ,
are to be determined.
1
a .
x-""oo
Thus for
x > X ,
(3.7.9)
If
f
is uniformly bounded above on
inequality implies that zero.
Consequently If
f
g'
g' (x) L
2
(X ,
[X,
00)
then the latter
is uniformly bounded away from which is a contradiction.
00)
is not uniformly bounded then there exists an
increasing sequence
{x} n
with
x
n
-""
00
along which
183
f(x )
-+
n
n
00
(3.7.9) then implies that
If(x)
I
-+
00
•
> 0
for
x > Xi'
say,
and hence
If'
(x)g' (x)
I
>
!
lal
Integrating the latter over find
r
If' (x)
f(x) [Xi' x
g'
(x) I dx
and letting
n]
n
-+
00
we
00
Xi
a contradiction, by the Schwarz inequality, since both f' , g'
E
L
2
Hence the conclusion is that
(a, (0)
CD => SLP .
Again, in general, this implication is irreversible [17, p. 313].
Thus
DI => CD => SLP => LP .
(3.7.10)
We now interpret these results for three-term recurrence relations, the theory having been developed in the case of ordinary differential expressions.
§3.8
DIRICHLET CONDITIONS FOR THREE-TERM RECURRENCE RELATIONS: Let
c
n
> 0
(c)
for all
(b) n
n
n.
Let
be real sequences and suppose that (a) n
be a sequence of real numbers
184
where
a
0
(t) n sequence of real numbers defined by n
for all
n.
Let
be an increasing
n=O,l, ...
where t
n
+
c_
l as
00
and
> 0
n
+
t_
is fixed.
= a
l
v, a
We also assume that
00
Now define step-functions both
(3.8.1)
be constant on
v, a
-
0)
C
n = 0 , 1 , ...
[t n- l ' t n )
that these have discontinuities at the
a(t ) - o I t. n n
by requiring that
(t
and
only, given by
n)
n=O,l, ...
n +c n- I-b n ,
and v(t)
n
We
-
v(t
n
also suppose that
- 0)
v, a
-a
n
n=O, 1, . . . .
are both continuous at
(3.8.2-3)
a
and
that neither have a jump at infinity. Let
be summable and consider the differential
equation £ [y] (x )
(x)
X
E
[a,
00)
(3.8.4)
where
v, a
are defined above.
Rewriting the solution of the
above as the solution of a Volterra-Stieltjes integral
185
equation we see that, using the methods of Chapter 1, the solution
y(t)
then
satisfies the recurrence relation
-c
n
y
n+l
- c
is linear on
n-l
y
n-l
+b
Y
n
[t n- l ' tn)
and if
yn
= y(t n )
n=O,l, ...
n
(3.8.5)
where
= ¢(t n )
¢n
Thus the domain
V
of the operator
generated by the
L
generalized differential expression above consists of polygonal curves, i.e.
continuous and linear on Moreover the space space
.Q, 2 ( I a I )
V
each function in
is absolutely
for n == 0 , 1 , ... n- l ' tn) L 2(V; I) becomes, in this case, the
i. e.
[t
f
E
.Q, 2 ( 1 a
if
1 )
00
I [a n Ilf n 12
Since the domain
g
n
- g
n
c
n
l'>f
n
189
(3.8.19)
whenever either of (3.8.18-19) exists.
From the latter also
stems the relation
lim f (x)
x-H
g' (x)
lim c n+ co
O
(3.8.20)
f fig n n n
DEFINITION 3.8.3: The difference operator
L
is said to be in the
Strong Limit-Point case at infinity if for all
lim c
n+ co
L
all
n
f
n
fig
n
(=
exists
0)
•
f, g
E
V
(3.8.21)
is said to be in the Limit-Point case at infinity if for f, g
E
V
1 im c { f g- f -g 1 n n n+l n+l n ' n+ co
o .
(3.8.22)
The latter is consistent with the usual definition of limit(See for example,
point for a three-term recurrence relation. [3, pp. 498-99],
[32, p. 425, Theorem 2].)
We note, in passing, that the theory developed in section 3.6 also includes the difference operators as special cases.
Thus (3.8.22) holds for all
a certain difference operator
La
f, g
E
V if and only if
defined by
190
,Q, [f]
L f
ex
f
E
(3.8.23)
V
and
o is
J-self-adjoint in the Krein space
for all
n,
then
L
(3.8.24)
,Q,2(lal)
If
a
> 0
n
is self-adjoint and consequently every
ex
Lex
symmetric extension of
must coincide with
L
(This
ex
statement is also true in the Kreln space setting.)
When
(3.8.22) is satisfied it implies the self-adjointness of the "maximal" operator
L
[3, p. 499], it then follows that
Lex = L Moreover the implications in (3.7.10) are valid and generally irreversible.
EXAMPLE 3.8.1 : Let
c
n
n
1
In
Yn
Define
b
where
1 n
a
1
n
and let
if
n
2
if
n
2
m
some
m > 0
m
by
n
b
1
,
cnY n + l + cn-1Y n- l
n
Yn
o
say.
n
=1
, 2 , •••
A computation shows that
Y E,Q, n
2
and
191
if
zn
is a linearly independent solution then we must have
const n
n
=1
, 2 , •••
by the discrete analog of the Wronskian identity. z
n
Thus if
the Schwarz inequality applied to the latter identity
E
would produce a contradiction since the left side would be finite while the right side diverges.
c n Yn+l +c n-l Yn-l - b n y n
is
LP.
Thus
o
(3.8.25)
However
lim n y toy n n n-+ co does not even exist.
lim inf n y toy n n n-+ co
In fact,
-1
Thus (3.8.25) is not SLP.
lim sup n y toy n n n-+ co
o .
Other examples may be found to
show that, in general, the implications (3.7.10) are irreversible even for three-term recurrence relations.
The
next result follows from remarks 1, 2 of the preceding section.
THEOREM 3.8.1: A necessary and sufficient condition for
192
-c
to be
LP
for all
n
y
- c
n+l
in the f , g
E
Aan y n
y +b y n-l n-l n n
(3.8.26 )
£2(lal)-sense is that (3.8.22) should hold
V
(here
I
> 0 > 0
a
"0)
n
.
COROLLARY 3.8.1: Let for all
la
n
for all
n
and that
independently of the coefficient
Proof:
If we let
f
n
< M
n. Then (3.8.26) is limit-point in the
Thus
0 < c
+
n
0
as
f
n
V,
E
+
b
2
(I a I ) -sense
n
then
for every
00
£
bounded (3.8.22) holds for all
f
f, g
E
E
V. V.
Since the
care n The result now
follows. If we let
c
n
a
1
n
in (3.8.26) we find that the
equation
/:, 2 y
is always
LP
n-l
+b y n n
in the
Ay
n
n=O,I, •..
£ 2 -sense (see [32, p. 436] and [3,
p. 499]).
Unlike the results in Chapter 2, the limit-point,
193
limit-circle theory of difference equations differs substantially from the analogous theory for differential equations.
One reason for this appears to be related to the
general limit-point criterion (3.6.29) and its interpretation (3.8.22) for recurrence relations. and let
f
=
(f ) n
c
should exist and be zero. automatically satisfied.
n = an = 1
lim f
then it is necessary that
£2
E
For if we set
n
Consequently (3.8.22) is On the other hand if we consider
the differential equation
y" + b(x)y
"Ay
X
E
[0, (0)
then the maximal domain of the operator generated by the expression is a subset of
L 2 (0, (0)
Thus if
f
E
L
2
,
f
need not tend to a limit at infinity and can be essentially unbounded.
Hence (3.6.29) is far from being satisfied and
thus conditions have to be imposed upon that, say, if
f
E
V
then
f
and
f'
b(x)
to ensure
have limits at
infinity and (3.6.29) be satisfied. In the following
theorem we show that it is possible
to strengthen the conclusion of Corollary 3.8.1 under the same set of hypotheses.
THEOREM 3.8.2: Let
a
n
Corollary 3.8.1.
0,
and
c
n
satisfy the hypotheses of
194
Then
n = 0 , 1 ,
(3.8.27)
has the Dirichlet property at infinity.
Proof:
According to Definition 3.8.1 it is necessary to show
(3.8.15-16) . Let Since
la
n
I
f
E
£
2
(I a I)
be such that
then
> 0 > 0
f
E
£2
square-summable in the usual sense. £2
i.e. the sequence Since
f
=
(f ) n
f
n
is
is in
then
(M ) n
and the hypothesis
c
n
< M
implies then that
co
for
f
E
V •
The same argument shows that
I
[c n +c n- Illf n
2
1
0
on
[a, b]
q (x)
E
L
l oc
(a, b)
is such that
the problem
-(p(x)y')' + q(x)y
y(a)
y(b)
Ay
o
(4.2.1)
(4.2.2)
admits a denumerable number of eigenvalues having no finite point of accumulation. that both 8.4.6].)
p
-1
,q
E
(Conditions which guarantee this are
L(a ,b)
,
see [3, p. 215, Theorem
In this case the eigenfunctions form a complete
orthonormal set in
L 2 (a , b)
Associated with (4.2.1-2) is the "indefinite" boundary problem -(p(x)z')
I
+ q(x)z
Ar(x)z
(4.2.3)
213
z(a)
where
r(x)
such that
(4.2.4)
is a real-valued function defined on r(x)
measure.
o
z (b)
[a, b]
and
takes both signs on some subsets of positive
The "Indefinite case" is characterized by the fact
that both
q, r
have a variable sign in
[a, b]
(see [53,
p. 288] and not being equal a.e.
LEMMA 4.2.1: Let
f
eigenvalue
A
I: Proof:
be an eigenfunction corresponding to some of (4.2.3-4).
(p
Then
If' I 2 + q If I 2) dx
We multiply (4.2.3) by
over the interval
t a
[a, b]
(pf ' )' f dx + A
A
I
I
b
a
r If
1
2
(4.2.5)
dx.
and integrate both sides
to find
I
b
a
r If
1
2
dx =
I
b
a
q 1f
1
2
dx .
Integrating the first integral by parts and applying the boundary conditions (4.2.4) the result follows.
LEMMA 4.2.2: Let
A,
eigenfunctions
A f, g
be two non-real eigenvalues with respectively.
Then
214
b
Ja i.e.,
f, g
are
o
r(x)f(x)g(x)dx
(4.2.6)
J-orthogonal in the Krein space
L 2(lrl)
and
t a
Proof:
{p (x) f ' (x) g' (x) + q (x ) f (x) g ( x ) } dx
o .
(4.2.7)
We have - (pf')'
+ qf
Arf
(4.2.8)
- (pg ')
+ qg
flrg
(4.2.9)
along with
f(a) = f(b) = g(a)
(4.2.8) by
g
and (4.2.7) by
g(b) f
we obtain, upon integration over
t a
=
O.
Multiplying
and subtracting the results [a, b]
,
r(x)f(x)g(x)dx = fb{(pg')'f- (pf')' g}dx a
and integrating the latter integral by parts we find b [p(g'f - gfl)] a
o because of the boundary conditions.
A"
•
This proves (4.2.6) since
215
Multiplying (4.2.8) by
g
and integrating over
[a, b]
we obtain
\/\ fb
rfg dx .
a
Integrating the first term in the left by parts we see that
-fa
b
g
(pf')'
r a
pf'g' dx .
Thus
\ Jb
/\
rfg dx
a
o by (4.2.6).
This completes the proof.
Associated with (4.2.1) is the differential operator A
defined in
V(A)
{y E
where for
L
2
(a, b)
y , py'
L 2 (a , b)
y
E
V(A)
V(A)
E
V(A)
AC (a, b) l oc
,
Ay If we let
with domain
-(PY')'
be defined by
+ qy .
defined by
and
Ay
E
L 2 (a , b) }
216
{y
E
V(A)
: y(a)
o}
y(b)
(4.2.10)
and let
Ay then
- is a restriction of A
Y
E
A to
(4.2.11)
V(A)
V(A)
A is, in fact,
and
a symmetric operator [34, §4.11, Theorem 1]. The following lemmas are part of the theory of the regular Sturm-Liouville equation and can be found in [34], thus we omit the proofs.
LEMMA 4.2.3: a)
The regular Sturm-Liouville operator
A
above, is bounded below, i.e. there exists a constant
defined y
E
m
such that
(Af , f)
where
> Y(f , f)
,
(4.2.12)
is the usual inner product in b)
The operator
A
L
2
(a, b)
•
has at most a finite number of
negative eigenvalues.
Proof:
For part a) see [34, §5.17, Ex. 5.3 0 and §6.7,
Corollary].
For
Part b) is proved in [34, §5.8, Theorem 2].
f
E
V(A)
the expression
(Af, f)
defines a
217
quadratic functional with values
f
V(A)
E
This is immediate if we follow the argument leading to (4.2.5).
LEMMA 4.2.4: We define
{y
D (Q)
D(Q)
E
L
2
by
co
L I x . I I (y
(a , b)
o
co Q(y)
where
(le
j
(4.2.1-2)
)
,
L 1e·1 (y J
(epj)
2
J
2
(a, b)
, ep.)
J
I2
0
a.e.
of the quadratic functional D(Q)
=
D(Q') (b
Proof:
y
E
(Ay, y)
Q(y)
,
Q' (y)
are identical, i.e.
and
J {p I y' a where
the extensions
D(Q')
2
I
+ q Iy
2
I }dx
Io 1e·1 J
(y , 0
a.e.
on
[a, b]
,
q
L(a, b)
E
as in the hypotheses following (4.2.3-4).
and
The eigen-
value problem (4.2.3-4) possesses at most a finite number of non-real eigenvalues.
M
If we let
the number of pairs of distinct non-real eigenvalues of (4.2.3-4),
N
the number of distinct negative eigenvalues of (4.2.1-2)
(which we know is finite by Lemma
4.2.3(b)), then (4.2.15)
M < N .
Proof:
We let
A ' A ' ... , A 1 N_l o
be the negative eigenvalues
of (4.2.1-2) arranged in an increasing order of magnitude. Let
N . Then it is possible to choose the
is orthogonal (in the
2
L (a, b)-sense)
to
(e. )
J
¢0 '
For it is necessary that
(f , ¢.)
J
and so
o
j = 0 , 1 , ... , N-l
so that
f
221
M-l
L
o ,
e.(z., cp.)
i=O
1
1
]
=0
j
The latter constitutes a set of
N
, 1 , ... , N-l •
linear equations in
unknowns where
M > N.
solution
not all zero which we fix.
(e ) j
Thus this system has a non-trivial
necessary that, for such a choice of
Q' (f)
It is then
(e. ) ]
(4.2.19)
> 0
because of a preceding remark.
Q' (f)
M
Moreover,
Q'{Ie.z.} ]
b
]
z.)
, - -'
p (Ie. z .) (Ie. z,) + a (Ie. z .) (Ie. ]] 11 ]] 11
Ia
M-l
L
i, j=O
But since
e.e. ]
1
b
Ia
for all
{ p z ".
]
o
z.1
< i
+ q z . Z. }dx . ]
, j
0
I
+ qy
and continuous on
Consider the equation
[a, b]
ous and is negative in a subinterval of continuous on
[a, b]
L
(I r I i
[a, b l ) = H
functions
f
q(x)
[a, b]
with the property that
sign at least once in 2
(4.2.20)
"Ary
[a, b]
is continu-
and r (x)
r(x) changes
In this case the space
defined by those (equivalence classes of)
such that
is a Krein space with the indefinite inner product given by
f
E
H
(see Appendix III and the references therein for more discussion on these spaces). Let
U (y) i
is
U 2(y)
be the linear forms defined in
Appendix 1.4, equation (1.4.1).
U,,(y)
We denote the relationships
0
223
by
o .
U(y)
(4.2.21)
The problem
Ly
7T
Ary
Uy
o
then defines an eigenvalue problem, i.e., A
E
C
(4.2.22)
seek values of
such that (4.2.22) has a non-trivial solution satisfy-
ing (4.2.21). With some loss of generality, we shall say that the eigenvalue problem
7T
is formally J-self-adjoint if
[f , Lg]
f , g
for all
E
2
C (a, b)
U(f)
For a
[Lf , g]
which satisfy
U(g)
o .
J-self-adjoint problem, non-real eigenvalues mayor
may not exist but, in any case, if they do exist, their number appears to be finite for general boundary conditions also, because of the preceding theorem. results in [53, formulate
Combining the
§4] with the preceding theorem we can
224
THEOREM 4.2.2: The for.nally J-self-adjoint problem
Ly
>..ry
y(a)
y(b)
o
has a finite number of non-real eigenvalues, in some cases none at all, and on
IAI > A
has only real eigenvalues, with
no finite point of accumulation, clustering at minus infinity and plus infinity.
The second part of this theorem is due to Richardson [53, p. 301, Theorem VII].
It would seem plausible that
Theorem 4.2.2 remains true for arbitrary "J-self-adjoint" boundary conditions though we shall not go into this at the present time.
Theorem 4.2.1 extends, with appropriate
changes in the argument, to the general even order formally self-adjoint differential equation
(_l)n(p y(n)) (n) + (_l)n-l(p y(n-l)) (n-l) + ••. +p Y o 1 n y (j) (a)
where
Po > 0
[a , b] that
where Pk
E
y (j) (b)
and i
p. (x ) l
o
=0
, ••• , n-l
changes sign at least once in
is in the range
C (n-k) (a , b)
j
Ary
0 < i
T .
A.
239
Then tla(t) - AV(t)
I
T
and consequently Theorem 2.1.4 implies that (5.1.17) is nonoscillatory for such
A.
On the other hand let (5.1.15) hold
and suppose that (5.1.17) is non-oscillatory for all
A.
Suppose that, on the contrary,
lim t t+ oo
By our hypothesis
I v (t) I -
v(t) < 0
t
7-
0
•
and so
r c t t.) + nlv(t)
t(a(t) - AV(t))
We now choose
a
I .
so large that
to (t) >
a
-2"
t > T
and tlv (t)
Then, for
t > T,
I
2
t > T
A > 0 ,
t(a(t) - AV(t)) >
.
Since by hypothesis (5.1.17) is non-oscillatory for all
A we
240
can choose
A
so large that a 2
-(A - 1)
where
E >
0
1 > - + 4
is some fixed number.
E
Thus for such a choice of
t(a(t)-Av(t))
Thus
a(t) - AV(t)
t
if positive for
t
>
T .
T,
such A.
An
application of Theorem 2.2.1 shows that (5.1.17) is oscillatory for such
A
This is a contradiction and thus
a
=
O.
This
completes the proof.
In particular when
=0
a(t)
we obtain the result
of Kac and Kreln [38, p. 78, Proposition 11.9°]. original see p. 97, superfluous.
(2), of [38].)
(For the
Again (5.1.15) is not
The latter result had extended a theorem of
Birman [23, p. 93, Theorem 7] since we can let absolutely continuous and then, when
a(t)
= 0,
v
be (5.1.13) is
equivalent to -y"
where
p(x) > O.
Ap(X)y
X
E
(5.1.19)
[a, (0)
Glazman [23, §29] calls this case the
"polar" case though the latter is usually connected with the sign indefiniteness of
p(x)
in (5.1.19).
Because of
Theorem 5.1.0, other criteria for the finiteness of the
241
negative part of the spectrum can be obtained via the nonoscillation theorems of Chapter 2.1.
Moreover because of the
applications to recurrence relations, we therefore obtain some criteria for the finiteness of the negative part of the spectrum of difference operators.
Example 1: and let
c
If we let n
= 1
a (t)
for all
n
and define
0
-
t
or
n
= n
v(t)
for all
by (3.8.3) n
,
then
(5.1.0) includes the difference equation
-
where
a
n
/':,2
Yn-l
n=O,l,
AanYn
by hypothesis.
> 0
...
(5.1.20)
The discrete analog of Birman's
theorem (above) is that the spectrum of (5.1.20) is discrete if and only if co
lim n n-+co
whenever
La
n
0 ,
A(x) - /E"B(x)
r
> 0 •
Hence 2A(x)B(x)
Inserting this in (5.2.12) we obtain
II ox
2 f lido
I;
II
{ ( + 1] A2 (x) + EB 2 } • C (x)
248
where
C
is the quantity (5.2.1).
Replacing
s
by
siC
we
find 2
2
C(s)A (x) + sB (x) where 1 + C s
C (c )
This completes the proof.
When
is absolutely continuous the above lemma can
a
be found in [18, p. 339, Lemma 1] in the case when Our proof appears to be simpler than the case Consequently, if we choose f
E
s
p= 1 .
p= 1
of [18].
1 = 2" we find that for each
V , (5.2.13)
where
C'
C
•
LEMMA 5.2.3: For every
Proof:
f
E
V ,
f'
lim f (x)
lim
x-" oo
x-" oo
E
f
L
2
(0,00)
(x) f' (x)
and
a .
This can be shown as in Lemma 2 of [18].
For if
249
by (5.2.13).
Since
lim x-+ oo
f
E L
2
we must have
(0, 00)
x
00
J0
(5.2.14)
A simple calculation also shows that
((Lf) (t)f(t)dt
C
{!f'1
2dt+
IfI
2dCJ(t)}
(5.2.15a)
However, since
f,Lf
E
L 2 (0,00)
we must therefore have
lim f (x) f' (x) x-+oo
00
But by taking real and imaginary parts in (5.2.15a), and noting that
CJ
is real, the latter equation is clearly impossible.
This contradiction proves that
f'
E
2 L ( 0 , 00) •
Hence (5.2.13)
implies that
(5.2.15b)
Thus (5.2.15a) implies that
f(x)f' (x) -+ a ,
as
x -+ 00.
But
250
since that
If
(x)
12
-+ S,
I f(x) I f
Since
E
L
2
(0, (0)
as 2
If
r
1
f
f E
00
( 0)
If
I
:
2
(x) f
The following relation implies
(x
_
(5.2.16)
+ J 2 r e ( ff ' ) d t o The lemma is proved.
I
I
(x)
II do (x) I
(x) f' (x)
V.
x -+
o .
o
for all
O.
,
We also obtain Thus
=
a
ff'EL(O,oo)
0 •
The rest of the argument now follows that in Chapter 4, §2. For, an adaptation of Lemma 4.2.2 shows that, if
A,
are
non-real eigenvalues of
Af
Arf
f(i) (a)
o
i
=
0 , ••• , n-l ,
(III.4.6) where
r (x)
is, say, continuous on
[a, b]
and changes sign
306
at least once there, and
A
then
o
f(x)g(x)r(x)dx
(III.4.7)
o where
f, g
(III.4.8)
are the eigenfunctions corresponding to
A,
respectively. Thus we let (111.4.6)
0 '
u,
such that
eigenfunctions
be the non-real eigenvalues of
••• ,
¢0
'
J
1
¢1
'
••• ,
with
, Since
¢M-l
¢. (x) 1
E
D(A)
we
have P
for
n-r j
.
I
¢ j) 1
i = 0 , .•• , M- 1
1
2
dx
Thus we let M-l f (x)
L
j=O
e. ¢. (x)
(III.4.9)
J J
and, as in Chapter 4.2, we see that if e.
possible to choose the coefficients k=O, ... , N - l .
(Af , f)
> 0
M > N J
then it is
such that
This would then imply that
•
But by substituting (111.4.9) in the latter relation and
307
expanding the form, we shall find that
(Af ,f)
on account of (111.4.7-8).
0
This contradiction then proves
the result.
*Note:
The problem here is the following:
Richardson's
idea is to approximate the eigenvalues of the continuous problem, (py , )
,
+
(q
y(O)
+ Xk ) y
y(l)
0
0
by the eigenvalues of the discrete problem,
m
i
2
I1 (p. l1y. 1
= 0 , 1 , 2 , •••
points
i/m
,m
1-
1)
+ q.1 y.1 + Xk 1, Y 1.
where the values of
are denoted by
0,
y, p , q ,k k.
1
The claim appears to be that for large values of
at the
respectively. ill
the
eigenvalues of the discrete problem with
o are approximations to the eigenvalues of the above continuous problem.
However it is not at all clear that if the discrete
308
problem has non-real eigenvalues then these must necessarily approximate non-real eigenvalues in the continuous case. it is conceivable that these limits may be real.
For
It does not
seem as if enough information is provided in [53] to exclude the latter possibility.
In fact in some cases no non-real
eigenvalues may exist and so one. needs to establish some criteria on the coefficients which will guarantee their existence.
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Subject Index
Conditional Dirichlet,
180
Difference Equation, Dirichlet property,
15 180
Generalized derivatives,
120 ff.
Generalized ordinary differential expressions,
123 ff.
Generalized ordinary differential operators, 156 ff., 225 Green's function, 25-27, 273 ff. Indefinite weight-function, J-self-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, 180-181 Sturm comparison theorem, 10 ff. Sturm separation theorem,
4, 22 ff.
Three-term recurrence relation, 16 Volterra-Stieltjes integral equation, 29 Weyl classification,
129 ff.
