Table Of ContentBounded and Almost Periodic Solutions of
Nonlinear Operator Differential Equations
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.S.R.
Yu. 1. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.
M. C. POL YVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Volume 55
Bounded and Almost Periodic
Solutions of Nonlinear Operator
Differential Equations
by
A.A.PANKOV
Institute ofA pplied Problems in Mechanics and Mathematics,
Ukrainian Academy of Sciences,
L'vov, U.S.S.R.
~.
"KLUW ER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging-in-Publication Data
Pankau, A. A. IAleksandr Andreevlch)
[ Or g ran I c hen n y e i po c h tIP e rio d i c h e ski ere she n i fa n e I I n e 1n y k h
oifferentsial 'no-operatornykh uravnenil. EnglIsh]
Bounded and almost periodic solutions of nonlinear operator
differentIal equatIons r by A.A. Pankov ; [translated from the
Russian by V.S. Zajackovski and A.A. Pankov].
p. cm. -- (Mathematics and its apolicatlons (Soviet series)
v. 55)
Expanded and revIsed translatIon of, Ogranlchennye i pochtl
per I od I :hesk I e reshen I fa ne II ne 1 nyk h d i fferenrs i a I 'no-operatornyk h
uravnenli.
Inc I uoes bib II ograph I ca I references.
ISBN 0-7923-0585-X
1. Operator equaTlons--Numerlca' solutions. 2. DIfferential
eauatlons~ Non-l1near--Nurner1cal sclutlons. ~. TTtie. II. Series:
Matnerr.atlcs ane Its ao"11cations IKluwer AcademIc PublIshers).
Soviet series; v. 55.
QA329.4.P3613 1990
515' .355--d:20 89-48125
ISBN 978-94-011-9684-0 ISBN 978-94-011-9682-6 (eBook)
DOI 10.1007/978-94-011-9682-6
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
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,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
This is the expanded and revised translation of the original work
OrPAHIr14EHHbiE Ir1 n04H1 nEPIr10111r14ECKI-1E PEWEHI-151
HEJlYtHEVlHblX llWI><I>EPEHUYtAJlbHO-onEPAI9PHbIX YPABHEHI-1V1
Published by Naukova Dumka, Kiev, © 1985.
Translated from the Russian by V.S. Zajackovski and A.A. Pankov
All Rights Reserved
This English edition © 1990 Kluwer Academic Publishers
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.
To Tanya
SERIES EDITOR'S PREFACE
~Et moi, .... si j'avait su comment en revenir. One service mathematics has rendered the
je n'y serais poin t aUe.· human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non·
The series is divergent; therefore we may be sense',
able to do something with it. Eric T. Bell
o. lleaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non-
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
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 service logic has rendered com·
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'e1re of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to recxamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related ficlds does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate arc suddenly secn 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
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algcbras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still appiies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, sec, 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
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu·
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/ sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
viii SERIES EDITOR'S PREFACE
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. It is in the non-
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci-
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com-
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre-
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 scope of the series, which for various (sound) reasons now comprises five sub-
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis-
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and! or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
The theory of bounded, periodic, and almost periodic solutions of (systems of) ordinary differential
equations has a venerable history and a large literature. The present volume is concerned with the
same kinds of solutions but now for the infinite dimensional case such as equations with retarded
arguments, variational inequalities and evolution equations of the form u,+ A (/)u+ L (/)u = f
where L (/) is a linear unbounded operator and A (I) is a monotone nonlinear operator. It is in this
sense that the word operator in the title must be understood.
It is a pleasure to welcome in this series a book on this important topic by an author who has
made substantial contributions to the subject (for instance to almost periodic solutions for a general
class of abstract first order nonlinear evolution equations, nonlinear partial differential equations
and to cerlain questions of Lions concerning variational inequalities).
The shortest path between two truths in the Never lend books. for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
l'occasion de resoudre des problemes ... elle right than other people, but to be wrong for
nous fait pTcssentir la solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, August 1990 Michiel Hazewinkel
Contents
Series Editor's Preface. ......................................................................................... vii
Introduction.
Chapter 1.
Almost periodic functions. ..................................................................................... 5
1. Almost periodic functions and Bohr compactification. ....................... 5
2. Besicovitch almost periodic functions and harmonic analysis. .......... 10
3. Structure of Bohr compactifications and spaces of almost
periodic functions. ........................................................................................ 18
4. Stepanov almost periodic functions. ...................................................... 24
5. Weakly almost periodic functions ......................................................... , 28
6. Spaces of smooth almost periodic functions on RN. ............................ 37
Comments. .................................................................................................... 43
Chapter 2.
Preliminaries. ........ ........................................ ........................................................... 45
1. Some integral inequalities ........................................................................ 45
2. Composition operators. ........................................................................... 49
Comments. .................................................................................................... 55
Chapter 3.
Solutions of evolution variational inequalities bounded and almost
periodic in time. ....................................................................................................... 56
1. On variational inequalities. ..................................................................... 56
2. Bounded solutions. ..................................................... ............................. 66
3. Regularity and almost periodicity of bounded solutions. ................... 74
4. The use of compactness. ......................................................................... 87
5. Almost periodicity in the sense of Besicovitch. .................................... 94
6. Singular perturbation. ............................................................................. 106
7. Some examples and additional results. ................................................. 114
Comments. .................................................................................................... 120
Chapter 4.
Bounded and almost periodic solutions of certain evolution equations.
122
1. Abstract evolution equations. .. .............................................................. . 122
2. Applications. . .......................................................................................... . 134
3. Additional results .................................................................................... . 147
Comments .................................................................................................... . 157
x CONTENTS
Chapter 5.
Problems that are almost periodic in space variables. ..................................... 158
1. Nonlinear elliptic equations. .................................................................. 158
2. Almost periodic first order systems. ...................................................... 170
3. Symmetric hyperbolic systems with monotone nonlinearity. ............. 182
4. A nonlinear Schrodinger-type equation. ............................................... 189
Comments. .................................................................................................... 193
Appendix 1. On certain linear evolution equations. ............................................ 194
Appendix 2. On certain wave equations. .............................................................. 203
Appendix 3. Open questions. ................................................................................. 207
References. .. ............................................................................................................. 213
Subject Index. ........................................................................................................... 223
INTRODUCTION
The theory of almost periodic (a.p.) functions was initiated between 1924 and
1926. Subsequently, important contributions were made by A. Besicovitch, S.
Bochner, J. von Neumrum, V. Stepanov, and B. Levitan. We especially mention
those of von Neumrum, who was the first to consider a.p. functions on groups.
From the earliest days on, the theory of a.p. functions has been connected with
problems of differential equations. (Moreover, important work by P. Bohl and E.
Esclangon had appeared before Bohr's contributions.) In his fundamental paper
[126], J. Favard had introduced the notion of envelope of an a.p. differential equa-
tion (in modern interpretation, it is equivalent to considering a family of equations
parametrised by the points of the Bohr compactification) and had connected the
problem of existence of a.p. solutions with some separation properties of bounded
solutions of equations which belong to the envelope. This work of Favard is the
starting point of many further investigations.
In the present text, we are not concerned with the theory of ordinary (finite-
dimensional) a.p. differential equations (see, for example, [35, 41, 127] and the
references given there). Instead we fix our attention to infinite-dimensional evolu-
tion equations, bOtll abstract and involving partial derivatives. Some important
results for the linear case were obtained by Z. Amerio [100]. Starting from the
well-known results of S. Sobolev [78] on the homogeneous wave equation, he inves-
tigated the corresponding inhomogeneous equation. Further, using the notion of
weakly a.p. function introduced by him in [99], Amerio justified an infinite-
dimensional version of the minimax method of constructing a.p. solutions (due to
Favard) and obtained some generalizations of Favard's classical theorems. The
strongest results in tllis direction were proved by V. Zhikov. These are presented
systematically in [25, 41]. In the constant coefficient case an interesting approach
was suggested in [3Ia].
We also note that recently the theory of linear partial differential equations with
coefficients that are a.p. in all variables has gained interest (see, for example, [91,
94]).
Among the nonlinear a.p. evolution equations the class best understood is that
of the abstract parabolic equations. For such equations G. Prouse [145] has
obtained an existence theorem for a.p. solutions. Subsequently tllis result was
strengthened considerably by V. Zhikov [22] (see also [25, 41]).
A new important class of evolution problems, known as variational inequalities,
was introduced by J.-L. Lions and G. Stampaccia [140]. In 1969 Lions posed the
question about bounded and a.p. solutions of such problems (see [44], problems
