Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations

342 Pages·1993·28.665 MB·English

by  I. T. Kiguradze, T. A. Chanturia (auth.)

AsymptoticProperties ofSolutions of
Nonautonomous Ordinary DifferentialEquations
Mathematics and Its Applications (Soviet Series)
ManagingEditor:
M.HAZEWINKEL
CentreforMathematicsandComputerScience,Amsterdam,TheNetherlands
EditorialBoard:
A.A.KIRILLOV,MGU,Moscow,Russia
Yu.I.MANIN,SteklovInstituteofMathematics,Moscow,Russia
N.N. MOISEEV,ComputingCentre,AcademyofSciences,Moscow,Russia
S.P.NOVIKOV,LandauInstituteofTheoreticalPhysics,Moscow,Russia
Yu.A.ROZANOV,SteklovInstituteofMathematics,Moscow,Russia
Volume89
Asymptotic Properties of
Solutions of
N onautonomous Ordinary
Differential Equations
by
I. T. Kiguradze
Institute ofA pplied Mathematics,
Tbilisi, Georgia
and
T. A. Chanturia t
..
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Kiguradze. I. T.
[Asimptoticheskie svolstva resheni1 neavtonomnykh obyknovennykh
d i fferenfS i a I 'nykh uravnen i 1 . Eng I ish 1
Asymptotic properties of solutions of nonautonomous ordinary
differential equations / by I.T. Kiguradze and T.A. Chanturia.
p. cm. (Mathematics and its appl icat1ons. Soviet series
89)
Translation of: Asimptoticheskie svolstva reshen11 neavtonomnykh
obyknovennykh d i fferenfS 1a I 'nykh uravnen 11.
Includes index.
ISBN 978-94-010-4797-5 ISBN 978-94-011-1808-8 (eBook)
DOl 10.1007/978-94-011-1808-8
1. Differential equat1ons--Asymptot1c theory. I. Chanturifa. T,
A. (Telmuraz Ambros 'evich) 11. Title. Ill. Series: Mathematics and
its appl icat10ns (Kluwer Academic Publishers), Soviet series; 89.
QA372.K5413 1992
515' .35--dc20 92-37359
ISBN 978-94-010-4797-5
This is the translation of the original work
Asymptotic Properties of Solutions of
Nonautonomous Ordinary Differential Equations
Published by Nauka, Moscow, © 1985
Printed on acid-free paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
Spftcover reprint of the hardcover 1 st eidtion 1993
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 photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
SERIESEDITOR'SPREFACE
'Et moi,.., siJavaitsucommentenrevenir, One service mathematics has rendered the
jen'yserais:pointalit: humanrace. It hasput commonsenseback
JulesVerne when:it belongs, on the topmost shelfnext
tothedustycanisterlabelled'discardednon
Theseriesisdivergent; thereforewe may be sense'.
abletodosomethingwithit. EricT.Bdl
O.Hcaviside
MathematiCsisatoolfor thought. Ahighlynecessarytoolinaworldwherebothfeedback andnon
linearitiesabound. Similarly, all kinds ofparts ofmathematicsserveas tools forotherpartsandfor
othersciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics ...'; 'One servicelogic has rendered com
puterscience...'; 'Oneservicecategorytheoryhasrenderedmathematics ...'.Allarguably true. And
allstatementsobtainablethiswayformpartoftheraisond'ctreofthisseries.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumeshaveappeareditseemsopportunetoreexamineitsscope.AtthetimeIwrote
"Growing specialization and diversiftcation have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparateare suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometryinteractswith physics; the Minkowskylemma, codingtheory and thestructure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant tofiltering; andpredictionand electricalengineeringcanuseSteinspaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrablesystems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existingclassificationschemes. They
drawuponwidelydifferentsectionsofmathematics."
By and large, aU this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continueto try tomakesuchbooksavailable.
Ifanything, thedescriptionI gavein 1977is now an understatement.To theexamplesofinterac
tionareas oneshould add string theorywhereRiemannsurfaces, algebraicgeometry, modularfunc
tions, knots, quantumfield theory, Kac-Moody algebras, monstrousmoonshine(andmore) allcorne
together. And to the examples of things which can be usefully applied let me add the topic 'finite
geometry'; a combination ofwords which soundslikeit might not even exist, let alonebe applica
ble. And yet itis beingapplied: to statisticsvia designs, to radar/sonar detection arrays (viafinite
projectiveplanes), and to busconnectionsofVLSI chips(viadifferencesets). Thereseems to beno
partof(so-calledpure)mathematicsthatisnotinimmediatedangerofbeingapplied. And, accord
ingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics,
thetraditionalworkhorses, hemay needallkindsofcombinatorics,algebra, probability,andso on.
v
vi
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
extra mathematical sophistication that this requires. For that is where the rewards are. linear
models are honest and a bit sad and depressing: proportional efforts and results. Itis in the non
linearworld thatinfinitesimalinputs may resultinmacroscopicoutputs(orviceversa). Toappreci
atewhat I am hintingat: ifelectronicswerelinearwewouldhaveno fun with transistorsand com
puters; wewouldhavenoTV;infactyouwouldnotbereadingtheselines.
There is also no safety in ignoring such outlandish things as nODStandard analysis, superspace
and anticommuting integratioIi, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fr~
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading
fortunately.
Thus the original scopeof the series, whichfor various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), stillapplies. Ithasbeenenlargeda bittoincludebooks treatingof thetoolsfrom onesubdis
ciplinewhichareusedinothers.Thustheseriesstillaimsatbooksdealingwith:
a central concept which plays an important role in several different mathematical and/or
scientificspecializationareas;
newapplicationsoftheresultsandideasfromoneareaofscientificendeavourintoanother;
influenceswhich theresults, problemsand conceptsofonefield ofenquiry have, and havehad,
on thedevelopmentofanother.
1beshortestpath between two trothsin the Never lend books, for no one ever ~turns
~ domain passes through the complCll them; the only books I havein my Iibnuy
domain. arebooksthatolherfolk.havelentme.
J.Hadamard Anatole France
La physique ne nous donne paa seulement Thefunctionofanex.pertisnot tobemore
l'occasionde resoudre des problCm~... dIe right thanotherpeople,buttobewrongfor
nousfaitp=entirlasolution. moresophisticatedreasons.
H.Poincare DavidButler
MichielHazewinkd
Contents
Series Editor's prBface v
Preface xi
Basic Notation xiii
Chapter I. LINEAR DIFFERENTIAL EQUATIONS 1
§l. Equations Having Properties A and B 1
1.1. Classification ofnonoscillatory solutions 2
1.2. Conjugate points ofthe equation (1.1) 12
1.3. Necessary and sufficient conditions for the equation (1.1) to have prop-
erties A and B 20
1.4. Integral criteria for the equation (1.1) to have properties A and B 25
1.5. On general equations 30
Notes 41
§2. Oscillatory and Nonoscillatory Equations 42
2.L Some auxiliary assertions 43
2.2. Oscillation and nonoscillation criteria 52
2.3. On third, fourth and sixth order equations 57
Notes 67
§3. Oscillation Properties of Solutions of Equations with Strongly Os-
cillating Coefficients 68
3.1. Zeros ofsolutions on a finite interval 68
3.2. Oscillationofallsolutionsofequationswithstronglyoscillatingcoefficients 74
Notes 76
§4. The Subspace of Solutions Vanishing at Infinity 76
4.1. Some integral identities and inequalities 77
4.2. On functions satisfying the conditions (4.19) 81
4.3. On a related problem 82
4.4. Theorems on the dimension ofthe space VCn,lT)(Po,... ,Pn-d 84
4.5. Theorems on the dimension ofthe space VCn,lT)(p) 89
4.6. On solutions ofsecond order equations 96
Notes 106
vii
viii CONTENTS
§5. Bounded and Unbounded Solutions 106
5.1. Inequalities ofKolmogorov-Gorny type 107
5.2. Lemmas on the solvability ofrelated boundary value problems 109
5.3. On the space B(n)(p) 112
5.4. Existence theorems for unbounded oscillatory solutions 118
5.5. On second order equations 126
Notes 128
§6. Asymptotic Formulas 128
6.1. Statement ofthe main theorem 128
6.2. Auxiliary assertions 129
6.3. Proofofthe main theorem 131
6.4. Equations with almost-constant coefficients 134
6.5. Equations with asymptotically small coefficients 135
6.6. Equations asymptotically equivalent to two-term ones 136
Notes 143
Chapter II. QUASILINEAR DIFFERENTIAL EQUATIONS 145
§7. Statement ofthe Problem. Auxiliary Assertions' 145
7.1. Linear equations having the Levinson property 147
7.2. On a system ofnonlinear integral equations 150
§8. The Family of L-h Type Solutions ofthe Equation (7.1) 152
8.1. Necessary and sufficient conditions for the existence ofL-h type solutions 152
8.2. Asymptoticrepresentationsforsolutionsofarelateddifferentialinequality162
8.3. Stability ofthe family ofL-h type solutions 163
Notes 169
§9. L-~, C:' and L-h Type Equations 169
9.1. Lemmas on integral inequalities 169
9.2. L-~ type equations 171
9.3. L-'h' type equations 173
9.4. L-h type equations 176
Notes 178
Chapter III. GENERAL NONLINEAR DIFFERENTIAL
EQUATIONS 179
§10. Theorems on the Classification ofEquations with Respect to
Their Oscillation Properties 179
10.1. Comparison theorem 180
10.2. Equations having property A or B 184
10.3. Equations having property A B AZ, or Bic 194
k, k,
Notes 198
CONTENTS ix
§1l. Singular Solutions 198
11.1. Nonoscillatory first kind singular solutions 199
11.2. Nonoscillatory second kind singular solutions 202
11.3. Nonexistence theorem for singular solutions 205
Notes 207
§12. Fast Growing Solutions 207
12.1. Existence theorem 207
12.2. Asymptotic estimates 210
Notes 212
§13. Kneser Solutions 212
13.1. Existence theorem 213
13.2. Kneser solutions vanishing at infinity 215
Notes 220
§14. Proper Oscillatory Solutions 220
14.1. Existence theorems 221
14.2. Proper oscillatory solutions vanishing at infinity 229
Notes 234
Chapter IV. HIGHER ORDER DIFFERENTIAL EQUATIONS
OF EMDEN-FOWLER TYPE 235
§15. Oscillatory Solutions 235
15.1. Classification ofequations with respect to their oscillation properties 235
15.2. Existence ofproper oscillatory solutions 236
15.3. Proper oscillatory solutions vanishing at infinity 237
15.4. Oscillatory first kind singular solutions 238
Notes 242
§16. Nonoscillatory Solutions 242
16.1. Kneser solutions 242
16.2. Solutions with power asymptotics 245
16.3. Fast growing solutions 246
16.4. Second kind singular solutions 247
Notes 248
Chapter V. SECOND ORDER DIFFERENTIAL EQUATIONS
OF EMDEN-FOWLER TYPE 249
§17. Existence Theorems for Proper and Singular Solutions 249
17.1. Existenceofproper solutions 249
17.2. Existence ofnonoscillatory singular solutions 251
17.3. Existence ofoscillatory singular solutions 254
Notes 256
x CONTENTS
§18. Oscillation and Nonoscillation Criteria for Proper Solutions 256
18.1. Oscillation ofall proper solutions 256
18.2. Existence ofat least one oscillatory proper solution 261
18.3. Nonoscillation ofall proper solutions 263
Notes 265
§19. Unbounded and Bounded Solutions. Solutions Vanishing at
Infinity. 265
19.1. Bounded solutions 266
19.2. Solutions vanishing at infinity 266
19.3. Unboundedoscillatory solutions 269
Notes 270
§20. Asymptotic Formulas 270
20.1 Asymptotic representations for oscillatory solutions 270
20.2. Auxiliary assertions 276
20.3. Asymptotic representations for nonoscillatory solutions (the case ofa
negative coefficient) 287
20.4. Asymptotic representations for nonoscillatory solutions (the case ofa
positive coefficient) 295
20.5. Asymptotic representations for nonoscillatory singular solutions 303
Notes 305
REFERENCES 307
AUTHOR INDEX 329
SUBJECT INDEX 331