Global Properties of Linear Ordinary Differential Equations 0792312694

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Global Properties of Linear Ordinary Differential Equations

Mathematics and Its Applications (East European Series)

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Editorial Board: A. BIAL YNICKI-BIRULA, Institute of Mathematics, Warsaw Univerity, Poland H. KURKE, Humboldt Univeristy, Berlin, Germany J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia L. LEINDLER, Bolyai Institute, Szeged, Hungary L. LOVASZ, Bolyai Institute, Szeged, Hungary D. S. MITRINOVIC, University of Belgrade, Yugoslavia S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland Bl. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria H. TRIEBEL, University of Jena, Jena, Germany

Volume 52

GLOBAL PROPERTIES OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS

Frantisek Neuman Mathematical Institute of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia

KLUWER ACADEMIC PUBLISHERS DORDRECHT /BOSTON/ LONDON

Library of Congress Cataloging-in-Publication Data Neuman, Frantisek. Global properties of linear ordianry differential equations / by Frantisek Neuman. p. cm. - (Mathematics and its applications. East European series ; v. 52) Includes bibliographical references (p. ) and indexes. ISBN 0-7923-1269-4 (Kluwer Academic Publishers) l. Differential equations. 2. Transformations (Mathematics) I. Title. II. Series: Mathematics and its applications (Kluwer Academic Publishes). East European series ; v. 52. 91-14560 QA372.N48 1991 515'.352-dc20 ISBN 0-7923-1269-4

Scientific Editor Prof. RNDr. Michal Gregus, DrSc., Member of the Czechoslovak Academy of Sciences and Slovak Academy of Sciences Reviewer Doc. RNDr. Milan Gera, CSc. Published by Kluwer Academic Publishers. Co-published with Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr. W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, IOI Philip Drive, Norwell, MA 02061, U.S.A. Sold and distributed in Albania, Bulgaria, China, Cuba, Czechoslovakia, Hungary, Korear, Peoples Rep., Mongolia, Poland, Rumunia, U.S.S.R., Vietnam and Yugoslavia by Academia, Publishing House of the Czechoslovak Academy of Sicences, Prague, Czechoslovakia. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

©1991 F Neuman All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocoping, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in Czechoslovakia

To my highly esteemed and beloved teacher " Otakar BORUVKA

in warmest gratitude and friendship

Contents

Preface . . . . . . . . .

.

.

. .

. . . . .

. . .

. . . . . . .

Symbols defined in the text

xi xiii

1. Introduction with historical remarks . .

1

2. Notation, definitions and some basic facts

5

2.1 2.2 2.3 2.4 2. 5 2.6 2. 7

Generalities Maps . . . Topology. . Algebraic structures Vector spaces . . . Linear differential equations Functional equations

3. Global transformations

3.1 3.2 3.3 3.4

. .

Definition of the global transformations . . Smoothness of global transformations . . . Algebraic approach to global transformations . Fundamental problems. . . . . . . . . . .

4. Analytic, algebraic and geometrical aspects of global transformations

4.1 4.2 4.3 4.4

Some useful formulas . . . . . . . . . . . . . . . . . Global transformations of special classes of linear differential equations . . . . . . . . . . . . . . . . . . . Covariant constructions of linear differential equations Geometrical apprqach to global transformations . . .

5 6 8 9

16 18

22

25 25 33 38

46 49

49 52 57 69

CONTENTS

Vlll

5. Criterion of global equivalence . . . . . . . . . . . . . .

5.1 5.2

Boruvka 's criterion of global equivalence of the second order equations . . . . . . . . . . . . . . . . . Criterion of global equivalence of the third and higher order equations . . . . . . . .

Notation and preliminary results . . . . Preparatory results . . . . . . . . . . Subgroups of stationary groups with increasing elements Stationary groups with decreasing elements . . . . . . Complete list of stationary groups and characterization of the corresponding equations . . . . . . . .

7. Canonical forms

7.1 7.2 7.3 7.4 7.5 7.6 7. 7

. . . . . . . . . . . . . . .

Notion of canonical forms . . . . . . . . The Laguerre-Forsyth and Halphen fonns . C'.1.rtan's moving-frame-of-reference method . Hereditary property . . . . . . . . . . . Global canonical forms: geometrical approach. Global canonical forms: analytic approach . . List of canonical forms of the second and third order equations

8. Invariants . . . . . . . . . . . . . . . . . . .

8.1 8.2 8.3 8.4 8.5

Notion of invariant and covariant Covariants . . . . . . . . . Local invariants and covariants . Global invariants . . . . . . . Smoothness of coefficients as an invariant.

9. Equations with solutions of prescribed properties

9.1 9.2 9.3 9.4

80

Notation and definitions Representation of zeros Second order equations

88 89 99 I 06 109

123

123 124 133 140 146 159 162 165

165 167 169 172 173 .

Coordinate approach . . . . . . . . . Asymptotic properties of solutions of the second order equations . . . . . . . . . . . . . . . . . Periodic solutions of the second order equations . Geometrical a pp roach . . . . . . . . .

10. Zeros of solutions

10.1 10.2 10.3

74

87

6. Stationary groups

6.1 6.2 6.3 6.4 6.5

73

185

185 189 194 213 215

215 217 223

IX

CONTENTS

10.4 10.5 10.6

Third order equations . . . . . . . . Iterative nth order equations . . . . . Periodic solutions of nth order equations

230 248 250

255

11. Related results and some applications . . . . . . .

11.1 11.2 11.3 11.4 11.5

Asymptotic properties and zeros of solutions of second order equations . . . . . . . . . . Integral inequalities . . . . . . Affine geometry of plane curves . Isoperimetric theorems. . . . . Related results and comments, possible trends of further research . . . . . . . . . . . . . . . . . . . . .

12. Appendix: Two functional equations . . . . . . . . . . . .

12.1 12.2

255 262 264 278 285 293

Abel functional equation . . . . . . . . . . . . Euler functional equation for homogeneous functions.

293 298

Literature cited in the book and/or for supplementary reading

299

Subject index .

315

Index of names

. . . . .

.

.

. .

. . . . . . .

.

318

Preface

This monograph contains a description of original methods and results concerning global properties of linear differential equations of the nth order, n > 2, in the real domain. This area of research concerning second order linear differential equations was started 35 years ago by 0. Boruvka. He summarized his results in the monograph "Lineare Differentialtransformationen 2. Ordnung", VEB, Berlin 1967 (extended version: "Linear Differential Transformations of the Second Order", The English Univ. Press, London 1971 ). This book deals with linear differential equations of the nth order, n > 2, and summarizes results in this field in a unified fashion. However, this monograph is by no means intended to be a survey of all results in this area. It contains only a selection of results, which serves to illustrate the unified approach presented here. By using recent methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and linear differential equations of the second order, and by introducing several original methods, global solutions of problems which were previously studied only locally by Kummer, Brioschi, Laguerre, Forsyth, Halphen, Lie, Stackel and others are provided. The structure of global transformations is described by algebraic means (theory of categories: Brandt and Ehresmann groupoids ), a new geometrical approach is introduced that leads to global canonical forms (in contrast to the local Laguerre-Forsyth or Halphen forms) and is suitable for the application of Cartan's moving-frame-of-reference method. The results contain also a criterion of global equivalence of two linear differential equations which is in general effective for n > 3 and new global invariants of linear differential equations with respect to global transformations. This theory also provides effective tools for solving some open problems, in particular concerning the distribution of zeros of solutions. The theory of functional equations plays an important role in studying the asymptotic behavior of solutions. The book also contains applications to other fields of mathematics, especially to differential geometry and functional equations. Some related results and further possible areas of research are mentioned at the end of the book.

..

Xll

PREFACE

This monograph is written for mathematicians working with differential equations and systems and, due to the application of modern aspects of other fields of mathematics, also for those working in algebra, topology, differential geometry, functional-differential equations and functional equations. The description of some situations is quite easy and some answers are obtained without complicated calculation, hence these parts are appropriate for undergraduate students. This book can also be of use for specialists in computer science and in those areas where linear differential equations occur: physics, chemistry, engineering, biology, astronomy, and others. The book is self-contained: the notions and results needed in the text are mostly introduced and derived here. I wish to thank warmly Academician 0takar Boruvka for introducing me to this area of research and for many fruitful discussions about the subject. I am very grateful to Professors J. Aczel (Waterloo, Canada), W. N. Everitt (Birmingham, Great Britain), K. Kreith (Davis, CA, U.S.A.), M. Kuczma (Katowice, Poland) and J. Kurzweil (Praha, Czechoslovakia) for many useful comments, encouragement and support in this research. I also use this opportunity of expressing sincere thanks to Dr. M. Cadek for his review of the first draft and valuable comments, to him and to Professors 0. Boruvka, M. Gregus, M. Gera, 0. Dosly, V. Novak, J. Simsa, S. Stanek, V. Tryhuk and J. Vosmansky for their careful reading of the manuscript and for their kind advice. Warmest thanks go to my daughter Darina, who carefully typed the manuscript, and also to Ing. B. Kyselova, the Publishing House Academia, Praha, Czechoslovakia and to Kluwer Academic Publishers, for their collaboration. Frantisek Neuman

Symbols defined in the text

N, Z, iR, iR+, ~-' iRn, C, I, J, IJI, (a, b), [a, b], (a, b] ·ck, c~, cl, c 00 , Lv

cw

!?}omf &1lan f idM IM ~M

{h, t} ct (M) 1,·nt (M) ,n?om (P, Q)

~(P) A lp

a, p, y, a -1

t>, ...

0

a [k] d1, d2, d3, d4

E, En det (M) z, Y, ...

lrl Mn, G lln, §lln, On

H(c) sn-1

special sets of numbers definition, specification sets of functions real analytic functions domain off range off identity on M constant function I on M restriction off to M Schwarzian derivative of h at t closure of M interior of M set of morphisms stationary group of P category identity morphisms or transformations inverse to a composition kth iterate of a axioms of categories unit matrix determinant of matrix M vectors euklidean norm of y sets of matrices hyperplane unit sphere

2.1 2.1 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.3 2.3 2.4 2.4 2.4 and 3.3 2.4 2.4 or 3.1 2.4 2.4 2.4 2.4 and 3.3 2.5 2.5 2.5 2.5 2.5 2.5 2.5

SYMBOLS DEFINED IN THE TFX r

XIV

f

vector space euklidean space

2.5 2.5

antiderivative of y

2.5

vector product scalar product categories of certain linear differential equations second order equation linear differential equations linear differential expression adjoint equation to P coefficient Wronskian determinant of y at X Wronskian matrix of y at x global transformation of y global transformation of P global transformation of P intoQ stationary group of P E A?z fundamental group group of increasing elements of

2.5

X

y(s) 1 is also called a ck-diffeomorphism of I onto J. Evidently l is then -a ck-diffeomorphism of J onto I. For a number v > 0, Lv(M) denotes the set of all real or complex functions f A1 ~ ~ or C defined on a set M (M is mostly an open, half-open, or closed interval of ~) such that

r

f

lf(s)I' ds

M

exists and is finite.

2.3 Topology A topology on a set Mis given when a family of subsets of Mis given containing the empty set 0, the whole set M and (generally) other subsets of M, the family being closed under operation of arbitrary unions and finite intersections. The elements of the family are called open sets. A set N c M is compact (in a given topology) if each covering of N with open sets has a finite subcovering. A set N is closed if its complement M \ N is open. The closure ct (N) of set N is defined as the intersection of all closed sets containing N. The set M is connected if we cannot express M as a union of two subsets N 1 and N 2 of M such that N 1 # M, N 2 # M, both N 1 and N 2 being open and closed at the same time. An open set N containing a point a is called a neighbourhood or a vicinity of a and is often denoted by ~(a). A point p is an interior point of a set N if there exists a neighbourhood of p contained in N. The set of interior points of N is

2.3 TOPOLOGY

9

called the interior of N and denoted by (;?'it (N). The boundary of N c Mis the set ct(N) n ct(M\ N). A set N 1 is dense in N if ct (Ni) ::) N. A mapping of a set N 1 onto N 2 is called homeomorphism if it is a continuous bijection of N 1 onto N 2 with continuous inverse. The product topology on M 1 x M 2 is given by family of open sets formed as follows: all _N 1 x N 2 where N 1 and N 2 are open in M 1 and M 2, respectively, and all unions of these sets. The relative topology on subset M 1 of the set M is induced by a topology on M if open sets on M 1 are just intersections of open sets of M with M 1. A path (a curve) in M is a homeomorphism of an interval of reals into M. A point p E M is an accumulation point of set N c M if every neighbourhood of p contains other point of N different from p. In a discrete topology every set is open. By the usual topology on IR we mean the topology comprised of all open intervals (a, b) c IR and all their unions. The product topology on !Rn is then called the usual topology on !Rn.

2.4 Algebraic structures An equivalence relation on a set Mis given by a decomposition of M, that is a covering of M by a non-intersecting family of subsets of M. Two elements of M are called equivalent if they belong to the same subset. Any equivalence relation is reflexive, symmetric and transitive. In fact, these three properties characterize an equivalence relation so that if a relation has these properties then it is an equivalence relation. A relation on a set M that is reflexive, antisymmetric and transitive is said to be a partial ordering of M; it is often denoted by b, then the set Mis called linearly ordered with respect to relation 2. The stationary group ?(Pn) zs en+ 1-conjugate to a closed subgroup of a group listed in Theorem 6.2.5. Let

Pn

" Proof In fact, the stationary group ?(P n) is in general only a subgroup of the stationary group

?(P.-2/(n; I)) of the second order equation

because the conditions (iii) and (iii*) in Theorem 5.2.1 and Remark 5.2.2 may exclude some elements of this group. However, these conditions represent a finite number of equations involving continuous functions. Indeed, higher derivatives of elements h of ?(P n) up to the (n + 1)st order are expressible as continuous functions of h, h' and h" due to the third order Kumer equation (p, p) in which 2 p E c"- (1). Hence, they are continuous functions of the Cauchy initial conditions h(t0 ), h'(t 0 ) =I= 0, and h"(t 0 ) as Theorem 6.2.1 guarantees. According to Theorem 6.2.5, the group

is cn+ 1-conjugate to one of the groups listed in this theorem. Each group on this list is either the fundamental group ffe or a restricted subgroup of it, the

6.2 PREPARATORY RESULTS

99

restrictions are again in the form of a finite number of equations involving continuous functions. Hence the stationary group ?(Pn) is en+ 1-conjugate to a closed subgroup of a group listed in Theorem 6.2.5 ■

6.3 Subgroups of stationary groups with increasing elements In this paragraph we investigate equations P n E A~ 1 whose stationary groups ?(Pn) contain other increasing elements than the identity. Notation 6.3.1

Denote by ? + (P n) the subgroup of the group ?(Pn), P n E A~ 1, consisting of increasing elements,

Remark 6.3.2

Indeed, elements of?+ (P n) form a group since for h' > 0 and k' > 0 we have

(hok- 1)' = (h'ok- 1)/(k'ok- 1) > 0. Proposition 6.3.3

For each Pn or two.

E

A~1, ?+(Pn) is a normal subgroup of the group ?(Pn) of index one

Proof

If the stationary group ?(P n) consists of increasing elements only then ? + (P n) coincides with ?(Pn) and its index is one. If there is a decreasing element h_ in ?(P n) then

Indeed, if h Eh_ o ?+(Pn) then h = h_ oh+ for some h+ E ?+(Pn) and also h = (h_ oh+ o h= 1) oh_. Hence h E ?+(Pn) oh_. And also conversely, if h E? +(Pn) oh_ then h = h+ oh_ = h_ o (h= 1 oh+ oh_) Eh_ o? +(Pn). Hence ? + (P n) is a normal subgroup of ?(P n). Since each decreasing element h of ?(Pn) can be written in the form

the index of ?+(Pn) in ?(Pn) is two. ■ The following two theorems occur to be essential in classifying equations P n with respect to the groups ? + (P n)·

100

6 STATIONARY GROUPS

Theorem 6.3.4

Let Pn = Pn(Y, x; I) be an equation from A~1, n > 2. If there exists an element h E ?+(Pn) such that h -# id1 and h(x 0 ) = x 0 for some x 0 EI then Pn is an iterative equation. Proof

Suppose that h

E

? + (P n), h -# id 1 and h(x0 ) = x 0

E

I. The set

M == {x EI; h(x) -# x} is open and non-empty. Hence M is the union of at most denumerable number of open disjoint intervals and each of the intervals has at least one finite end-point inside interval I. Denote by (c, d) one of the intervals. Without loss of generality, let c E I. Then h(c) = c. Equation P n belongs to class A~ 1• According to paragraph 5.2, it can be written in the form .

_

Pn(Y, x, I) =

( /( l)) Pn- 2

n

+

[n]

3

+ rn_ 3(x) y (n-3) + ... + r0 (x) y =

0.

Since h E ? + (P n), condition (iii) of Theorem 5.2.1 gives

rn_ 3 (h(x)) (h'(x)) 3 = rn_ 3(x) Let us define the function Rn_/ I

on -+ ~,

I.

(6.3.1)

by the formula

xo x 0 being an arbitrary number from interval

relation (6.3. I), we get

I. Evidently Rn_ 3 E C 1(!) and from

Rn_ 3 (h(x)) - Rn_ 3(h(c)) = Rn_ 3 (x) - Rn_ 3(c), that is (6.3.2) because h(c) = c. Now, let Rn_ 3 be not a constant on the interval (c, d). Then there exist numbers c1 and d 1 from (c, d) such that Rn_ 3(c 1) -=/: Rn_ 3(d 1). Define two sequences

{ci}r: 1 and {di}r: 1 as follows.

6.3 SUBGROUPS WITH INCREASING ELEMENTS

101

If h(c 1) > c 1 and then necessarily also h(di) > d1, then we put

ci+l == h- 1(cJ

di+l == h- 1(di)

and

for i = 1, 2, .... If h(c 1) < c 1 and hence also h(d 1) < d1, we define

ci+ 1 == h(ci)

and

di+ 1 := h(di)

for i = 1, 2, .... Each of the sequences is converging to the number c, however, due to relation

(6.3.2) Rn-3(ci)

=

Rn_ici) #- Rn_ 3(di)

=

Rn_ 3(d;)

which contradicts to the continuity of the function Rn_ 3 E C 1(I) at the inner point c of interval /. Hence Rn is a constant function on the interval (c, d) and therefore rn_ 3(x) = 0 for all x E (c, d). Thus rn_ 3(x) = 0 on the whole set M, the union of intervals of this type. Let us return to condition (iii) of Theorem 5.2.1. Since rn_ 3(x) = 0 on M c I, we get rn_ 4 (h(x)). (h'(x))4 =

rn_ 4 (x) on

M.

Because h'(x) > 0 on /, we may write also

1rn_ 4 (h(x))l 114 • h'(x) = lrn_ix)l 114 , and

for

f X

R._ix) •=

lr._ 4 (t)l 114 dt,

fixed

x0 E

J.

xo

Again we have Rn_ 4 E C 1(I), and by the same arguments as above applied to the sequences {c;}~ 1 and {d;}~ 1 we get

Rn_ 4 = const that is

on

(c, d)

102

6 STATIONARY GROUPS

and consequently

Analogously we obtain ri(x) = 0

on

M =

{x

El; h(x) =I-

x}

for i = 0, 1, ... , n - 3. If the set M is dense in the interval 1 then the continuity of the r/s gives ri(x) = 0 on 1 for all i = 0, 1, ... , n - 3. If M is not dense in 1 then there exists an interval 11 c 1 such that h(x) = x for all x E 1 1. Take x 1 E 11. Evidently h(x.) = x 1, h'(x 1) = 1, h"(x 1 ) = 0. According to Condition (ii) of Theorem 5.2.1 and Corollary 4.2.4, the fucntion h E fl+ (P n) satisfies the Kummer equation (p, p) on the whole interval 1. Solutions of this equations are uniquely determined by the initial conditions as Theorem 6.2.1 states. Since id 1 satisfies the Kummer equation on 1, i~ complies with the above initial conditions of the function h, we have h = = id1, which was excluded from our considerations. Hence the case when the set M = {x E 1; h(x) =I- x} is not dense in 1 is impossible and the only possibility is that all r/s vanish on the whole interval 1. That means that equation P n is the iterative equation

Theorem 6.3.5

Let equation P n = P n(Y, x; I) E A~ 1, n > 2, be not an iterative equation and let there exist h E fl+ (P n), h =I- id 1. Then either fl+ (P n) is an infi.nite cyclic group generated by a function he E 1 E en+ (1), h~(x) > 0 and he(x) =I- X on 1' . or fl+ (P n) is en+ 1-conjugate to the group whose elements are translations {fc; C E IR}

fc:

IR

-+

!R,

fc(t)-t+c.

Proof

Let us notice that the group fl+ (P n) has at least two elements, id 1 and the function h, h =I- id 1. Hence fl+ (P n) contains an infinitive cyclic group. If any two different elements of the group fl+ (P n), say h 1 and h2 , intersect each other somewhere in the interval J, i. e.

6.3 SUBGROUPS WITH INCREASING ELEMENTS

103

and h1 =f. h2 , then the function h3 == h1 1 o h2 is an element of the group p +(Pn), it is not identity on I and h 3(x 0 ) = h1 1 (h 2 (x 0 )) =

x0

E

I.

According to Theorem 6.3.4, equation P n is an iterative equation and this was excluded from our considerations. Hence no two different elements of p + (P n) intersect each other in the interval /. The elements of the group p + (P n) can be linearly ordered in the following way: for

h1, h2

E

p + (P n)

write

h 1 ~ h2 ,

if h1(x 0 ) < h2 (x 0 ) for some (then any) number x 0 EI, or if h1 = h2. In this ordering the group p + (P n) is an archimedean group, since for h =f. id1 there is h(x) =f. x on I, and the sequences

=f.

{h[iJ(x0 )}~ 1 and {h[iJ(x0)}i~~ 1 converge to both ends of interval J for any number x 0 E I. Let us recall that function h[i] denotes the ith iteration of function h. According to Proposition 2.4.6 there exists an order preserving isomorphism of the group p + (P n) onto a subgroup of the additive group IR. Let this subgroup be an infinite cyclic group generated by a non-zero number e E IR and formed by all numbers of the form ie, i E 7L. Denote by he this element of the group p + (P n) that corresponds in the above isomorphism to the number e. Evidently he E en+ 1(1), h;(x) > 0 and he(x) =f. x on I because he belongs to the group p + (P n). In this case we have ? + (P n)

= {hfJ;

i

E

7L} ,

that means that the element he is a generator of the infinite cyclic group p + (P n). Now, let p + (P n) be not an infinite cyclic group. With respect to Theorem 6.2.10, p + (P n) is en+ 1-conjugate to a closed subgroup of one of the groups listed in Theorem 6.2.5. It can be shown that there exists an en+ 1-diffeomorphism 1/f such that one of the following relations holds for each h E p + (P n), h =f. id: l/f- 1h

1/f(t) = Arctan (k 2 tan t)

1/1- l h

1/f (t) = Arctan (tan t + 1) E ff ,

l/f- 1h

1/f(t) = t + wn

E

ff,

or

or E

ff.

104

6 STATIONARY GROUPS

Let h 1 =fa id 1 and h2 =fa id 1 be two different elements of the group?+ v'n) that do not belong to the same infinite cyclic group. Since no two iterates h/ 1J and hin2] (n 1, n2 E Z; nf + n~ =fa 0) intersect each _other, the union of gra~hs of all finite compositions of h 1, h 1 1, h2 , and h2 1 being of the above forms 1s dense. Then by using 0. Holder [I] theorem and the fact that the group ?+(Pn) is closed we may apply Theorem 1 of G. Blanton and J. A. Baker [I] which states: "Each group whose elements are en+ 1-diffeomorphisms of an interval / onto / and such that to each point (x 0 , y 0 ) E / x I there exists just one element h of the group satisfying h(x0 ) = y 0 , is formed by functions

g(g- 1(x) + c) where g is a en+ 1-diffeomorphism of~ onto I and c ranges through the real numbers." In our case we may write

where fc: ~ -+ ~, fc(t) = t + c, c E ~Details of the proof are given in [32] and [34].

■

Theorem 6.3.6

Let Pn = Pn(Y, x; I) E A~ 1, n > 2, and let the group? +(Pn) contain an element h E en+ 1(/), h'(x) > 0 and h(x) =fa x on I. Then the equation P n(Y, x; I) can be globally transformed into an equation Qn(z, t; ~) E A~ 1 with all periodic coefficients of the same period on ~Proof

Consider the following functional equation

1/f(h(x)) = 'l'(x) + c where c is a non-zero constant, 1/f is an unknown function, and h is the (known) function introduced in the assumptions of the theorem. Under the conditions on h, Theorem 2. 7.1 ensures the existence of a solution 1/f of the functional equation that satisfies 'I'

E

en+ 1(/),

and

1/f'(x) =fa 0

on I.

A construction of solution 1/f is also described in Appendix 12.1. In fact, the sign of f// is the same as the sign of c . (h(x) - x) =fa 0. This solution 1/f also maps interval / onto the whole ~- Indeed, the ith iterate of h, h[i], is defined for each positive and negative i E Z, because h is a bijection of/ onto /. Since 1

l/f(h[i](x 0 )) = 'lf(x 0 ) + ic

105

6.3 SUBGROUPS WITH INCREASING ELEMENTS

and c =/. 0, we have lim If/ (h[i] (x 0 ))

= + oo . sign c .

i ➔ ±r:tJ

Hence If/ (I) = [ij_ That means that If/ is a en+ 1-diffeomorphism of I onto For a en+ 1-diffeomorphism g == 1/1-l of [ij onto I we may write

[ij_

(6.3.3) Now, let us transform equation Pn(Y, x; I) E A~ 1 by means of the global transformation /3 =(EI g' I (l-n)/2,g)y into the equation Qn(z,t;J)EA~ 1• Since g([ij) = /, we have J = [ij_ Moreover, due to relation (6.3.3 ), the translation d: [ij --+ [ij, d(t) = t + c, is an element of the group fl+ (Qn). Indeed, the groups fl(Pn) and fl(Qn) are en+ 1-conjugate, ·

fl(Qn) = g-l

O

fl(Pn)

0

g,

according to Theorem 3.3.13, and increasing elements of fl(P n) are converted into increasing elements of the group fl(Qn),

fl+(Qn) = g-1

o

fl+(Pn)

o

g.

g- 1 o h o g = d ·E fl+(Qn), and the global transformation () = = (EI d' I (l-n)/2, d)z transforms equation Qn into itself. However, d'(x) = 1 for all x E [ij_ It means that for () = ( E, d)z we have

Hence

Qn * () = Qn' in other words, all coefficients with the same period c,

+ c) =

qi: [ij --+ [ij

of equation Qn

on

[ij,

i = 0, 1, ... , n - 2,

qn- 1(t) = 0

on

[ij

and, of course,

qn(t) = 1

on

[ij_

qi(t

qi(t)

Theorem 6.3. 7 Let Pn = Pn(Y, x; I) group

E

E

A~ 1 are periodic

.■

A~1, n > 2, and let fl+ (Pn) be en_+ 1-conjugate to the

Then equation P n(Y, x; I) can be globally transformed into an equation Qn(z, t; A01 with constant coefficients on [ij_ n

E

[ij) E

106

6 STATIONARY GROUPS

Proof In accordance with the assumptions of the theorem, let

where g is a en+ 1-diffeomorphism of~ onto J. Denote by rt the global transformation ( E I g'I (t-n)/2 , g )y that transforms equation P n(Y, x; I) into equation Qn = Qn(z, t; 1). Evidently J = g- 1(1) = ~ and Qn E A~ 1. Moreover, due to Theorem 3.3.13, the group ? + (Qn) is formed exactly by the translations he, c E E ~, where h: ~ ~ ~, h(t) = t + c. That means that each global transformation

( E I h~ I (l -n)/2, hc)z that is

(E,t + c)z foreach

CE~

globally transforms the equation Qn(z, t; are normalized by

R)

into itself. Since coefficients of Qn

and then uniquely determined by the fundamental solution z, we get

and each i = 0, 1, ... , n -

qi(t) = ci = const,

1. This shows that all the coefficients are constants for

i = 0, 1, ... , n - 2,

and, of course,

because QnEA~ 1•

■

6.4 Stationary groups with decreasing elements Theorem 6.4.1

Let P n = P n(Y, x; I) be an equation from A~ 1, n > 2. If there exists an element h E 9(Pn) such that h'(x) < 0 on I, then either ho h = id 1, or equation Pn is an iterative equation.

107

6.4 STATIONARY GROUPS WITH DECREASING ELEMENTS

Proof Since h'(x) < 0 on I, there exists a number x 0 E I such that h(x 0 ) = x 0 . Then h O h(x 0 ) = x 0 , the composition h o h is also an element of the group g(P n) and moreover, this element is an increasing function. If it is not identity on I, ho h =I= id 1, then Theorem 6.3.4. ensures that P n is an iterative equation. ■

Theorem 6.4.2 Leth be a en+ 1-diffeomorphism of I onto I, let h'(x) < 0 on I and ho h = id 1, n > 2. There exists a solution lfl= I ---t IR of the functional equation 'lf(h(x))=-lfl(x),

xEl

given by the formula (6.4.1)

'll(x) = ½(x - h(x)) which is of class en+ 1(1) and such that 1/f'(x) > 0 on I.

Proof Since h E en+ 1(1), also 1/1 given by formula (6.4.1) is of class en+ 1(1). Moreover, 1/f'(x) = ½(1 - h'(x)) > 0, because h'(x) < 0 on IR. Finally,

'lf(h(x)) = ½(h(x) - h(h(x))) = ½(h(x) - x) = -'ll(x) for x

E

I, thus 1/1 is indeed a solution of the functional equation.

■

Remark 6.4.3 If interval / in Theorem 6.4.2 is finite, I ½(b - a)). If interval I is unbounded then

= (a, b ), then 1/1(/) lfl(I) = IR.

(½(a -

b ),

Theorem 6.4.4 Let Pn = P n(Y, x; I) be an equation from A~ 1, n > 2, being not an iterative equation. Suppose that there exists an element h E p(P n) such that h'(x) < 0 on I.

Then equation P n can be globally transformed into equation Qn = = Qn(z, t; J) E A~ 1 such that -id1 E p(Qn). Moreover, if I = IR and the group ? + (P n) is an infinite cyclic group with the generators x + e, x E IR for some e E IR+, then equation P n can be also globally transformed into equation Qn = = Qn(z, t; J) such that J = IR, -id~ E p(Qn) and t + e, t E IR, are again the generators of the group p+(Qn(z, t; IR)).

108

6 STATIONARY GROUPS

Proof With respect to our assumptions, Theorem 6.4. l yields ho h = id 1. Theorem 6.4.2 asserts that the function 'lf(X) == ½(x - h(x)) satisfies the relation XE

1/f

E

cn+ 1(J),

I'

1/f'(x) > 0 on/. Hence the inverse function g to 1/f, g = l/f- 1,

satisfies g- 1 oh og(t)

g

E

=

-t

on

J

= 'If(/),

en+ 1(J), g'(t) > 0 on J. Now, the transformation 0 on lR. That means also, that k is conjugate to the generator he(x) = x + e by means of the element h,

h o he o h- 1 = k, all functions being elements of the same group ,q;(P,J Hence k is one of two generators, x + e or x - e, that gives

h(x + e) = h(x) + e, or

h(x + e) = h(x) - e . Since h is a decreasing function and e > 0, we have

h(x + e) < h(x) < h(x) + e That means that the first case, k(x) k(x) = x - e. Therefore

h(x + e) = h(x) - e.

x

+

e, is impossible, and we have

109

6.4 STATIONARY GROUPS WITH DECREASING ELEMENTS

Now we may write

fll(x + e) = ½((x + e) - h(x + e)) = = ½(x - h(x)) + e = fll(x) + e, that is

Hence t

+

e, t

E

IR are again the generators of the group fl+ (Qn)·

■

6.5 Complete list of stationary groups and characterization of the corresponding equations Let us remind that, due to Proposition 4.2.6 and Theorem 3.3.13, the stationary group ~(.i\) of any equation f\ E is conjugate to the stationary group of 1 an equation P n E A~ to which equation Pn can be globally transformed by means of a transformation y,

A!

This conjugacy is given by the formula

Furthermore, consider an equation Pn = P n(Y, x; 1) E A~1, its stationary group ~(Pn), and any transformation a of the form a = (Af, h)y where A is a non-singular constant n by n matrix, functions f and h satisfy conditions ( 1) and (2) with an open interval J c IR, see Definition 3.1.1 of global transformations. For such a global transformation a, the group

is realized as the stationary group of the equation Qn = P n * a. Moreover, according to Theorem 6.1.2, for any equation

Pn = Pn(Y, x; I)

E

A~

1

the elements of the stationary group ~(P n) are global transformations uniquely determined by the elements of the group fl(P n) of parametrizations that means, by a group whose elements are certain en+ 1-diffeomorphisms of the interval J onto I.

110

6 STATIONARY GROUPS

These are the reasons why the following list contains the groups ?(P n) for Pn E A~1, n ;:=: 2, up to cn+ 1-conjugacy. Each cn+ 1-conjugate group to any group ?(P n) listed here is in fact realized as the stationary group of a suitable equation globally equivalent to the equation Pn. Theorem 6.5.1 (List of stationary groups ?(P n))

The following list contains, up to en+ 1-conjugacy, all possible stationary groups ?(Pn) of equations P n E A~ 1 for arbitrary n > 2: 1. The functions h: !R ---+ !R, h(x)

a tan x

+b +d

= Arctan - - - c tan x

ad - be

+ 1'

that is the fundamental group ffe" which is a three-parameter group both with increasing and decreasing real analytic functions, bijections of !R onto !R.

2.

The bijections h of lR+ onto lR+, h(x)

a tan x

= Arctan - - - - b tan x

a

E

!R+, b

3m.

E

1/a

!R, a two-parameter group of increasing real analytic functions.

For each m h(x)

+

E

N, the bijections h of (0, mn) onto (0, mn ), a tan x

= Arctan - - - - b tan x

a E !R +, b functions.

+

1/a

!R, a group containing both increasing and decreasing real analytic

E

4m. For each positive integer m, the bijections h of mn - n/2),

(0,

mn -

n/2) onto

(0,

= Arctan

h(x)

(a tan x)

and

h(x)

= Arctan

(a cot x),

a E !R+, forming a group whose elements are both increasing and decreasing real analytic functions.

5. The bijections h: !R

=

h(x) for all a

E

x

!R.

+

a

or

---+

!R of the form h(x)

=

-x

+

a

6.5 COMPLETE LIST OF STATIONARY GROUPS ...

1 1I

6. The increasing functions from 5, that is the bijections h of IR onto IR having the form h(x)

= x + c for all c

E

IR,

a one-parameter group. 1. The bijections h of IR onto IR of the form h(x)

= x + i and h(x) = -x + i, iEZ.

8.

The increasing functions from 1.

9.

Two functions: id!R and -id!R.

10.

The identity on IR only: id!R.

Remark 6.5.2 Each of the groups listed in this theorem is a restricted subgroup of the fundamental group ff up to a en+ 1-conjugacy. A characterization of equations P n E A~ 1 with respect to their stationary groups p(Pn) is given in the following theorem, all cases mentioned there refer to the corresponding cases in the preceeding Theorem 6.5.1.

Theorem 6.5.3. If equation Pn = Pn(Y, x; I) E A~ 1, n > 2, is an iterative equation Pn iterated from the second order equation

u"

+

= (p)[n]

(p)

p(x) u = 0 ,

(that means that for n = 2 the equation P n coincides with (p)) and, moreover, the equation (p) is both-side oscillatory then and only then case 1 in Theorem 6.5.1 takes place for the stationary group p(Pn), (p) is one-side oscillatory then case 2 is valid for v(P n), and conversely, (p) is of the finite type m and of the special kind exactly when case 3m holds for p(Pn), the equation (p) is of the finite type m and of the general kind then case 4m is true for the stationary group p(Pn), and with exception of m = 1 also conversely. Case 5 in Theorem 6.5.1 occurs when equation P n is not an iterative equation and is globally transformable into an equation Qn(z, t; IR) E A~ 1 with constant coefficients on IR that are identically zeros by the (n - i)th derivatives of z with odd i, i. e. qn-i(t) = 0

on IR

for

i = 1, 3, ....

(6.5.1)

Case 6 is valid if and only if Pn is not an iterative equation and it is globally transformable into an equation Qn(z, t; IR) E A~ 1 with constant coefficients and

112

6 STATIONARY GROUPS

relations (6.5.1) are not satisfied, that means, there exists an odd i such that the coefficient qn-i is not identically zero on ~Case 7 occurs for p(P n) if and only if Pn is not an iterative equation, it is not globally transformable into an equation with constant coefficients on ~, 1 however it is globally transformable into an equation Qn(z, t; JR) E A~ with periodic coefficients of the same period on ~ satisfying the relations (6.5.2)

for all i = 2, ... , n, (qn = 1 and qn- l = 0 on ~ because Qn already belongs to A~ 1). Case 8 is valid for the group p(P n) if and only if P n is not an iterative equation, is not globally transformable into an equat~on with constant coefficients on ~, it is globally transfomable into an equation Qn(z, t; ~) E A~ 1 with periodic coefficients of the same period on ~, however relations (6.5.2) are not satisfied, i. e. there exists i E {2, ... , n} such that qn-i(-to)

=f:.

(-lf qn-i(to)

for some t0 E ~Case 9 in Theorem 6.5.1 is valid for the stationary group p(Pn) of equation Pn if and only if it is not of any form described above but it can be globally transformed into and equation Qn(z, t; iR) whose coefficients satisfy relation (6.5.2). Case 10 takes place for p(Pn) just when Pn is not of any form introduced in the preceding cases. Remark 6.5.4 Each iterative equation P n E A~ 1, especially each second order equation from A~, has at least one-parameter subgroup of increasing elements in p(P n)· With exception of the case when P n is iterated from a one-side oscillatory second order equation, each other iterative equation P n has both increasing and decreasing elements in its stationary group p(Pn). In the cases when, up to en+ 1-conjugacy, the group p(Pn) is listed only once in the list in Theorem 6.5.1, the corresponding assertion of Theorem 6.5.3 has the form of a sufficient and necessary condition. However, the group in case 4m for m = 1 is conjugate to the group introduced under 5. Indeed, the group in case 4m with m = 1 is formed by increasing and decreasing bijections h of (0, n/2) onto (0, n/2) of the form

h(x) = arctan (a tan x) and h(x) = arctan (a cot x), a E

~+ ,

the values of arctan being in the interval (0, n/2). It can be easily checked that the elements of this group are expressible in the form ~ -1 gonog ,

6.5 COMPLETE LIST OF STATIONARY GROUPS ...

113

where g == arctan o exp is a en+ 1-diffeomorphisin of ~ onto (0, n/2) for any n E N, and hare precisely the elements of the group introduced under 5 in Theorem 6.5.1, since for positive a E ~+

h(x) = arctan (a tan x) = = arctan (exp (In (a tan x))) =

= g(g-1(x) + In a) = go ho g- 1(x) for '1(t) = t +

a, a =

In a

E

~

'

and

h(x) = arctan (a cot x) = arctan (exp (In cot x +Ina))= = g(-g- 1(x) + In a) = go ho g- 1 (x) for h( t) = - t + a, a = In a E ~However, it will be proved that this conjugacy between the group in 4m with m = 1 and the group in case 5 is the only possibility of conjugacy between the groups introduced in Theorem 6.5.1. Proof of Theorems 6.5.l and 6.5.3

Suppose that Pn = Pn(Y, x; I) E A~ 1 for some n > 2. First let us consider the group ? + (P n) consisting of increasing en+ 1-diffeomorphims of the interval / onto I belonging to the stationary group p(P n). Let hEp+(Pn) be such that h =I= id 1 and h(x 0 ) = x0 El. ThenPnisiterated from and equation (p) E A~'n, according to Theorem 6.3.4. Due to the criterion of global equivalence of equations of an arbitrary order, Theorem 5.2.1 and Remark 5.2.2, applied to the iterative equation P n' that means in the case when conditions (iii) and (iii*) give no restrictions, the group 9(Pn) coincides with the group p( (p) ), briefly p(P) only. Hence, with respect to Theorem 6.2.5, the stationary group p(P n) is en+ 1-conjugate to one of the group introduced in cases 1, 2, 3m and 4m according to the type and kind of the second order equation (p ), as stated in Theorem 6.5.3. Suppose P n is not an iterative equation and ? + (p) is not trivial. Applying Theorem 6.3.5 we get that 9 + (P 11 ) is en+ 1-conjugate either to the group in case 6 of Theorem 6.5.1, or to the group in case 8 of the theorem. In the former case according to Theorem 6.3. 7, the equation P n can be globally transformed into an equation Q (z, t; ~) E A01 with constant coefficients on ~, in the latter one n n 01 Theorem 6.3.6 ensures that Pn can be globally transformed into Qn(z, t; ~) E An with all periodic coefficients of the same period on ~If neither of the above cases occurs then according to Theorem, 6.3.4 and 6.3.5 all possibilities are exhausted except the case when the group ;, + (P 11 ) consists of the identity id 1 only.

114

6 STATIONARY GROUPS

Now we come to decreasing elements of the group ?(Pn). Proposition 6.3.3 asserts that there are only two possibilities, either the group ?(P n) has only increasing elements, it coincides with the group?+ (P n) or, otherwise, the factor group ?(Pn)l?+(Pn) consists of exactly two elements. For cases 1, 2, 3m and 4m, that means, for iterative equations, the decreasing elements have already been considered and completely characterized. Hence, do not let P n be an iterative equation. Suppose that ?(Pn) is en+ 1-conjugate to the group in case 6 of Theorem 6.5.1, and h E ?(P n) such that h $ ? + (P n). We have shown that in this case Pn can be globally transformed into an equation Qn with constant coefficients on [ij, the group ? + (Qn) being exactly the group in case 6. All decreasing elements of ?(Pn) are converted by conjugacy again in decreasing elements of ?(Qn). Hence there exists h E ?(Qn) such that h ¢ i+ (Qn), h: ~ -+ ~- Equation Qn is not an iterative equation, otherwise Pn would be, hence condition (iii) of Theorem 5.2.1 gives (6.5.3) for the first non-zero constant 'n-i(t) = 'n-i = const -=I= 0, i = 3, ... , n. This is because all coefficients of the iterative operator whose first three coefficients coincide with the first three constants coefficients of Qn are also constants on ~Hence relations (6.5.3) implies

lh'(t)I = 1 on Since

Ii:

~

-+

[ij_

~ and

dh(t )/dt < 0, we get

h(t) = - t + c0 for a suitable c0

E [ij_

However, ?(Qn) is a group with increasing elements of the form t ~ t + c for arbitrary c E [ij_ Therefore the decreasing elements t ~ -t + c for any c E [ij also belong to ?(Qn). Hence we may summarize: if ? + (P n) is en+ 1-conjugate to the group in case 6 and, moreover, if ?(P n) contanins a decreasing element, then ?(P n) is en+ 1-conjugate to the group in case 5 of Theorem 6.5.1 and Pnis globally transformable into an equation Qn(z, t; [ij) E A~ 1 with constant coefficients on ~ for which -idlR E ?(Qn), that means, qn-i = const = 0 for odd i, because qn = 1 (and qn-l = 0) as stated in this case in Theorem 6.5.3 in the form of relations (6.5. 1). Now, let the group ?+(Pn) be cn+ 1-conjugate to the group in case 8 of Theorem 6.5.1, and moreover, let the factor group ?(Pn)l?+(Pn) be non-trivial. We have shown that in this case P n can be globally transformed into an equation 2;

3. if P 2 = (l(O,mn)) is y" + y = 0 on (0, mn), or Pn = (l(o, mn)En] for n > 2; 4m. if P2 = (l(o, mn: _ n:/i)), or Pn = (l(o, mn: _ n/i))[n] for n > 2,

5. if P 4

= y 1v +

6. if P 3

=

ym + y

7. if P4

=

y1v + (cos 2nx)y = 0 on ~;

8. if P4

=

y 1v

+ y' +

(cos 2nx)y

9. if P 5

-

yv

+

+

(sinh x) y

=

y 1v

+

+

(sinh x) y = 0 on ~-

10. if P 4

y

y' y'

= 0 on ~; =

0 on ~;

= 0 on~;

= 0 on ~;

117

6.5 COMPLETE 11ST OF STATIONARY GROUPS ...

The examples of equations given for cases 1, 2, 3m and 4m are obtained directly from the corresponding cases in the characterization in Theorem 6.5.3. Due to the same theorem, the group in case 5 of Theorem 6.5.1 is the stationary group ?(P 4 ) for P 4 = y 1 v + y = 0 on ~, since this equation is not iterative, it has constant coefficients on ~ and p 4 _ 1 = p4 _ 3 = 0. The equation P 3 - y 1II + y = 0 on ~ is not an iterative equation, it has constant coefficients on ~. but p 3 _ 3 = p0 = 1 on ~- According to Theorem 5.2.1,if hE?(P 3 )then p0 (h(x)).

(h'(x)) 3 = p 0 (x) on~.

In our case however, p 0 = 1 = const, hence h'(x) = 1 on~ and the group ?(P 3 ) has no decreasing elements. Therefore the stationary group ?(P 3 ) is the group introduced in case 6 of Theorem 6.5.1. The equations y 1v + (cos 2nx) y = 0 on~ and y 1 v + y' + (cos 2nx) y = 0 on ~ serve as examples for cases 7 and 8 in Theorem 6.5.1, respectively. Indeed, neither of them is iterative, both have only periodic coefficients of the same period on ~, and relations (6.5.2) are satisfied by the coefficients of the former equation: cos 2nx = cos ( - 2nx ), while the stationary group of the latter equation has no decreasing elements h due to coefficient 1 of y' and Theorem 5.2.1. Moreover, neither of these equations can be transformed into an equation Q4 = z1v + c2z" + c 1z' + c0 z = 0 from A~ 1 with constant coefficients on ~, because otherwise

hence c 2 = 0 and (6.5.4) or (6.5.5) for the global transformations

- (E I g'I I (l-n)/2 '

al -

g l ) Z'

or

rx 2 --

(E I y'2 I (I -n)/2 '

g2) z

that transform Q4 into the first equation or into the second one, respectively, according to Theorem 5.2.1. However, this is impossible, since relation (6.5.4) cannot be satisfied for any g I E C 5 (~) with g 1(x) # 0 on ~ as required from the form of a 1, and relation (6.5.5.) gives g 2(x) = a 1 on ~, that is a0 , a I being constants,

118

6 STATIONARY GROUPS

and Q4* a 2 is an equation with constant coefficients and not the required equation y 1Y + y' + (cos 2nx) y = 0 on ~Each element h of the stationary group 9(P 5 ) for P 5 y v + y' + + (sinh x) y = 0 on ~ satisfies the condition

(h'(x)) 4 = 1 that follows from Theorem 5.2.1 in which p 11 _ 2 = P3 = 0, Pn-3 = r,,_3 = = s11- 3 = p2 = O' and pn- 4 = r11- 4 = sn- 4 = p 1 = 1 on ~- Hence h(x) = + x . + c0 on ~- However, only for c0 = 0 the transformations A I h' I (1- 11 )12 , h )y = ( A, h )y transform the P 5 into itself, and 9(P 5 )


2, consider P n = P n(Y, x; I) E A~. If h E 51(P,i) then, according to Proposition 4.1.2 where Pn-l = qn- I = 0 and Pn-2 = qn-2' we get p11 _

2

= p11 _ 2 (h) . h, 2 +

(n +3 1) {h,

x

}

on I.

This is in fact the Kummer equation

see also Remark 5.2.6, and each solution h defined on I such that h(I) = I belongs to class C3(I) and h'(x) -:/- 0 on I, see Theorem 6.2.1. Such a solution h also belongs to the stationary group

of the second order equation

+

y"

p._ 2 (x) y / ( n ;

1 )

=

0

on I,

according to Corollary 4.2.4. Hence each h

E

51(P 11 ) belongs to

the latter group being C3-conjugated to a (closed) restricted subgroup of the fundamental group ff, Theorem 6.2.5, and the proof is completed. ■

Remark 6.5.10 We proved here that each stationary group 51(P 11 ) of any equation P n from A~ is again a restricted subgroup of the (maximal, three-parameter) fundamental group ff. However, the problem of complete characterizations of such subgroups consists in the following. Whereas for equations· P n from A~ 1 each solution h: I ---+ ~ of the Kummer equation

is of class C 11 + 1(!) according to Corollary 4.2.4, because p 11 _ 2 E C"- 2 (1), in the case when Pn E A~ we have in general only p11 _ 2 in class C 0 (1) that implies h

E

c 3 (1)

6.5 COMPLETE LIST OF STATIONARY GROUPS ...

121

due to the same corollary. If really h is not smoother that means, if h is not of the class en+ 1(I), this h does not belong to the stationary group ?(P n) nevertheless it satisfies the above Kummer equation. This is the strong implication of Theorem 3.2.3 requiring f E cn(I) for a factor fin any global transformation between any pair of equations from An and hence h E en+ 1(1) for any parametrization of a transformation of equations from A~ where f = = c . lh'l(l-n)/2 according to Corollary 4.2.3. However, there may exist a solution h E en+ 1(1) of the Kummer equation

with only continuous Pn- 2 E c0 (!) \ C 1(!); it may happen that such an equation Pn EA~ has a non-trivial h =I=- id 1 in its stationary group ?(P n). For example, each equation P E An (even in An \ A~) with all periodic coefficients Pi E 0 1 E c (~)\C (~) of the same period d,

P;(x + d) = plx) on

~

i = 0, ... , n - 1, d being a positive constant, is globally transformed into itself by the transformation

(E, g)y

=I=- lp

where g: ~ ~ ~, g(x) = x + d on ~, and g is of class C 00 ( ~ ) nevertheless P; $ C 1(~ ). Then, of course, the stationary group ?(P n) is at least an infinite cyclic group since each element g[iJ, i E Z, belongs also to it. But, in fact, there must not be too many solutions h of class en+ 1(/) of the Kummer equation

in this case. If the stationary group ?(P n) contains a subgroup of increasing non-interseting elements such that their graphs cover the open square I x I then it can be proved that Pn- 2 E cn- 2(!), see G. Blanton and J. A. Baker [I].

7 Canonical forms

Here we shall study the next important problem mentioned in paragraph 3.4, the problem of canonical forms of linear differential equations of the nth order. Two special forms of ordinary differential homogeneous equations have been introduced in the literature, the so-called Laguerre-Forsyth forms and the Halphen forms. The former or the latter are sometimes called canonical. As G. D. Birkhoff [ 1] pointed out already in 1910, in the case of the third order equations, the Laguerre-Forsyth form is not global in the sense that not every linear differential equation can be transformed in an equation of the form on its whole interval of definition. However, we need not go to the third order, the second order is sufficient to show that even here the Laguerre-Forsyth form is not global. The Halphen forms are not global either. Moreover, neither of the forms becomes global by restricting ourselves to linear differential equations with sufficiently smooth or even analytic coefficients.

7.1 Notion of canonical forms In Chapter 3 it is shown that the set A of ordinary differential linear homogeneous equations is an Ehresmann groupoid where morphisms are global transformations, and each class of globally equivalent equations has a structure of a Brandt groupoid. Some special objects, representatives, of a full subcategory A* of the (or a, if you do not consider only the category of linear differential equations) Ehresmann groupoid A are often called canonical. Definition 7.1. l

Let A* be a full subcategory of an Ehresmann groupoid A, and let A* denote its decomposition into the equivalent classes with respect to morphisms of the groupoid. If in each class of the equivalence there is exactly one special object from a family C(l of the special objects then the set C(l is called the set of canonical forms of A* in the sense of Mac Lane and Birkhoff [I].

124

7 CANONICAL FORMS

However, in some cases the requirement on the uniqueness of a cannonical object in each class of equivalence occurs to be too restrictive. Even in the matrix theory the Jordan canonical forms are not unique, since permutations of blocks along the diagonal give similar matrices. Hence we introduce more refined definitions. The reason is not only a historical one, since neither the LaguerreForsyth, nor the Halphen forms are canonical in the Mac Lane-Birkhoff sense, but also because under a "natural" hereditary condition 7.4.1 it is impossible to have global and unique canonical parametrizations of functions, Theorem 7.4.7. Hence neither Halphen approach nor Cartan's moving-frame-of-reference method can be improved to get unique and at the same time global canonical forms of linear differential equations of the third and higher orders. Definition 7. 1.2

A set C€ c A* is said to be the set of unique canonical forms for A* if there is at most one object of the set C€ in each equivalence class 8 EA*. In the case of linear differential equations, a set C€ is a set of unique canonical forms for A*, if each equation from A* can be globally transformed into at most one equation in C€. We say that a set C€ c A* is a set of global canonical forms for A* if there is at least one object of the set C€ in each equivalence class 8 EA*. For linear differential equations a set C€ is a set of global canonical forms for A* if each equation from A* can be globally transformed in an equation in C€.

Remark 7.1.3 These definitions does not exclude some trivial cases, e. g., when C€ consists only from one object and then evidently it is the set of unique canonical forms, or C€ coincides with A* and then it is the set of global canonical forms. However, we are naturally more interested in the cases of the opposite nature.

Remark 7.1.4 A set C€ c A* is a set of canonical forms in the sense of Mac Lane and Birkhoff exactly when the set C€ is a set of canonical forms that are both unique and global.

7.2 The Laguerre-Forsyth and Halphen forms The only known canonical forms for linear differential equations are the Laguerre-Forsyth and the Halphen forms.

125

7.2 THE LAGUERRE-FORSYTH AND HALPHEN FORMS

Definition 7.2.1 The Laguerre-Forsyth canonical forms are the equations Pn(Y, x; I) E A characterized by the vanishing of coefficients of the (n - 1)st and the (n - 2)nd derivatives, that means the equations

y(n) + Pn_ 3 (x) y(n- 3 ) + ... + p0 (x) y = 0

on I

c

IR,

n > 2, whose first three coefficients form the succession

1,

0,

0,

and other coefficients are arbitrary continuous functions Pi E C 0 (!), i = 0, ... , n - 3.

Remark 7.2.2 The Laguerre-Forsyth canonical forms for the second order equations are equations

y" = 0 defined on arbitrary intervals I c IR. At the beginning of Chapter 5 it is shown that neither of these equations can be globally transformed into the equation

y"

+

y = 0

on IR .

Since equation (0 1 ) is of the type 1 and general if I =I= IR. or special if I = IR., then applying Theorem 5.1.1 we see that precisely those second order equations can be globally transformed into one of the Laguerre-Forsyth forms whose solutions have at most one zero on their interval of definition. However, on the basis of Theorem 5.2.1 we can formulate more general result for equations of an arbitrary order.

Theorem 7.2.3 An equation P = P n(Y, x; I)

E

A~

1 ,

p = y(n) + Pn-ix) y(n-2) + ... + Po(x) y = 0'

n > 2, Pn- 2 E cn- 2 (1), pi E C(I) for i = 0, ... , n - 3, can be globally transformed into one of the equations of the Laguerre-Forsyth forms defined on an interval J if and only if the second order equation

126

7 CANONICAL FORMS

Pn- 2 E cn- 2(1), is of the type l, or equivalently, if and only if each non-trivial solution of this equation has at most one zero on the interval I. In addition, J =I= IR or J = IR exactly if the second order equation

is of the general or special kind, respectively.

Proof

If equation P

E

A01 n '

can be globally transformed into an equation of the Laguerre-Forsyth forms, i.e., into an equation z(n)

+

qn_ 3(t)

z(n- 3 )

+ ... +

q0(t) z = 0

on J .'

then condition (ii) of Theorem 5.2.1 implies that the second order equation

is globally transformable into the equation v" = 0

on J,

since qn_ 2(t) = 0 on J. Hence, according to Theorem 5.1.2, equation

is of the same type and kind as equation v" = 0 on J. However, this equation is always of the finite type equal to 1 and moreover, its kind is general if and only if J =I= IR. Conversely, let equation

(P.-2/(n; I)) with Pn- 2 E cn- 2(1) be of the type 1. According to Theorem 5.1.2 there exists a global transformation of the form (BI h' 1- 112 , h)u that converts the equation into

v" = 0

on J

127

7.2 THE LAGUERRE-FORSYTH AND HALPHEN FORMS

if and only if the last equation is of the same type and kind. The type of this equation is always 1, its kind being general or special exactly when J i= IR or J = IR, respectively. Then, due to Theorem 5.2.1, the transformation ( A I h'

I (i -

n )12,

h) Y

globally transforms equation P n(Y, x; I) into and equation from A~ 1 whose first three coefficients are 1, 0, 0, that means, into one of the equations of the Laguerre-Forsyth forms defined on J. ■ Corollary 7.2.4 The set of the Laguerre-Forsyth canonical forms of linear differential equations is not global for any order n, n > 2, even for equations with real analytic coefficients only. Proof

This is a direct consequence of Theorem 7.2.3, since for any n > 2, no equation y(n)

+ Pn- 2 (x)

y(n- 2 )

+ ... + p 0 (x)

y

= 0 on I

with Pn- 2 E c"- 2 (1) can be globally transformed into an equation of the Laguerre-Forsyth forms if

is either of a finite type greater than 1 or of the infinite type, for example, if Pn- 2 = 1

on IR

■

Definition 7.2.5

The set of equations of the forms

/)I + l(p)[n](y, x; /)I + l(P)[n](y, x;

l(P)[n](y, x; /)I +

e3y(n-3)

+ 'n-4(x) y(n-4) + 'n-s(x) y(n-5) + e4y(n-4) + 'n-s(x) y(n-5) + esY(n-5)

+ r0 (x) y = 0, + r0 (x) y = 0, + r 0(x)y=0,

+

l(P )[ n J(y, x; /)I + l(P)[n](y, x; /)I 0

for all n > 3, all J C IR, and all p E c"- 2 (1), ri E c (J), i = 0, ... , n - 4, 8 . being + 1 for odd i and + 1 or - 1 for even i, is called the set of the Halphen c~nonical forms, see for example J. E. Wilczynski [ 1].

128

7 CANONICAL FORMS

Remark 7.2.6 Let us remark that the differential operator of an equation of one of the Halphen forms is the sum of an iterative operator,

and a linear differential operator of an order < n - 3 starting with the coefficient + 1 or - 1, y(n-i)

+ rn-i-l(x) y(n-i-1} + ... + ro(x) y'

if n - i > I. In the last two cases, the second term in the sum is either + y only or it vanishes at all and the Halphen form is just an iterative equation. Let us recall also that each equation of any of the Halphen forms belongs to class A~ 1•

Theorem 7.2.7 Let P = P n(Y, x; form

I)

be an equation from A~ 1, n > 3. Let us consider it in the

Equation P can be globally transformed into one of the equations of the Halphen forms if and only if either P is an iterative equation, or the function r1-. belongs to en(!) and r;.(x) # 0 for all x E J where i* is the maximum of these i = 0, ... , n - 3 for which ri is not identically zero on I.

Proof If equation P can be globally transformed into the Halphen form introduced as the last in Definition 7.2.5 then, according to Theorem 5.2.1, P is an iterative equation. If P E A~ 1 can be globally transformed by a transformation ( Af, h) into one of the equations of the other Halphen forms then Theorem 5.2.1 c~n be applied since this equation belongs to A~ 1 as well. Let the equation be of the form

for some i*, 0 < i* < n - 3. Condition (ii) of Theorem 5.2.1 gives rlh(t)) (h'(t)t-i = 0

on J

129

7.2 THE LAGUERRE-FORSYTH AND HALPHEN FORMS

that means rlx) = 0 on I, for i = n - 3, ... , i* For the function r;* we get

+

1, because h'(t)

-=I-

0 on I.

Since his a en+ 1-diffeomorphism of J onto I and en-i* = const -=I- 0, we have r;.(x) -=I- 0 on I and r;* E en(!), i* being the maximal index i = 0, ... , n - 3, for which r;* is not identically zero on I. Conversely, if P is an iterative equation then, in fact, it is already in the Halphen form introduced as the last in Definition 7.2.5. Hence, let us assume that P is not an iterative equation, that means that P can be written in the form

r;* being non-vanishing on I and r;* global transformation

E

en(!). According to Theorem 5.2.1, a

(AI h, I (1- n )/2' h) y transforming P into Q, Q

= l(q )[n ](z, t; 1)1 +

sn_ 3 (t) z(n- 3 )

+ ... + s0 (t)

z

= 0

yields the following relations

and

on the interval J

==

h- 1(1). For s;.(t)

r;.(x) = en-i* (k'(x)r-i*,

en-i* and k

=

:=

h-

1

we have

I.

XE

Put ei-i* = 1 for odd n - i*, and

en-i*

sign (r;.(x))

for even

n -

i*.

Then

f X

k(x) = k1 +

(en-i' r;,(s)) 1/(n-i') ds,

xo

k 1 being a constant. Evidently k is a en+ 1-diffeomorphism of I onto k(I) = J because r;* E en(!) and r;* is always positive or always negative on the whole I.

130

7 CANONICAL FORMS

Hence k- 1 = h is such a en+ 1-diffeomorphism of J onto I for which the tranforma tion

globaly transforms P into an equation of the Halphen forms.

■

Corollary 7.2.8 The set of the Halphen forms is not a set of global canonical forms for linear differential equations of any order n, n > 3, even if only equations with real analytic coefficients are considered.

Proof follows immediately from Theorem 7.2. 7. It is sufficient to consider the equation

l(p)[n] (y, x; /)I +

X •

y(n- 3) = 0,

on an interval/ containing zero, 0 E

/, p

n > 3,

being an arbitrary function from class

. cn- 2 (!). For any n > 3, the equation cannot be globally transformed into any of the Halphen forms since the first non-zero coefficient ri, rn_ 3(x) = x on I,

changes sign on the interval of definition.

■

Definition 7.2.9 Let us introduce a special subset OIL of the set of the Halphen canonical forms defined in 7.2.5. First, consider an equation of the Halphen form

l(P)[nJ(y, x; I) I + y(n-i) + rn-i-l(x) y(n-i-1) + ... + ro(x) y = 0 for an odd i, 3 < i < n, p E cn- 2 (!), rj E c 0 (J) for j = 0, ... , n - i - 1. Each equation obtained from this equation by translation that means, by a global transformation of the form

where h(x) = x + d, d being a constant, is defined on the interval {x + d; x EI} and it is again of the Halphen form. All translations define a decomposition of the set of equations of the above Halphen forms. We choose just one equation of each class of the decomposition to be an element of the special set 0/i.

Secondly, let an equation be of the Halphen form

l(P)[n]I + eiy(n-i) + rn-i-1(x)

Y(n-i-l)

+ •·· + ro(x) Y = 0

7.2 THE LAGUERRE-FORSYTH AND HALPHEN FORMS

131

for even i, 3 < i < n, B; E { 1, -1 }, p E c(n- 2 )(/), rj E CO(/) for j = 0, ... , ... , n - i - 1. We consider all global transformations ( A, h )y where

h(x) = x

+ d,

x

E

I

or

h(x) = -x

+ d,

x EI

for all real constants d E IR! that means, we consider all translations and symmetries of the reals. The set of linear differential equations of the above Halphen form is decomposed in the classes with respect to the translations and symmetries. We take just one representative of each of the classes to be an element of the special set 0/L. Finally, we consider an equation of the Halphen form

p E

c"- 2(1),

that means an iterative equation. The second order equation

u" + p(x) u = 0 on I

(p)

is of certain type and kind, see Definition 5.1.1. From the whole class of globally equivalent equations to (p) with coefficients of class cn- 2 on the corresponding intervals that means, from the set of all second order equations A~" that are of the same type and kind as equation (p ), choose just one representative to be an element of the special set 0/L. Do the same for each class of equivalent equations from A~"- For example, as the representatives we may select the countable set of the equations u" + u = 0 on the intervals (0, rc/2 ), (0, re), (0, 3rc/2), ... , (0, mrc - rc/2), (0, mrc), ... , (0, oo ), and (- oo, oo ). Let us claim that there are no other equations selected as elements of such a special set 0/L. This subset 0/L of the set of Halphen canonical forms of linear differential equations chosen in such the manner will be called a set of unique Halphen forms.

Remark 7.2.10 A justification of the adjective 'unique' is given in the following theorem that thesserts that a set of unique Halphen forms is a set of unique canonical forms for linear differential equations of all orders n, n > 3, in the sense of Definition 7.1.2. However, any set of unique Halphen forms is not a set of canonical forms for linear differential equations with respect to global transformations in the sense of Mac Lane and Birkhoff, Definition 7. 1. 1, since it is not global.

132

7 CANONICAL FORMS

Theorem 7.2.11

The set au of unique Halphen forms of linear differential equations is a set of unique canonical forms. Proof

Consider a set au constructed in Definition 7 .2.9. With respect to Definition 7 .1.2 of a set of unique canonical forms, it is sufficient to show that there are no two equations in the set au that are globally equivalent. Hence, let P and Q be two equations from au. Suppose that P can be globally transformed into Q by a transformation ( Af, h Hence, let P be of the form

>v·

p

=

l(P)[n](y, x;

/)I +

eiy(n-i)

+

rn-i-l(x) y(n-i-l)

+ + r0 (x)

y

= 0 on I .

Then equation Q is of the same order and of the form

Q

= l{q)[n]{z, t; J)I +

ejz(n-J)

+

+ .. . + s0 ( t) z =

sn_ _ 1(t) z(n-J-l) 1

0

on J .

The criterion of global equivalence of the nth order equations, Theorem 5.2.1, gives i = j and e;(h'(t)f =

ej =

e~,

= 1 = e~ and h(t) = t + d. Due to the construction of the set au in Definition 7.2.9, equations P and Q coincide. If i is even then both ei and e~ are of the same sign, i being an integer, 3 < i < n. If i is odd then ei

h(t)=+t+d,

and equations P and Q again coincide, since just one equation is selected for the set au from the set of all equations of the Halphen form that differ by translations, t + d, and symmetries, - t + d. Finally, let P be an iterative equation

that means that Pis listed as the last of the Halphen forms in Definition 7.2.5. Due to Theorem 5.2.1, any equation globally equivalent to P is an iterative equation with a coefficient of class cn- 2 that means

7.2 THE LAGUERRE-FORSYTH AND HALPHEN FORMS

133

Moreover, the second order equations

u"

+ p(x)u =

v"

+ q(t)

0

on!

and

v = 0

on J

are globally equivalent as follows from conditions (ii) of Theorem 5.2.1. However, from each class of globally equivalent equations of the second order there is just one representative in the set 0//. Hence P and Q coincide that has to be proved. ■

7.3 Cartan's moving-frame-of-reference method Our geometrical approach to global transformations of linear differential equations is explained in Chapter 4, par. 4. Especially Theorem 4.4.1 asserts that the fundamental solution of any equation globally equivalent to an equation Pn(Y, x; I) with the fundamental solution y is obtained as a section of a centroaffine image of the cone K K

:=

{cy(x);

CE [ij+'

X E

J}

C

Vn

in a certain parametrization. Hence when we are looking for a special equation in a class of globally equivalent equations containing the equation P n(Y, x; I), we may seek some special section of the cone K. Moreover, we may try to find this section of the cone K intrinsically with respect to the centroaffine transformations of the space Vn, that means that for each centroaffine mapping A, the curve Au is the special section assigned the centroaffine image AK of the cone Kif u is the special section of the cone K. If we succeed to find such intrinsic sections then the set of the corresponding linear differential equations can even be considered a set of the unique canonical forms. In fact, all special sections of all centroaffine images AK, A E E G ll_n, of a fixed cone Kare the curves Au, A E G [Ln; hence all linear differential equations with solutions being coordinates of all curves Au coincide; in other words, in each class of globally equivalent equations there is at most one special, canonical equation. It is natural that in such a situation we try [19] to adopt Cartan's [l] moving-frame-of-reference method. This method fits in the case when we want to find special sections with coordination intrisically and invariantly chosen on the cones (manifolds) in the space Vn with the general centroaffine group of

134

7 CANONICAL FORMS

transformations G ILn. For this purpose, consider a cone K = K(c, x) = {cy(x ); c E IR+; x E I} c Vn, where y is the fundamental solution of an equation Pn(Y, x; I) E An, n > 2. The frame (u 1, ... , un), corresponding to the point K(c, x) E Vn, has its origin at K(c, x), the vector u 1 is identified with the vector K(c, x ), and u2 is tangent to the cone K. Then using the Cartan method for choosing a unique frame of reference, we obtain the following system of differential equations for cones. For two-dimensional space V2 we have u'I

, U2 -

For three-dimensional space V3 we obtain

u;: J -. !R 3, i = 1, 2, 3; k 1: J -. IR being a function, called invariant. For four-dimensional space V4 we get ,

= u; = , U3 = U1

3k1U1 3k2U1

2u 1

u'4 ui: J -. !R4, i

+ + +

U2

k1U2

+

4k2U2

3u2

U3

k1U3

+

+

3k2U3

U4

3k1U4

1, 2, 3, 4; k 1: J -. IR and k 2 : J -. IR are functions, called

invariants. Let us demonstrate the derivation of the system in the three-dimensional space V3. Our reasoning is similar to that applied by J. Favard [I], however, in his case the group is unimodular whereas we have the general centroaffine group GIL 3. Consider the cone K = {cy(x): c E IR+, x EI} where y is a fundamental solution of a linear differential equation of the third order P 3(y, x; I). In accordance with E. Cartan, let the frame (u 1, u2, u3) correspond to the point K (c, x) = (c y1(x), c yi(x), c y 3(x))T on the cone K: IR+ x I -. V3, and

u1 is the vector K(c, x ), and u2 is tangent to K at the point K(c, x ).

135

7.3 CARTAN'S MOVING-FRAME-OF-REFERENCE METHOD

If w 1 w 2, and

w

0

1

w3 =

dK

=

w

'

3

are the principal forms then

=

du 1

w 1u 1

du 2

W2U1

du 3

= W3U1

1 1

2 W1,

w2

w!,

+ w 2u 2 2 + W2U2 + 2 + W3U2 +

w3 1

0,

3 W2U3

3

W3U3.

The structural equations for the centroaffine group are

dcd =

w1

cd1 + w 2

/\

I\

cd2 i, j

Since the integrability condition gives w 1

/\

w{

+

= 1, 2, 3 .

w 2 I\ w~ = 0,

we have

w 2 I\ w~

0,

this is

Hence 2 1 2 w1 I\ w{ + w~ I\ w~ + w~ I\ w~ = da I\ ro + a(w I\ wf + w I\ w~),

that implies w~ I\ aw

2

+

2 aw I\ w~

da I\ w 2 + a(w 1 I\ w 2 + w

2

I\ w~),

and w 2 I\ ( - 2aw~ + aw~ + aw + da) = 0 . 1

We can conclude that da - 2aw~

+

aw~

+

2 1 aw = -bw .

The variation with respect to the secondary parameters vanishes, 8a - 2ae~

+

ae~ = 0 ,

that means 8a = a(2e~ - en.

136

7 CANONICAL FORMS

The special case a = 0, i. e. w~ = 0 , implies that the cone K lies in a plane. We exclude this case from the consideration since cones obtained in our geometrical approach to the third order linear differential equations are not situated in a plane (that would then necessarily have gone through the origin) because the Wronskian determinants of the fundan1ental solutions, curves, on the cones are not vanishing. Putting a := 1 , we obtain

w~ = w 2 ,

and

+ kw 2 •

2w~ - w~ = w 1

The integrability condition yields

that means

db - 3w1

+ 3w~ - bw~ + bw 1

-3ew 2

•

'

thus

ob - 3ei + 3e~ - be~ = 0 . Putting b

:=

0, we find that Wl

2

The exterior differentiation gives

that is de - 2ew~

+

2wj

+

2ew 1

2ii.w 2

'

and

De - 2ee~

+ 2ej = 0.

Put e := 0 ; then w1 = w~ and

Wj

ii.w 2• The latter of the relations implies

=

2

w I\ (3ii.w 1 + dii. - 3aw~) = 0, dii. - 3ii.w~ + 3aw 1 = - 36w2'

and

Ba - 3ae~

0.

7.3 CARTAN'S MOVING-FRAME-OF-REFERENCE METHOD

137

In general case (of cones of a general type) we can put a == 1 , which yields

wl = w2 and w~ = w1 + 6w2 . The integrability condition gives

w2 I\ (w1 - db) oP - e1 = o.

kw 2

= 0,

Putting 6 == 0, we obtain w~ = w 1 and system of differential equations: dK

= du 1

w 1u 1

du 2

-k1W2U1

du 3

w 2u1

+ +

w1 =

'

k 1w2• Hence we get the following

w 2u 2 W1U2

k1W2U2

+ +

w 2u 3 1

W U3,

where dw 1 = dw 2 = 0, that means that w 1 and w 2 are total differentials. Moreover, the integrability condition gives dk 1 /\ w 2 = 0, hence k 1w 2 is also a total differential. Putting w 1 = ds and w 2 = dt, we find that u 1 ds

-k 1u 1 dt

u1 dt

+ u2 dt + u2 ds + u3 dt - k 1u2 dt + u3 ds

where a function k 1 depends only on t. For w 1 = ds = 0, i. e. for some fixed s = c0 E ~+' we obtain the system u'1 u'2

where t E J c ~, u;: J --+ V3, t 1--+ ui (c0 , t). Especially u 1 is a curve lying on the cone K, it is a special section of the cone in a special parametrization t. This is exactly the system introduced above for the three-dimensional space V3. The first vectors u1 in the moving frames always end on the cones. Hence the curves

are sections of the cones and they satisfy certain differential equations obtained from the above systems when eliminating other vectors of the frames.

138

7 CANONICAL FORMS

For n = 2 we have

hence

u'{ + u1 = 0 on J

c

Ill .

For n = 3 (and k 1 E C1(J)) we get

that is J

C

Ill.

(7.3.1)

+ k 1 + k2 E C2(J), k2 E c0 (J)) we obtain u{v - lO(ki + k1 + k2) u'{ - lO(ki + k1 + k2)'u 1 - 5u 1 +

For n = 4 (and kf

+ (9(kf + k1 +

k2)2 - 3(kf

+ kl + k2)") U1 + 3k2U1 = 0. (7.3.2)

Hence the special equations obtained for the second order equations are y"

+

y = 0

on

J

c

IR .

The set of all these equation (for all J c Ill) is a set of global canonical forms for A2 in the sense of Definition 7.1.2, since each second order equation can be globally transformed into one of the equations due to Theorem 5.1.2 or its consequence, Theorem 5.1.5. For the third order equations in the general case we get y"'

+

2py'

+ (p' +

1) y = 0 ;

(7.3.3)

after the change x == -t, y(x) == u1(-t) and p(x) == k 1(-t) in (7.3.1). Evidently, these third order equations can be written in the form

l(p/2)[3JI +

y

=

o, .

since the third order iterative operator l(p/2)[ 3 ] (y, x; 1)1 is in fact y"'

+

2py'

+

p'y'

see Remark 4.3.12. Hence the special third order equations obtained by Cartan's moving-frame-of-reference method are equations in one of the Halphen canonical forms, namely for n = 3 and e3 = 1 in Definition 7.2.5. These equations do not form a set of global canonical forms even for A~w, see Corollary 7.2.8.

139

7.3 CARTAN'S MOVING-FRAME-OF-REFERENCE METHOD

A similar situation occurs also for the fourth order equations. By putting q := - (kf + k 1 + k2 ), s0 == 3k 1 and by writting z for coordinates of u 1, Eqn (7.3.2) becomes

z'v q E C

2

+

(J),

1, 0,

lOqz" So E

+

lOq'z'

c0 (J).

+

(9q 2

+

3q") z - 5z'

+

s0 z = 0,

According to Remark 4.3.12,

(n+3 1) q, 2 (n +4 1) q,' (n +5 1)( 3q, + 5n 3+ 7q

2)

'

are coefficients of the iterative operator l(q)[n](z, t; J)I starting from the coefficient of z(n)_ Hence Eqn. (7.3.2) can be written in the form

l(q)[4 J1 - 5z' + s0 (t)

z

= 0 on J.

(7.3.4)

With respect to Theorem 5.2.1, Remark 5.2.2, and Theorem 7.2.7, the global transformation

with x = h(t) of the form

=

-5 113 t

+ c,

cbeingaconstant, h(J)

=

I, convertsanequation

l(p)[4 J(y, x; J)I + y' + r0 (x) y = 0 on I into Eqn (7.3.4). Indeed, for -5 = s 1(t) = sn_ 3 (t) # 0 on J, r n- 3 (x) = r 1(x) = 1 on /, condition (iii) in Theorem 5.2.1 implies

and

(h' (t) )3 = - 5 , that is

h(t) = -5 113t

+

const.

Moreover, due to condition (ii), the iterative operator l(q)[4 ]1 is transformed into the iterative operator l(P)[4 J1, and condition (iii*) gives a continuous coefficient r0 E C0 (J). Hence we come to the conclusion that the special fourth order equations obtained by Cartan's moving-frame-of-reference method are, after a suitable global transformation, again equations of one of the Halphen canonical forms, namely that introduced in Definition 7.2.5 for n = 4 and e3 = 1. These equations again do not represent a set of global canonical forms either for A~w, see Corollary 7.2.8.

140

7 CANONICAL FORMS

Summarizing. our investigations concerning the Cartan method of specification of moving frames in two, three, and four-dimensional spaces for finding special sections of certain cones in order to obtain canonical forms for linear differential equations of the corresponding orders, we come to the following result. Theorem 7.3.1 Cartan 's moving-frame-of-reference method applied to the second order equations gives the equations y"

+

y = 0

on

I

c

lR ,

that are global canonical forms for A2. The same Cartan 's method gives the Halphen canonical forms for the third and fourth order equations that are not global, either for A~w or A~w.

Remark 7.3.2

Hence we see that a very natural tool for the situation, Cartan's moving-frameof-reference method, fails in finding global canonical forms. In the next paragraph we show that not only the Cartan method but any other procedure satisfying a certain natural "hereditary" property as the Cartan method does, cannot give unique and at the same time global canonical forms of linear differential equations of the third and higher orders. And, in fact, it concerns much wider class of objects than linear differential equations, since many constructions of special objects in special parametrizations satisfy the hereditary property. Remark 7.3.3

In spite of the fact that Cartan's method led to the forms of linear differential equations that are not global as shown above, see also [19], it helped to formulate in 1984 the criterion of global equivalence of linear differential equations of orders greater then or equal to three.

7 .4 Hereditary property There are several situations when special parametrizations of certain objects are used. For example, when the length parametrization of curves is considered, when canonical forms oflinear differential equations are dealt with, when Cartan\ moving-frame-of-reference method is used. In general, these specifications of parameter cannot be applied on the whole domain of definition of the objects as we have seen, e. g., in Corollary 7.2.8. A natural question arises whether it is

141

7.4 HEREDITARY PROPERTY

possible to find in certain cases a special parametrization of considered objects on their whole domain of definition. The above examples of special parametrizations of certain special, canonical objects have a common property that can be formulated as follows. Consider an Ehresmann groupoid A, each object P of which is defined on ~om P, i. e., on the domain of definition of P. For each P E A, let a certain set &l(P) of objects be assign to Pas its representatives, canonical forms, such that Yf om (P, R) #- 0 for each RE &l(P) and &l(P) = &l(Q) for each Q EA with :Ye om (Q, P) #- 0. Moreover, let each morphism a E Yf om (P, R), a reparametrization of the object P leading to one of its canonical forms, R, correspond to the bijection ~ of ~om Ponto ~om R. Furthermore, let PjD, D c ~om P, denote the restriction of the object P to the domain D, which may or may not be an object of the category A. Definition 7.4.1

We say that a choice &I of representatives, canonical forms, in an Ehresmann groupoid A satisfies the hereditary property Yf if &l(PID) is formed just by restrictions Rlli(D)

whenever P E Yfom

E

A,

D

c

~om P,

PID

E

A,

for all R

E

&l(P),

and a

E

(P, R); in other words

a restriction of an object has a canonical form obtained as a restriction of a canonical form of the original object. Write also &l(S) instead of &l(P) if PE SE A. Definition 7.4.2

Let k be a positive integer, and let e0k be a category whose objects are all real continuous functions defined on all open intervals of reals, i. e. the set c0, and morphisms are defined as follows: for f E c0 and g E c 0 , the set Yf om (J, g) contains exactly those Ck-diffeomorphisms 'lfJOf ~om g onto ~omffor which f

O

'IIJ = g.

Then, of course, the category e0k becomes an Ehresmann groupoid. Two functions from the same class of equivalence will be called k-equivalent. For each positive integer r define erk as the full subcategory of e0k whose objects are functions from cr, that means real functions defined on open intervals of reals with continuous derivatives up to and including the order r. Furthermore, let e~ denote the full subcategory of erk whose objects are functions from C':+-,-that means objects from the category erk with nowhere vanishing first order derivatives. Let us recall that both erk and are again Ehresmann groupoids, see Proposition 2.4.1.

e!

142

.

7 CANONICAL FORMS

Theorem 7.4.3

Let k be a positive integer. Any set of global and unique canonical forms for the category ck_/: that satisfies the hereditary property Jf is formed by all restrictions to open interals I C IR of an arbitrary chosen (fixed) function X E c~ having f!Aan x = IR. Each such a restriction xl 1 is the unique representative of just one class of k-equivalent functions from C~ having the same range as xii• Proof

Let k > 1 be a fixed integer. Consider the category

c"; and its decomposition

ck_/: into classes of k-equivalent functions. Each of the classes,

B E er;, is umquely determined by the range, interval i(B) of its elements. Indeed, if two functions, f and g, belong to the same equivalent class B, then, according to Definition 7.4.2,

f

O

'IIJ = g

for a Ck-diffeomorphism 'IIJ of ~om g onto f!Aan f

=

~/J't

g =:

~om

i(B).

Conversely, if two objects f and g of the category

fE

C~,

g EC~,

o

(J l o g) =

er; have the same range, i. e.

!Ran f = PAan g ,

then there exists the inverse function J 1 to f, composition J 1 o g is well-defined. Evidently ; of ~om g onto ~om f, and f

f; hence

1

o

=

J

1 ,

and the g is a C -diffeomorphism

fJl/an

g

~om 11

g.

Hence functions f and g are k-equivalent. We can also observe that to each open interval I c IR there exists just one class B of k-equivalent functions from C~ such that i(B) = I. In fact, the identity function on interval /, id 1, is an element ◊f the class B. Now, any choice of a fixed function taken from each class B c ck_/:, independently from one class to another, may form a set of global and unique canonical forms for in the sense of Definition 7.1.2. However, the hereditary property Jf has not been yet considered. Under this condition the selection of a unique representative from B must be a suitable restriction of the unique representative of the unique class, say B 00 , consisting of all functions from whose range is the whole IR that means for which i(B 00 ) = IR. Chose arbitrarily an element, a function x from B 00 and define

er;

c\

143

7.4 HEREDITARY PROPERTY

Then, due to the hereditary property :Yf,

~(B) = {xlx-I(i(B))} for every BE

cf.

■

Remark 7.4.4 If the identity function idrR on IR is taken for x then, up to translation and possibly up to symmetry, we get the specification of the parameter used in the length parametrization of curves, employed in the Halphen canonical forms of linear differential equations or applied in Cartan's moving-frame-of-reference method. In fact, in all these cases, special parametrizations x are subject to the condition

x'

=

1 or

Ix' I

=

hence, up to translation,

1,

x

id or

x

+id.

Remark 7.4.5 If k+ -equivalence is introduced between objects, elements of C~, in such a manner that only increasing morphisms 1/fJin Definition 7.4.2 are admitted, that means that the morphisms are increasing Ck-diffeomorphisms, then all unique and global representatives of classes of k+ -equivalent functions from C~ that comply with the hereditary property :Yf are all restrictions to open intervals of two functions x1 and x2, x1 E C~ and x2 E C~, one of which is increasing and the other decreasing, xi . x2 < 0. For example, one may take idrR and - idrR. Such a situation may happen when reparametrizations of certain objects should keep the orientation.

Remark 7.4.6 One may think that our restriction to objects from the class C~ when considering k-equivalent functions is too restrictive. However, the next result shows that in general we cannot remove either the requirement on the order of differentiability being the same as the order of required equivalence, or the condition on non-vanishing first derivatives of the functions in question.

Theorem 7.4. 7 Let k be a positive integer. There does not exist a set of global and unique canonical forms for C';k satisfying the hereditary property :Yf if m < k. Neither there exists a set of global and unique canonical forms for ckk complying with :Yf.

144

7 CANONICAL FORMS

Proof

First consider the category c:k where m < k. The objects are functions from class Hence there exists- a function .f: (a, c) --+ ~ and a number b E E (a, c) ~ ~ such that ~(a,b) E Ck(a, b), ~(b,cl E Ck(b, c), f'(t) =I= 0 on (a, c), and f ¢: Ck(a, c); we may also suppose f E C - 1(a, c). Suppose that there exists a set fffi of global and unique canonical forms for c:\ m < k, satisfying property .Yf. Denote by 8, 8 1, and 8 2 the classes of k-equivalent functions containingf,~(a,b)' and~(b,c)' respectively. Since ~(a,b) E EC~ and ~(b,c) EC~, Theorem 7.4.3 guarantees the existence of a function x E - .~n x = ~, such that xl 1 I and 1 2 are unique representatives of the classes 8 1 and 8 2 determined by the conditions

c:.

xl

c\,

xl11 f!llan xl1i f!llan

= f!llan ~(a, b)' = f!llan -~(b, c)

.

Indeed, the assumptions of Theorem 7.4.3 are satisfied for the subset fffi 1 of fffi formed by functions from f!Jl of class since fffi I is a set of global and unique canonical forms for C~ and satisfies the-hereditary property as well as fffi. Put b == x- 1(f(b)). Due to the hereditary property .Yf and contunuity of X, 1 iu{6}ul2 is the unique representative of class 8. However, function! cannot be k-eqmvalent to the representative xl 11 u{6}ui2 that belongs to C~, because f $ Ck; that contradicts to our assumption on the existence of a set fffi of global and unique canonical forms for m < k, satisfying .Yf. Secondly, consider the category ckk and supose that there exists a set fffi of unique and global canonical forms for ckk satisfying .Yf. Take the function

c\,

xl

c:\

f:

~ --+ ~,

f(x) = x3 .

Evidently f E Ck(~) and is an object of the category ckk_ Both restrictions of it to ~+ and ~- belong to Denote by 8, 8 1, and 8 2 the classes of k-equivalent functions containing-!, ~R+' and ~R-' respectively. The subset fffi 1 of the set fffi formed by functions of fffi with non-vanishing first derivatives is a set of global and unique canonical forms for C~ satisfying again the hereditary property as fffi does. Hence Theorem 7.4.3 ensures the existence of a unique function x E C~, f!llan x = ~, such that

c\.

are unique representatives of classes 8 1 and 8 2, respectively, the intervals J 1 and J 2 being determined by the conditions

7.4 HEREDITARY PROPERTY

145

Put 6 = x- 1(0). Due to property :Y't and continuity of X, function x (on the whole ~) is the unique representative of class 8. However f(x) = x3 on ~ cannot be k-equivalent to x since k > 1, x' =fa 0 on ~, l/11 is a Ck-diffeomorphism of ~ onto lR according to Definition 7.4.2 and thus l/lJt) =fa 0 on ~, whereas f' vanishes at 0. That contradicts to our supposition on the existence of the set fl/i. ■

Remark 7.4.8. In fact, Theorem 7.4.7 shows that as long as the hereditary property .Yf is satisfied, the local character of unique canonical parametrizations cannot be avoided, and each method of this type gives only local results. However, it does not mean that we cannot enlarge class C~ of functions for which we can have a set of unique and global canonical forms if we do not require the hereditary condition :Yf. We can arrange it, for example, in the following way. First we specify a certain set fl/i of representatives and then we add other functions that are k-equivalent with functions from fl/i. We have only to choose functions in the set fl/i of representatives in such a manner that no two of them are k-equivalent, because fl/i should be the set of unique canonical forms. For example, put flt == flt 0 fl/i 0 == {id 1; I c ~ be open intervals}.

All k-equivalent functions with representatives from fl/i 0 form exactly the set C~. Enlarge the set fl/i 0 by all functions t

1---+

a

+

b sin t ,

tE(0,n)

for all a E ~, b E ~ + forming the set fl/i 1. Consider two functions from fl/i 1, t 1---+ a 1 + b 1 sin t and t 1---+ a 2 + b2 sin t, t E (0, n ). They are k-equivalent, k > 1, if there exists a Ck-diffeomorphism l/1 of (0, n) onto (0, n) such that a 1 + b 1 sinl/1 (t) = a 2 + b2 sin t on (0, n ). However, fort tending to 0 + we get a 1 = a 2 and then max {b 1 sin l/f(t)} = max {b 2 sin tE(O, 1l)

t}

tE(O, ?l)

gives b 1 = b 2 . Hence any two k-equivalent functions from fl/i 1 coincide. Moreover, neither of the functions form fl/i 1 is k-equivalent to a functions id 1 from fl/i 0 for k > 1, since the first derivative of each of the functions from fl/i 1 vanishes at n/2 whereas the derivative of id 1 vanishes nowhere. Hence

146

7 CANONICAL FORMS

may be considered as a set of unique and global representatives of the set of all k-equivalent functions to any of the function from the P/i. This set is larger than C~. Of course, the hereditary condition is not satisfied for this set P/i of representatives. Remark 7.4.9 As follows from Remark 7.4.4, Cartan's moving-frame-of-reference method and the Halphen canonical forms of linear differential equations depend on a canonical paramitrization of functions that, due to the fact that they satisfy the hereditary condition, cannot be global and at the same time unique, Theorem 7.4.7. Hence a natural question arises, whether we can construct a set of unique and global canonical forms for functions without the hereditary property :Yf. If we had such a set of unique and global canonical functions for c0• n+ 1 which, moreover, is closed under addition of constants, then we would be able to produce a set of unique and at the same time global canonical forms for linear differential equations from A~ 1 on the basis of the criterion of global equivalence, Theorem 5.2.1, by putting the ith power of the first derivatives of these representatives on the place of e; in the Halphen forms defined in 7.2.5. More precisely, we formulate the following open problems. Problem 7.4.10 For each integer n, find a set P/i of unique and global canonical forms for the category c0n, i. e., select from each class of n-equivalent continuous functions from c0 just one function, one representative of the class, one function in a special parametrization, one canonical function such that this selection @I satisfies the following Shifting property: If a function f belongs to the set fYi, then also f + c is a member of @i for any real constant c, that means that the set fYi is closed under addition of any constant. Remark 7.4.11 Of course, we know that for such a selection @i required in Problem 7.4.10 the hereditary property £ cannot be satisfied. For our purpose of global and at the same time unique canonical forms oflinear differential equations it would be sufficient to have a solution of Problem 7.4.10 for the category c 1n, a full subcategory of c0n restricted to the objects from C 1 only.

7.5 Global canonical forms: geometrical approach There have been several attempts to define canonical forms of linear differential equations, all of them known in the mathematical literature are described in the preceedings paragraphs. However, they give canonical forms which are not

7.5 GLOBAL CANONICAL FORMS: GEOMETRICAL APPROACH

147

global. Global canonical forms for the second order equations were obtained in the sixties by 0. Boruvka [2]. Then also global canonical forms for linear differential equations of an arbitrary order occured. In 1972 we introduced [11] a construction of certain global forms on the basis of the geometrical approach explained in paragraph 4.4. The present paragraph is devoted to this construction. For other global canonical forms we [24] made use of the criterion of global equivalence of linear differential equations of an arbitrary order, Theorem 5.2.1. These forms are described in the next paragraph. Construction 7. 5. l

Our geometrical approach to global transformations of linear differential equations is expressed in Theorem 4.4.1. On the basis of this theorem we obtain all fundamental solutions of all equations globally equivalent to a given equation Pn(Y, x; I} with the fundamental solution y: I ~ lR" as suitable sections of an centroaffine image of the fixed cone K,

in suitable parametrizations. The cone K is considered in n-dimensional vector space Vn, each fundamental solution is represented as a curve in the space whose coordinate functions are linearly independent solutions forming the fundamental solution and the independent variable is the parameter of the curve. By a suitable section of the cone AK and its suitable parametrization we mean a curve A . f(t). y(h(t))

where functionsf(determining the section) and h (choosing the parametrization) comply with requirements (1) and (2) of global transformations, Definition 3.1.1. Thus if we want to have a special, canonical equation in the class of equations globally equivalent to a given equation Pn(Y, x; I), we may specify a section of the cone K and a parametrization of the section. Any centroaffine image of this section with the same parametrization gives a curve whose coordinates are again linearly independent solutions of the same linear differential equation. We may do it in the following way. First we make the vector space Vn to be the euclidean space En, that means that we introduce the scalar product of two (column) vectors a and b from Vn as (matrix multiplication). Moreover, the (euclidean) norm a vector

aE

V,1'

lal is defined as (aT. a) 112 .

148

7 CANONICAL FORMS

Now, we take the central projection v of the curve y onto the unit sphere sn-1 := {a E E; lal = ] } C En, that means

v(x)

:=

y(x )/ly(x )I,

XE

I'

see Fig. 7. I. This is possible since jy(x )I -# 0 for all x E I, as it follows from the fact that the Wronskian determinat of the fundamental solution y is nonvanishing on the whole I and hence the first column, y(x ), of the Wronskian matrix of y is not the zero vector for any x E I.

0

Fig. 7.1

Moreover the vector function (the curve) v belongs to class en( I) because y E en(!) and IY I : J --+ IR, IYI E en(!). Hence 1/IYI E en(!) and is not vanishing. According to Proposition 2.6.1. the Wronskian determinant of v is non-vanishing. Let us mention that we would have the same curve v if we started from any curve f(x) . y(x ), f(x) > 0, on the cone K. In this sense we may say that we have specified the factor in the section v. Let us proceed to specify a parametrization of the curve v on the sphere Sn-I· From many other possibilities, see Theorem 7.4.3 and Remark 7.4.4, let us choose the length parametrization of v. That means that we are looking for a curve u,

u(t) = v(h(t)),

7.5 GLOBAL CANONICAL FORMS: GEOMETRICAL APPROACH

t

E

149

J, h(J) = I, such that lu'(t)I = 1

for all t E J. Of course, we need to check under what conditions it is possible. If such a reparametrization h: J ~ I exists then 1 = lu'(t)I = lv'(h(t))I . lh'(t)I, hence lh'·(t)I = 1/lv'(h(t))I

on J,

because lv'(x )I =I=- 0 due to the non-vanishing Wronskian determinant of v on the whole I. Thus there exists the inverse function k to h that satisfies lk'(x)I = lv'(x)I, that gives

f X

k(x) = +

lv'(s)I ds +

c .

xo

Hence, define k by this relation with arbitrary but fixed x 0 E I, c E IR and one of the signs, let x run through the whole interval I. This function k belongs to en(!), k'(x) =I=- 0 for all x EI. Put J == k(I) and h == k- 1• Then his a cn-diffeomorphism of J onto I. Hence actually we may define U := VO

h

so that u == J ~ sn- l

C

En ' and

lu'(t)I = lv'(h(t))I . lh'(t)I = 1 on J. Due to Proposition 2.6.1, the Wronskian determinant of u is non-vanishing on J. With respect to the fact that

u(t) = f(t). y(h(t)), f(t) = 1/ly(h(t) )I

t E J,

0 on J, f E cn(J), curve u is of class cn(J) with non-vanishing Wronskian determinant on J, hence its coordinates can be considered as an n-tuple of linearly independent solutions of the unique linear differential equation of the nth order, Qn(u, t; J). According to Definition 3.1.1, Qn(u, t; J) is globally equivalent to the original equation Pn(Y, x; 1). =I=-

150

7 CANONICAL FORMS

Since the above described construction requires no conditions on the smoothness of coefficients of the original equation P n(Y, x; I) and on interval of definition of the equation, it describes a selection [!Jig (of certain Representatives chosen by geometrical means) of special linear differential equations in each class of globally equivalent equations from An. In other words, fl//g is a set of global canonical forms for the category A of all linear differential equations of all orders n, n > 2, with respect to global transformations. An equation Qn(u, t; J) from the set [!Jig is completely characterized by the property that it admits an n-tuple of linearly independent solutions u = (u 1, ... , un)T: J ~ !Rn such that

lu(t )I = 1 and iu'(t )I = 1,

t

E

J.

We may sumarize the above considerations in the following theorem. Theorem 7.5.2

Any n-tuple y of linearly independent solutions of any linear differential equation P n(Y, x; I) E An can be globally transformed into an n-tuple v of linearly independent solutions of an equation from An satisfying

lv(x)I = 1 on I and, moreover, into an n-tuple u of linearly independent solutions of an equation Qn(u, t; J) E An complying with the relations

lu(t)I = 1 and lu'(t)I = 1 on J. The set [!J/ 9 of all equations Qn(u, t; J) constructed in this way for all equations Pn(Y, x; I) from An is a set of global canonical forms for An. Geometrically these global transformations are equivalent to the central projection of y, considered as a curve in n-dimensional euclidean space En, onto the unit sphere Sn-I that gives curve v, and to the length parametrization of the projection v to get curve u. Especially for the second order equations we have Corollary 7.5.3

Any couple y of linearly independent solutions of any second order linear differential equation from A2 can be globally transformed both to ( - sin t, cos t) ,

and to (sin t, cos t)

7.5 GLOBAL CANONICAL FORMS: GEOMETRICAL APPROACH

151

where t runs through a suitable interval J c [ij_ In other words, y: I -+ E2 can be globally transformed into an arc of the unit circle S 1 in the euclidean plane E2. This arc can be obtained as the central projection of the curve y onto S 1 and its length parametrization in either clock-wise or anticlock-wise orientation. Generalized Frenet Formulas, 7.5.4 Now we derive some formulas for solutions of equations belonging to the set f!J/g of global canonical forms obtained by Construction 7.5.1. Consider and equation Qn(u, t; J) E Pllg and its n-tuple of linearly independent solutions, the curve u: J -+ [ijn such that u E cn(J),

lu(t)I = 1 ,

lu'(t)I = 1 on J

and the Wronskian determinant w[u ](t) is non-vanishing on J. To each point u(t) of curve u we assign the orthonormal frame consisting of n mutually orthogonal unit vectors, u 1(t), ... , un(t ), defined in the following manner. For arbitrary, but fixed t E J, consider n-tuple of vectors

u(t), u'(t), ... , u(n-l)(t).

(7.5.1)

Due to the non-zero Wronskian determinant of u at t, these vectors are linearly independent. No we apply the Gram-Schmidt orthogonalization process to this ordered n-tuple of vectors to obtain another n-tuple of mutually orthogonal unit vectors (7.5.2) This process consists of identifying u 1( t) with the unit vector parallel to the first vectorof(7.5.l),inourcase,infact, u1(t)==u(t), since lu(t)I = 1 according to our assumption. The second vector, u2 (t ), is chosen as a linear combination of the first two vectors from (7. 5.1 ), u( t) and u' (t ), that is orthogonal to u 1( t) and of the unit norm. In general, having u 1(t), ... , ui(t) from (7.5.2), the vector ui+ 1(t) is a linear combination of the first i + 1 vectors from (7.5.1 ),

u(t), u'(t), ... , u(i)(t), that is unit and orthogonal to the i-dimensional space spanning the vectors u(t), u'(t), ... , u(i-l)(t) that is in fact the same as the vector space of linear combinations of the vectors u 1(t), u 2 (t), ... , ui(t).

152

7 CANONICAL FORMS

For the orthonormal frame (7.5.2) considered for t E J, we may write the following system of linear differential equations, called also Generalized Frenet Formulas:

u; =

u'n

=

(7.5.3)

where ki

E

cn-i-- 1(1), ki(t) -# 0 on J for

i . 1, ... , n - 2,

ui E cn-i+l(J) for

i = 1, ... , n. We may always choose p06itive ki by suitable orientation of U;.

Indeed, since u1(t) = u(t), we have u 1 E cn(J),

lu~(t)I = lu'(t)I =

1;

hence we may put ui(t) == u~ (t) to get the first of the relations in (7.5.3 ). Then u2 E cn- l(J),

u2(t) = u{'(t) = u"(t). Then u" can be expressed as a linear combination of u1(t), ui(t), and n > 2) u3 (t),

(if

U2(t) = C21(t) U1(t) + C22(t) U2(t) + C2 3 (t) U3(t). Due to orthonormality of u 1 E cn(J) and u2 following relations for the scalar products:

E

cn- t (1), we may write the

and

Hence -1

smce u'1

'

153

7.5 GLOBAL CANONICAL FORMS: GEOMETRICAL APPROACH

and

If n > 2, we obtain

that gives

lc 23 (t)I = lu 2(t) + u1(t)I. Since U2 cn- 1(1) and some t 0 E J, then

U1 E

cn(J), we have

C23 E

cn- 2(1). If c23(to) = 0 for

that means

and this contradicts to the non-vanishing Wronskian determinant of u at t0 for n > 2. Hence c23 (t) =I= 0 on J, and due to continuity of c23 on J and the possibility of choosing orientation of u3, there is

By putting k 1 == c23 we get the second relation in (7.5.3 ). Suppose that the firstj relations of (7.5.3) are valid, j < n, ki ki > 0 on J for i = l, ... , j - l. Since

u1 = -k1_ 2(t) u1_ 1

+

E

cn-i- 1(J),

k1_ 1(t) u1+ 1 ,

kj-2 E cn-j+l(J) c C 2(J), kj-1 E cn-f(J) c C 1(J), k1-1(t) > 0 on J, uj E 2 3 E cn-j+l(J) c C 2(J) , u.1-l E cn-1+ c C (J) , u~1+1 can be expressed as a linear combination of ~1_ 1, u1_1, u'1, and u1~U_sing the first (j - 1) relations of (7.5.3) where k; E cn-i- 1(1) and ui E cn-t+ 1(1), we can express u;' as a linear combination of the derivatives of u 1 of orders < j + l. If j + l = n, then u\n) is a linear combination of u 1, u1, ... , u\n-l) because u = u 1 and each component of u satisfies the linear differential equation of the nth order, Qn(u, t; J). Hence UJ+l = u~ = cn 1(t)u 1

+ ... +

cnn(t)un.

If j + l < n, then each derivative of u 1 or u of order < j combination of u 1, u2, ... , u1+ 2·

+

l is a linear

154

7 CANONICAL FORMS

Hence u;+l

= cj+l,1(t)

U1

+ ... +

cn-i+ 1(J) c c 1(J) for i for i -=I= m, we have

U; E

cj+l,j+it)

U_;+2 ·

= 1, ... , j + 1, and [ U; . uJ = 1, [ U;.

um]

= 0

and

For m < j

+

1 we already have

cm,j+l(t) = 0 If j

+

for

m

2 be an integer. Consider equation Qn(u, t; 1) from An whose n-tuple u of linearly independent solutions satisfies

lu(t)I = 1 and lu'(t)I = 1,

t E 1.

There exists an orthonormal moving frame u 1, ... ,unattached to u, u', ... , u(n-l) by the Gram-Schmidt orthogonalization process such that the system (7.5.3) is satisfied in w~ich U1 = u, ki: 1 --+ IR, k; E cn-i- 1(1), ki(t) > 0 on 1, ui: 1 --+ !Rn, Ui E cn-1+

1(1).

On the basis of this theorem it can be proved the following theorem describing the explicit construction of equations from the set Pllg of global canonical forms for A defined by Construction 7.5.1 in Theorem 7.5.2. Theorem 7.5.6 Let n > 2 be an integer. The set Pllg of global canonical forms for linear differential equations of the nth order, An, n > 2, is formed by all equations Qn(u, t; J) whose coefficients depend on (n - 2) arbitrary positive functions ki 1 E cn-i- (1), i = l, ... , n - 2, defined on arbitrary intervals 1 c IR obtained as coefficients of the corresponding derivatives of u 1 when eliminating all other vectors u2 , ... , un from Generalized Frenet Formulas (7.5.3). Proof

Consider equation Qn(u, t; 1) E Pllg. According to Theorems 7.5.2 and 7.5.5, assign a system (7.5.3) to the equation. Due to sufficient differentiability of k; E cn-i- 1(1) and u; E cn-i+ 1(1), and because of the special form of system (7.5.3) in which all ki(t) #- 0 on 1, it is possible to eliminate all vectors U; for i = 2, ... , n from (7.5.3) to get one relation on Jin which u{n) is expressed as a linear combination of u 1, u1, ... , u{n-l) with continuous coefficients formed from certain admissible derivatives of (n - 2) functions k 1, ••• , kn_ 2 . Since u = u 1 is formed by n-tuple of linearly independent solutions of Qn(u, t; J), the coefficients in the relation expressing the linear dependence of u 1, u1, ... , u{n) on J normalized by the unit coefficient of uln) are exactly the coefficients of the corresponding derivatives of u in the equation Qn(u, t; J). ■ Corollary 7.5.7 For the set of second order linear differential equations, A2, we get the following global canonical forms in {!Jig: u"

+u =

0

on (different) J

c

IR.

156

7 CANONICAL FORMS

For the third order linear differenatial equations, A3, there are the following global canonical equations in rF/g: k'(t) 1 u"' - - - u" k 1 (t)

+

k'(t)

+ kf(t)) u' - - 1- u = k 1 (t)

(1

for arbitrary intervals J on J.

c

~,

0

arbitrary functions k 1 E C1(J) such that k 1(t) > 0,

Proof

According to Theorem 7.5.6, for the second order equations we have to consider the following system (7.5.3) u")...

on

J

c

~-

Hence

that gives u" + u = 0

on

J

c

~ .

For the third order equations, the corresponding system (7.5.3) reads as follows: I

=

U1

U2

+k 1(t) u3

u2 = -u 1 U3 = k1

E

C

1

-k1(t) U2 '

(J), k1 (t) >

0 on J,

U1 E

C 3(J),

Hence u3 = ((u 2 + u 1)/k 1)' = -k 1u 2 that gives

and

U2 E

C 2 (J),

U3 E

C 1(1).

7.5 GLOBAL CANONICAL FORMS: GEOMETRICAL APPROACH

157

Remark 7.5.8 We have seen in Theorem 7.5.6 that the global canonical forms from f1,ll for the nth order equations from An depend on n - 2 rather arbitrary fu~ctions, k 1, ... , kn_ 2 (of course, also on the interval of definition J, however, the interval of definition of a function is always attached to the function in our considerations without explicit mentioning it). Let us notice here that we cannot expect any better choice of global canonical forms with respect to the number of arbitrary functions occurring in their coefficients since any linear differential equation P n(Y, x; I) of the nth order from A 11 generally depends on n functions, its coefficients p0 , ... , p 11 _ 1 that are arbitrary continuous functions from c0 (!) for an arbitrary interval / c !R1, and any global transformation

(Af, h)y depends on two rather arbitrary functions, f and h (a matrix A does not change the equation obtained after the transformation). Hence only two conditions may be imposed on the form of canonical equations that can decrease the number of arbitrary functions by two, that means on n - 2 that was really achieved in our construction.

Remark 7.5.9 In spite of the fact that we have the minimum of arbitrary functions in the form of global canonical equations from [1,/lg, the set [1,llg is not a set of unique canonical forms. Even for the second order equations we may immediately observed that, e. g.

u"

+

u = 0

on

(0, n/3)

is globally equivalent to

u"

+

u = 0

on

(0, n/2) ,

since the equations are of the same type and same kind, Theorem 5.1.2. The reason for that can be explained as follows. Cartan ·s moving-frame-of-reference method gives special sections on cones Kin the vector space V11 intrinsically with respect to the whole centroaffine group GIL n of transformations of Vn and hence it introduces special and unique linear differential equations, in fact, the Halphen canonical equations. It complies with the hereditary property .Ye, and thus essentially the method is not global and the corresponding canonical equations are not global. In our approach we consider the cones and their sections in the euclidean space En. Special sections satisfying Generalized Frenet formulas exist without any restriction on smoothness of coefficients or on intervals of definition of

158

7 CANONICAL FORMS

linear differential equations, hence the corresponding canonical equations are global. However, these section-; :ire chosen intrinsically only with respect to the orthogonal group On of tran-,1: ,rmations, thus in general if u corresponds toy in our Construction 7.5.1, then Ou corresponds to Oy for any OE On and system (7.5.3) is the same for u and for Ou, but Ou need not correspond to Ay for A E ![;[l_n and hence the corresponding systems (7.5.3) need not be the same if we start from y and from Ay. In fact, let u1, ... , un be another orthonormal frame satisfying the same system (7.5.3) as u 1, ... , un- Define the n by n matrices n2 d n2 U: J -+ ~ an U: J -+ ~ , UT: =

(u 1,

... ,

un) and

OT== (u 1~ ... , unJ.

Evidently

U' = K(t) U and 0' = K(t) 0 on J where K is the matrix of system (7.5.3). Hence

(uTO)' = uT,a + uTa, = uT KT(t) a + uT K(t) a = uT (KT(t) + K(t)) 0 = 0 (null matrix), since K(t) is antisymmetric, KT(t) = -K(t). We obtain

uT(t) . O(t) = o 0 being a constant n by n matrix. Both U(t) and O(t) are orthogonal matrices for each t E J, hence

where O = UT(t 0 ) O(t 0 ) is an orthogonal matrix. That gives

u 1(t) = 0 . u 11t) for all t E J and some OE On. Conversely, if f!T(t) = oOT(t) for a constant matrix OE On, and U' = K(t) U, then (U(t))' = (U(t)O)' = U'(t) 0 = K(t) U(t) O = K(t) O(t). Hence system (7.5.3) determines the orthonormal moving frame u 1, ... , un up to a fixed orthogonal transformation uniquely. Finally, the central projection of y onto the unit sphere Sn- l in En is given by y/lYI = y/[y. y] 112 =: v, and the central projection of Oy, 0 E On, is

Oy/[ Oy . Oy Jl/2 = Oy/(yToT . Oy)l/2 = Oy/(yT y)l/2 = Ov.

159

7.6 GLOBAL CANONICAL FORMS: ANALYTIC APPROACH

7.6 Global canonical forms: analytic approach There is also another possibility how to obtain sets of global canonical forms for linear differential equations of an arbitrary order. This approach is based on the criterion of global equivalence of these equations described in Theorems 5.1.2 and 5.2.1. Definition 7.6.1 1

Let n > 2 be a positive integer. For a fixed function q E C: (1 0 ) define 0/a, n(q) to be the set of all linear differential equations of the nth order from A0II whose first three coefficients form the sequence

where J is an arbitrary open subinterval of J 0, other coefficients being arbitrary continuous functions on J. Theorem 7.6.2 Let n > 2 be a positive integer and q belonging· to second order linear differenatial equation

cn- 2(1 0 )

be such that the

(7.6.1) is both-side oscillatory. Then 0/a, n(ii) is a set of global canonical forms for linear differential equations of the nth order from A~, i. e.,for equations P n(Y, x; I) whose first three coefficients form the sequence 1,

Pn-1 E cn-1(1), Pn-2 E cn-2(1)

for arbitrary I c lR. Even a subset 01!, n(ii) of 0/a, n(ii) is a set of global canonical forms for A~, when intervals of definition J of equations from 01~, n(q) run only through a countable set of open subintervals of J O for which the equations (7.6.1) restricted on J achieve each of (countable many) types and each kind (two possibilities) defined in 5.1.1. Note 7.6.3 We use notation t?li a, n(ii) or t?Ji*a, n(q) with the subscript a for sets of global I representatives of linear differential equations from An that are obtained by the analytic method described here to distinguish there sets from the set t?li 9 of global representatives derived by the geometrical approach in the preceding paragraph.

160

7 CANONICAL FORMS

Proof of Theorem 7.6.2 ([24]) Let q satisfy the assumptions of the theorem. Due to Proposition 4.2.6, each equation from A~ can be globally transformed into an equation P n(Y, x; I) from A~ 1 whose first three coefficients are

Pn- l = 0,

1,

Pn-2

E cn-2(1) .

According to the criterion of global equivalence given in Theorems 5.1.2 (n = = 2) and 5.2.1 (n > 2), P,z{.v, x; I) can be globally transformed into an equation Qn(z, t; J) E A~ 1 with iin- 2 E c 11 - 2(J) if the second order equations

v"

+ q,, _2 ( t) v/ ( n ;

1 )

(7.6.2)

= 0 on J .

and

u" + p,,_ (x)u/(n; 2

1 ) = 0

(7.6.3)

on/

are globally equivalent, or equivalently, if they are of the same type and kind. Since equation (7.6. 1) is both-side oscillatory on J O then, due to Theorem 5.1.5, there exists a restriction of q to J c J O such that Eqn. (7.6.2) with ij 11 _ 2 := iil 1 is of the same type and kind as Eqn. (7.6.3) and the equation Qn(z, t; J), defined on the interval J, belongs to the set Bf:, 11 (q). Hence we have shown that each equation from A~ can be globally transformed into an equation from n (q). With respect to Definition 7.1.2 it means that n (ii/ as well as Bf a, n (q), containing n (q), are sets of global canonical forms for A11 • ■

Bf:,

Bf:,

Bf:,

Corollary 7.6.4

The set of all linear differential equations of all orders n, n > 2, of the form y(n)

+

y(n-2)

+

rn_3(x) y(n-3)

on arbitrary open intervals I

c

+ ... + ro(x)

y = 0

IR, with arbitrary functions ri

E

c 0 (J),

is a set of global canonical forms for A 1. Proof follows from Theorem 7.6.2 ifwe put n > 2. Indeed, the equation

q:

IR~ IR,

q(x)

=

1 on J 0 = IR for each

7.6 GLOBAL CANONICAL FORMS: ANALYTIC APPROACH

161

is both-side oscillatory on IR for any integer n, hence 00

is the set of equations of arbitrary orders characterized by its first three coefficients forming the sequence

1,

0,

1.

■

Remark 7.6.5 Since the Laguerre-Forsyth canonical forms are characterized by the first three coefficients giving the sequence

1,

0,

0

we can see that if Laguerre and Forsyth had taken 1 instead of their O as the coefficient of the (n - 2)nd derivative of the dependent variable they would have obtained global canonical forms instead of their local ones.

Remark 7.6.6 Let us note that for each n > 2 global canonical equations in both sets, Bf a, n(q) and Bf:, n(q), depend on (n - 2) functions, coefficients of the derivatives of yup to and including the (n - 3)rd order, the coefficient of the (n - 2)nd derivative, q, is fixed (up to suitable restrictions). Again we observe the same minimal number, n - 2, of arbitrary functions in the canonical forms, see Remark 7.5.8. In fact, this is the correct formulation, namely, that a special, canonical form of linear differential equations of the nth order with respect to global transformations depend on n - 2 arbitrary functions, by two arbitrary functions less than the number of arbitrary coefficients of an equation in a general form. These functions occur in formulas for coefficients of the canonical equations, see Bfg, or occur as coefficients that happened for Bf a, n· However, it does not imply that two coefficients may vanish.

Corollary 7.6.7 Let n > 2 be an integer. The set of linear differential equations of the nth order characterized by the first three coefficients:

1, 0,

( n ; 1)'

162

7 CANONICAL FORMS

the other coefficients being arbitrary continuous functions on the following countable set J of intervals of definition: (0,n/2), (0, n), (0, 3n/2), (0, 2n), (0, 5n/2), ... , and

(0, oo), (-oo, oo), is a set of global canonical forms for A~. Proof

follows from Theorem 7.6.2 if we put

31)

q(x) = ( n +

1 0 = IR,

for all x E IR. According to Theorem 5.1.2, u" + u = 0 achieves each possible type and kind when J runs through the set J. Hence·

is a set of canonical forms for A~.

■

Remark 7.6.8 In spite of a local character of the Halphen canonical forms and Cartan's moving-frame-of-reference method giving the same canonical forms, these results and methods together with the criterion of global equivalence enable us in better understanding the role of the hereditary condition Jf in a special parametrization of functions. And similarly the fact that the Laguerre-Forsyth canonical forms are not global led first to applying the criterion to discover exactly which equations can be globally transformed into the Laguerre-Forsyth forms and then to find global canonical forms by using the criterion of global equivalence.

7. 7 List of canonical forms for the second and third order equations Here let us give a survey of old and new canonical forms of the second and third order linear differential equations with their characterization. If nothing else is said, I c IR means that I runs through all open intervals of reals,· r E c 0 (J) expresses that r runs through all functions in c0 (I).

y" = 0

on

I

c

IR;

the set of the Laguerre-Forsyth canonical forms, they are not global for A2, neither for A~w. y"

+

y = 0

on

I

c

IR;

163

7.7 LIST OF CANONICAL FORMS ...

the set of global canonical forms for A 2; these forms were considered by 0. Boruvka, they are obtained by Cartan 's moving-frame-of-reference method, as well as by geometrical and also by analytical approaches explained in this chapter; even only y" + y = 0 on the countable set of intervals (- oo, oo ), (0, oo) and (0, mn/2)

for m

E

N;

is the set of global and unique canonical forms for all second order equations, A 2, already introduced by 0. Boruvka. y"' + r(x) y = 0 on I c lR, r E c0 (I); the set of the Laguerre-Forsyth canonical forms, they are not global for A 3, neither for A~w.

y

111

+ 4p(x)y' + 2p'(x)y + y =

0,

and

y

111

+ 4p(x) y' + 2p'(x) y =

I

on

0

lR, arbitrary p E ct(!);

c

the set of the Halphen canonical forms, equations obtained by Cartan's method as well, they are not global for A 3, neither for A~w. Y

k'(x)

111 -

- -

k(x)

y"

+ (1 +

k'(x)

(k(x))2) y' - -

k(x)

y = 0

on

I

c

lR,

arbitrary k

E

ct(!), k > O·

'

the set of global canonical forms for A 3. For a fixed p0 E C 1 (I 0 ) such that y" + p0 (x) y = 0 is both-side oscillatory on I 0 :

y

111

+ 4p 0(x) y' + r(x) y =

0

on

arbitrary I

c

I O and arbitrary

r E

C0 (I);

the set of global canonical forms for Aj; in particular for I 0 == lR, Po == 1/4,

y

111

+ y' + r(x) y =

0

on

I

c

lR, r

E

c0 (I);

is the set of global canonical forms for Aj; even only

y

111

+ y' + r(x) y =

0

(0, mn) for m

E

on the countable set of intervals ( - oo, oo ), (0, oo ), and

N with r running through continuous functions there; is the set of global canonical forms for A j.

8 Invariants

Invariants and covariants of linear differential equations with respect to transformations have been studied from the early beginning of consideration of transformations in papers of Laguerre, Brioschi and many others till the present time. Constructions of invariants and covariants were mostly connected with considerations leading to the Halphen canonical forms or were directly derived from the Halphen forms. However, these constructions satisfy the hereditary condition£ and hence they are all essentially local. We illustrate this approach in paragraphs 8.2 and 8.3 where we give also quotations of some papers devoted to this study even from the recent period. Fortunately together with global investigation of transformations of linear differential equations that started in the fifties, also certain types of global invariants have been discovered. We present these results in paragraphs 8.4 and 8.5.

8.1 Notion of invariant and covariant First, let us give some definitions, since some notions have sometimes different meanings in the literature. Consider and Ehresmann groupoid A being a collection of Brandt groupoids Bi, i EM, each of the groupoids consisting of mutually equivalent objects of A with respect to their morphisms. Definition 8.1.1

A non-constant mapping 'P defined on the set of objects of A with values in a set X 'P: A

~

X ,

such that 'P restricted to B; is a constant mapping for any fixed i

'P(P) = 'P(Q)

if

P

E

Bi

and

Q

E

Bi

for the same i

E

E

M,

M, i. e.,

166

8 INVARIANTS

is called an invariant mapping, or simply an invariant on the Ehresmann groupoid A (of course, with respect to the morphisms of the groupoid). A collection 'I' of invariant mappings 'Pj, j E L, ('1' is, in fact, again an invariant mapping) is called complete, if for two different Brandt groupoids Bi and Bi' of A i =I- i', there exists j E L such that

'P/BJ

=I-

'P/Bi') , i.

e.,

'l'(Bi) =I- 'l'(Bi,) for i =I-

i' .

Now, let A* be a full subcategory of the category of all linear homogeneous differential equations of an arbitrary order n, n > 2, as objects and global transformations as morphisms. Then this category A* is an Ehresmann groupoid, see Theorem 3.3.9 and Proposition 2.4.1 ~ An invariant mapping on the set A* in sense of Definition 8.1.1 will be called a global invariant on A*. Let x be a mapping that assigns a real function x[P] E c 0(/k), x[P]: Ik-+ IR, an integer k > 1, to each equation P = Pn(Y, x; I) from the full subcategory A01 of A.

Definition 8.1.2 The mapping xis said to be a covariant mapping, or simply, a covariant on A01 if (8.1.1)

(t 1, ... , tk) E Jk, for equations P = Pn(Y, x; I) and Q = Qn(z, t; J) from the subcategory A01 such that P is globally transformable into Q by means of the transformation

If k = 1 and, instead of relation (8.1.1 ), the following relation is satisfied for m E Z,

x[P] (h(t)). (h'(t)r = x[Q] (t),

t

E J

(8.1.2)

then X is called a relative covariant mapping, or a relative covariant of dimension m.

Remark 8.1.3 The notions of invariant and absolute invariant are also used for covariants as ' well as canonical, or fundamental, or relative invariants stand for relative covariants in the mathematical literature.

167

8.1 NOTION OF INVARIANT AND COVARIANT

Some authors also require for covariants and relative covariants a special form of the mapping X, namely x is supposed to be a function of coefficients and some of their derivatives taken from equations under consideration, see e. g. E. J. Wilczynski [l The notion of invariant and covariant will be also used for other objects, e.g. curves in Chapter 11.

J.

8.2 Covariants Let us consider the class A~ 1 of linear differential equations for some integer n, 1 n > 3. According to Remark 4.3.12, each equation P = P n(Y, x; I) from A~ can be written in the form (8.2.1) p E

cn- 2(1), 3: A~ 1

3[ P]

r; E

--+

c0 (1)

for i

= 0, ... , n

- 3. Define the mapping 3

Co ,

== r n-]

E

0

C (1)

where the function r n- 3 is the coefficient of y(n- 3 ) in (8.2.1 ). Moreover, put

f X

0[P] (x, x0 ) •=

(rn_ 3(s)) 113

ds.

xo

Theorem 8.2.1 For any integer n, n > 3, the mapping 3 is a relative covariant of dimension 3 on 1 A~ 1, and e is a covariant on A~ •

Proof

If two equations P and Q from A~ 1 are globally equivalent, i. e., P * ex = Q for some ex of the form (8.2.2)

where P is written in the form (8.2.1) and

Q

=

l(q)[n] (z, t; 1)1

+

sn_it) z(n- 3 )

+ ... +

s0 (t) z = 0

on J,

168

8 INVARIANTS

then the functions rn_ 3 Theorem 5.2.1, and

c0 (I)

E

and sn_ 3 E

c0 (J)

satisfy the condition (iii) of

That means that

9[P] (h(t)). (h'(t)) 3 = 9[Q] (t) on J, and 9 is a relative covariant of dimension 3. Then, of course, h(t

(r._ 3 )(s)) 113 2. In fact, each equation Pn from A 1 can be globally transformed into an equation Pn(Y, x; I) from A01 . And

_

P n(Y, x;

I) = Y(n) + Pn-2(x)

2

Y(n- )

can be written in the form

Then the following theorem holds.

+ ··· + Po(x) Y

O,

173

8.4 GLOBAL INVARIANTS

Theorem 8.4.2 Each of the mappings that assign to equation the equation

Pn from

A~ the kind or the type of

in the sense of Definition 5.1.1 is a global invariant mapping on A~for every n > 2.

Proof follows immediately from condition (ii) of Theorem 5.2.1.

■

Remark 8.4.3 Theorem 8.4.2 does not provide a complete global invariant on A~ for n > 3 as Theorem 8.4.1 does for A 2. The reason is in the fact that condition (ii) of Theorem 5.2.1 is not sufficient for global equivalence of equations of the third and higher orders if both of the equations are not iterative.

Remark 8.4.4 Let us note that contrary to the form of invariants, or better covariants in our terminology, considered in the last century, the values of which having been expressed as certain functions, the values of the global invariant mappings in Theorems 8.4.1 and 8.4.2 are positive integers, in finite type case, or the infinitive type, and one of two kinds: general or special and one-side or both-side oscillatory. Another sort of invariants is given in the next paragraph. They concern the order of smoothness of coefficients of the studied equations. Remark 8.4.5 On the basis of the Kummer equation and Proposition 4.1.2 it can also be shown that the mappings given in Theorem 8.4.2 are global invariants even for class A~.

8.5 Smoothness of coefficients as an invariant Each second order linear differential equation y"

+

p 1(x) y'

+

p0 (x) y

= 0

with real (and possibly only) continuous coefficients Po and p 1 on an open interval J of the reals can be globally transformed on its whole interval of definition into another equation of the same form with real analytic coefficients, in particular into the equation

z"

+z=

0

on a suitable interval J

c

lR, as follows from Corollary 7.5. 7.

174

8 INVARIANTS

A natural question arises whether for a given linear differential equation of the nth order, n > 2, a global transformation exists that converts this equation into an equation with more regular coefficients, e. g., belonging to Ck for some k > 0, or k = oo (infinitely differentiable functions), or even k = w (real analytic functions). Such kind of results would be very desirable especially in view of the form of the criterion of global equivalence, 5.2.1, and consequently the results concerning stationary groups that require a certain order of differentiability of coefficients of considered equations of higher orders. The situation described above for the second order equations is of course correct, but a bit misleading. We shall prove in this paragraph that e. g., the order of differentiability of the coeffcient Pn-f of y(n-1) in an equation of the nth order cannot be improved if p 11 _ 1 E ck \ C + 1 and k < n - 2, whereas if k > n - 2 then there exists a suitable global transformation that the corresponding transformed coefficient is even real analytic. Evidently, this critical case k = n - 2 occurs for continuous coefficients, k = 0, when second order equations are considered, n = 2. In what follows, Theorem 3.2.3 plays an important role because it guarrantees a certain order of differentiability of functions occurring in any global transformation between any single pair of equations of the nth order. First, let us derive certain auxiliary results.

Lemma 8.5.1 Let n > 2 be an integer, and let the following relation be satisfied

z(t) = f(t) y(h(t)),

(8.5.1)

lvhere real functions y: I ~ [ij, z: J ~ [ij belong to classes C"(I), C"(J), respectively, and f: J ~ [ij, f E C'(J), f(t) =/. 0 on J, and his a C'-diffeomorphism of J onto I for some integer r > n. Then

y(x) = Aoo2(g(x)), y'(x) = A 1o2(g(x))

+

A11 z'(g(x)),

y"(x) = A20z(g(x))

+

A21 z'(g(x))

+

A22 z"(g(x)),

175

8.5 SMOOTHNESS OF COEFFICIENTS AS AN INVARIANT

where g is the inverse function to h and Au, 0 < i < n, 0 < j < i, are rational expressions inf and hand their derivatives such that the real functions x ~ Ai .(x) are of class cr-(i-j)- 1(!) for j > 0, and cr-i(I) for j = 0. Moreover, at ~ost the order i - j of derivatives off occurs in Au, and Aii(x) =I- 0 for all x E / and all i, 0 < i < n. Proof

From relation (8.5.1) we have y(x) = (1/f(g(x))) z(g(x)). Thus A 00 = 1/f(g) E er(!) = cr-i(I) for i = 0 and j = 0. Also A 00 (x) #- 0 on I and A 00 contains no derivatives off Further,

E

y'(x) = A 00 (x) z(g(x)) + A 00 (x) g'(x) z'(g(x)), that is

y'(x) = A 10 (x) z(g(x)) + A 11 (x) z'(g(x)) where Aw All

E

E

cr- 1(!)

cr- 1(1)

= cr-i(J) for i = 1 and j = 0, and

= cr-(i-j)-l

for

i

= 1 and

j

= 1.

Also A 11 (x) #- 0 on I and A 1i contain derivatives off of orders< 1 - i, i = 0,1. Suppose by induction that

Au E cr-(i-j)- 1(!) for j > 0, AiO E cr-i(J), Aii(x) =I- 0 on/, and Au contains no derivatives off of orders > i - j, 0 < j < i < n. Then y(i+ l) = A;0 z(g)

+ (Am g' +

A;1) z'(g) +

(Ailg'

+ A;2 ) z"(g) + ...

+ (Ai,j- 1g' + A;j) zU)(g) + (Aijg' + A;,j+ i) zU+ 1l(g) + + ...

+ (Ai,i-lg' + A;J z(i)(g) + Aug'

z(i+l)(g)

= A;+1,oz(g) + A;+1,1 z'(g) + ···

... + Ai+l,j zU)(g) +

Ai+l,j+l z(j+l)(g)

··· + Ai+l,i z(i)(g) +

Ai+l,i+l z(i+l)(g) ·

+ ...

=

176

8 INVARIANTS

Evidently A i+ 1, 0

E cr-(i+l)(J)

'

and for1·, 0 < 1· < i, we have

Ai+l,J E cr-(i-(J-1))-1(/) n cr-1(1) n cr-(i-J)-2 = cr-(i-J)-2(1) =

= C' - (i + 1 - J)- 1( J) . Moreover, A;+ t,J contains derivatives of f of orders less than or equal to max {i - (j - 1), i - j + 1} = (i + 1) - j. Finally, A;+t,i+ 1(x) = Aii(x) g'(x) # 0 on I. ■ Lemma 8.5.2

Let n and r be integers, 2 < n < r. Let an equation P n(Y, x; I)

be globally transformable into an equation Qn(z, t; J) Qn(z, t,. J) -_

Z (n)

+ qn-1 ( t )

Z (n-1)

E

E

An,

An,

+ ... + qo (t ) Z -- O ,

by means of a transformation ( Ef, h )y where functions f and h satisfy the assumptions in Lemma 8.5.1. Then the coefficients qi are expressible in the following way: (qn =

1)

qn-1(t) = Bn-1,n(t) + Bn-1,n-l(t) Pn-1(h(t)), qn-2(t) = Bn-2, n(t)

qi(t)

Bn-2, n-1 (t) Pn-1 (h(t))

+

Bn-2, n-2(t) Pn-2 (h(t)) ,

= Bi,n(t) + Bi,n-1(t) Pn-1(h(t)) + Bi,n-2(t) Pn-2(h(t)) +

... +

qo(t)

+

BiJ(t) p1(h(t))

+ ... +

B;;(t) P;(h(t)) ,

= B0n(t) + Bo,n-i(t) Pn_ 1(h(t)) + ... + BiJ(t) p/h(t)) +

... + B00 (t)

p 0 (h(t)),

whe~e. t f---+ B;/t) for O < i < n - ~ and i < j < n are functions of class cr+i-1 - 1(J)for i > 0 andofclassc'-1(J)for i = 0. Moreover, B;;(t) # 0 onJ for all i = 0, ... , n - 1, and BiJ are rational expressions inf, hand their derivatives, the derivatives off being at most of the order j - i.

177

8.5 SMOOTHNESS OF COEFFICIENTS AS AN INVARIANT

Proof

Using the notation of Lemma 8.5.1, we may write n

Y(n)(x) + Pn- l (x) y(n-l)(x) + ... + p0(x) y(x) =

L p)x) yU)(x)

=

j=O n

j

LL

pAx)Aj;(x)z(i)(g(x)).

j=Oi=O

Hence the coefficient q;(t) of z(i) in Qn(z, t; 1) is n

q;(t) = (Ann(h(t)))-l

L Aj;(h(t)) Pih(t)). j=i

Thus

For i > 0, Aii belongs to C_r-_(i-i)- 1(/). Since h E cr(l), r > n, ~nd Ann(x) # 0 on I, we _have Bu E cr-(J-r)- 1(1). Furthermore, A.0 E cr-1(!) and hence B0i E cr-1(1) for j = 0, ... , n. Finally, B;;(t) = A;;(h(t) )/ Ann(h(t)) # 0 for tEl. ■ Theorem 8.5.3

Let n > 2 be a positive integer and let equation P n(Y, x; I) from An be globally transformable into an equation Qn(z, t; 1) E An. If

for some k, 0 < k < n - 3, then

that means that this kind of smoothness is an invariant property with respect to global transformations; in other words, the validity of the above relation is a global invariant on A11 whenever O < < k < n - 3. Proof

Suppose that Pn(Y, x; I) E An is globally transformable into Q11 (z, t; 1) by means of (Af, h)y- According to Theorem 3.2.3, f E C 11 (l), f(t) # 0 on 1, h E C11 (l),

178

8 INVARIANTS

and h'(t) # 0 on J. Hence f'/fE cn- 1(1) and h"/h' E cn- 2 (1). If Pn-1 for some O < k < n - 3, then Proposition 4.1.2 ensures that qn-l = Pn-J(h) h' - nf'/f -

C)

E

ck(J)

h"/h'

is ot class cs(JJ where s = min {k, n - 2} = k. However, if Pn'- 1 E ck+ then qn-l ¢ C + 1(1), because for qn-l E ck+ 1(J) we would get Pn- 1(x)

= (h' (t) J- 1 (qn-1 (t) + n f'(t )/f(t) +

belonging to class cr(J), r = min {k to our assumption. ■

+

C)

1, n - 2} = k

+

1

(1)

h"(t )/h'(t) )1,-h-l(x) 1, that would contradict

Theorem 8.5.4

Let equation P n(Y, x; I) E An be given, n > 2. If Pn- l E Ck(J) for some k > n - 2, then there exists a global transformation ( Ef, h )y with f E ck+ 2(J) and h E ck+ 2(J) that globally transforms P n(Y, x; I) into an equation Qn(z, t; J) E An with qn-1 E cw(J), in particular with qn-1 = 0 on J' i. e., Qn(z, t; J) EA~. Proof

For n > 2, consider equation Pn(Y, x; I) E An with Pn- l belonging to ck(J), k > n - 2. Choose an arbitrary interval J c ~, a function 'In-l from cw(J), and a non-vanishing function J E ck+ 2(J). Suppose that a function h: J ~ I exists such that h E C 2 (J), n'(t) # 0 on J, and the following relation is satisfied on J:

(8.5.2) Then 1

2

fi'(t) = c~(t)i{ -•l/ exp {P(fi(t)) /

C)}

f t

exp { -

q,,_ 1(s) ds /

C)}

to

where c is a non-zero constant and P denotes an anti-derivative of Pn- l Hence PE ck+ 1(1), and by putting ·

E

Ck(J).

, 179

H, SMOOTII NUSN 011 OIW fll IBNTS AS AN INVARI ANT

ond

(()

WS

t,

k+ 1(/), (/

F

e

ck 1 2(.1),

and

P(fi(t)) fi'(,) - 0(1) on J

(8.5.3)

where F is positive on / und o is a non-vanishing function on J . By intogrotion of relation (8.5.3) from

f

t0

e / to t we obtain

I

S(li(t)) - S{/i(l0 ) )

g(s) ds

-

= :

G(t) ,

(8.5.4)

to

where S is an unti-dcriva tive of F, S e ck-+ 2(1), and G is an anti-derivative of g, G e Ck + J(J). Since S' = F is positive on / , there exists the inverse function s- t lo S defined on f S(/). Evidently s- 1 e ck +2(f). Now, suppose in addition that the functions J and 4n- t are chosen in such a way thnt the function G maps J onto IR. Indeed it is possible, since e. g. for J IR, the functions J(t) = 1 and q11 _ 1(t) = 0 on IR comply with the requirement, because in this case 1=

,

G(t)

=

,

f

g(s) ds

f

=

c ds

10

= c(t

-

10) ,

C

:/= 0 ,

10

thus G(IR ) = IR. Hence, if G(J )

= IR,

we may define the interval J of those

l

e J for which

G(t) + S(fi(t0 )) E f , because G'(t)

= g(t)

=I=

0 on J. On the basis of relation (8.5.4) we may write

for all t e J . Now, it is easy to observe that for given functions Pn- t E Ck(/), q,, _1 e cw(J), Je ck+2(J), ](t) =I= 0 on J, where k ~ n - 2, such that the function G given in (8.5.4) maps J onto Ill, the function h: J -. I defined by

180

8 INVARIANTS

:= {t E J; G(t) + S(h ) E S(I)} for a fixed h0 E /, is of class 0 g(t)/S(h(t)) #- O on J, and h(J) = s- 1(l) = I. Hence his a ck+ 2-diffeomorphism of J onto I. With respect to relation (8.5.2) and Proposi-

on the interval J

ck+2(1), h'(t)

=

tion 4.1.2, the transformation

(Ef, h)y globally transforms the equation P n(Y, x; I) E An into equation Qn(z, t; J) where f and qn-l are restrictions of Jand qn-l to the interval J, respectively. Evidently f E ck+ 2(J), h E ck+ 2(J) and qn-1 E cw(J). In particular, for f = 1 on J there exists a C + 2-diffeomorphism h of J onto I such that qn- l = 0 on J. ■ Theorem 8.5.5

Let equation Pn(Y, x; I) E An be given, n > 2, and let Pn- l E Ck(!) and Pn-2 E ck- 1(!) for some k > n - 1. Then Pn(Y, x; I) can be globally transformed into an equation Qn(z, t; J) E A~ 1 with qn-1 = 0 on J and qn-2 E ck- 1(J). Proof

If Pn- l E Ck(!) for k > n - 1 then Theorem 8.5.4 ensures the existence of a global transformation (Ef, h)y with f = 1 on J and h E ck+ 2(J) such that the transformed equation Qn(z, t; J) belongs to A~, i. e., its coefficient qn- l is identically zero on J. Moreover, due to Lemma 8.5.2 in which r = k + 2 > > n + 1, we get the following relation for the coefficient qn_ 2,

where B ij..

E

cr+(i-j)- 1(1) for i > 0 and B ~. E cr-j(J) • Hence for n > 2

and for n = 2 (for even stronger result see Corollary 7.5.7)

Remark 8.5.6

Let us recall that Theorems 8.5.4 and 8.5.5 are refinements of Proposition 4.2.6.

181

8.5 SMOOTHNESS OF COEFFICIENTS AS AN INVARIANT

Theorem 8.5.7

Let equation P n(Y, x; I) E An, n > 2, be globally transformable into Qn(z, t; J) An. Let i be a positive integer, i < n - 1. Then

E

E

Pn-1 E ci-1(1), Pn-2 E ci-I(J)' ... , Pi+l E ci-1(1) if and only if qn-1 E Ci-l(J), qn-2 E Ci-l(J)' ... ' qi+l E Ci-l(J). In this case

and moreover, Ck(!) \ ck+ 1(1) for some k, 0 < k < i - 2 , if and only if qi E ck(J) \ ck+ 1(J)' P;

E

that means that the existence of such a pair (i, k) is an invariant property with respect to global transformations. Proof

Suppose that the following conditions Pn- I E ci- 1(1), ... , P; + 1 E ci- 1(1) hold for coefficients of a given equation Pn(Y, x; I) for some i, 1 < i < n - 1. Due to Theorem 3.2.3, Lemma 8.5.2 can be applied for r = n that gives qn-1

= B11-l,n +

E

3

C 11 -

(J)

= ci- 1 (1), qi+l = Bi+J,n

··· +

B11-l,n-l pn-1 (h) E cn- 2(1)

(1

C"- 2(J)

n

because also

Ci-l(J) n C

11

c

n

-

11

(J)

1

(J) n ci- 1 (1) n

1

-

n

Ci-l(J) =

n - 2 > i + 1,

+ Bi+l.n-1 Pn-1(h) + Bi+Ln-2Pn-2(h) +

B; + 1, j P; (/1)

n+ i + ··· + B ;+ 1, ; + 1 P; + 1(I1 ) E e

1-

1- n - I

(J)

n

n cn+i+l-(n -1)-l(J) n Ci-l(J) n cn+i+l-(n-2)-l(J) n n Ci-l(J) ... n C'H-i+l-j-l(J) n Ci-l(J) n ...

... n cn+i+ !-(i+l)- 1(1) n ci--l -= ci- 1(1)'

because

i

+

j

> 1.

182

8 INVARIANTS

In this case if also pi Indeed

E

for some k, 0 < k < i -

Ck(!)

qi = Bin+ Bi,n-1Pn-1(h)

I, then

qi

E

Ck(J).

+ Bi,n-2Pn-2(h) + ... + Biipi(h) E cn+i-n-I(J) n

n cn+i-(n-1)-l(J) n Ci-l(J) n cn+i-(n-2)-I(J) n

since i > 0 and k < i However, if Pn-1 E

ci-1(/), ... , Pi+l

I.

E

ci-1(!)

and

Pi 1= ck+I(I)

for some k, 0 < k < i - 2, then also qi ff= ck+ 1(1). Otherwise, due to Lemma 8.5.2 we have Bdt) =I- 0 on J, and hence pi(h) can be expressed as a linear combination of the functions qi, Pi+I' P;+ 2 , ... , p11 _ 1 of classes ck+I(J) and ci- 1(1) with the coefficients

of class ci- 1(1). Since k + I < i - 1 < n - 2 and h is a cn-diffeomorphism of J onto /, Pi E ck+ 1(/) that contradicts to our assumption. The converse is true due to the symmetry of our assumptions on equations P11 (y, x ; I) and Qn(z, t; J). ■ Corollary 8.5.8

Let Pn(Y, x; I)

E

An be globally transformable into Qn(z, t;

J)

E

An, n > 2. If

O 0 then h can be considered a phase of equation (p0 ) with coefficient p0 given by formula (9.1.2).

Remark 9.1.3 The phase his not uniquely assigned to P 2(y, x; I) since neither the canonical equation (1 1 ) is uniquely determined by P 2 as Remark 7.5.9 shows, nor the transformation cc satisfying P 2 = (1 1 ) * cc is unique as follows from Theorems 3.3.16 and 6.5.1. However, by any C 3-diffeomorphis1n has a phase, the equation P 2 is uniquely determined, its coefficient p0 is given by formula (9.1.2), and its fundamental solution y is expressible in the form (9.1.1 ).

Theorem 9.1.4 Let P i(y, x; I) be a second order equation from A~ and let h denote one of its phases. Equation P 2(y, x; I) is non-oscillatory, or one-side oscillatory, or both-side oscillatory on 1 if and only if interval h(I) is bounded, or one-3idl' :111hounded, ur h(I) = IR, respectively.

Proof follows immediately from formula (9.1. l ), the form of the fundamental solution of equation P 2(y, x; I). Theorem 9.1.4 is also a direct consequence of the fact that equation P2(y, x; J) is globa11y equivalent to equation (1 1 ) and hence they both have the same oscillatory character, evidently given hy the form of interval J = h(l). ■

}89

9.2 ASYMPTOTIC PROPERTIES OF SOLUTIONS OF THE SECOND ORDER EQUATIONS

9.2 Asymptotic properties of solutions of the second order equations In this chapter we shall consider the second order equations from A~, i.e., equations of the Jacobi form

y"

+

p(X) y

(p)

= 0,

In accordance with Definition 9.1.1 each equation (p) E Ag is coordinated by (any of) its phase h being a C3-diffeomorphism of interval / onto a suitable interval JC IR. First, we shall deal with bounded solutions of the second order equations.

Theorem 9.2.1 (Bounded solutions, [18]) Each solution of a second order equation (p) from Ag is bounded on I if and only if for a phase h assigned to (p) the function 1/lh'I is bounded on I, the definition interval of equation (p ).

Proof Leth be a phase of an equation (p) from Ag. In accordance with Definition 9.1 .1, his a C3-diffeomorphism of I onto some open interval J = h(I) c IR such that formula (9.1. I) establishes the couple y 1, y 2 :

y1(x) = lh'(x)l- 112 cos h(x), y2(x) = lh'(x)l - 112 sin h(x)

(9.2.1)

of linearly independent solutions of equation (p) on /. ( ::::> ). If each solution of (p) is bounded then the solutions y 1 and y 2 are bounded on/, and the function (y 1(x)) 2 + (yi{x))2 = lh'(x)l - 1 is bounded on / as well. ( ). If each solution of (p) E A~ tends to zero as the independent variable x approaches a or b, then also lim (yf(x)

+ y~(x)) = lim lh'(x)l- 1 = 0

as x tends to a or b, respectively. ( 0 on IR+, and

flh'( 00

x )1-1 dx < oo'

Xo > 0,

xo

then the equation

y"

+

p 0 (x) y = 0

on IR+

with the coefficient p0 given by formula (9.1.2) is in the limit circle case at oo. If, in addition, 1/lh'I is not bounded on the interval [ x 0 , oo) then there exists a solution of equation (p0 ) that is unbounded on the interval [ x 0 , oo) as follows from Theorem 9 .2.1.

Fig. 9. l

194

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

However, such a function h actually exists, because there exists a positive function g E C 2 ( [ x 0 , oo)) such that

f 00

g(x)dx < oo

xo

and still g is not bounded on the interval [ x 0 , oo ), see also Fig. 9.1. Then it suffices to put

f X

h(x ):

ds/g(s ), x

E

[xo, oo ).

■

xo

Remark 9.2.7 A stronger result than Corollary 9.2.6 concerning arbitrary b and ensuring the existence of equation (p 0 ) EA~ in the limit circle case at b having every nontrivial solution unbounded on (x 0, b) was derived in [18]. There are also numerous interesting results concerning limit circle and limit point classification in papers of W. N. Everitt [l, 2], W. T. Patula and J. S. W. Wong [I], M. K. Kwong [I], J. Walter [l], and also in [8, 12, 13, 18] where the classification is studied with respect to oscillations and transformations as well. Essential contributions to problems concerning coexistence of periodic solutions of second order equations with a parametr, to bounded solutions of equations with periodic coefficients, to Liouville transformation of such equations and to other interesting problems in this area were made by F. M. Arscott [1], M. Bartusek [1-4], 0. Boruvka [2, 3, 4], J. H. Guggenheimer [1-4]. N. A. Izobov [1-5], V. A. Yakubovich and V. M. StarzhinskiI [I], N. P. Erugin [1-3], W. Magnus and S. Winkler [I], S. Stanek [1-8], and the author [1-13].

9e3 Periodic solutions of the second order equations In this paragraph we adopt another approach, analytic in its essence, to construct linear differential equations with solutions of prescribed properties, the approach, that differs from the coordinate approach introduced and illustrated above. Namely, when a certain property of solutions is given we consider first only some (simple) equations with solutions of this property and then we modify these equations to get all required ones. We derive here the second order equations from A~ with some or all periodic solutions. These results published in 196 7-1968 [3, 4 had important conse-

J

195

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

quences also in differential and integral geometry of plane curves and led to generalizations of Blaschke's and Santal6 's isoperimetric theorems, as presented in Chapter 11. The same approach was also successfully used for constructing certain non-diagonalizable second order linear differential systems having again close connection with some problems in the differential geometry, see L. A. Besse [ 1] and the author [21].

Proposition 9.3.1 Let y: [ij ~ [ij be a nontrivial periodic or half-periodic solution with period n of the second order equation from A~ y"

+

p (X) y

(p)

= 0,

Suppose that there exists at least one zero of the solution y. Then there exists a periodic function g E C2 (1R) lVith period n such that g(a;) = g'(a;) = 0 for i = 1, ... , k and the solution y can be written as

y(x) = u(x) exp {g(x)} where u:

[ij ~

IR, k

k

u(x) = efI b;(sin (x - a))- 2 )- 112 sign (TT sin for x

= 0

(x -

a;))

i= 1

i= 1

=I=-

a 11

+ mn,

n = 1, ... , k, m

E

Z

otherwise,

a1 < a 2 < ... < ak are all zeros of y in the interval [O, n) and b; : = 1/ly'(a;)I,

i = 1, ... , k, e : = sign y'(ad. Proof

Under the assumption and notation of the proposition we have y(x + rr) = y(x) on IR1 when k is even and y(x + n) = - y(x) on [ij if k is odd. Define the function s: IR1 ~ IR, k

k

s(x) =

e

y(x) ( L sin (x i= 1

-

1

a;) )- (

L bf L (sin (x i=l

for x 1 elsewhere.

k

-

a) )2 )112

j=l j =I= i

=I=-

a11

+ mn,

n

1, ... , k, m

E

7L,

196

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

Then function s is continuous on ~, since lim

s(x)=

x---.an+mn

Because e = sign y'(ad and y'(an) y'(an+ 1) is always negative, the sign of y'(an) is equal to e( - l Hence

t-n.

Thus we get s(x) = l,

lim x---.an + mn

that means that function s is continuous on ~- Moreover, function s is periodic with period n and positive on the interval [O, n ). In fact, c:y(x) and the product k

n sin (x -

aJ

i= 1

have the same sign on each of the intervals (an, an+ i), n = 1, ... , k; ak+ 1 == == a 1 + n, because they both are positive on the interval (ak, a 1 + n) as follows from the inequalities

and O < x - a;< n for xE(ak,a 1 + n) and each i = 1, ... ,k. Due to periodicity, function s is positive on the whole ~Furthermore, for n = 1, ... , k we have lim s'(x) =

.

= hm t---.an

(

. SIO

y(x)

(x - an)

)' ( k

L b7 [Ik

i=l

)1/2 ( [Ik sin (x - a;))-1

. (sm(x - a}) 2

j=l

i=l

j#i

i#n

+

197

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

However, by using l'Hospi tal rule we get

y(x)

lim (

because y

!~°:,

)' = O

sin (x - an)

x-+an

E

C2([ij) is a solution of equation (p) that vanishes at an- And also k

( ( ;~i

k

bf}] (sin (x -

a))

1/2 (

2)

k }]

j#i

)-

sin (x - a;)

1),

= o.

i#n

Since y E C2([ij), the function s belongs to C1([ij) and s'(an + mn) = 0 for n = 1, ... , k and m E "lL. For the existence and continuity of the second derivative of s it is sufficient to establish the existence of the limit

for n = 1, ... , k, because the expression k ( ;~i

bf

k j~

j#i

(sin (x - a))

2) 1/2 (

g sin (x k

)- 1

a;)

i#n

has derivatives of all orders in a vicinity of an. Evidently function s has a continuous derivative of the second order at each point x # an + mn, n = 1, ... , k, m E "lL, because y is a solution of the second order equation (p) and hence it belongs to C2 ([ij ). Thus by applying }'Hospital rule we obtain

That means that the function s belongs to C2([ij). Let us recall the properties of s:

198

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

s E C 2(~), s(x + n) = s(x), and s(x) > 0 for x s'(an + mn) = 0 for n = I, ... , k, m E 7l.., and

Ct bf Ct bf

y(x) = e s(x)

e s(x)

112

(sin (x -

~ 1

a)}2)-

E ~,

s(an

+ mn)

b

sin (x - a;)

1,

=

1

)rl

(sin (x - a;)J-

= s(x) u(x) for x

2)-l/2sign

C~

sin (x - a;)) =

E ~ .

Let us introduce the function g:

~ ~ ~

by putting

g(x) : = In s(x) . Evidently g E C 2(~ ), g(x + n) = g(a;)

=

In s(a;)

=

g(x) on

~,

0,

g'(a;) = s'(a;)/s(a;) = 0

for

z

= 1, ... , k,

and ■

y(x) = u(x) exp {g(x)} on ~. Proposition 9.3.2 ~ ~ ~

Let a function y:

be given by the formula

y(x) = u(x) exp {g(x)} ,

(9.3.1)

where functions g: ~ ~ ~ and u: ~ ~ ~ satisfy the following properties

g(x + n) = g(x) on g(a;)

=

g'(a;)

=

0 for

~,

O < a 1 < ... < ak < n, k < l ,

and k

u(x) = e ( i~t

bf (sin (x

)-1/2 sign (Esin (x -

- a;)J- 2

k

s = 1 or - 1, and b; being arbitrary positive constants.

)

a;) ,

199

9.3 PERIODrC SOLUTIONS OF THE SECOND ORDER EQUATIONS

Then y is a solution of the second order linear equation (p) from A~ with the coefficient p E c 0 ([ij) , p(x) = 1 - g"(x) - (g'(x)) 2

( ) 'i..J°' k

- 2g' x

+3

(

(

i=l

k i-I LL i=lj=l

-

a; )

COS ( X -

b;2 - - - - (sin (x - a;)) 3

)

(

b2; Lk - - - -2)- I + (sin (x - a;))

i=l

b?b? (sin (a. - a-)) 2 I}

I

L

(sin (x - a;) sin (x - a)) 4

i=l

)- 2

b?

)( k

}

I

(sin (x - a;)) 2

(9.3.2)

The solution y is periodic for even k: y(x + n) = y(x ), and half-periodic for odd k: y(x + n) = - y(x ), it vanishes at a; and y'(a;) = e( - 1t-i/b 1 • Proof

Under the suppositions of the proposition

y"(x)/y(x) = g"(x) + (g'(x)) 2 + 2g'(x) u'(x)/u(x) + u"(x)/u(x) for x #- an + mn, n = 1, ... , k, m we have k

b?

i=l

(sin (x - a;))

L .

(u(x))-2 =

'

E

"ll... Now, except of the points an + mn,

2

and

u'(x) (u(x))- 3 =

k bf cos (x - a;) L - - - -3

i=l

(sin (x - a;))

·

Hence

u'(x)/u(x) = (

i bf i=l

a;)) ( i

cos (x (sin(x - a;))

i=I

2)-I

bf

(sin (x - a;))

and

u"(x )/u(x) _

-

(

k

;~1

=

(u'(x )/u(x ))'

bf (-

+ (u' (x )/u(x) )2

(sin (x - a;))

2

-

=

3 (cos (x - a;)J2)

(sin (x - a;))

4

i i=l

bf (sin(x - a;))

+ 2

200

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

-1 - 3

k

(

+

i-I

L L

i=l j=l

22( (cos(x-a;))2 b.b. -------1 1 (sin (x - a;)) 4 (sin (x - a))2

(cos (x - a))

+

2 )

(sin (x - aj)) (sin (x - a;)) 2 4

_

2

k

;- 1

LL

i=lj=l

2 2 cos (x - a; ) cos ( x - aj ) ) bibj . . 3 (sin (x - a;) sin (x -:- a))

(

'°' k

b2

i=1

(sin (x - a;))

~

;

·

)-

2

2

For isolated points an, n = 1, ... , k, let us consider the following limits: lim g'(x) u'(x)/u(x)

and

lim u"(x)/u(x).

Since g'(an) = 0 and u'(x )/u(x) = r(x )/sin (x - an) in a vicinity of the point an and r is a non-vanishing continuous function on the vicinity, the existence of the first limit follows from l'Hospital rule. The explicit formula for u"(x )/u(x) guarantees the existence of the second limit as well. Hence the function y satisfies the relation

y"(x)

+ p(x) y(x) =

0

on the whole [R where the continuous function p is given by (9.3.2) The properties of y follow immediately from the formula by which it is defined. ■ Theorem 9.3.3 (Criterion of periodicity)

A second order linear differential equation (p) from A~ defined on periodic (half-periodic) solutions with period n if and only if

[R

admits only

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

201

it has a nontrivial oscillatory periodic (half-periodic) solution y with period n and the following relation is satisfied,

f 1l

(y(x)J- 2

-

r(x)) dx

= 0

(9.3.3)

0

where k

r(x) ==

L (y' (a;) sin

(x - a;) )- 2,

i= 1

being all zeros of yon the interval [O, n ). The periodic or half-periodic case occurs exactly when the integer k is even or odd, respectively. Proof

Let (p) have only periodic (half-periodic) solutions on IR. Then equation (p) is both-side oscillatory on IR. Denote by y 1 a non-trivial oscillatory periodic (half-periodic) solution of (p) with period n. Let y 1(x0 ) -:f- 0 for x 0 E IR. Due to Proposition 2.6.5, there is a finite number of zeros of y 1 on the interval [ x 0, x 0 + n ). Let us denote them by a 1 < a2 ..• < ak, k > 1, and let us put also a0 == ak - n and ak+ 1 == a1 + n. On every interval J on which y 1 is non-vanishing, the function X

y* : x ,_. y 1(x)

(J

2

(y 1(s)J- ds

+ c).

x*

E

J

x*

is a solution of (p) on J. Moreover y 1 and y* are linearly independent on J. If the interval J is finite, then the function y*, and its first and second derivatives have limits from left (right) at the right (left) end-point o·f J. Choose points d; E E (a;, a;+i) for i = l, ... , k - l, and put d0 == x 0 E (a0 , a 1 ) and dk == x 0 + + n E ( ak, ak + 1). Define the following function y2 on k

U (a;, ai+ 1)' i=O X

.Yi : x

,_. Y1(x)

(J

(y 1(s)J- 2 ds +

c),

d;

c; being constants, i = 0, ... , k. We show that there always exist constants c;, i = 0, ... , k, such that y2 is a restriction of a solution of equation (p) to

202

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

For this it is sufficient and necessary to require that the following relations are satisfied: (9.3.4) lim .Vi(x) =

lim y2(x) ,

(9.3.5) (9.3.6)

for i = l, ... , k. If relation (9.3.4) is satisfied then also (9.3.6) is valid, because for x =t= ai one has y'2(x) = - p(x )yi(x) and the function p is continuous on ~As y2 is always a solution of equation (p)'on each interval (ai, ai+i), all the above limits exist. Let us consider relation (9.3.4 ):

f X

x~1.::- y 1(x) (

1 2

(y (s)J- 0, both relati~ns (9.3.8) and (9.3.9) are satisfied just when ck = c 0 . Let us return to relation (9.3. 7J As y 1(a;) =/:- 0 for i = 1, ... , k, we divide the relation by y 1(a;) and sum them for all i = 1, ... , k. We obtain k

0 = c0

-

ck

+

I

(y 1(a;))- 2 (cot (di-I - a;) - cot (d 1

-

ai))

+

i= 1

Hence c0

;~, f

= ck occurs exactly when

d·

k

I

((y 1(s)t

2

-

(yJ(a;) sin (s - a;)t

2

)

ds

+

di-I

k

+

I

(y 1(ai))- 2 (cot (d;_ 1

ai) - cot(d; - a;)) = 0.

-

(9.3.10)

i= 1

Let us rewrite the last relation into a simplier form. The following identity holds

fJ d;

2

(yJ (aj)) sin (s - aj))- ds +

di-I j#i

k

+

I j=l j#i

(y 1(aj))- 2 (cot (di -

aj) -

cot (di-I -

a)) =

0.

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

207

If we subtract these identities for i = 1, ... , k fron1 relation (9.3.1 O) we get

Jf d;

((Y1(s)t

2

-

J

(y;(aj) sin (s - a)t 2 ) ds

+

d;-1 k

k

L L (Yi(a)- 2 (cot (di-l

+

- a:;) - cot (di -

a)) = 0.

i=l J=l

However k

k

L L (Yi (a)- 2 (cot (d;_

1 -

a) -

cot (d; - ai)) =

i=l J=l

k

k

L (y 1(a))- L (cot (d;_ 2

=

J=l

1 -

a) -

cot (d; - ai))

=

i=I

k

L

=

(Y1(a))- 2 (cot (do - a) - cot

(dk - a;))

=

J=l k

L

=

(Y1(a;))- 2 (cot (xo - a) - cot (xo + n - a)) = 0 .

.i=l

Thus we may conclude that c0

= c

11

occurs exactly when

that means x 0 + rr

f

((y 1(xWc -

r(x)) dx = 0

xo

where

r(x) stands for

k

L

(y;(a)

sin (x - a;))- 2 .

j= I

Since both (y 1(x) )2 and r(x) are periodic functions with period n, the last relation is satisfied exactly when relation (9.3.3) holds where y 1 stands for y. ■

208

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

Remark 9.3.4 Let us note that the integral in relation (9.3.3) is convergent since the limits in (9.3.4) always exist. This convergence can also be verified directly by setting y(x) = gi(x) sin (x - a;) in vicinities of points a;, i = 1, ... , k and showing that the integrand k

(y(x))- 2

-

I

(y 1(a;) sin (x - a;))- 2

i= 1

has finite limits at each a; continuous function on ~-

+

mn, that means that it can be extended to a

Theorem 9.3.5

Every solution of a linear differential equation of the second order (p) on ~ from A6 is periodic (half-periodic) with period n and with exactly k zeros on the interval [O, n) if and only if k > 1, k is even (odd), the coefficient p is given by formula (9.3.2) where g E C2 (~), g(x + n) = g(x) on ~, g(a;) = g'(a;) = 0 for O < a 1 < ... < < ak < n, bi are positive constant for i = 1, ... , k, and the following relation is satisfied n

f(I

k

0

i=l

.

b~1

)

(sm (x - a.))

2

(exp { - 2g(x )} - 1) dx

= 0.

(9.3.11)

I

The solution y that vanishes at ai and satisfies y'(a;) ... , k is given by (9.3.1 ).

=

s( -1 t-i/b; for i

=

1, ...

Proof

A6

Suppose that an equation (p) from defined on ~ has only periodic (half-periodic) solutions with the period n. According to Theorem 9.3.3, equation (p) is both-side oscillatory on ~ and it admits a non-trivial periodic (half-periodic) solution y with period n that satisfies the relation (9.3.3 ). Proposition 9.3.1 guarantees that y can be written in the form (9.3.1 ). Once solution y is of this form the coefficient p of equation (p) is given by formula (9.3.2) as Proposition 9.3.2 asserts. ( => ).

209

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

Relation (9.3.3) for solution y of the form (9.3.2) reads as follows n

0 =

f

f n

((y(xW

2

-

r(x)) dx =

0

((u(x)t 2 exp {-2g(x)} - r(x)) dx

0

n

f(

Lk .

i=I

0

b;

)

1

(sin (x - a.)) 2

(exp {-2g(x)} - 1) dx,

I

that is exactly condition (9.3.11 ). ( 1, and positive constants bi, i = 1, ... , k, there always exists an equation (p) from A~ with only periodic (half-periodic) solutions and positive coefficient p on !R. An example of such an equation can be obtained by putting g = 0 on !R in Theorem 9.3.5:

Example 9.3. 7 Let k > 1 be an integer and O < a 1 < ... < ak < n, b; > 0 for i = 1, ... ... , k. The differential equation

y"

k

i-l

I L

+( 1 + 3(

b;b? (sin (a. - a.)) 2 I

.I

I

J

b?

)( k

i=lj=t(sin (x - a;) sin (x - a)) 4

L

)-

I

i=t(sin (x - a;)) 2

2 )

y=0

(with the positive coefficient on !R) admits only periodic or half-periodic solutions with period n. The function k

y(x) = /] sin (x - a;)

(

k

k

;~i } ]

(sin (x - a))

)-1/2

2

j #i

is the solution of the equation that vanishes at = (-1t-i/b; for i = 1, ... , k.

ai

and satisfies

y'(a;)

210

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

We introduce also another special class of equations from A~ with only periodic or half-periodic solutions, namely those that admit a non-trivial solution having consecutive zeros at the same distances and, moreover, with the same absolute value of the first derivative at each of its zeros. We deal with these equations in more detail since they play an important role in affine geometry of closed plane curves as explained in the next Chapter. Theorem 9.3.8

An equation (p) from A~ has only periodic (half-periodic) solutions with period n and, moreover, it admits a non-trivial solution y such that for a positive integer k, the numbers a

+

mn/k,

mE"1L

are all zeros of y and ly'(a + mn/k )I = 1/b ,= canst for all m the coefficient p of equation (p) is given by the formula p(x) = k 2 where g: g(a

g"(x) - (g'(x)) 2

~ [ij, g E C2 ([ij ), g(x

[ij

+

-

mn/k) = g'(a

+

-

2kg'(x) cot (k(x - a)),

+

n) = g(x ),

mn/k) = 0 for all

m

E

E

"1L if and only if

(9.3.12)

"1L,

and n

_ex_p_{-_2g_(x_)}_-_1 dx =

J 0

(sin (k(x - a)))

2

0

.

(9.3.13)

The solution y is expressible as y

e

= - exp {g(x)} sin(k(x - a)), kb

e = 1 or -1 .

The periodic case takes place exactly when k is even. Proof ( [3, 9 ])

Without loss of generality, let a + (i - 1) n/k =: ai and 1/ly'(ai)I = i = I, ... , k. According to Proposition 9.3.1 solution y is of the form (9.3.1) ( => ).

= bi = b = const for

y(x) =

k

el

exp {g1(x)}

(

L . i=l

b2

(sm (x - a))

)-1/2 sign (n sin (x k

2

i=l

aJ)

211

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

2 where gl E C ([ij), g1(x + n) = g1(x) on [ij and g1(a;) = gaa;) = 0 for i = 1, ... , k, e1 = 1 or - 1. Moreover, due to Theorem 9.3.3, the following integral condition is satisfied 7t

f(

(y(x))- 2 -

0

±

2

b ) dx = 0. ; = 1 ( sin (x - a;) )2

Now consider the function v: [ij

-+

[ij

1 v(x) = - sin (k(x - a)).

kb

This function is a solution of the equation y" + k 2y = 0 that has only periodic (half-periodic) solutions with period n with zeros exactly at points a + mn/k, m E Z. Moreover, lv'(a + mn/k )I = 1/b. Hence, according to Theorem 9.3.3, relation (9.3.3) holds that, in our case, gives the identity

f

n ((v(x)J- 2 -

±

2

b ) dx = 0. i=l (sin (x - a;))2

0

Due to Proposition 9.3.1, function vis expressible in the form v(x) = e2 exp {g 2(x)}_ (

Ik . i=l

b2

(sin (x - a;))

)-1/2 ( sign 2

n sin (x k

)

a;) ,

i=l

e2 = 1 or - 1, g2 E C 2 ([ij ), gix + n) = g2(x) on [ij and g2(a;) = g 2(a;) = 0 for i = 1, ... , k. Hence

where

y(x) = e1 exp {g 1(x)

+

e3 g 2(x)} v(x) = e1 exp {g(x)} -

1

kb

sin (k(x - a)), (9.3.14)

e3 = 1 or -1, g == g 1 + e3g2 , g E C2([ij), g(x + n) = g(x) on [ij and g(a;) = g'(a;) = 0 for i = 1, ... , k. Moreover, by subtracting the integral where

conditions we get

f 7t

0 =

0

fe? 7t

((y(x))-2 - (v(x))-2) dx = k2b2

{-2g(x)} - 21 dx

(sin (k(x - a)))

0

212

9 EQUATIONS WITH SOLUTIONS OF PRESCRIBED PROPERTIES

that establishes relation (9.3.13 ). The coefficient p can be evaluated as - y"(x )/y(x) by using formula (9.3.14) that gives (9.3.12 ). ( ¢:: ). If pis given by formula (9.3.12) and condition (9.3.13) is satisfied then equation (p) admits a solution of the form (9.3.14) with consecutive zeros at the distances equal to n/k and the same absolute value of the first derivative at the zeros and, moreover, equation (p) has only periodic (half-periodic) solutions. This follows from Theorem 9.3.3, since condition (9.3.13 ), after adding the above integral identity concerning v, gives relation (9.3.3 ). ■

Remark 9.3.9 Coefficient pin formula (9.3.12) and the integrand in condition (9.3.13) become continuous functions by defining their values at points a + mn/k, m E Z, as their limits there, that, due to the properties of g, always exist. For k = 1 we get the following important consequences of Propositions 9.3.1, 9.3.2 and Theorem 9.3.8.

Corollary 9. 3. 10 Equation (p) from A~ admits a half-periodic solution with period n and exactly one zero on [O, n) if and only if its coefficient p is of the form p(x) = 1 - g"(x) - (g'(x)) 2 where a

E

-

2g'(x) cot (x - a)

1R and function g: IR ~ IR satisfies the following conditions

g(x Function y: IR

~

+ n) = g(x) on

IR and

g(a) = g'(a) = 0.

IR given by the formula

y(x) = exp {g(x)} sin (x - a) is a half-periodic solution of period n of the equation. In such a situation every solution of equation (p) is half-periodic with period n if and only if 7t

f 0

exp {-2g(x)} - 1 - - - - - - dx = 0. (sin (x - a) )2

Remark 9.3.11 This corollary was proved in 1964. Then by putting g(x) == -½ In (1 - c sin 2(x - a) (sin (x - a))2), c E (-1, 1) in Corollary 9.3.10, we established in [2] the cardinality of the set of equations from A~ having only half-periodic solutions. _

9.3 PERIODIC SOLUTIONS OF THE SECOND ORDER EQUATIONS

213

Example 9.3.12

Equations (p) p (X )

E A~

with p:

[ij

~ lR,

sin 4(x - a) + c(sin (t - a)) 4 = 1 - 3c - - - - - - - - - - - - - , (1 - c sin 2(x - a) (sin (x - a))2)2

a E lR, c E ( -1, 1), admit only [O, n ). Hence the cardinality of equations from A~ defined on [R period n and exactly one zero on numbers.

half-periodic solutions with just one zero on the set of all second order linear differential and having only half-periodic solutions with [O, n) is the same as the cardinality of all real

9.4 Geometrical approach Another approach to constructions of linear differential equations with solutions of prescribed properties is based on the representation of these equations as curves in n-dimensional vector space Vn whose coordinates are linearly independent solutions as introduced in Chapter 3. Some special properties of solutions can be reformulated in terms of behavior of the curves and from these curves the corresponding equations can be constructed. It concerns namely the asymptotic behavior of solutions. An extensive investigation was especially made concerning oscillatory or non-oscillatory behavior of solutions. Hence we devote to this subject the whole next chapter.

10 Zeros of solutions

Problems concerning distribution of zeros of solutions, like conjugacy and conjugate points, disconjugacy and other oscillatory properties of solutions, were perhaps one of the most studied areas in the theory of linear differential equations from the beginning till now. The reason seems to be in wide applications and many important consequences of oscillatory behavior of these equations. Let us remind at least the Separation Theorem for the second order equations and a close connection of zeros of solutions of the nth order equations with factorization of the corresponding linear differential operators. Let us mention at least some of many authors: C. Ascolli [l ], N. V. Azbelev and Z. B. Caljuk [l ], J. H. Barrett [l ], E. Barvinek [1-4], L. M. Berkovich [l ], G. D. Birkhoff [l ], 0. Boruvka [2-4], T. A. Chanturija [l ], W. A. Coppel [l ], J. M. Dolan [l ], I. I. Endovickij [l ], G. J. Etgen [l ], M. Gera [1-8], M. Gregus [1-6], H. W. Guggenheimer [1-4], G. B. Gustafson [1-3], G. B. Gustafson and S. Sediwy [l ], M. Hanan [l ], I. T. Kiguradze [l, 2], C. A. Lazer [l ], A. Ju. Levin [1-3], G. Mammana [l ], Z. Mikulik [l ], M. Rab r3], R. Ristroph [l ], G. Sansone [l, 2], V. Seda [1-7], T. L. Sherman [l ], M. Svec [1-5], C. A. Swanson [l ], W. F. Trench [l ], A. Wintner [l ], A. Zettl [l, 2], M. Zlamal [1-4]. The essence of our approach to questions on distribution of zeros of solutions is based on a certain geometrical representation published in 1971 [ 11] and explained in this chapter. Using this method we can sometimes see without a lengthy calculation what is possible and what i1npossible, simply by drawing a curve only. Some complicated constructions or proofs can be easily understood and hints for possible new results can be obtained by this approach. This technique offers also suggestions to possible direction of investigation of open problems in this area as explained in the following paragraphs.

10.1 Notation and definitions Let us recall some notions concerning zeros of solutions and their distribution as introduced in the mathematical literature.

216

10 ZEROS OF SOLUTIONS

A zero x 1 E

/

of a non-trivial solution y of an equation P n(Y, x; I)

E

An,

n > 2, has the multiplicity k if

y(x 1)

= 0, y'(x 1) = 0, ... , y(k-l)(xi) = 0, y(k)(xi) =I- 0.

Evidently 1 < k < n - 1 . A non-trivial solution y of Pn(Y, x; I) E An, I = (a, b ), - oo < a < b < oo, is said to be oscillatory as x tends to b (from the left) if for each x 0 E / there exists x1 E (x 0 , b) such that y(x 1) = 0. Otherwise the solution y is non-oscillatory for x approaching b. Analogously we introduce oscillatory and non-oscillatory solutions for x -+ a+. Let ff denote the set of all solutions of an equation P n(Y, x; I) E An, n > 2, I = (a, b ), - oo < a < b < oo, considered as an n-dimensional vector space. Suppose a linear subspace ff 1 of solutions from ff. We say that ff 1 is (i) non-oscillatory as x -+ b _ if every non-trivial solution from !/ 1 is non-oscillatroy for x -+ b _, (ii) weakly oscillatory as x -+ b _ if the subspace ff 1 contains both oscillatory and non-oscillatory solutions for x -+ b _, (iii) strongly oscillatory for x -+ b _ if each non-trivial solutions from ff 1 is oscillatory as x -+ b _, (iv) oscillatory as x -+ b _ when ff 1 is either weakly or strongly oscillatory for x -+ b _. The equation P n(Y, x; I) itself is called non-oscillatory, weakly oscillatory, strongly oscillatory, or oscillatory as x -+ b _ according to its whole n-dimensional space ff of solutions. Analogous definitions are introduced for the case when x approaches a, the left end-point of the interval I of definition of P n(Y, x; I). A differential equation Pn(Y, x; I) E An, n > 2, is called disconjugate (on I) if none of its non-trivial solution has more than n - 1 zeros (on I} including multiplicities. Consider equation Pn(Y, x; I) E An, n > 2, I= (a, b), a number d E E (a, b) and a positive integer k. Let ff(d) denote the set of all non-trivial solutions of P n(Y, x; I) such that each of them vanishes at d and has at least k + n - 1 zeros on the interval [d, b) including multiplicities. For any such a solution y E ff(d), denote by d = dl < d2 < ··· < dk+n-1 its consecutive zeros (including multiplicities). If ff(d) =I- (/J then rpk(d) == inf {dk+n-l; y

E

ff(d)}

is called the kth conjugate number to d. For ff(d) being the empty set we say that the kth conjugate point to d does not exist.

IO.I NOTATION AND DEFINITIONS

217

A linear differential equation of the third order P 3(y, x; I) e A 3, I = = (a, b ), is said to be of class/ or II if for each point x0 e / and arbitrary solution y the following implication holds:

if

then y(x) > 0

for all x

E

(a, x0 ) or (x 0 , b ), respectively.

10.2 Representation of zeros Let us remember the geometrical approach to linear differential equations, introduced in Chapter 4 and already used in Chapter 7. We consider an equation Pn(Y, x; I) E An, n > 2, and its fundamental solution y: I __. [R" being a vector function with coordinates formed by an n-tuple of linearly independent solutions. Now we represent the fundamental solution y as a curve in n-dimensional vector space V11 , the independent variable x serving for parameter of the curve. It is evident that for any curve y: I __. V11 whose coordinates are of class C11 (J) and with non-vanishing Wronskian deter1 minant w[y] on /, there always exists an equation in An whose fundamental solution is y. Denote by c a non-zero constant vector from V11 , n > 2,

The set of vectors 11 = (17 1,

... ,

11n)T

E

V11 satisfying the relation

forms an (n - 1)-dimensional subspace of V11 • This subspace of codimension 1 is a hyperplane going through the origin O of Vn. We denote it by

to express explicitly its dependence on the constant vector c. We also say that k hyperplanes

218

JO ZEROS OF SOLUTIONS

are independent if the n by k matrix (c 1, combination of the hyperplanes

for some constants l 1,

••. ,

... ,

cd

has the rank k. Also a linear

Ak is considered as the hyperplane

H(l1c1 + ... + lkck) if

A1C1

+ ... + AkCk ¥= 0.

We say that, for an integer k > 0, curve y: I -+ Vn has a contact of the order k with a hyperplane H(c) at the point y(x 0 ) of parameter x 0 if CT. y(xo) = 0, ... ' CT. y(k)(xo) = 0, CT. y(k+l)(xo) ¥= 0. Each point y(x 0 ) of contact of curve y with H(c) lies in the hyperplane since, even for k = 0, we have 0 = CT . y(xo) = C1 Y1 (xo)

+ ··· + en Yn(xo) ·

For each linear differential.equation Pn(Y, x; I) E An, n > 2, and its fundamental solution y: I -+ !Rn considered as a curve inn-dimensional vector space Vn define the following mapping of the set of all non-trivial solutions of P n(Y, x; I) onto the set of all hyperplanes in Vn passing the origin. Definition I 0.2.1

For any non-trivial solution y of P n(Y, x; I) solution y, i. e.,

E

An, n > 2, with the fundamental

r(y) == H(c). Theorem I 0.2.2. (Representation Theorem)

Let Pn(Y, x; I) E An, n > 2, be an equation and y: I -+ !Rn be it's fundamental solution considered as a curve in Vn- Then the mapping r of all non-trivial solutions of Pn(Y, x; I) onto all hyperplanes in Vn going through the origin d~fined in I 0.2.1 is a linear mapping converting any system of linearly independent solutions into a system of independent hyperplanes such that the following property is satisfied: a non-trivial solution y of P n(Y, x; I) has a zero x 0 E J of the multiplicity k, 1 < k ( < n - l) if and only if the hyperplane r(y) has a contact of the order k - l with the curve y at the point y(x 0 ), 0 < k - 1 < n - 2.

10.2 REPRESENTATION OF ZEROS

219

In particular, a non-trivial solution y of P n(Y, x; 1) has a zero x 0 exactly when the hyperplane r(y) intersects the curve y at the point of the same parameter x 0 .

Proof

First let us prove that the mapping r is linear. Indeed, for constants l I, and two solutions y, y of Pn(Y, x; 1) such that ).y + Xy is not trivial, we have y=cTy,

y=cTy,

r(y) = H(c),

r(y) = H(c),

and hence

Moreover, when y ranges through the set of all non-trivial solutions of Pn(Y, x; I), the image of y by r goes through the set of all hyperplanes in V11 passing the origin 0. And any choice of a set of linearly independent solutions is mapped by r into a set of independent hyperplanes passing the origin since the independence or dependence in both of the cases depend on the same fact, namely whether the corresponding constant vectors e's are independent or not. Now, suppose a non-trivial solution y of P 11 (y, x; J) has a zero x 0 E I of the multiplicity k, k > 1,

Evidently k < n - 1, otherwise y is identically zero. Since y can be written in the form y

=

CTY

we have CT

y (Xo )

=

CT

y '( Xo ) -_ ... -_

CT .

y(k-1)( Xo ) -_ Q

and CT y(k)(xo) =I= 0 ' in other words, the hyperplane H(c) = {11 E V11 , CTfl = O} has a contact of the order k - 1 with the curve y at the point y(x 0 ) of parameter x 0 .

220

10 ZEROS OF SOLUTIONS

And also conversely, if x 0 is parameter of a point y(x 0 ) having a contact of the order k - 1 with a hyperplane H(c) then the non-trivial solution y determined by the non-zero constant vector c, namely

has a zero at x 0 of the multiplicity k. ■ In addition, suppose that then-dimensional vector space Vn is the euclidean space En with the norm lvl of the vector v = (v 1, ... , vn)T E Vn defined as

IV I == (v21 + ... +

2)1/2

Vn

,

and the unit sphere Sn- t is formed by all unit vectors in En. Great circles on the unit sphere are intersections of hyperplanes H(c) with Sn_ 1,

y(c)

==

H(c) n

y(c)

Sn-t .

In the next theorem we prove that for studying distribution of zeros of solutions of any equation from An it is sufficient to consider behavior of curves and great circles on the unit sphere only and not necessarily in the whole space Vn as the preceding result shows. For this reason, we introduce also the following

Definition 10.2.3 Let rO be the mapping of all non-trivial solutions y of P n(Y, x; I) onto the set of all great circles on Sn-t given by the formula

Theorem 10.2.4 (Representation on the unit sphere)

Let Pn(Y, x; I) be an equation from An n > 2, and y: I --+ !Rn be its fundamental solution, a curve in the euclidean space En. Then for the central projection v of the curve y onto the unit sphere Sn_ 1, V

==

YIIYI '

the following situation occurs: a non-trivial solution y of Pn(Y, x; I) has a zero x 0 E J of the multiplicity k, 1 < k ( < n - 1) if and only if the great circle r0 (y) has a contact of the order k - 1 with the curve v at the point v(x 0 ) of the same parameter x 0, 0 < k - 1 < n - 2, everything taking place on the unit sphere only.

221

10.2 REPRESENTATION OF ZEROS

Remark I 0.2.5

Let us point out that for this representation only central projection of y onto the unit sphere Sn- l is considered without any change of parametrization as used in Construction 7.5. l. Proof of Theorem l 0.2.4

With respect to Theorem 10.2.2 it is sufficient to show that if curve y has a contact of the order k - 1, 1 < k < n - 2, with hyperplane H(c) at a point y(x 0 ) then and only then its central projection y/lYI has the same contact at the point of the same parameter with the great circle y(c) c Sn- l. By a simple calculation we obtain the following relations for the central projection

v(x) = y(x )/ly(x )I . First

1/ly(x )I > 0 on I,

1/IYI

E

cn(I),

and then

= y'(x)/ly(x)I +

v'(x)

v"(x) = y"(x )/ly(x )I +

A 10 (x) y(x) A 21 (x)

y'(x) +

A 20 (x)

y(x)

k-2

v(k-l)(x) = y(k-l)(x)/ly(x)I +

L

Ak-1,;(x)

Y(i)

i=O

k-1 v(k)(x)

= y(k)(x )/ly(x )I +

L

Ak, ;(x)

Y(i) ,

i=O

where A;/ I ---+ ~, l < j < k, 0 < i < j - 1, are certain functions. Hence, if at some x 0 E J for some non-zero constant vector c we have

we also obtain C T V (Xo )

=

CT V '( Xo ) -_

... -_

CT V (k-1)( Xo )

and, of course, the converse is true as well.

■

222

10 ZEROS OF SOLUTIONS

Remark 10.2.6

Due to the form of global transformations it is clear that the distribution of zeros including their multiplicities of solutions of all globally equivalent linear differential equations from An is the same up to a cn_reparametrization and in fact,

r(y)

Sil- I Fig. 10.1

according to Theorem 10.2.4, it can be estimated by studying intersections of just one curve, v on the unit sphere with great circles on Sn-I· Hence oscillatory behavior of all equations from An can be globally described by considering certain curves in a compact topological space only, namely the unit sphere Sn_ 1, see Fig. IO. I. Let us introduce some simple consequences of the precedings theorems. We keep the notation of the theorems that means that y stands for a curve in Vn representing the fundamental solution of a given equation Pn(Y, x; I) E An, n > 2, and v denotes the central projection of y onto the unit sphere Sn- I in En= Vn.

Corollary I 0.2. 7 Equation P n(Y, x; I) is non-oscillatory on I if and only if no hyperplane going through the origin (no great circle on Sn_ 1) intersects curve y (curve v) at points of infinitely many parameters. Equation P n(Y, x; I) is strongly oscillatory on I if and only if each hyperplane passing the origin (each great circle on Sn_i) intersects curve y (curve v) at points whose parameters form an infinite set.

10.2 REPRESENTATION OF ZEROS

223

Corollary 10.2.8

Equation P n(Y, x; I) = 0 admits a non-vanishing solution on I if and only if there exists a hyperplane H passing the origin O (a great circle y* on S11 _ 1) such that curve y (curve v) lies in an open half-space in V11 determined by H (an open hemisphere with boundary y*). Corollary 10.2.9

Equation P11 (y, x; I) = 0 is disconjugate on I if and only if no hyperplane passing the origin in Vn (no great circle on Sn-I c E 11 ) intersects curve y (curve v c S11 _i) at points corresponding to more than n - 1 parameters including multiplicities (that are counted as orders of contact plus one each). Remark 10.2.10

Several further consequences of Theorems 10.2.2 and 10.2.4 could be introduced that would visibly describe oscillatory behavior of solutions of linear differential equations in large. For example, P. Hartman's principal solutions of these equations may be easily characterized by special hyperplanes or great circles in our correspondence given by mappings r or r O that have a certain order of contact in the end-points of the corresponding curves. Some applications of the above geometrical representation of zeros of solutions are given in the following paragraphs.

10.3 Second order equations Here we give some applications of our geometrical approach to distribution of zeros of the second order equations. The first of them plays an essential part in the new proof of Boruvka's criterion of global equivalence of the second order equations given in paragraph 5.1, then the Separation Theorem is presented as an obvious fact from elementary plane geometry, some of the other consequences of the preceding paragraph introduce, from the new point of view, a relation between certain elements of stationary groups and phases already derived by 0. Boruvka [2]. This relation, the Abel functional equation, enables to describe a connection between distribution of zeros and asymptotic behavior of oscillatory solutions of the second order equations in the Jacobi form in the next chapter. We keep the notation of mappings r and r0 from the preceding paragraph. The next theorem is a direct consequence of Theorems 10.2.2. and 10.2.4 for n = 2. Theorem 10.3.1

Consider a second order equation P 2(y, x; I) E A 2, and a couple y 1, y 2 of its linearly independent solutions being coordinates of a plane curve y the euclidean plane E2.

224

10 ZEROS OF SOLUTIONS

The central projection v of y is an arc on the unit circle S 1, and the following situation happens: a non-trivial solution y of Pi(y, x; I) has a zero x 0 E / if and only if the straight line r(y) going through the origin intersects the arc v at point of the same parameter x 0. Moreover, for linearly independent solutions y and y of P 2(y, x; I), the corresponding lines r(y) and r(y) are independent lines passing the origin, and also conversely.

0

Fig. 10.2

Looking at Fig 10.2 we may observe the following evident fact: when an arc

v of the unit circle S 1 in the euclidean plane E2 encircles the origin O (in positive or in negative direction) then between any two of its consecutive intersections v(x 0 ), v(x 2), x 0 < x 2, with a straight line H 1 going through the origin (if they exist) there is just one intersection v(x 1), x 0 < x 1 < x 2, with a straight line H 2 passing also the origin and independent to H 1• This simple fact is exactly the contents of the well-known Separation Theorem when geometrical representation given in Theorem 10.3.1 is applied: Corollary 10.3.2 (Separation Theorem)

Between any two consecutive zeros of a non-trivial solution of a second order equation there is exactly one zero of each linearly independent solution of the equation. Using the notation of type and kind of second order linear differential equations, Definition 5.1.1, we have the following direct consequence of Corollary 10.3. l that, in fact, gives again Theorem 9.1.4.

10.3 SECOND ORDER EQUATIONS

225

Corollary I 0.3.3

Let PiY, x; 1) be a second order linear differential equation and let y denote its fundamental solution. Denote by v the central projection of y considered as a curve in E2 onto the unit circle S 1• Equation P iY, x; I) is of finite type m and of general kind if and only if the euclidean length e(v) of the arc v satisfies

(m - 1) n < e(v) < mn. Equation P iY, x; I) is of finite type m and of special kind if and only if

e(v) = mn. Equation P 2(y, x; I) is one-side oscillatory if and only if the arc v runs infinitely many times around the origin only in one (positive or negative) direction as the parameter x ranges through the whole interval I. Finally, equation P 2 (y, x; I) is both-side oscillatory when v encircles the origin infinitely many times in both directions as x runs through the whole I. Now, let us recall from the coordinate approach presented in Chapter 9, that each second order equation P 2(y, x; I) E A2 (not necessarily in the Jacobi form) can be coordinated by couple

where (I 1 ) is the second order equation

u"

+u= 0

(I1 )

on J

globally equivalent to P 2(y, x; I), u(t) = (cost, sin t)T for t E J represents the fundamental solution of equation (1 1 ), and a = x1

and

y1(x) = O} ,

as follows from definition of conjugate points in paragraph 10.1. Due to the form of solution y 1 and relation (10.3.4 ), for such a number x 2 the following relations are satisfied:

h(x 2 )

+

c 21

= mTC + TC if h is increasing

h(x 2 )

+

c 21 = mTC - TC

or if h is decreasing .

228

10 ZEROS OF SOLUTIONS

Since h'(x) =/:- 0 on I, we have

h(x 2) + c21 = mn + n sign h' that together with relation (10.3.4) gives

h(x 2 )

-

h(x 1) = n sign h'.

If 'P(x 1 ) stands for x2, and x denotes any of those x 1 that admit the first conjugate point in /, we get

h(qJ(x)) = h(x) + n sign h', that implies (10.3.3) because h has the inverse.

■

Corollary 10.3.8 Let a second order equation Pi(y, x;

I)

E

A2 ,be coordinated as

P 2(y, x; I) = [(1 1 ), (Af, h)u] where u is formed by a couple of linearly independent solutions of (1 1 ). Then the dispersion qJ of equation P 2 (y, x; I) satisfies the Abel functional equation

h(qJ(x)) = h(x) + n sign h',

(10.3.5)

and it has the following properties:

'P(x) > x on I 1,

qJ E

C2(J 1) ,

qJ

If the parametrization h is of the class

'(x) > 0 on I 1 .

Ck(I)

then also

qJ E

(10.3.6)

ck(! 1).

Proof Under the assumptions of the corollary, Theorem 10.3.7 assures that relation (10.3.3) for h and qJ holds that gives the Abel functional equation (10.3.5). Relation 'P(x) > x on I 1 follows from Definition 10.3.4. If the parametrization h is of the class ck(J), then k > 2 and h is a ck-diffeomorphism of I onto J. Hence dispersion qJ belongs to ck(/ 1) because it satisfies relation (10.3.3). Finally, due to the Abel functional equation (10.3.5), we have

that gives there. ■

qJ'

> 0 on I 1 for either always positive h' on I or always negative h'

Remark 10.3.9 The dispersion qJ of an equation P 2(y, x; I) E A2 is uniquely determined, however the global transformation ( Af, h) u in coordination of the equation is not unique,

229

10.3 SECOND ORDER EQUATIONS

see Theorem 6.2.5. Hence, nevertheless a parametrization h is not uniquely determined by a given equation P 2 (y, x; I) from A2, for each such h the Abel functional equation (10.3.5) holds with the same dispersion ((J. For equations in the Jacobi form, these and further results were obtained by 0. Boruvka [2, 3 in the fifties together with the following refinement.

J

Theorem 10.3.10 Let

(p) be an equation of the Jacobi form from y"

+

p (X) Y

= 0,

p

E

CO (I) ,

(p)

and h denotes its phase. Then the dispersion functional equation

h(((J(x)) = h(x)

+

A~,

(f}:

I 1 ~ J of (p) satisfies the Abel

n sign h'

and

(10.3.7) Proof

In the case of a second order equation in the Jacobi form, a parametrization is a phase of this equation and it belongs to class C 3 (J), see Theorem 9.1.2. Hence by applying Corollary 10.3.8, relation (10.3.7) follows from (10.3.5) and (fJ E ck(/1) for k = 3. ■ We finish this paragraph with a fundamental theorem derived in 1961 by E. Barvinek [l] who extended B. Choczewski's result [l]. On the basis of the theorem we may construct second order equation with prescribed distribution of zeros of their solutions. This result ensures that in general there are no other properties of dispersions of second order equations than those given in Theorem 10.3.10. Theorem 10.3.11 Let

(f}:

I 1 ~ !R be a function defined on an open interval I 1

((J(x) > x

on/ 1,

((JEC 3(Ji)

and

(f}

1

(x) > 0

c

~

satisfying

on 1 1 .

Then there exists an equation of the second order of the Jacobi form, P 2 (y, x; I) EA~, defined on the smallest interval I containing I 1 u (fJ (I 1), whose dispersion is function ((J.

Proof

Suppose that (f}: 11 ~ !R satisfies conditions in Theorem 10.3.11. Consider the Abel functional equation

h(((J(x)) = h(x)

+

n,

230

10 ZEROS OF SOLUTIONS

for an unknown function h. Due to Theorem 2.7.1 proved in Appendix, under the assumptions on (f), there exists a solution h: I-+ IR of the functional equation such that I is the smallest interval containing I 1 u (f) (I 1), h E C 3(J), and h'(x) > > O on J. With respect to these properties, his a C 3-diffeomorphism of I onto an open interval J := h(I). Hence, according to paragraph 9.1, his a phase of the second order equation

y"

+

p0(x) y = 0

on J

(10.3.8)

whose coefficient p0 is given by formula (9.1.2). This equation, however, has a dispersion uniquely determined by

where h is a phase of the equation, see Theorem 10.3.10. This ensures that the dispersion of equation ( 10.3.8 ). ■

(f)

is

10.4 Third order equations There is an extensive literature concerning oscillatory behavior of the third order linear differential equations, see for example a nice surveys in C. A. Swanson's book [I] from 1968, in J. H. Barrett's paper [I] from 1969, or a very interesting monograph of M. Gregus [ 6] from 1981. Nevertheless of the fact, this area of research is still very alive and give arise several open problems. In this paragraph we use our geometrical representation of zeros of solutions to give an easy explanation of some results without lengthy calculation and we demonstrate also its capability to offer suggestions for possible ways of solving open problems. In 1948 G. Sansone [2] proved the following result, see also V. A. Kondrat'ev [2]: Theorem 10.4.1

There exists a linear differential equation of the third order on interval (-oo, oo) each non-trivial solution of which is oscillatory as the independent variable x tends to oo, in other words, this equation is strongly oscillatory as x -+ oo. Sansone's result shows that the set of the third order linear differential equations with only constant coefficients is not a sufficient model for studying oscillatory behavior of solutions of all equations in A 3 with variable coefficients, as it is the case of the second order equations. Indeed, with respect to Remark 10.2.6, distribution of zeros of solutions of any second order linear differential equation is given (up to C 2-diffeomorphism) by intersections of an arc v on the unit circle in the euclidean plane E 2 with straight lines going through the origin.

10.4 THIRD ORDER EQUATIONS

231

Of course, in three-dimensional space E3 the unit sphere S2 is a two-dimensional manifold and a curve v on it can vary much more than an arc on the unit circle S 1 in E 2. Let us return to Sansone's example. With respect to Representation Theorem 10.2.4 or its Corollary 10.2. 7, a strongly oscillatory third order equation P 3(y, x; I) exists if we find a curve v: I ---+ E 3 on the unit sphere S2 c E3, lvl = 1, such that v is of class C 3(J)

,

the Wronskian determinant of vis non-vanishing on/, and each great circle on the unit sphere S 2 intersects the curve vat points whose parameters form an infinite set. Since the curve v should be considered on the sphere S2, condition on non-zero Wronskian determinant at v(x 0 ) would be satisfied if v(x 0 ) is not a point inflexion of v on S 2 .

Fig. 10.3

A draft of such a curve v is on Fig. 10.3. This sufficiently smooth curve (v E C 3(I)) without points of inflexions (w[v ](x) #- 0 for x EI) is some kind of a "prolonged cycloid" encircling the origin on the unit sphere along its equator infinitely many times (for example periodically) as the parameter x runs to the right-end (to the left-end or to both ends) of the interval I. This ensures that, nevertheless each great circle of S2 may have a finite number of intersections (however, at least one) with the curve v, the set of the parameters of the intersections is infinite. We do not state that this description of such a curve v represents a proof of Theorem 10.4.1, however, it certainly points on the existence of a strongly oscillatory third order equation that can be explicitely given (if needed) by

232

10 ZEROS OF SOLUTIONS

writing coordinates of such a curve and evaluating coefficients of the equation whose fundamental solution is the curve v. We can also see that the length of interval I is not essential. The existence of a strongly oscillatory equation implies that there does not exist factorization of every third order linear differential operator IP3 (y, x; 1)1 into linear differential operators of the second IQ2(y, x ; 1)1 and the first IS 1(y, x ; /)I = y' + s0(x) y orders.

IP3I = IQ2I IStl neither near an end point of interval I , otherwise each third order equation P3(y, x; I) would admit at least one non-oscillatory solution on I according to Proposition 2.6.4. In connection with factorization of the third order linear differential operators one may also ask for factorization of P3 in the reverse order,

into linear differential operators of the first and then second orders. Due to Proposition 2.6.4, if such a factorization is possible then for the adjoint operators l~I, IS~I, and IQ~I to IP3 1, 1S11, and IQ 2 1, respectively, the following factorization holds

l~I = IQ~I IR~I. Hence a natural question arises whether an equation of the third order exists which is strongly oscillatory together with its adjoint equation. This problem was proposed by J. M. Dolan [l]. We know the affirn1ative answer to the question:

Theorem 10.4.2 There exists a linear differential equation of the third order which is strongly oscillatory together \Vith its adjoint equation.

Proof was given by V. A. Kondrat'ev [l , 2]; here we illustrate our geometrical reasoning of the affirmative answer. For the existence of such an equation we consider again a curve v: I -+ S, on Fig. l 0.3 which is the fundamental solution of a strongly oscillatory third order equation P 3(v, x; I). Moreover, suppose that the parameter in the curve v is the length parameter that, as we know from Construction 7.5.1 , can always be arranged on the whole interval of definition of v without any additional assumptions. Hence lv(x )I = l and lv'(x )I = l for all x E I , and the vector product u(x) := v(x) x v'(x) is the unit vector, that means that the curve u: I-+ E3 lies also on the unit sphere S2. According to Proposition 2.6.3, coordinates of v°(x) = v(x) x v'(x)/w[ v ](x) are three linearly independent solutions of the

J 0, for which the conjugate points qJk(a) are defined only for finite number of k. Fig 10.9 shows how such an equation may be constructed. In the plane 17 1 = 1 (not passing the origin) of E3 with coordinates (r, 1, r, 2, r, 3 )T there are two "h

(1,0,0)

(1,-1,0)

. . __

-

---~---------...,..------,----------:------"12 I, ...........

=

-1

I

......

..... ..... ....__

Fig. 10.9

lines: r, 2 = -1 and r, 2 = - 3. Curve y: I ---+ E3 lying in the plane and being of the shape as drawn in Fig. 10.9 which passes the point (1, 0, 0) as x = a, may serve as an example of a fundamental solution of the required equation. Of course, what is said is not a proof, however it suggests how to construct such an equation explicitly, see [l l]. The following problem was also proposed by J. H. Barret in [l ]. "If equation P iY, x; (a, oo)) is oscillatory as x ---+ oo then is its adjoint equation ~(ya, x; (a, oo )) also oscillatory for x ---+ oo?" M. Hanan [I] and M. Svec [5] answered this question in the affirmative for equations in classes I and II. However, in general, the answer is negative.

Theorem 10.4.6 There exists an oscillatory third order equation P 3(y, x; (a, oo )) as x that its adjoint equation is non-oscillatory on (a, oo) for x ---+ oo.

---+

oo such

Proof was given in 1969 by A. Ju. Levin [3] who used factorization of linear differential operators. Here we mention another proof based on our geometrical ap-

10.4 THIRD ORDER EQUATIONS

239

proach, see also [11]. Calculation of coefficients of required equation is done from coordinates of curve y obtained as the vector product u x u', where u is a curve in the plane 17 1 = 1 in E 3 coordinated by (17 1, 17 2, 17 3 )T. Curve u is of

(1,0,0)

Fig. 10.10

the shape in Fig. 10.10 and it is chosen in such a way that u is not intersected infinitely many times by any plane passing the origin in E3, and moreover, curve y = u x u' is intersected by a plane passing the origin in points of infinitely many parameters. Then Theorem 10.4.2 shows that, if u is sufficiently smooth and without points of inflexion, curve y may serve as the fundamental solution of a required equation P 3(y, x; (a, oo) ). ■

Fig. 10.11

240

10 ZEROS OF SOLUTIONS

The same curve u in Fig. 10.10 may also serve for demonstration of another fact that cannot occur for equations of the second order, namely:

Proposition 10.4. 7 There exists a linear differential equation P 3 (u, x; (a, b)) of the third order which is non-oscillatory on (a, b) and still there does not exist a number c E (a, b) such that this equation is disconjugate on (c, b ).

Relation between disconjugacy and non-oscillation of the third order equations has been studied by many authors. However our geometrical approach often offers correct results without complicated constructions that may include incorrectness especially when some c - t5 proofs are too long. For example, Fig. 10.11 shows that it holds

Proposition 10.4.8 There cannot happen for any third order linear differential equation that zeros of every couple of its linearly independent solutions separate each other.

Fig. 10. 12

In Figs. 10.12 and 10.13 you can observe that for equations of the third order it may really often happen that by a number c and the first conjugate number ({) 1(c) to it, the solution y that vanishes at c and at ({) 1(c) is, up to a non-zero constant factor, uniquely determined. Such a solution y is also called the extremal solution. However, there are exceptions as Fig. 10.14 shows. G. D. Birkhoff [l ], Z. Mikulik [l] and others studied self-adjoit third

241

10.4 THIRD ORDER EQUATIONS

order linear differential equations, that coincide (for the third order) with iterative equations,

+

y"' with p

E

4p(x) y'

+

2p'(x) y = 0

C 1(J). They obtained also the following result:

r(s)

Fig. 10.13

r(_v)

Fig. 10.14

242

JO ZEROS OF SOLUTIONS

Theorem 10.4.9

Equation (p )[3] has always a non-vanishing solution on the whole interval of definition. If a non-trivial solution has a zero then either all zeros of the solution are simple, or all zeros of the solution are double. Let us give a visible description of the situation. According to Corollary 4.3.5 and Theorem 4.3.6, any iterative equation (p )[ 3 ] has three linearly independent solutions uf, u 1u2 , u~, and hence also

uf, j2u 1u2, u~, where u 1, u 2 is any couple of linearly independent solutions of the second order equation u"

+

(p)

p(x) u = 0 .

Due to Theorem 9.1.2, solutions u 1 and u 2 of

(p)

can be expressed as

u 1(x) = lh'(x )1- 112 cos (h(x)) u2(x) = lh'(x)l- 112 sin (h(x)),

XE

I'

where a phase his a C 4-diffeomorphism of I onto an interval J = three linearly independent solutions of (p )[3] can be written as )21h'l- 1 cos h . sin h,

lh'l- 1 (sin

h(I). Hence

h) 2 .

The curve yin E 3 that corresponds to the solutions lies on the same cone as the curve v: J -+ E3

v(t) = (v 1(t), vit), v3 (t))T, V 1 ( t)

= (COS t )2 ,

v2(t) = J2sint.cost, t

E

J

= h(I).

Since lh'I =I= 0 on I and lh'I E C 3(J) and h is a C 4-diffeomorphism of I onto J, distribution of zeros of solutions of equation (p )[ 3 ] is up to this reparametrization h given by behavior of curve v with respect to planes in E3 going through the origin, as follows from Theorem I 0.2.2 and Remark 10.2.3. However curve v lies in the plane 77 1 + 77 3 = 1 of E 3 and, moreover, it is on the unit sphere S2, because

vi(t)

+ v~(t) + v~(t)

= ((cos

t) 2 + (sin t}2) 2

=

1.

243

10.4 THIRD ORDER EQUATIONS

Hence v: J

----+,

S 2 is a circle (or an arc on it according to the length of interval

J) in a plane not passing the origin, and evidently:

there exists a plane going through the origin that does not interesect v (e. g. the plane 17 1 + 17 3 = 0), once a plane passing the origin intersects curve v then either this intersection is simple and all other intersections are simple or the plane touches the circle, the curve vat some point and then all other intersections are contacts of order 1. With respect to Theorem 10.2.4, this gives exactly Theorem 10.4.9. Zeros of solutions of iterative equations of arbitrary orders are studied in the next paragraph. Till now we have demonstrated the capability of our geometrical method to show the existence of equations of some prescribed properties of zeros of their solutions. The solution of the following problem proposed by J. M. Dolan in [I] shows how the geometrical representation can be used together with other methods of topology of dynamical systems to prove the non-existence of a certain equation. His question reads as follows: "Does there exist a linear third order equation with the property that every two-dimensional subspace of its solutions is weakly oscillatory?" The negative answer to it is given in the next lemmas of a geometrically-topological nature, [ 15]. First some definitions and notation. Topology on S 2 is the relative topology induced by the euclidean space E3, e"nt(U) and ct(U) mean the interior and the closure of set U c E3, respectively. With accordance to notation introduced in paragraph 10.2, let H(c) and y(c) = H(c) n S 2 denote the plane going through the origin and the great circle on S 2, respectively. The angle {: (c 1, c2 ) of two vectors c 1, c2 E E 3, c 1 # 0, c2 =I= 0, is defined as

-t:(c 1, c2 ) = arccos ([c 1 • c2]/(lcil. lcl)) E [O, n],

cf.

where [ c 1 . c 2 ] denotes the scalar product of vectors c 1 and c 2, i. e. c 2. The (angular) deviation d(H(ci), H(c2 )) of two planes H(c 1), H(c 2) (or d(y(ci), y(c 2)) of great circles y(ci), y(c 2)) is defined as the angle -r (c 1, c2). For a non-empty set U c S2 define its circular radius r(U) to be r ( U)

:=

inf {SU p { -r (V,

C) ; V E

U} ;

C E

S 2} .

Evidently r(U) = 0 if and only if U is a point on S2 . Further denote a spherical cap cafi (a, (2) with the centre a (2 as

E S2

and radius

244

10 ZEROS OF SOLUTIONS

Since for a given non-empty set U

c

S 2, the mapping

c ~ sup { ~ (v, c), v E U},

defined on the compact set S2 is continuous, there always exists such a point c0 E S2 that

sup { ~ (v, c0 ); v E

U} = r(U),

that means ~ (v, c0 ) < r(U) for all v EU. This point c 0 is uniquely determined, otherwise, for two different c 01 and c 02 (with the same r(U)) we would get contradiction with the definition of r(U). Just for the sake of brevity let us introduce the following notation for special sets in E3 and on S 2:

S_(c) == E_(c) n S2 . Let U be again a non-empty set, U c5 of U as

c

ct(S + (c0 ) ). Define the angular diameter

Evidently c5(U) = 0 if and only if U is a subset of a great hemi-circle on S 2 (i. e. an arc on a great circle of the length < n ). For a continuous curve u: [ a, b) ----+ S2, u E c 0 ([ a, b)), define its limit set Q(u) for x ----+ b in accordance with the theory of dynamical systems as

Q(u):=

n

ct{u(x); xE(c,b)}.

cE[a, b)

Each point p E Q(u) is called a limit point of u. Let us note that our curve u may intersect itself that is not the case for dynamical systems. Nevertheless the following lemmas hold.

Lemma 10.4.10 For each continuous curve u E connected, and closed.·

c-0([ a,

b)) on S 2 the limit set Q(u) is non-empty,

Proof

is essentially the same as for dynamical systems due to the compactness of S 2, see e.g. P. Hartman [2]. Self-intersections of u do not play any role in the proof. ■

10.4 THIRD ORDER EQUATIONS

245

Lemma 10.4.11 0

Let u E C ( [ a, b)) be a continuous curve on S 2, u: [ a, b) -+ S 2 . For every point PE S 2, let there exist two great circles on S2, y(cp) and y(c;), both containing point p, such that u intersects y(cp) at points of finite many parameters, i.e. {x E [ a, b ); u(x) E E y(cc)} is finite, and u intersects y(c;) for infinite many parameters, i.e. {x E [ a, b ); u(x) E y(c*)} is infinite. P Then the angular diameter of the limit set of u as x -+ b zs zero,

a(Q(u)) = 0. Proof

Choose p E S2 . Since {x that either

E [ a,

b) ; u(x)

E

y(cp)} is finite, there exists

xP such

(10.4.1) or

Without loss of generality, let cP be always oriented such that ( 10.4.1) holds for every p E S 2. Hence we have Q(u) c ct(S+(cp)) for every p E S 2. First, if the circular radius of Q(u), r(Q(u)}, is zero then Q(u) represents a point on S 2 and a(Q(u)) = 0. Secondly, let r(Q(u)) = r0 > 0, where r0 Q(u). We already have Q(u) c ct(S+(c 8 ) keeping notation C8 from Lemma 10.4.11. Hence Q(u) c cafi (e, r0 ) n ct(S+(c8 )). Take arbitrary f E ,:nt(cafi (e, r0 ) n n ct(S + (c 8 ) ). If y(c,) n (t·nt(cafi(e, r0 )) n y(c 8 )) # 0 or if S _ (c,) :::> :::::J (caft(e, r ) n y(c 8 )) then there exists another cap, cafi(e 1, r 1 ), containing 0 Q(u) with r 1 < r 0 that contradicts to r(Q(u)) = r 0 . Hence we have

and moreover, for every f E int(caft(e, r0 ) n ct(S+(c8 ))) the closed hemi-sphere ct(S+(c,)) contains the arc (cafi(e, r0 ) n y(c8 )) on the great circle y(c 8 ). Since point f can be chosen arbitrary close to point e, and the limit set Q(u) is always contained in the set caft(e, r0 ) n ct(S+(c8 )) n ct(S+(c,)) where r0 > 0, the value a(ct(S+(c 8 )) n ct(S+(c,))) = d(y(c8 ), y(-c,)) can be arbitrarily small. Hence a(Q(u)) = 0 also in case 0 < r(Q(u)) < n/2, which finishes the proof of the lemma. ■

246

10 ZEROS OF SOLUTIONS

Lemma 10.4.12

Under the assumptions of Lemma 10.4.11, limit set Q(u) is a closed arc of a great circle of S 2 of the length 2r(Q(u)) < n, i. e. either (i) Q(u) is a point on S 2 (r(Q(u)) = 0), or (ii) Q(u) is a closed arc of a positive length < n, or (iii) Q(u) is a closed great hemi-circle (r(Q(u)) = n/2).

Proof is a direct consequence of Lemmas 10.4.10 and 10.4.11, since Q(u) is a non-empty, closed and connected set on a closed hemi-sphere of S 2 having its angular diameter zero. That means that Q(u) is a (non-empty) closed arc of a great circle on S 2 of the length twice of its angular radius r(Q(u)) < n/2. ■ Now we prove the following lemma

Lemma I 0.4.13 There does not exist a curve u complying with the assumptions of Lemma I 0.4.1.

Proof proceeds such that we show that neither of the cases (i), (ii) and (iii) in Lemma I 0.4.12 can occur. Hence, suppose there exists a curve u satisfying the assumptions of Lemma

I 0.4.11. Case (i). Q(u) is a point, say p E S 2 . Let a point q, q =I= p, be chosen on the circle y(cp), still in keeping notation of Lemma I 0.4.11. Consider all great circles on S 2 containing point q. Let y denote one of them. If p ¢ y then there is a finite number of intersections of curve u with y since y does not contain any limit point from limit set Q(u) that is, in fact, point p. However, if p E y then again the set {x E [ a, b); u(x) E y} is finite because y = y(cp), cf.definition of the great circle y(cp) in Lemma 10.4.11. Hence we get a contradiction with our assumptions on curve u concerning great circles on S? containing point q. Case (ii). De~ote by y0 the great circle of S 2 containing the closed arc Q(u) being of positive length less than n, p and q denoting the end-points of Q(u). a. Let the set of parameters of intersections of u with y 0 be infinite. Choose a point r E Q(u), r =I= p, r =I= q, and consider the set of all great circles on S 2 containing point r. Let y be one of the great circles. If y coincides with y 0 then the set {xE

[a,b);

u(x)Ey}

is infinite due to our assumption on y 0. If y =I= y 0, then again the set of parameters of intersections of u with y is infinite since both p and q are limit

10.4 THIRD ORDER EQUATIONS

247

points of u and lie in opposite hemi-spheres determined by great circle y. Hence also suitable neighbourhoods "Y(p) and 1',,.(q) of p and q on S2, respectively, lie on opposite hemi-spheres and curve u(x) as x approaches b goes infinitely many times from "Y(p) into "Y(q) and back producing, because of continuity, an infinite set {xE [a,b); u(x)Ey}. Thus case (ii)a. is impossible, since curve u does not satisfy the requirement concerning great circles going through point r. b. Let the set {x E [a, b ); ~(x) E y0} be finite. Choose a point r E y0 such that r ¢ Q(u) and r ¢ -Q(u). Every great circle containing r has only finite number of intersections with curve u since either it coincides with y 0, or it does not contain any point from Q(u ). But this contradicts to the assumption on great circles that is not satisfied at this r. Case (iii). It remains to consider Q(u) to be a hemi-circle of the great circle 1° on S2 with end-points p and q being, of course, opposite each other, p = - q. Consider y(c;), the great circle corresponding top in accordance with notation in Lemma I 0.4.11. Evidently q = -p E y(c;). Choose a point r on y(c~) such that r -=I- p and r -=I- q. If y is an arbitrary great circle on S2 containing point r, then either it coincides with y(c~), or it separates some vicinities oflimit points p and q on the unit sphere. In both cases, y is intersected by curve u at points with infinitely many parameters, which contradicts to the existence of great circle y(c,) that has to admit only finite number of parameters of intersections with curve u. Hence neither case (iii) of Lemma 10.4.12 can occur. Since all possible cases were exhausted, Lemma I 0.4.13 is proved. ■ Lemma I 0.4.13 states in fact more than we need to answer J. M. Dolan's problem. For our purpose it is sufficient to have the following consequence of the proceeding lemma. Corollary I 0.4.14

There does not exist a curve v: [a, b) ~ S2 c E 3, v E c3(J), w[ v ](x) -=I- 0 on [a, b) with the property that for every point p on S2 there exist two great circles on S2 containing p such that one of them intersects v at points whose parameters form a finite set, whereas the second great circle intersects vat points corresponding to an infinite set of parameters. Due to Representation Theorem I 0.2.4, this corollary is equivalent to the negative answer to the proposed question: Theorem I 0.4.15

Let (a, b) be an interval, - oo < a < b < oo. There does not exist a linear differential equation of the third order on (a, b) with the property that every two-dimensional subspace of its solution contains both oscillatory and non-oscillatory solutions as the independent variable x tends to b.

248

10 ZEROS OF SOLUTIONS

10.5 Iterative nth order equations Zeros of solutions of linear differential equations of the nth order have been studied by many authors, some of them are mentioned at the beginning of this . chapter. Here we restrict ourselves only on demonstration how some results can be obtained on the basis of the geometrical representation given in paragraph l 0.2. Similar kind of results were already obtained using another approach by V. Seda [4] in 1963. First we shall deal with oscillatory or non-oscillatory behavior of iterative equations of the nth order. Theorem 10.4.9 describes zeros of iterative equations of the third order, here we deal with the general case. Consider an iterative equation of the nth order, (p)[n] = (p)[n](y, x; I), iterated from the second order equation

(p)

u" + p(x)u = 0, Let h denote a phase of equation

(p ).

Theorem 10.5. l

For odd n, lh'l(l -n)/2 :

/ ~

IR

is a non-vanishing solution of iterative equation (p)[n]_ Moreover, there always exist n linearly independent non-vanishing solutions of (p)[n]_ If n is even then the oscillatory character of each solution of (p)[n] is the same as the oscillatory character of (any of) solutions of (p ), namely if(p) is non-oscillatory on I then (p)[n] is non-oscillatory on I, if(p) is one-side (both-side) oscillatory on I, then (p )Ln j is strongly one-side (strongly both-side) oscillatory on I. Proof Let n > 2 be an integer. According to definition of a phase h of equation (p) and iterative equation (p )[n J, Yi(x) = lh'(x )l(t -n)l2(cos h(x) r-i (sin h(x) )i-l , i

= 1, ... , n, are n linearly independent solutions of (p)[nJ, h being a cn+l_diffeo-

morphism of I onto J = For odd n we have

h(I). -

lh'(x)l(l-n)/2 = = lh'(x)l(l-n)/2((cos h(x)) 2 + (sin h(x)) 2)(n-l)/2 =

249

10.5 INTERATIVE NfH ORDER EQUATIONS

en

= lh'(x)1 0, if for each x belonging to D also x + d and x - d are elements of D. The empty set is a periodic set with an arbitrary period.

Theorem 10.6.2 Let k > 0 be an integer and a curve u: lR ➔ Vn not passing the origin O E Vn be of class Ck(lR).

251

10.6 PERIODIC SOLUTIONS OF NTH ORDER EQUATIONS

Each hyperplane H(c ), 0 =I= c E Vn, going through the origin O intersects u at points whose parameters form a periodic set D(c) always with the same period d > 0 not depending on c, not all D(c) being empty, if and only if curve u is of the form

u(x) = p(x) exp {g(x)}

!Rn is a periodic or half-periodic vector functicn with period d > 0, Ck(J), g: [R -+ [R and g E C\!R).

where p: !R p

E

(10.6.1)

-+

Proof

(~ ). Take a non-zero vector c E Vn such that the periodic set D(c) of parameters of intersections of hyperplane H(c) with curve u is non-empty. Choose arbitrarily x 0 E D(c). Then x 0 + d E D(c) and the rank of the matrix

is l. Indeed, 0 is excluded by requirement u(x) =I= 0 for all x rank is 2, then the system of two linear equations vT. vT .

E

!R and if the

u(x0 ) = 0 u(x0 + d) = 1

has a solution v =I= 0. However, this contradicts to the assumption that D(v) is periodic with the same period d because then the right side of the second equation of the system would be zero. Through each point u(x ), x E !R, there goes a hyperplane H(c) for some c, 0 =I= c E vn. Hence ianl (u(x),

for all x that

E

u(x + d)) =

1

!R. That means that there exists a scalar funcition s: !R

s(x) u(x) = u(x

+ d) on

!R.

Hence

ls(x)I = lu(x + d)l/lu(x)I > o because ju(x )I > 0 for all x

s(x)

=

= £

r(x) ,

E

!R, and we may write

-+

!R such

252

JO ZEROS OF SOLUTIONS

where r == Isl is a positive function on lR of class Ck(lR) and e 1 or always - 1 on lR. For function G: lR ---+ lR defined by

G(x)

==

=

signs is always

In r(x) on lR

we have G E ck(rR) and u(x

+ d) = e u(x) exp {G(x)}.

Consider the following Abel functional equation

g(x + d) = g(x) + G(x)

(10.6.2)

for a unknown function g: lR ---+ lR. According to Theorem 2. 7.1, there always exists a solution g: lR ---+ lR of the functional equation that is of class Ck(lR ). Let g denote one of the (in fact, infinitely many) solutions of (10.6.2) of class Ck(lR ). For p == u exp {-g} we get

and

p(x + d) = u(x + d) exp { -g(x + d)} = = e u(x) exp {G(x)} exp {-g(x) - G(x)} = e p(x) for every x E lR. Hence p is periodic or half-periodic with period n on lR, and we may write u(x) =

p(x) exp {g(x)} ,

as required in Theorem 10.6.2. ( 2, defined on iR. Suppose that zeros of each non-trivial solution of the equation form a periodic set with period d > 0, the period being the same for all solutions.

253

10 PERIODIC SOLUTIONS OF NTH ORDER EQUATIONS

Then every n-tuple of linearly independent solutions y can by written in the form

f X

y(x) = q(x) exp {- ~

p,,_ 1(s) ds}

(10.6.3)

XO

where q: lR -+ [Rn is a periodic or half-periodic vector function of period d belonging to class C 1(rR) and Pn-t is coefficient of y(n-I) in equation P 11 (y, x; lR). If the vector function y: lR -+ [Rn given by formula ( 10.6.3) is of class C11 (lR) and the Wronskian determinant w [y] is non-vanishing on lR then also the converse is true.

Proof

Let y: lR -+ [Rn denote any n-tuple of linearly independent solutions of equation Pn(Y, x; lR) written as a column vector function. Put u == y and k == n in Theorem 10.6.2. Since all assumption of the theorem are satisfied, we have

y(x) -= p(x) exp {g(x)} where p: lR Then

-+

[Rn,

p(x

+ d) = + p(x) on

lR, p

E

cn(lR ), g: lR

f

-+

lR, g E C/1(~ ).

X

0

,f-

w[y ](x) =

e"Y(x)

w[p ](x) = c0 exp {-

p,,_ 1(s) ds}

xo

for a suitable non-zero constant c0 due to Propositions 2.6.1 and 2.6.2 where w[p] is a periodic or half-periodic non-vanishing function of period d on ~ belonging to class ct(~). Hence we may write eg(x)

= (

Co

w[p ](x)

)t/n exp{-~fx p _

11

(s)ds}.

1

n XO

If we put

q(x)==(

co

)l/11p(x)

w[p ](x) we get formula (10.6.3) with q: ~-+ ~'1, q E ct (rR), q(x + d) = +q(x), which finishes the proof of the first part of Theorem l 0.6.3 .. 11 Under the assumptions y E C11 (lR), w[y] =I= 0 imposed on y: lR -+ lR satisfying (10.6.3), the converse is a direct consequence of Theorem 10.6.2 and

254

10 ZEROS OF SOLUTIONS

the existence of an equation from An having this y as its n-tuple of linearly independent solutions. ■

Remark 10.6.4 Formula (10.6.3) estimates the rate of growth of all solutions of equation Pn(Y, x; !R). The following consequences of Theorem 10.6.3 are obtained by special requirements on coefficient Pn- 1•

Corollary 10.6.5 If zeros of each non-trivial solution of equation y(n)

+ Pn-i(x) y(n-l) + ... + p0(x) y = 0

n > 2, P;

E

on 1R

cO(IR) for i = 0, ... , n - 1, form a periodic set of period d > 0

with the same d for all solutions, and if

f X

p._,(s) ds is bounded on some I

c

IR

xo

then every solution of the equation is bounded on I. If, in addition,

f

x+d

p._,(s) ds = 0 for all x

E

IR

X

then each solution is periodic or half-periodic on 1R with period d. In particular, ifin the above situation coefficent Pn-l is identically zero on~, i.e., the equation is from class A~, then each solution of such an equation is periodic or half-periodic on 1R with period d.

11 Related results and some applications

The methods and results presented in the preceding chapters have been successfully applied in several situations, see e.g. [2, 5-15]. Some of the related results will be mentioned here. In the first paragraph we deal with a relation between asymptotic properties of solutions of the second order equations and distribution of zeros of their solutions. The Abel functional equation plays an important role in this investigation. Then we derive certain integral inequalities useful in the sequel. The third paragraph is devoted to the affine differential geometry of the plane curves. Then we prove a generalization of Blaschke's and Santal6's isoperimetric theorems for dosed curves and introduce some related results. Finally we comment results in other areas of mathematics where either methods or results of the preceding chapters were applied: functional-differential equations, linear differential systems and their transformations, the second order systems and their relation to the differential geometry, generalized linear differential equations with quasi-derivatives, the theory of functional equations. An interesting area of research is based on studying special algebraic structures generated by linear differential equations and their transformations.

11.1 Asymptotic properties and zeros of solutions of second order equations Consider linear differential equation of the second order in the Jacobi form y"

+

p (X) y

=

0,

(p)

I= (a, b), - oo < a < b < oo. Suppose always in this paragraph that equation (p) is osci11atory as x tends to the right end b of interval J. Let (f}k denote the dispersion of order k of equation (p), see Definition 10.3.4 and Remark 10.3.6. In fact, (f}k(x 0 ) is the kth conjugate point to x 0 E J. Due to osci11atory behavior of equation (p ), (f}k is defined on the whole interval J for each positive integer k.

256

11 RELATED RES .LTS A D SOME APPUCATIO S

Theorem 11. 1. 1

Let k > l be a fixed integer and let the dispersion 'Pk of order k of equation (p) satisf) one of the follol, ing conditions (i)

0 (ii) everJ solution of equation (p) is bounded on (c, b) (iii) b = oo and even non-trivial solution of (p) is unbmmded on (c b) respectively. Proof Denote by h one of the phases of equation (p) defined in 9.1.} . According to paragraph 9.1 h is a C 3-diffeomorphism of interval I onto an interval J and

lh'I- - cos Ji,

lh._'1- 112 sin h

are linearly independent solutions of (p ). Hence

x el

(11.1.1)

with arbitrary constants c1 and c2, is the general solution of equation (p). Let 'Pm denote the dispersion of order m m ~ I , of equation (p ). Oscillatory charac·ter of equation (p) implies that 0 on IR.

+ 1

holds for every non-negative t and v E (0,1 ], the equality occurringjust for t = 1 (and arbitrary v E (0,1 ]). Hence, with respect to relation (I 1.2.3), we may write

f k2 ( f 71:

71:

r(x))v

0


in the assumption of Theorem 11.4.1 then

f 7t

2n ltr(v)I = V

p(t) dt

0

is not bounded for a fixed volume V > 0, since J~p(t) dt is not bounded from above on the set of coefficients p of corresponding equations (p ). This was shown in [9] on equations from Example 9.3. 7 with k = 2, see also Theorem 11.4. 7. Let us note that for v = 1/2 relation ( 11.4. l) does not depend on V. Moreover, due to Theorem 11.3.2 and Remark 11.3.4, we have also the following consequence of Theorem 11.4.1.

Corollary 11.4.3 Let v: I --+ V2, v E C 2 (J) with det (v, v') =I- 0 on I be a closed curve with index m and having Property !ff>. Then for arbitrary non-singular constant 2 by 2 matrix A, Boruvka 's centroaffine length t 112 satisfies the following relation

282

11 RELATED RESULTS AND SOME APPLICATIONS

the maximum being achieved just for ellipses centred at the origin and considered as closed curves with index m.

Remark 11.4.4 Let us recall here Isoperimetric Theorem of L.A. Santal6 [2: p. 96], that states: "Let ls denote the unimodular centroaffine length (in the sense of Santal6, see Remark 11.3. 7) of a convex centrosymmetric simply closed curve containing the origin. If V denotes volume of the area bounded by the curve then (11.4.2)

and the equality is valid only for ellipses with centres at the origin." For curves v that admit a parametrization of class C 2, Santal6's result is a consequence of Theorem 11.4.1. Indeed, due to Remark 11.3. 7 and relation ( 11.3.1 ), 18 = ½lt 1(v;x0,xi)I, where radius vector v exhibits just one orbit around the origin when x runs from x 0 to x 1. Hence v is closed with index 1. Curve v is also cen trosymmetric that means that it intersects each line passing the origin in two points opposite to each other, volume of both of parts of interior of the curve divided by the line being the same. Hence curve v has Property &' for which, in fact, would be sufficient to have only one line of this position. Applying Theorem 11.4.1 for m = 1 we get

the equality holding just for ellipses centered at the origin, that implies (l 1.4.2):

½lt 1(v)I

V

< n2

including the condition on equality.

Remark 11.4. 5 Isoperimetric Theorem 11.4.1 is also a generalization of W. Blaschke's result [1: p. 61 "If 18 denotes the unimodular affine length (in the sense of Blaschke, c. f. Remark 11.3.6) of a convex simply closed curve v, and if Vis volume of the area bounded by the curve, then

J:

(11.4.3)

and the equality occurs just for ellipses."

283

11.4 ISOPERIMETRIC THEOREMS

This result can be derived from Theorem 11.4.1 as follows. W. Blaschke already assumed that v E C 2(J). Because his length /B is an unimodular affine invariant of convex closed curves as shown in Remark 11.3.6, we may cho~se the

~ O' \ \

\

' /

/

--- ----

/ I

b* l

O*

Fig. 11.4

origin O inside the curve. Then curve v satisfies the assumptions of Proposition 11.3.9 and it admits a reparametrization u = v oh corresponding to a second order equation (p) in the Jacobi form. Since v is a convex simply closed curve, its index is 1 and coefficient p is non-negative. Moreover, there exists a straight line passing the origin O (inside curve v) such that it divides the area bounded by curve v into two parts of the same volume. Indeed, if H0 is a fixed straight line passing the origin then either it has the above property, or the divided parts of interior of curve v are of different volume. Due to continuous dependence of volume of one of the parts on the angle 0. As a background for a generalization we may consider the fact that the change x 1---+ x + d of the independent variable in this case does not change the equation or the system (supposing there are defined on the whole [ij ). For a generalization of the Floquet Theory to nth order linear differential equations from A 11 we [20] may take the requirement on the existence of a global transformation (not necessarily only translation x 1---+ x + d) that globally transforms an equation from An into itself, see Chapter 6. That means that a generalized Floquet Theory can be.considered for those equations from An whose stationary groups are not trivial. Several authors dealt with generalizations of the Floquet Theory to different types of equations, M. Laitoch [l ], J. Moravcik [2], S. Stanek [6-8], and others. From this respect, there were extensively studied especially the second order equations in the Jacobi form by using 0. Boruvka's results [2-4]. Invariant differential equations on homogeneous manifolds were investigated by S. Helgason [ 1], changes of the independent variable, 'motions', that leave equations from An invariant were studied by J. Posluczny and L. A. Rubel [1]. Problems concerning a possible generalization of Floquet Theory were considered by D.R. Snow [1]. Linear differential systems were studied in depth also

290

11 RELATED RESULTS AND SOME APPLICATIONS

by N. A. Izobov [1-5], V. A. Yakubovich and V. M. Starzhinskii [l], M. I. Elshin [I] and N. P. Erugin [1-3]. Another very interesting field of research is a factorization of linear differential operators with respect to their transformations, especially a study of linear differenatial equations that can be transformed into equations with constant coefficients. A serious contribution to this area was done by L. M. Berkovich [1], A. Ju. Levin [1-3], J. Moravcik [1-5], V. Seda [3] and others. Linear differential equations of the nth order occurred also in characterization of functions F: I x J -+ IR of two variables that admit a decomposition into the form of a finite sum of products of functions!; and 9; of just one variable, k

F(x, t) =

I

J;(x) 9;(t)

i= 1

for an integer k. For a sufficiently smooth function F the condition for such a decomposition reads as follows: the determinant of the Wronskian type is identically zero on the whole rectangle I x J,

0 on 11, functions has the inverse s- 1 defined on an open interval / 2 = s(J 1) and s- 1 E Ck(/ 2 ). If x 1 E [x 0 , s(x 0 )) then m = 0 and x 1 = x*. If x 1 > s(x 0 ) then s- 1(x;) is defined and, due to condition s'(x) > 0, we have s- 1(x 1) > x 0 . Either s- (x 1) E [x 0 , s(x 0 )) = /* and then 1 m = 1 and x* = s- (x 1), or s- 1(xi) > s(x 0 ). Then s- 2 (xi) = s[- 2 J(x 1) is defined and we may proceed as above to find m E 7l__ such that sE -mJ(xi) = x* E /*. Such an integer m E 7L actually exists since it cannot happen that for all positive integers m the following relation holds s[ -mJ(x 1 )

> s(x 0 )

because s[ -mJ(x 1 ) would be defined for all m = 0,1, ... and it would form a decreasing sequence with respect tom and bounded from below by s(x0 ). Hence it would have the limit lim s[ -mJ(xi) = x 2 m-+oo

E

11 ,

295

12.1 ABEL FUNCTIONAL EQUATION

and due to continuity of s, we would get

11 3 x 2 = lim s[-m](x 1) = s( lim i-m-l](x 1)) = s(x 2 ) m ➔ oo

m ➔ OO

that would contradict to assumption s(x) > x on / 1• Analogously, if x 1 < x 0 then there exists a positive integer k such that s[k](x 1) = x* E /*. The reasoning is the same as in the proceeding case x 1 > s(x0). Hence for -k == m we get relation (12.1.3). Now, let us show that each x 1 El can be expressed in the form (12.1.3) uniquely. Indeed, if for some x 1 E l we have x 1 = s[m,](x*) = s[m 2J(x*),

x* El*,

and hence s[m2-m1J(x*) = x* E /* then m1 = m2 because s(x) > x implies i m1(x) #- x on I 1 for m Now, define the function F: l --+ IR in the following way: for each x E l uniquely expressible as m

E

=f:.

0.

Z.,

m= 0 F(x) == F0(x*) + r(x*) + ... + r(s[m](x*)) for m > 0 F 0 (x*) - r(s[-l](x*)) - ... - r(s[m](x*)) for m < 0, F 0 (x*)

for

(12.1.4) see Fig. 12.1. Since each x E l can be uniquely expressed in the form (12.1. 3), function F is well-defined on interval /. Moreover, function s is defined on 11,

Fig. 12. l

296

12 APPENDIX

hence for each x

E /1

considered in the form x = sCm J(x* ), x*

E

/*, we have

s(x) = sCm+IJ(x*) and thus F(s(x))

=

F(sCm+tJ(x*))

=

F(s[mJ(x*)) + r(s[mJ(x*))

=

= F(x) + r(x) for m > 0 or

F(x)

F(sC111 J(x*)) = F(sEm+tJ(x*)) - r(s[mJcx*)) = F(s(x)) - r(x) for m < 0 .

=

=

Hence function F, defined on I by (12.1.4), is a solution of the Abel functional equation (12.1.1 ). Let us return to function F O E Ck(/*), /* = [ x 0 , s(x 0 ) ). With respect to definition of function F by formula (12.1.4), the following limits hold at each x 1 = s[m J(x*) E / x*1 E /* ' x*1 1..J.. x O•· ' 1 ' lim F(x) = X-+X1

+

lim F'(x)

lim F(x) = F(x 1) X-+X

=

1-

lim F'(x) X-+X

X-+XJ +

F'(x 1)

I-

X-+X

X-+X) +

=

I-

Hence function F is continuous and has continuous derivatives up to the order k inclusively at each number x E /, x # sEmJ(x0 ). Now, consider function F at a vicinity of number x 0 . We may observe that lim F(x) = X-+XQ+

lim F0 (x) = F0 (x 0 ) = F(x0 ) X-+XQ+

and lim F(x) = X-+XO-

Jim

(F(s{x)) - r(x))

=

X-+XO-

=

lim (F0 (s(x)) - r(x)) = X-+XQ-

lim F 0 (s(x)) - r(x 0 ); X-+XQ-

hence, due to condition {12.1.2), function Fis continuous at x0 • Moreover lim F'(x) = X-+XO+

lim F 0(x) = F 0+(x 0 )

X-+XO+

297

12.1 ABEL FUNCTIONAL EQUATION

and lim F'(x) X-+XQ-

=

lim (F(s(x)) - r(x))'

=

X-+XQ-

lim (F 0 (s(x)) - r(x))'

=

X-+XQ-

lim F 0(s(x))s'(x 0 )

-

r'(x 0 )

X-+XQ-

that, with respect to boundary condition (12.1.2), ensures that the first derivative of F is continuous at x 0• Analogously we derive that x-+xo+

X-+XQ-

= lim p(i)(x) for

i < k

X-+XQ-

that means that all derivatives of F up to the order k inclusively are continuous at x 0 . However function F satisfies the Abel functional equation ( 12.1.1) with functions s E Ck(/i), r E Ck(/ 1), and s'(x) > 0 on / 1. Hence F has continuous derivatives up to the order k inclusively at any number s[m J(x 0 ) belonging to interval /. Summarizing our considerations we may conclude that function F given by formula ( 12.1.4) is a solution of class Ck(!) of the Abel functional equation ( I2.1. 1). Now, in addition to the above assumptions on functions, specify r to be a positive constant. Under this additional requirement, it is possible to choose function F O on /* = [ x 0 , s(x 0 )) such that besides the above properties, F 0 e C\I) and relations (12.1.2), the first derivative of F 0 is positive on/. Such a choice is always possible on the basis of Whitney's Theorem, because r = const > 0 in ( 12.1.2 ). For such a choice of function F 0, the first derivative of solution F of ( 12.1.1) given by formula ( 12.1.4) is positive on the whole definition interval /. Indeed, function F satisfies the following relation F(s[-mJ(x)) = F(x) - mr and hence

for all x and s[ -m J(x) belonging to /. Since each x (12.1.3) x*

E

/*

E /

can be written in the form

298,

12 APPENDIX

for a suitable m

E

"ll.., and

(i -m])' is positive whenever it is defined, we obtain

F'(x) = F'(x*) (i-m])'(imJ(x*)) = F 0(x*)/(im])'(x*) > 0, which finishes the proof of Theorem 2. 7.1.

■

12.2 Euler functional equation for homogeneous functions F(ar, as) = ak F(r, s)

( 12.2.1)

This equation was investigated and solved by L. Euler in 1755, 1768 and 1770. The general solution F of this equation under different assumptions on a, k and the domain of definition of Fis described in J. Aczel's monograph [I]. Here we prove Theorem 2. 7.2 where a > 0, and F: IR+ x IR -+ IR. By putting a:= 1/r for arbitrary r >. 0, equation (12.2.1) becomes F(l, s/r)

= r-k F(r, s),

hence solution F is of the form

F(r, s) = l g(s/r). Conversely, if a function F: IR+ x IR -+ IR g: IR -+ IR is an arbitrary function, then

(12.2.2) is of the form (12.2.2) where

F(ar, as) = (arl g(as/ar) = ak,k g(s/r) = ak F(r, s). Hence function F: IR+ x IR -+ IR given by (12.2.2) with arbitrary g: IR -+ IR is the general solution of equation ( 12.2.1) under the conditions of Theorem 2.7.2. ■

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