Almost Periodic Functions and Differential Equations

Almost periodic functions and differential equations

Almost periodic functions and differential equations

B.M.LEVITAN & V.V.ZHIKOV Translated by L. W. Longdon

CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge, The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia.

© Moscow University Publishing House 1978 English edition © Cambridge University Press 1982 Originally published in Russian as Pochti periodicheskie funktsii differentsial' nye uravneniya by the Moscow University Publishing House 1978 Assessed by E. D. Solomentsev and V. A. Sadovnichii First published in English, with permission of the Editorial Board of the Moscow University Publishing House, by Cambridge University Press 1982 -

Printed in Great Britain at the University Press, Cambridge Library of Congress catalogue card number: 83 4352

British Library Cataloguing in Publication Data Levitan, B.M. Almost periodic functions and differential equations. 1. Periodic functions I. Title II. Zhikov, V.V. uravneniya. English 515.8 QA331 ISBN 0 521 24407 2

III. Pochtiperiodicheskie funktsii i differentsial'nye

Contents

Preface

1 1

2 3 4 5 6

2 1 2 3

4 5 6

Almost periodic functions in metric spaces

1

Definition and elementary properties of almost periodic functions 1 Bochner's criterion 4 The connection with stable dynamical systems 8 Recurrence 9 A theorem of A. A. Markov 10 Some simple properties of trajectories 11 Comments and references to the literature 12

Harmonic analysis of almost periodic functions Prerequisites about Fourier—Stieltjes integrals Proof of the approximation theorem The mean-value theorem; the Bohr transformation; Fourier series; the uniqueness theorem Bochner—Fejer polynomials Almost periodic functions with values in a Hilbert space; Parseval's relation The almost periodic functions of Stepanov Comments and references to the literature

3 Arithmetic properties of almost periods 1 Kronecker's theorem 2 The connection between the Fourier exponents of a

3

ix

function and its almost periods Limit-periodic functions

14 14 17

21 25 31 33 36 37 37 40 45

vi

Contents

4 Theorem of the argument for continuous numerical

4

1 2

complex-valued almost periodic functions Comments and references to the literature

48 51

Generalisation of the uniqueness theorem (N-almost periodic functions)

53

Introductory remarks, definition and simplest properties of N-almost periodic functions Fourier series, the approximation theorem, and the uniqueness theorem Comments and references to the literature

5 Weakly almost periodic functions 1 Definition and elementary properties of weakly almost

2 3

periodic functions Harmonic analysis of weakly almost periodic functions Criteria for almost periodicity Comments and references to the literature

A theorem concerning the integral and certain questions of harmonic analysis 1 The Bohl—Bohr—Amerio theorem 2 Further theorems concerning the integral 3 Information from harmonic analysis 4 A spectral condition for almost periodicity. 5 Harmonic analysis of bounded solutions of linear

53 59 62 64

64 68 70 76

6

equations Comments and references to the literature

7

Stability in the sense of Lyapunov and almost periodicity

Notation 1 The separation properties 2 A lemma about separation 3 Corollaries of the separation lemma 4 Corollaries of the separation lemma (continued) 5 A theorem about almost periodic trajectories 6 Proof of the theorem about a zero-dimensional fibre 7 Statement of the principle of the stationary point

77

77 81 87 91 92 96

98 98 98 101 105 107 109 113 116

Contents 8 9

Realisation of the principle of the stationary point when the dimension m -._ 3 Realisation of the principle of the stationary point under monotonicity conditions Comments and references to the literature

8 Favard theory 1 Introduction 2 Weak almost periodicity (the case of a uniformly convex space) 3 Certain auxiliary questions 4 Weak almost periodicity (the general case) 5 Problems of compactness and almost periodicity 6 Weakening of the stability conditions 7 On solvability in the Besicovitch class Comments and references to the literature

9 1 2 3 4

10

1 2 3 4

11

The method of monotonic operators General properties of monotonic operators Solvability of the Cauchy problem for an evolution equation The evolution equation on the entire line: questions of the boundedness and the compactness of solutions Almost periodic solutions of the evolution equation Comments and references to the literature Linear equations in a Banach space (questions of admissibility and dichotomy) Notation Preliminary results The connection between regularity and the exponential dichotomy on the whole line Theorems on regularity Examples Comments and references to the literature

vii

117 121 123 124 124 127 130 134 135 140 142 147

149 149 153 157 161 165

166 166 166 170 172 176 181

The averaging principle on the whole line for

parabolic equations 1 Bogolyubov's lemma 2 Some properties of parabolic operators

182 182 183

viii

Contents 3 4 5 6

The linear problem about averaging A non-linear equation The Navier—Stokes equation The problem on the whole space Comments and references to the literature

186 189 193 195 199

Bibliography

200

Index

208

Preface

The theory of almost periodic functions was mainly created and published during 1924-1926 by the Danish mathematician Harald Bohr. Bohr's work was preceded by the important investigations of P. Bohl and E. Esclangon. Subsequently, during the 1920s and 1930s, Bohr's theory was substantially developed by S. Bochner, H. Weyl, A. Besicovitch,, J. Favard, J. von Neumann, V. V. Stepanov, N. N. Bogolyubov, and others. In particular, the theory of almost periodic functions gave a strong impetus to the development of harmonic analysis on groups (almost periodic functions, Fourier series and integrals on groups). In 1933 Bochner published an important article devoted to the extension of the theory of almost periodic functions to vector-valued (abstract) functions with values in a Banach space. In recent years the theory of almost periodic equations has been developed in connection with problems of differential equations, stability theory, dynamical systems, and so on. The circle of applications of the theory has been appreciably extended, and includes not only ordinary differential equations and classical dynamical systems, but wide classes of partial differential equations and equations in Banach spaces. In this process an important role has been played by the investigations of L. Amerio and his school, which are directed at extending certain classical results of Favard, Bochner, von Neumann and S. L. Sobolev to differential equations in Banach spaces. We survey briefly the contents of our book. In the first three chapters we present the general properties of almost periodic functions, including the fundamental approximation theorem. From the

x

Preface

very beginning we consider functions with values in a metric or Banach space, but do not single out the case of a finite-dimensional Banach space and, in particular, the case of the usual numerical almost periodic functions. Of the known proofs of the approximation theorem we present just one: a proof based on an idea of Bogolyubov. However, it should be noted that another instructive proof due to Weyl and based on the theory of compact operators in a Hilbert space appears in many textbooks on functional analysis. Chapter 4 is devoted to the theory of N-almost periodic functions. In comparison with the corresponding chapter of the book AlmostPeriodic Functions by B. M. Levitan (Gostekhizdat, Moscow (1953)), we have added a proof of the fundamental lemma of Bogolyubov about the structure of a relatively dense set. Chapter 5 is concerned with the theory of weakly almost periodic functions developed mainly by Amerio. Chapter 6 contains, as well as traditionally fundamental questions (the theorem of Bohl—Bohr about the integral, and Favard's theorem about the integral), more refined ones', for instance, the theorem of M. I. Kadets about the integral. We mention especially Chapter 7 whose title is Stability in the sense of Lyapunov and almost periodicity. The two chapters that follow it are formally based on it. Actually, we use only the simplest results, and when there is a need to refer to more difficult propositions we give independent proofs. Therefore, Chapters 6-11 can be read independently of one another. Chapter 8 contains Favard theory, by which we mean the theory of almost periodic solutions of linear equations in a Banach space. In Chapter 9 the results from the theory of monotonic operators are applied to the problem of the almost periodicity of solutions of functional equations. In Chapter 10 we give another approach to the problem of almost periodicity. Finally, Chapter 11 is slightly outside the framework of the main theme of our book. In it we give one of the possible abstract versions of the classical averaging principle of Bogolyubov. Chapters 1-5 were written mainly by B. M. Levitan, and Chapters 6-11 by V. V. Zhikov. The authors thank K. V. Valikov for his assistance with the reading of the typescript.

Translator's note This translation has been approved by Professor Zhikov, to whom I am grateful for correcting my mistranslations and some misprints in the original Russian version. Professor Zhikov has asked me to mention that the theory of Besicovitch almost periodic functions is not reflected fully enough in the book, since this theory has recently been applied in spectral theory and in the theory of homogenisation of partial differential equations with almost periodic coe ffi cients. The additional references are, in the main, concerned with this theme.

L. W. Longdon

1

Almost periodic functions in metric spaces

1 Definition and elementary properties of almost periodic functions Throughout the book J denotes the real line, X a complete metric space, and p = p(xl, x2 ) a metric on X. Let f(t):J -.* X be a continuous function with values in X; we denote the range of f, that is, the set {x e X: x = f(t), t E J}, by Rh and its closure by af;

Definition 1. A set E c J of real numbers is called relatively dense if there exists a number 1> 0 such that any interval (a, a + 1) c f of length 1 contains at least one number from E.

Definition 2. A number T is called an e-almost period of f :1 -* X if (1) sup p(f(t +r), f(t))--- 6. teJ

Definition 3. A continuous function f : j -+ X is called almost periodic if it has a relatively dense set of 6-almost periods for each 6 >0, that is, if there is a number 1 =1(e)>0 such that each interval (a, a +1)c J contains at least one number T -= T E satisfying (1). Every periodic function is also almost periodic. For if f is periodic of period T, then all numbers of the form nT (n = E 1, ±2, . . .) are also periods of f, and so they are almost periods of f for any e >0. Finally, the set of numbers nT is relatively dense. It is easy to produce examples of almost periodic functions that are not periodic, for instance, f(t)= cos t + cos t.s12.

2

Almost periodic functions in metric spaces

We prove some of the simplest properties of almost periodic functions; these are straight-forward consequences of the definition. Property 1. An almost periodic function

f :J *X is compact in the

is compact. sense that the set Proof. It is sufficient to prove that for any e > 0, Rf contains a finite e-net for Rf. Let 1=1(e) be the length in Definition 3 corresponding to a given E. We set Rfa =

e

f: X =

Pt),

//21.

From the continuity of f it follows that the set Rhi is compact; we show that it is an E-net for the set RI.. Let to € J be chosen arbitrarily, and take an e-almost period 7 = re such that to + T 1/2, that is,

to+ //2. Then P(f(t0+ 7 ), fit0))= 8.

Because to -FT E [-1/2, 1/2], the set Rfa is an e-net for Rf, as we required to prove. Remark. For numerical almost periodic functions (that is, when X = R 1 ) and for almost periodic functions with values in a finitedimensional Banach space, Property 1 reduces to the following: if f is an almost periodic function, then R f is bounded. Property 2.

Let f :J > X be a

continuous almost periodic function. Then f is uniformly continuous on J. Proof. We take an arbitrary e >0 and set E l = e/3 and 1 = 1(0. The function f is uniformly continuous in the closed interval [—I, 1+1], that is, there is a positive number 8 = 3(E 1 ) (without loss of generality we may assume that 8 < 1) such that (2)

P(As"),Ps 1))<E1

whenever Is"— s'l< 8, s', s"EJ. Now let t', t" be any numbers from J for which It' — CI< 8. We take a 7 = TEI with 0 /, that is, tT El —t'+1. Then t'' + 7E , e [ 1, 1+1]. We set s' = t'+ re, and s"= t" + rE,. From (1), (2) and the triangle inequality we have -

PU (t" ), fit'))=P(Pt"), fis"))+P(Ps"),Ps1)) +P(fis'), f(e))< E.

Definition and elementary properties

3

Property 3. Let fn : J X, n = 0, 1, 2, . . . , be a sequence of continuous almost periodic functions that converges uniformly on J to a function f. Then f is almost periodic. Proof. We take an arbitrary e > 0 and let n = ne be such that -

sup p(f(t), f,(t))- ---5 E/3. te j

(3)

Let T = T[ file ] denote an (e/3)-almost period of the function fne. Then it follows from (1), (3), and the triangle inequality that p(f(t + r), f(t)).--. p(f(t + r), fne (t + r)) +P(fne (t ± 7), fne(t))± to (fnE (t), fit)) "-... E

for all t € 1. This proves that f is almost periodic because the set of almost periods r[fne ] is relatively dense. Property 4. Let x = f(t) be a continuous almost periodic function with values in a metric space X, and y = g(x) be continuous on 0—if with values in a metric space X 1 . Then g[f(t)] is an almost periodic function with values in X1. Proof. Since the set 22 c is compact and the function g(x) is continuous on -f, g(x) is uniformly continuous on 0.-/f. Therefore, for all 6 >0 there exists a 8 = 8(e)>0 such that for all x', x"E with p(x' , x")---- 8 we have p i (g(x"), g(x'))--Ç E. Therefore, if T is a 8-almost period for f(t), then P(f(t +7), f(t)) = (5, and so p i(g(f(t + r)), g(f(t)))---5 E. Corollary. Let f be a continuous almost periodic function with values in a Banach space X. Then l[f(t)li k is a continuous numerical almost periodic function for all k > 0. Property 5. Suppose that fis an almost periodic function with values in a Banach space X. If the (strong) derivative f' exists and it is uniformly continuous on J, then f' is an almost periodic function. Proof. The proof uses the concept of an integral of a vector-valued function. In the case of continuous functions this is very simple because the Riemann integral exists with the usual fundamental

4

Almost periodic functions in metric spaces

properties (see, for example, G. E. Shilov, Mathematical Analysis. Functions of a Single Variable, Part 3, Ch. 12, § 12.5). By hypothesis, the derivative f' is uniformly continuous, and so for all s >0 there is a ô = 8(e)>0 such that It(e) — f (r)ii< s whenever le— t"1 < S. Therefore, if 1/n 0 and the function d(zo(t), A) is nonincreasing, z 1 also does not belong to A. We extract from the sequence {t,n } a subsequence {t'm } for which the sequence zo(t'm +7) is fundamental for any rational 7. Because a translation on Z is continuous for t....- 0, the sequence zo(t' m +7+77) is a fundamental sequence for every n. . -.- 0, that is, zo(t' m + t) is a fundamental sequence for every t E j. Therefore we have convergence to some trajectory z i (t): zo(t + tim)-> zi(t) (t €J).

Markov's theorem

11

Since the function d(x o(t), yo(t)) is non-increasing, we have

d(x i (t), y i (t))= lim d(x 0(t + t'in ), yo(t + t' in )) const., m-»CO

that is, z 1 e A. The contradiction proves that A = Z. From our conclusion that A = Z it follows easily that S(to)xi 0 S(to)x 2 for x 1 0 x2(t--.- 0), that is, through any point from X a unique trajectory passes. It is also easy to conclude that the mapping 5 (to) : X -+ X is 'onto', that is, the inverse mappings S -1 (t0) are continuous. This proves the theorem. The next proposition is proved by a similar argument.

Proposition 1. Suppose that on a compact metric space K there is defined a non-contractive operator T :K -+ K, that is,

P(Txi, Tx2) --.- P(xi, x2). Then TK = K.

6

Some simple properties of trajectories

1. The results of this section are purely subsidiary. We consider some general properties of the so-called continuous semigroups. A semigroup S(t) (t---- 0) is called continuous if every semitrajectory of it is a continuous function r --> X, where r denotes the semiaxis [0, co).

Proposition 2. Suppose that S(t) is a continuous semigroup on a compact metric space X. Then when t ranges over a finite interval on the open semiaxis (0, 00), the transformations S(t) are equicontinuous. Proof. We set Z = X x X, and consider the space B of all continuous scalar functions 0(z) on Z, and an obvious semigroup of linear operators on B:

O t (z)= 0(S (t)z) (here z = (x i , x2)). Since the trajectories are continuous, we easily see that the function 0 t :I + --> B is measurable. But then, as is well known from the theory of semigroups of linear operators (see Dunford & Schwartz [40], p.616), the function 0 t is continuous on (0, oo). Hence, by putting 0(z) = p (x i , x 2) we obtain the required result. It follows from Proposition 2 that a compact semitrajectory of a continuous semigroup is uniformly continuous on the semiaxis J,

12

Almost periodic functions in metric spaces

and that trajectories belonging to a compact invariant set are uniformly continuous on the whole axis.

2. To the concept of a recurrent trajectory (see § 4) there corresponds the obvious concept of a recurrent function. Let K be a complete metric space, and let 0(K) denote the set of all continuous functions J -> K with the topology of uniform convergence on each finite segment. For f(s) e 0(K) we set f t = f(s +t). A function f(s)e 0(K) is called recurrent if the trajectory f t is recurrent in 4P(K). There is a natural connection between recurrent functions and recurrent trajectories. Let x t be the recurrent trajectory of a continuous semigroup defined on a complete metric space X, and let 0 :X -+ IC be a given continuous function. Then it follows easily from Proposition 2 that f(t)= 0(x t ) is recurrent. In particular, if 0 is a scalar function, then since every semitrajectory is everywhere dense in a minimal set, it follows that sup f(t)= sup f(t)= sup f(t). teJ

Comments and references to the literature

1. The definition of an almost periodic function and its simplest properties for numerical functions is due to Bohr [17] and [22]. Long before the publication of Bohr's work, Bohl [15] and Esclangon [120], [121] had discussed a special case of almost periodic functions §

which are now known as conditionally periodic (or sometimes, quasiperiodic) functions. In contrast to Bohr's definition in which the only condition on almost periods was relative denseness, the definition of Bohl and Esclangon imposed further conditions. The latter definition is as follows: A continuous function f is called conditionally periodic with periods 27r/A 1 , 27r/A 2, , 27r/A„, if for every E > 0 there is a 3 = 3(E ) > 0 such that each number r satisfying the system of inequalities

iikk71< (5 (mod 27r),

k = 1, 2, . . . , m,

also satisfies the inequality sup Ilf(t + tEJ

E,

that is, it is an E-almost period for f(t). The position of conditionally periodic functions in the class of continuous almost periodic func-

Comments and references

13

tions is discussed in Chapter 3, § 3, and the role of the system of inequalities (8) in the theory of almost periodic functions is considered in Chapter 3, § 2, and Chapter 4, § 1 (Bogolyubov's theorem). The extension of the theory of almost periodic functions to vectorvalued (abstract) functions is due to Bochner [27]. Bochner's work was preceded by an important article by Muckenhoupt [93] who considered essentially a special class of abstract almost periodic functions with values in a special Hilbert space. It is interesting to note that the concept of a Bochner measurable and summable function, which is being widely extended at the present time, had its origins in Bochner's investigations on abstract almost periodic functions. § 2. The compactness property of an almost periodic function was discovered by Bochner [25]. Lyusternik's theorem was published in [83]. §§ 3 and 4. The connection between almost periodicity and stable dynamical systems is well known (see the monograph of Nemytskii & Stepanov [95], Ch. 5); we have presented only the most elementary facts. § 5. The following result is due to Markov: a compact Lyapunov stable trajectory of a dynamical system is almost periodic. In fact, the result which we have called Markov's theorem says slightly more. Proposition 1 was first stated in a paper by Brodskii & Mil'man [30], and then in a more general form in a book by Dunford & Schwartz ([40], p. 459).

2

Harmonic analysis of almost periodic functions

1 Prerequisites about Fourier-Stieltjes integrals 1. Let o-(A), A GI, be a numerical (complex-valued) function of bounded variation on the real line. The Fourier Stieltjes transform of o-(A ) is the function f(t) defined by -

+00

f(t)= f co exp (iAt) do-(A). Let A 1 , A2, .. . denote the points of discontinuity of cr(A) in any order, and d1 , d2, ... the corresponding jumps, that is, d, = cr(À, + 0) a-(À, -0) (7) = 1, 2, ...). -

We set d(A)= E d, Av 0 and s(A) is a continuous non-decreasing function.

2 Proof of the approximation theorem 1. The approximation theorem. For every continuous almost periodic function f: J --> X and for every E > 0 there is a trigonometric polynomial Ne

Pe (t)= E b,,e exp (iA„, st)

v=i

(b„,e EX, yk i,„ el)

3 It is proved, for example, in the book by G. E. Shilov (referred to in footnote I), p. 404.

18

Harmonic analysis

such that

sup 'if (t) tEJ

P

<E

Remark 1. We can deduce from the approximation theorem other

basic theorems in the theory of almost periodic functions, and so it must be regarded as a principal result of this theory. Remark 2. The function a exp (iÀt) is periodic of period 27r/IA I for all a EX and for all A € J. Property 6 (Chapter 1, § 2) implies that every trigonometric polynomial is an almost periodic function, and it follows from Property 3 (Chapter 1, § 4) that every uniform limit of trigonometric polynomials is an almost periodic function. The approximation theorem asserts that all almost periodic functions can be obtained in this way. Proof of the approximation theorem. For any E >0 we choose numbers 1 = 4E18) and 5 = 5(E18) in accordance with Definition 3 (Chapter 1, § 1) and Property 2 (Chapter 1, § 1), respectively. Then in any interval of length 1 there is a number T such that

sup Ilf(t +7)-f(011-- 618 tEJ

(9)

and lif(t") — f(t 1 )11 ,- . s18 for any t', tu J with It"

—

(10) < 3. We cover J with the intervals

Jn = (nl, (n +1)1), n = 0, ±1, ±2,

From (9) and (10) it follows that each interval Jn contains a subinterval An = (rn — 8, rn + 8) whose points are (s/4)-almost periods of Pt), that is, for every T I E An sup I f(t + 7') —

E 14.

(11)

We define a function Ks (s), s E j, by Ks (s ) =

1123 0

for s E An, for sit An.

It has the following obvious properties, which we shall need later on: (1) (1/2n/) f_nni / Ks (s)ds = 1 (n = 1, 2, ...). (2) For any s E J and any natural number m mi+s 1 K8 (r) dr =1+ 77(s), 2m1

19

Proof of the approximation theorem where

Iii(s)I 11m. (3) For every fixed 8 the set of functions \ 1f v (u +s) ds, ." 8‘8 2T T = Tn = nl, n = 1, 2, .. . , —co< u < is uniformly bounded and equicontinuous. Hence, by the classical theorem of Arzela we can find a subsequence Tk = Lk of the sequence Tn such that the limit 1f Tk 0 8(U) = lirn KS(S)K8(t1 S) ds (—co< u 0, and s(A) is continuous, non-decreasing and bounded.

2. Next we choose natural numbers m and n arbitrarily and set ml+s 1 = K8(S)K8(r)f(t — s +r) dr} ds. fs,mn(t) 2 f _ni 4mn/-m/+s From (11) we see that if K8(S)K8(r) 0, then —s + r is an (6/2)-almost period of f. Therefore, it follows from properties (1) and (2) of K8 (s) that

r te7

(13)

20

Harmonic analysis

where F = sup teJ

ilf(t)11*

We set nl = T and m/ = R to give

f 8,m,n(t) 1 f T -1 R+s { f IC8 (s)K8 (r)fit – s + r) dr} ds =— 2 T —T 2R -n+s TT

{

2-1 R f R R Ks(S)Ks(U + s)f(t + u) du} ds.

Thus by setting T = Tk (see property (4) in the preceding subsection) and taking the limit, after using properties (3) and (4) of Ks (s) and the equality (12), we obtain d lk i310 Arnmik (0 f8,„,(t)Lf

f R R f(t + u)08 (u) du =— J 1 fR — E ai, exp HAM — 2R _R f(t + u) x exp DA (t + u)] du 1 fR f(t + u)h(u) du, R

R

where h(u)=

(Utz) ds(A. ).

J

Hence, from (13) with n = mk, in the limit as k -* 00 we obtain

(14)

ilf(t)– fs,„,(t)II- el2+Tim.

To complete the proof of the approximation theorem we need the following simple lemma.

Lemma 3. Suppose

that a function f: J X has a compact trajectory.

Then the function

F (T) = — 1 JTf(t) dt

2T

_T

also has a compact trajectory.

(T

> 1)

Proof of the approximation theorem

21

Proof. It is easy to see from the definition of a Riemann integral that F(T) belongs to the closed convex hull of gif: But, as is well known, the latter is compact together with Now we can complete the proof of the approximation theorem. From Lemma 2 it follows that

_.00 1121R

R ii

R -Rf(t

f

+ u)h(u) dull \ 1/2

liM ( -2-1 Rf l R R I f(t + U)II2 du) R->c0 X lirn ,1 — 7., f R th (u )1 2 du =0. R--.00 zn -R

(15)

Thus, if a l , a2, . . . , aN are chosen so that 00

E ay < ear,

v=N+1

then for all R> 0 we shall have

1

co

R

E a,— ff(t

v=N+1

+ u) exp [i.A. (t + u)] dull< 6/2.

By Lemma 3 there is a subsequence limits exist for all v = 1, 2, , . . , N:

Rk = 771k1

(16)

such that the following

def 1 j'Rk A z, = klina co 2,.ko,r k -Rk f(t + u) exp [iA.(t + u)] du

1 = lim-– ,„_, 1,00 znk

R

fk -

f(t) exp (ikt) dt.

(17)

Rk

It follows from (14), (15), and (16) that Mt) -E Avav exP ( – iAut)ii < 6 , and this completes the proof of the approximation theorem.

3 The mean-value theorem; the Bohr transformation; Fourier series; the uniqueness theorem 1. We make the important point that in proving the approximation theorem we have used only the definition of an almost periodic function and the elementary Property 2 in Chapter 1. On the other hand, as we are going to show in this section, basic properties of an almost periodic function can be deduced comparatively simply from the approximation theorem. Clearly, there is no need to derive once again the properties we have already mentioned, but as an example

22

Harmonic analysis

we consider the theorem about a sum of almost periodic functions. Let f(t), g(t) be two almost periodic functions. For every E > 0 there are trigonometric polynomials PE12(t) and Qe12(t) such that sup WO – P./2(t)ii < 6/2, teJ

sup IIg(t)— Qs/2(t)ii< e/2. teJ

Consequently, sup Ilf(t)+g(t)—[Pe /2(t)+•28,2(t)]Il<e, tEl

and so f(t)+g(t) is an almost periodic function (see Chapter 1, § 1, Property 3). We show that other basic theorems in the theory of almost periodic functions can also be deduced from the approximation theorem. 2. Property 1. The mean value theorem. For every almost periodic -

function f(t), the mean value T def 1 liM ,,,, — j. f(t) dt = M{f} r--.00 z i —T

exists. In addition, the limit 1 j.T+a lirll ,,, — f(t) dt = MID T -4• 03 Z

(18)

1 —T+a

exists uniformly with respect to a. Proof. For all a e J and for all T > 0

1 r T+a 2T

1

if A = 0,

exp (iAt) dt =

J_T-Fa

exp (iAa)

sin AT T

ifA00.

Therefore, for all A E j, T+a

1

def {1 ifA=0, lim exp (in) dt = tif(A ) = T-.002 T —T+a —j . 0 if A 0 0

(19)

uniformly with respect to a. From (19) it follows that (18) holds uniformly in a for every trigonometric polynomial N

P(t)= E ak exp (ukt) k=1

and N

mIP(t)1=

E aktkAk).

k=1

(ak € X),

The mean-value theorem

23

Next let f be an arbitrary almost periodic function; then for every e >0 there is a trigonometric polynomial Pe (t) such that sup lif(t)— Pe (t)il< E. tej

We set def 1 is T+a f(t)dt = MU; T, al. 2T —T+a

Then we have

IIM{f; T', a } —Mff; T", all' IIMIf PE ; T', T', a} —M{P E ; T", — f; T", 011 IIMIP.; 7", al—M{Pe; T", cal+ 2E. Therefore, for every e> 0 there exists a 1 6 > 0 such that for T', T"> and for any a EJ we have —

IIMIf ; r, al—Mlf; T",

< 3 s,

as we required to prove. Notice that for any almost periodic function f and for all A e j the function f(t) exp (—i itt) is almost periodic. Hence, the function a(A ; f) = WM) exP ( — in )} is defined for every A E J; a(A. ; f) is called the Bohr transformation of f.

The next property is fundamental to the theory of almost periodic functions.

Property 2. The function a(A. ; f) is non-zero for at most a countable set of values of A. To prove this property let {Pk (t)} be a sequence of approximating polynomials for which

su? Ilf(t)—Pk(t)11--- 1/k (k =1, 2,

Suppose that nk

Pk(t) =

ak, m=1

and {AO

U Ak,m.

k,m

exp

• •

•)•

24

Harmonic analysis

The set {gn} is not more than countable. We show that a(A ; f) = 0 for A 0 gn. In fact, iia(A ; pii = ila(A.; Pk ) + a(A ; f — Pk)II =11a(A. ; f — Pk)11-. 11 k (k = 1, 2, . ..). Since we can take k arbitrarily large, we conclude that a(A ; f)= 0 for A 0 p. The set { An } of all those A for which a(A; f)00 is called the spectrum of f; obviously, {An} g- {An}. Let

an -= a (An ; f)• With each almost periodic function f we associate (formally, for the time being) the Fourier series

f(t)— E an exp

(Un

n

a

means that there is a relation between the an and f (t) and conveys no implication of convergence.) The elements a n EX are called the Fourier coefficients and the numbers { An } the Fourier exponents of f. The next property follows easily from the proof of the approximation theorem and from the mean-value theorem (see (18)). (^'

Property 3. The Fourier exponents of approximating polynomials nE

Pe(t) = E

k=1

ak, e

exp

(iikk e t)

can be chosen from those of f(t); the Fourier coefficients of approximating polynomials can be regarded as the products of the Fourier coefficients of the function and certain positive numbers (depending on e and the Fourier exponents of the function). As a simple consequence of Property 3 we obtain the following important result.

The uniqueness theorem. Let f(t) and g(t) be two almost periodic functions. If a (A ; f) a (A ; g), then f g. Proof. If a (A. ; f) -=- a (A ; g), then a (A. ; f — g) O. Therefore, we can assume that the approximating polynomial P8 (t; f g) 0 for every e >0. Consequently, f(t)—g(t)O. —

Property 4. For any almost periodic function f we have lim n _...:, an = O. In fact, let Pe (t) be a trigonometric polynomial for which sup WO — P.(t)II-. tef

E,

25

The Bohr transformation

and ne be such that M {PE W exp (-iAnt)} = 0 for n > ne. Then for n > ne

Ilmlf(t) exp ( - U.0111 = 11Mtl[fit) -Pe (t)] exp ( - iAnt)}11-

lian= E.

4 Bochner - Fejer polynomials 1. Let f be a 27r-periodic function with a Fourier series +0, f(t) - E ak exp (ikt), k=—oo

1 Ir ak = — f f(t) exp (-ikt) dt. 27r _, We set n

E ak exp (ikt), so(t)= ao,

sn(t)=

k=—n

and call sn (t) a partial sum (segment) of the Fourier series. Next we write un(t). so (t )+si(t ) + • • • + sn _1( t) n n

= E [1 - —11(1] ak exp (ikt). k

g

n

—

The sums crn (t) are called Fejer sums. For every continuous periodic function f, the Fejer sums converge uniformly to f(t) as n -> oo. On the one hand, the Fejer sums are the arithmetic means of the partial sums of the Fourier series of f, and on the other hand they can be obtained from the Fourier series by introducing into the series the multipliers 0 1(n)

rk ----{(

ikI ) — n

for iki 0 the trigonometric polynomial Qe (t; f) whose Fourier exponents belong to f3 and for which sup

— Q.(t ;

(29)

E.

tEl

By the properties of the Fejer composite kernel, for every m we have sup IIPm(t;

Pm(t; Q.)11= suP1I-Pm(t; f — 2.)II sup VW—Q 6(t )11-.5e'

(30)

tEl

In view of (28), for a fixed 6, there is a sufficiently large M = M(s) such that (31)

sup WE (t; f)— Pni (t; te'

for m M. Then for every t J and m M, it follows from (29), (30), and (31) that Ilf(t)—Pin(t;

as we required.

Qe(t; Dii+ 1142E4; +11P,z (t; Qe )— Pm (t;

— Pin.(t; QE)11

Functions with values in a Hilbert space

31

Almost periodic functions with values in a Hilbert space; Pars eval's relation 1. We are especially interested in the case when the space in which a function takes its values is a Hilbert space. Let X be a complex Hilbert space, x, y be elements of X, (x, y) be a scalar product in X, and 11x11= (x, x) l/2 be the norm of x e X. 5

Theorem. For every almost periodic function f(t) - E an exP (iAnt):J -4X, n

Parseval' s relation holds: 00

(32)

Mt{(f(t), f(t))} =E . 1 (an, an).

Proof. We take arbitrary elements c l , c2, ... , cn e X and arbitrary numbers Mi, M2,. . . , An cf and consider the function d = d(c i, c2 , . . . , cn)= Mtlilf(t) — Enk=i ck exP (4041 2 1. We call NI d the

deviation of the sum n

s(t)= E ck exp (itikt) k=1

from the almost periodic function f(t). We are going to find min ciex d(ci, c2,. . . , cn). In fact, it is easily obtained from an identity which is derived as follows: n

n

d = Mtl(f(t)— E ck exp (itint), f(t) - E ci exp (440)1

k1 =

1=1

n

= mtl(f(t), fit)» - E (ck, )W at) exP (-iiIkt)l) k=1

n

- E (Mat) exP ( — ikLit)}, 1=1 n

ct)

n

+ E E (ck, ct)MtlexP

(ip.kt) exp (—ittit)}

k=1 1=1

n

= MICRO, PO» - E (ck, a(t4k; D) k=1

n

n

E (ck, - E (a(ihk; f), ck)+ k=1 =

n

k1

Ck) n

+ E (a(ktk; A aCuk; D) — E (a(uk; f), a(Ak; A k =1 k1 =

Harmonic analysis

32

n

tk D) = Mtl(fit), PO» — kE 1 (a(lu k; f), a(k; n

+ E (ck—aCak; D, k=1

Ck — a(/uk;

f)).

(33)

It is clear from this identity that min d is attained when and only when ck = aCuk; D = Mtlf(t) exP ( it)}, —

that is, if Pk does not coincide with any of the Fourier exponents of f, Ck if ,uk coincides with one of the Fourier a(/.Lk; f) 0 0 exponents of f. This property is called the minimal property of the Fourier coefficients. When we set uk = Ak and ck = a(Ak ; f)= ak in the identity (33) we obtain 0

d = Mtl(fit), f(t))}-

(34) k=1

Since the left-hand side of this last identity is non-negative, we obtain Bessel's inequality, namely: n

---Aftl(f(t), f(t))}. E=1 Ilak112 k

(35)

Here n is arbitrary so that we deduce, in particular, the convergence of E c:=1 ilak112. Now we are going to prove Parseval's relation (32); for this we take any E > 0 and let ne P(t) = E bk e exp (iA kt) k=1

be a trigonometric polynomial for which6 sup II fit) — P8 (t)11- e. (36) tei It follows from (36) and the minimal property of Fourier coefficients that 0 -.5. d(a i , a2, • • • , ane )--- d(bi e, b2E, • • • , bne ) = Altfil fit) — Pe(t)11 21 -- e2. 6

We recall that the exponents of an approximating polynomial can be selected from the Fourier exponents of the function (see § 2, Property 3).

Functions with values in a Hilbert space

33

By combining this last inequality with the identity (34) we find that

ilaki1 2 --. s 2,

0 --Mt{lif (t)112} -

and therefore since s was chosen arbitrarily, we have proved Parseval's relation. 6 The almost periodic functions of Stepanov 1. Let F:J-4X (X is a Banach space) be measurable .in the sense of Lebesgue-Bochner. V. V. Stepanov suggested a generalisation of the concept of almost periodicity for this class of functions which is fully justified. Subsequently, Bochner pointed out that by using a very simple construction, a Stepanov function can be reduced to a Bohr function, which is vector valued even when the original is a scalar function; we reproduce this construction below. Let Y "(X) be the Banach space of measurable functions 4: i = [0, 1] X with the norm -

„1 ( J0

‘

110(n)11' dn)

lip

(p...-1).

Clearly, for every t EJ the function f(t +n) (n E A) is a measurable function from A into X. We now state a definition of almost periodicity in the sense of Stepanov which takes account of the observation of Bochner mentioned above.

Definition. We say that a function f(t):J - > X is almost periodic in the sense of Stepanov if f(t)= {fit +n), n E Al is almost periodic as a function J .' (X). More fully, f(t) is almost periodic in the sense of Stepanov if for every e >0 there is a relatively dense set of numbers {Te} satisfying the inequality7 1

lip

sup (i. ilf(t +71 +7)- f(t +n)F dn) tEl

--. E.

0

The simplest properties of Stepanov almost periodic functions can be derived from the corresponding ones of Bohr almost periodic functions. For instance, since a Bohr almost periodic function is 7

The continuity of f(t+ n ) as a function from J into .'(X) follows from the continuity in the mean of a function integrable in the sense of Lebesgue—Bochner.

34

Harmonic analysis

bounded, we have 1 sup flifit +,,o tel 0

r d,„ 0 77-1-h

liM cn ( T, a

T-*o3

) = -jLh j.h

an(s) ds

-

(41)

( 71 e [0, 1 - hp

uniformly with respect to a. In (41) we fix a and let h almost all n E A an,(n) = lim cn(T, a) = an. T.«) Therefore we can rewrite (41) as

0; then for

lim cn(T, a) = an

T-> co

uniformly with respect to a. Thus, we have proved that bn(71) X exp (-iAnn) is independent of n, that is, bn(71) exP ( - iAnn) = an €X and

1 = lim — T-4.00 2T

T+"

ff(t) t) exp (-iA nt) dt,

uniformly with respect to a. The an are called the Fourier coefficients of the Stepanov almost periodic function f(t). Thus, i c an exP [iAn(t +7)1.

.T.(t)= fit

n=i

By applying the Bochner summability method to .i(t) we obtain 1 lim sup j.

m -»°° tEl

f (t + n)

-

0

E

k=1

um kak exp (iAkn) exp (iA kt)11 dn =0, (42)

where the timk are convergence factors that depend only on the spectrum {AO. Obviously we can rewrite (42) in the form

E

k=1

,um kak exp (iA kt)1

=0, /11P (X)

36

Harmonic analysis

which is an 'approximation theorem' for Stepanov almost periodic functions.

Comments and references to the literature § 1. The simple facts about Fourier—Stieltjes integrals given in this section are well documented in mathematical analysis. The reader can find a detailed exposition in Bochner's book [24]. § 2. The first application of the Bochner—Khinchin theorem to the proof of basic theorems in the theory of (numerical) almost periodic functions is due to Bochner [24], who used this approach to derive Parseval's relation, from which it is comparatively simple to derive the approximation theorem (see, for example, Levitan [76]). Bochflees proof carries over easily to abstract almost periodic functions with values in a Hilbert space. Parseval's relation is meaningful for these functions (see § 5), but not for almost periodic functions with values in a Banach space (which is not a Hilbert space). The idea of the proof of the approximation theorem we have given goes back to one of the proofs of Bogolyubov (see Bogolyubov & Krylov [9]), who considered numerical almost periodic functions. It occurred to us that the proof of the approximation theorem could be considerably shortened by combining the basic idea of Bogolyubov's proof with the Bochner—Khinchin theorem. Amerio succeeded in carrying over Bogolyubov's proof to abstract almost periodic functions without using the Bochner—Khinchin theorem (see Amerio & Prouse [2]). § 4. Bochner—Fejer polynomials were first introduced by Bochner [25], who also extended them to abstract almost periodic functions [27].

§ 6. The Stepanov functions for numerical almost periodic functions were introduced by Stepanov in [104]. The special case corresponding to p =2 was also discussed by Wiener [33]. The reduction of Stepanov functions (numerical and abstract) to abstract almost periodic functions in the sense of Bohr was suggested by Bochner [27].

Arithmetic properties of almost periods

3

1 Kronecker's theorem

The following fundamental theorem of Kronecker about consistent solutions of a system of inequalities plays a significant role in what follows.

Kronecker's theorem. Let A1, A2,.

..

,

A n and 01, 02, . . . , On be arbitrary

real numbers. For the system of inequalities jAkt — Oki< 8 (mod 27)

(k = 1, 2, . . . , n)

(1)

to have consistent real solutions for any arbitrarily small positive number 8, it is necessary and sufficient that every time the relation 1 1A 1 + l2À 2 + • • • + ln A n = 0 holds, where 11, 12, ... , l n are integers, we have the congruence l 1 0 1 + 1202 + • • • + lnen .--- 0 (mod 27).

(2)

Proof. (a) Necessity. This .is proved very simply. We assume that for every 6>0 there is a t8 e J. such that (1) holds. We rewrite these inequalities in the form — 8 00

(A € Of).

(15)

Proof. From the equality

sup P(fit + t.), f(t + ta)) = SUP P(f(t + tin — ta), f(t)) tel

tEJ

we see that the f-normality of the sequence {4,2 } is equivalent to the sequence Tm,n = tm — tn being f-increasing. On the other hand, from the equality exp (iAtn)—exp (Atm ) = exP (inn )11 — exp (iA (t. — tn))} it follows that the limit (15) is equivalent to (14) for T m ,n = tm — tn. Thus, Theorem 4 is reduced to Theorem 3. Now we make some observations regarding Theorem 4. For an almost periodic function f:J-4X we consider the family ge = k(f) (see Chapter 1, § 3). Each element f(s) E k is obtained as a limit f(s) =limm ,o, f(s + tm ). With each J(s) we associate the function 0(A) = m-,00 lim exp (Wm ) (A E nf), (16) A (s) there corresponds the function 0(A ) exp (iAt), Here, to the shift P that is, a shift on Ye goes into the multiplication of 0(A) by exp (Mt). This correspondence is one-to-one. For if we assume that two functions 0(A) and 0*(A ) correspond to f(s), then there are two sequences {tm } and {t* m } such that lim f(s + tm ) = lim f(s +t*m )= Af(s), ,,,..0 m-.00

= lim exp (it*). lim exp (Wm ) = 0(A )*0*(A ) m-sco nz-..0 3

This means that cAi (x)=4/1 ( y)(i =1, 2, ...) implies that x = y (x, y are any two points of gf).

44

Arithmetic properties of almost periods

It is obvious that the combined sequence {h., el, t2, t 2,. . .} is fnormal, but it does not satisfy (15). Thus, the correspondence (16) is one-to-one. It can be studied in greater detail if we place additional arithmetical conditions on the module 9Rf. We restrict our discussion to the important case when Ef has a finite integer basis pi, ,R2, . . • , Pn. It is clear that in this case the convergence

exp (iA 4,, ) -00 (A ) (A

E Ef)

is equivalent to exp (/3j tn, ) -0 6(j= 1, 2, . . . , n ). Thus, we can associate with an J E k(f) a finite set of numbers =lei, f2,. 4,1. From Kronecker's theorem we obtain that in the case of a finite integer basis, a shift on k(f) is isomorphic to a standard shift on the n -dimensional torus.

e

..,

3. From Theorems 3 and 4 it follows, in particular, that if two almost periodic functions f and g have the same modules (Ef = Mg ), then they have one and the same set of increasing and of normal sequences. We are going to prove that if nf 0 Eg, then the sets off-increasing and g-increasing sequences are not the same.4 Suppose that Ef 0 Eg, for example, Eg E

nf.

(17)

We prove that if (17) holds, then there are arbitrarily exact almost periods of f(t) that are not arbitrarily exact almost periods of g(t). When (17) holds there are two possible cases: (1) among the Fourier exponents of g(t) there is a p, k0 which is not representable as a linear combination of the Fourier exponents {A k } of f(t) with rational coefficients, that is, Ako and 1A 1 , A2, . . . , An, . . .1 are linearly independent; (2) there are no such Fourier exponents, that is, every Fourier exponent kt k of g(t) is representable in the form Nk 1 kAk := 1=1

(18)

where /k > 0, the aki are integers with no common divisor, and there exists at least one exponent il k. for which lko > 1 in every representation (18) (if 1k = 1, then Eg 4 Thus, for two almost periodic functions f(t) and g(t) to have the same increasing sequences, it is necessary and sufficient that 204= Eg.

Fourier exponents and almost periods

45

In the first case, by Kronecker's theorem the system of inequalities lAkTl< 8 (mod 27r) (k = 1, 2, . . (19) (mod 27r) Lukor – bri 0. Therefore, if 3 is sufficiently small and N sufficiently large, then T is an s-almost period of f(t) (s can be arbitrarily small). By the last inequality (19), T cannot be an almost period of g(t) (with arbitrarily high accuracy). We consider the second case. Let V I, denote the smallest of the possible integers in the representation (18) for ktko : Nk 0

I I kop,jc 0 = E a/co 1 =1

x

(18')

By our condition, l',>1. We consider the system of inequalities

lAkTi< 8 (mod 27r) (k = 1, 2, . . . , N), (20) li.4k0 7- – 27r/V kol < 8 (mod 27r), where 8 > 0 and N are arbitrary. By Kronecker's theorem, for the system (20) to be solvable it is necessary and sufficient that from every equation Nk lko ktk o =

E akiA ;, J=1

(21)

where 1ko > 0 and the aki are integers, follows the congruence

2 lko — IT E 0 (mod 27r). / 1 k0 Next we are going to prove that this congruence holds. Every lko in the representation (21) is divisible by a minimal l' ko. For suppose that this is not so; then lko = mk0/ 1 ko + hko , where 1 --_. hko 0 there is an e €1 such that FI

(0) , t2 (0) , ... , tn (°) )

— Pal < E.

We choose 3 = 3(6) so that VF (t' ],

t'2 , . . . , t',,)

—

F (t" 1, t"2, . . . , ral< E

(22)

for le i — ril< 8 (i = 1, 2, . . . , n). Now we use Kronecker's theorem. By hypothesis the numbers 27r/qi are linearly independent, and so there

47

Limit-periodic functions

is an 6 and integers k b k2,. , 127T i• 217 (0

2 k
0 and a finite set of numbers cob 0)2, — . , con such that every 7 satisfying the system (3) lexP (itokr) - 11 0, then x o eg—g+U—Ucg—g+ V. For if 0(x 0) >0, then we have K(to)> 0 and K(t o + xo) >0 for some to, that is, to E g ± U and to + X0 E g + U, and the required property follows immediately. Let the set of numbers a l , a2, . . . , a, be a (5/4)-net of [0,41]. Then for every x e J at least one of the numbers x +a1 , x + a2,. . . ,x+ a, belongs to certain of the intervals Ak. Hence, a E

K(x+ ai ).- 1 for all x e J.

(5)

i=1

From this inequality it follows that

i Cx+ai)=MtiK(t) i K(t+x +ai)}=M{K(t)}.

i=1

i=1

Since M{K(t)} > 0, by taking the mean of both sides of the last inequality we obtain M{0(x)}> O. In the representation (4) we assume that co o =0, that is, A o = MIch >0. Next we set ( 11(X) =

08(x) =

1 j. T liM n Fr, C gX T-pco z / — T

= liM

T _..

1 fT 0

2T —T

+ 1)0 (t) dt

(X ± 00(0 dt.

From the properties of Fourier—Stieltjes integrals (see Chapter 2, § 1) we have 11/ (X)

1 rr, = liM 7e 1T->co zi

I:T —T

co

a(t +x)a(t)dt = E An2 exp (iconx). n=0

The function 0(x) has the property that from 0(x 0) >0 it follows that xo e g—g+g—g+ V; this is proved in the same way as a similar property of 0(x). Then we choose n sufficiently large that ..

z Ak2 L.5.A02 /2. k=n-F1

Suppose that s 0, k=1

that is, TEW-4-1-W-W+ V. This proves Bogolyubov's theorem, since for a sufficiently small ri every solution of the inequalities (3) satisfies (6) (conversely, for a sufficiently small E, every solution of (6) satisfies the system (3)). Let f(t) be an N-almost periodic function in the sense of Definition 2. Then for all E > 0 and for all N > 0 there exists a relatively dense set Eg; N of E, N-almost periods of f(t). It follows from the continuity of f(t) that there is a positive number 8 = (5(s, N) 0 there exists a to E J satisfying the inequalities 1t 0 - in 0 there is an ri = n (3) such that every solution of the inequalities

17'1 ---71, 17- N/21 ---71

(mod 277-)

N -almost periodic functions

59

is a 8-almost period for 0 (t). We choose an arbitrary N >0 and set k =mini t i NO(t). Then for Iti -...N and 8 < k

10(t + 7)– COI =

104

8

04)04 + r)

k(k –8) .

Therefore, if 8 = min (k/2, sk 2/2), then 81 k(k sequently T is an e, N-almost period for OW.

- ô) -.--. E , and

con-

2 Fourier series, the approximation theorem, and the uniqueness theorem 1. For N-almost periodic functions (even bounded ones) the mean value M {f( t)} = lim f(t) dt T_..,—J 2T -T , does not necessarily exist. Nevertheless, as we shall prove in this section, under certain natural conditions, with every N-almost periodic function we can associate a Fourier series. This correspondence is not one-to-one, because there are N-almost periodic functions with which one can associate a set of Fourier series. It is remarkable that in spite of this non-uniqueness, each of the Fourier series of an N-almost periodic function determines that function uniquely. Let f(t) be an N-almost periodic function, and A 1 , A2, . . . be a countable set of real numbers corresponding to f(t) by Definition 3. We denote by Ef = MA1 5 A25 • • .) = (A15 11'25 . . .) the smallest module for the set A 1 , A2, .. . , A,,. (see Chapter 3, § 2). We show how to construct a Fourier series for an N-almost periodic function satisfying the condition ..

1. i _To0 2-1T f7, Rfit)Ildt 00; we denote it by {A,,} = {A, ii , il.,,,2, .. . }, where Ami, = Ak E X. Thus, with each N-almost

60

Generalisation of the uniqueness theorem

periodic function (satisfying (13)) we can associate a Fourier series (14)

f(t) —E Ak exP (ittict).

We call the elements Ak E X ** the Fourier coefficients of f(t). It is possible to produce simple conditions for the Fourier coefficients to belong to the original Banach space X, and by taking the (strong) limit with respect to some sequence Tn —> 00, to obtain lim

Ak

1

Fr,

rn

n —Tn

f (t

exp ( - iktkt) dt (ktk

For example, for this it is sufficient that A(T) (T trajectory in X for all g E

E

nf). 1) has a compact

2. For N-almost periodic functions the approximation theorem is valid in the following form. Approximation theorem for N almost periodic functions. Let f(t):J -> X be an N -almost periodic function satisfying condition (13) and with a Fourier series (14). For all e >0 and for all N > there is a positive integer Q = Q(E, N) and real numbers ak = ak(E, N), k = 1, 2, . , Q, such that -

f(t)- E akAk exp (1/ k t) k=1

for ItN. Proof. By Definition 3 every T EJ satisfying the system of inequalities (2) is an E, N-almost period of f(t). We construct on the interval [-7r, 7r] a smooth non-negative, even periodic function OM with 0(0) > 0 and OM = 0 for VI 8. Then we define an almost periodic function OM by =

1 n H 0(Ait),

a i=1

where

a = lim

T-.00 2T

TT 01 (A it)) dt.

It is obvious that OM can be expanded as an absolutely convergent Fourier series:

=E ak exP (ip.kt)

(1-4 E Ef).

(15)

Fourier series; approximation and uniqueness theorems 61

Let Ta > co be a generalised sequence used to determine the Fourier coefficients il,,. We set Ta 1 f,(t)= lim f(t + u)0(u) du Ta co 2 Ta f Ta Ta 1 = lim f W04 — 21) du Ta -..02T,, Ta -

J ._

=E akAk exp (i,ukt).

(16)

k

Then we have

1 .ra f(t)0(u) du .0271, -Ta

lif(t) —fe MI = Talim -.

1

Ta Ta f(t

+ u)111(u) dul 2 Ta .1. 1 ya , urn o (17) — fit + u)110(u) du. Ta _.., 2 Tc, _Ta Ilfit) If 0(u) 0 0, then Ilf(t)— f(t + u )11 = E for VI --. N. Therefore, it follows from (17) that — 1l. Trl Ta -›oo

-.--.

1 f Ta (18) tku) du = E. If (t) — f(t)II - - E lim Ta -».0271, J—Ta Next, the absolute convergence of the Fourier series (14) implies that there is a Q such that CO

E laki= El r,

k =Q+1

where 1" =limT ›oo (1/2T) I-T TIM' dt. Therefore -

1

CO

E k=Q+1

akA k exp (ip.kt)0-... E.

(19)

Then from (16), (18) and (19) it follows that Q akAk exp (ip,kt) ---2E, E f (0 k=1

as we required to prove. 3. Now let f(t) and g(t) be two N-almost periodic functions satisfying (13). By combining the sequences {4} in Definition 3 for f(t) and g(t), we can make the modules nf and Eg coincide.4 4

These modules do not have to be the smallest ones.

62

Generalisation of the uniqueness theorem

In constructing the Fourier series for f(t) and g(t) we are going to consider only common generalised sequences Ta.

Uniqueness theorem for N almost periodic functions. If f(t) and -

g(t) have the same Fourierseries for at least one generalised sequence Ta, then they are identical. In fact, since in constructing Fourier series we are considering only common generalised sequences, and since by assumption the Fourier series for f(t) and g(t) are the same, it is clear from the construction of the approximating trigonometric polynomials that these polynomials for f(t) and g(t) are also the same. Therefore, f(t) = g(t), as required.

Comments and references to the literature §1. Definition 2 and the basic properties of N-almost periodic functions are due to Levitan ([73 1 and [74]). He was the first to show that the use of these functions leads to an interesting generalisation of a theorem of Favard about the solutions of ordinary differential equations with almost periodic coefficients (see Levitan [73] and [76]). Definition 3 belongs to Marchenko [86], while Bogolubov's theorem was first published in [6]. § 2. Levin & Levitan [72] constructed an example of an N-almost periodic function with a non-unique Fourier series; the construction is given in Levitan's book [76]. Other examples of the non-uniqueness of a Fourier series, as well as conditions for uniqueness, are to be found in a deep paper of Levin [71]. Our construction of a Fourier series for an abstract N-almost periodic function is due to Zhikov. The method of summing Bochner-Fejer series can be extended to N-almost periodic functions (for bounded N-almost periodic functions this was done by Marchenko [85], and by Levin [71] in the general case). We also mention that for an N-almost periodic function f(t) satisfying the condition

rT lim —T j -T Ilf(t)11 2 dt < co, 1

one can easily extend the proof of the approximation theorem in Chapter 2, § 2. A new class of generalised almost periodic functions was introduced by Bochner [28] in an interesting paper; they are called

Comments and references

63

Bochner almost-automorphisms. Almost automorphic functions have been studied in detail by Veech [34]. A numerical bounded uniformly continuous function f(t) is called almost automorphic if for every sequence Itm l for which we have the local convergence io, At), f(t + t m )----4

the 'returning' also holds: A ioc (t— tin ) --->>f (t).

Boles [10] and Reich [100] proved independently that an almost automorphic function is a bounded N-almost periodic function. These papers contain a more general definition of almost automorphic functions, and it is proved that the class of almost automorphic functions (in the sense of the slightly more general definition) coincides with the class of N-almost periodic functions. Baskakov [4] has proved that every scalar N-almost periodic function is the uniform limit of ratios of almost periodic functions.

5

Weakly almost periodic functions

1 Definition and elementary properties of weakly almost periodic functions 1. Let X be a Banach space and X* the dual space. We denote the value of a continuous linear functional x* E X* at x E X by x*(x) or (x*, x). From the linearity of elements of X* it follows that for arbitrary x * 1, X * 2, . . X * n E X * , arbitrary X1, X2, • • • , Xn e X, and any complex numbers al , a 2 ,. . • , an; Pi, 02, • • • On n n

E

kx*k,

(k =1

E E ak0;(x*k, E Aix.) = k=1 j=1

j=1

Let {xn}nm=1 be a sequence of elements from X; {xn } is said to be weakly convergent (weakly fundamental) if the numerical sequence x*(xn ) is convergent (fundamental) for all x*e X*. If in addition there exists an X E X such that liM n->oo

X * (X n ) = X * (X)

for all X* E X*, then the sequence {xn } is said to converge weakly to x, and x is called the weak limit of {xn }. We shall denote this by lim* xn = x

or xn

X.

n -4.00

It follows from the Hahn—Banach theorem that the weak limit is unique. We call a Banach space X in which every weakly convergent sequence is weakly convergent to an X E X, weakly complete. Reflexive Banach spaces are weakly complete (since the dual spaces are weakly complete). In particular, every Hilbert space is

65

Definition and elementary properties

weakly complete. On the other hand the space C[0, 1] of numerical functions continuous on [0, 1] is not weakly complete. Definition. A function f:j - > X is called weakly almost periodic if the numerical function x*(f(t)) is a continuous almost periodic function for all x*e X*. This definition is similar to that of a weakly continuous or weakly measurable (summable) function.

2. We now consider certain elementary properties of weakly almost periodic functions. Property 1. f(t) is an almost periodic function

f is weakly almost

periodic. This follows from the estimate

lx * (f( t + T ) - x *(f (t))1 -- Ilx*IIIIN + T) -POI. )

Rf is bounded Property 2. f(t) is a weakly almost periodic function and separable.' Proof For all x* e X* the numerical function x*(f(t)) is almost periodic, and consequently bounded. Therefore, it follows from the

Banach-Steinhaus theorem that sup lif(t)11< co. teJ

This proves that Rf is bounded. To prove separability we consider the countable set {f(r)}, where the r are rational points of J. For every to e J there is a sequence rl , r2,. . . , rk, . . . (rk -> to) of rational

numbers such that lim* f(rk)= f(t0)

k-*co

(f(t), being weakly almost periodic, is weakly continuous). Therefore, there is a sequence of linear combinations Nr,

Yn = E aknxk

(xk =Prk), akn

k=1

1

Recall that g21 = {x e X: x = f(t), t E j}.

E

C)

Weakly almost periodic functions

66

which converges strongly to f (to) as n -> 00. 2 Without loss of generality we may assume that all the akn are rational. Hence the set of all Yn is countable and so Rf is separable. Property 3. Suppose that a sequence tfn ( t )1 of weakly almost periodic functions converges weakly to f(t) uniformly in t El. Then f(t) is weakly almost periodic. Proof. For all x*e X* we have lim x*(fn (t)) = x* (f(t)),

uniformly in t e J. Therefore x*(f(t)) is an almost periodic function and so f(t) is weakly almost periodic. Property 4. Let f be a weakly almost periodic function and Isnl a sequence of real numbers for which

lim* f(t + sn )= g(t) for all t e j.

(1)

Then:

(i) the convergence is uniform in t e f ; (ii) if f2f denotes the convex hull of AS then 12f =

(2)

(iii) suPtEJ V(t)II = suPtEJ WWII.

(3)

Proof. (i) x*(f(t)) is a numerical almost periodic function for all X * E X. Therefore, if x*(f(t + sn )) is convergent for all t e J, then the convergence is uniform (see Chapter 1, § 2, the proof of the sufficiency in Bochner's thereom). (ii) By definition, 12-f is the closure of the set 12f, that is, of the set

of points P

Z = E pif(0, ti, t2, . . • , i=1 P

tp E I,

E pi =1, p>0.

1= 1

We consider an arbitrary point of 12g : Y = i p(t). k= 1

2

See, for example, L. A. Lyusternik & V. I. Sobolev, Elements of functional analysis, `Nauka', Moscow, 1965, p. 216 English translation, Frederick Ungar, New York, 1961, p. 123.

Definition and elementary properties

67

Obviously y

=lim* i Pkf(tk + sn) =lim* zn, n ->oo

n-*

k=1

where p Zn = E pkf(tk

+sn) G f2f.

k=1

Since the set Ar is closed and convex, by a theorem of Mazur it is also weakly closed. Hence y e flf and so il-g g_ flf. Now we observe that for every fixed x* E X* lim x*(f(t + s n )) = x*(g(t)) n->00

uniformly in t E J. Therefore, for every E > 0 there is a natural number ne (depending also on x*) such that sup tEl

for n > ne ; hence it follows that sup tej

that is lim x*(g(t — s n )) = x*(f(t)) n -> co

uniformly in t e J, or lim* g(t —

Sn)

= f(t).

(4)

n ->00

Therefore, [if g_ fig,

and so (ii) is proved. (iii) From (i) and a standard result of functional analysis 3 it follows that

iim l[f(t + sm)I1= sup Itf(t)II, 3

See, fer example, Lyusternik & Sobolev (footnote 2), p. 217; English translation p. 123.

68

Weakly almost periodic functions

and similarly from (4) we have lif(t)ii = sup Mg(t)119 tEl which proves (3).

2 Harmonic analysis of weakly almost periodic functions In this section we shall assume that the Banach space X is weakly complete (for instance, a reflexive space). We are going to prove certain additional properties of weakly almost periodic functions that are connected with their harmonic analysis. Property 5. f(t) is a weakly almost periodic function

the mean

value

a(A) = Ar{f(t) exp (-iAt)} == Tlifoo * 2 1T

exists for all A

E J.

1 liM * e,— I Fr

(5)

Moreover, the mean value T+s

,

T-0, C70

T . T j f(t) exp (-iAt) dt

f(t) exp (-iAt) dt

(6)

Z 1 f-T±S

exists uniformly with respect to s E J. Proof. For all A e J the function f(t) exp (-i AI), being weakly continuous, is Riemann integrable on every finite interval. Then the following mean value exists for all x* e X*: lim j - j.TT x*(f(t)) exp Hilt) dt T-..«, 2T -

f TT

f(t)exp (-iAt) dt).

(7)

Since x* E X* is arbitrary and the space X is weakly complete, it follows from (7) that the weak limit

1 fT a(A.) = lim*n— ,T, f(t) exp (-Ott) dt T-,,,, 2T .1 —T = M*1f(t) exp (-iAt)} exists. Then for all x* E X* the limit

1 f T+s x*(f(t)) exp (-iAt) dt = x*(a(A)) e.,,,, li rn -T _,,,,, z i -T +8

Harmonic analysis

69

exists uniformly with respect to s. Therefore, the limit (6) exists uniformly with respect to s. By analogy with the earlier terminology a(A) is called the Bohr transform of f(t).

Property 6. f(t) is a weakly almost periodic function a(A)= 0 except for at most some sequence {A}. Proof. By property 2 of § 1 Rf g XCI g X, where X0 is a separable subspace of X. It is obvious that a(A ) e Xo . Since X0 is separable, there exists a determining sequence (with respect to X0) of functionals {x* r} c X*. 4 From a property of determining sequences of functionals it follows that (8) iia(A)ii= sup ker, a(A))i. r For every fixed r, (x*,., f(t)) is a numerical almost periodic function, and so (for every fixed r) (e r, a(A )) = 0 except for some sequence {AO. It follows from here and (8) that a(A) = 0 except for at most a sequence {An } = U Akr , kr

as we required to prove. We set an =a(An),

(9)

and associate with f(t) the Fourier expansion f (t) —E an exp (iAn t).

(10)

n

Let B = { gn } be a rational basis for the sequence {An } (see Chapter 2, § 4). We are going to extend the Bochner—Fejer summation procedure to weakly almost periodic functions. Let 1V7n

Pin(t)=

E

tkinkak

exp (iAkt), 0 --. kt in k ---1,

k=1

be a Bochner—Fejer polynomial for f(t) (see Chapter 2, § 4).

lim* Pm (t)= Property 7. f(t) is a weakly almost periodic function f(t) uniformly on J. Proof. First observe that B is also a basis for the Fourier exponents of the functions (x*, f(t)) for all x* E X * . In fact, it follows from (7) 4

See, for example, Dunford & Schwartz [40].

Weakly almost periodic functions

70 that

a (A ; (x*, f(t)))= M{x*, f(t)) exp ( — iAt)} = (x*, a(A)).

The last expression is zero for A 0 {A n }. Furthermore, Islm

(x*, Pm(t)) =

E il in k ( x * , a(Ak)) k=1

exP (iAkt),

that is, (x*, Pm(t)) is a Bochner-Fejer polynomial constructed in terms of the basis B and the function (x*, f(t)). Therefore, 1imn_».0 (x*, Pm(t)) = (x*, fit)) uniformly on J as we required to

prove. Property 8. f(t) is a weakly almost periodic function and a (A ) z----. 0 f(t) = 0 (the uniqueness theorem for weakly almost periodic functions). Proof. If a(A) — 0, then Pm(t)0 for all m. Therefore, (x*, f(t)):::--. 0 for all x* € X* and hence it follows that f(t) ---= O.

9. Bochner's criterion. Let f(t) be weakly continuous. For f(t) to be weakly almost periodic it is necessary and sufficient that from each sequence {s n } we can extract a subsequence {s'} such that {fit + s'n)} is weakly convergent uniformly on J. Proof. The sufficiency of the condition is obvious. We shall prove the necessity. In the proof of Property 7 we remarked that for all x* E X* the Fourier exponents of the numerical almost periodic function x*f(t)) are contained in a fixed countable set {An }. It follows from this and Theorem 4 of Chapter 3 that it is enough to distinguish a sequence Is n' } satisfying the condition: the following limits exist for any k(= 1, 2, .. lim exp (is'nikk) =

Ok;

n -* c0

this is clearly possible.

3 Criteria for almost periodicity The following theorem gives a general criterion for almost periodicity.

Theorem 1. For a bounded function f :J -+ X to be almost periodic it is necessary and sufficient that: (1) For each x* from a set D everywhere dense in X* the scalar function (x*, fit)) is almost periodic;

Criteria for almost periodicity

71

(2) f(t) is compact in the sense that the closure of the set of its values is compact. In particular, for a weakly almost periodic function to be almost periodic it is necessary and sufficient that it is compact. Proof. The necessity of both conditions is obvious. We shall prove

the sufficiency. From the boundedness of f(t) and condition (1) it follows that f (t) is weakly almost periodic. In fact, for any x* E X * we can find a sequence of elements x*n e D such that Ile- enli -> O. Then

RX * ) fit» — (x*, f (t))1 .-ç-IIX * — X * nil sup lif Mil, t EJ

from which it follows that (x*, fit)) is almost periodic. 6 We proved earlier that the set R.!. is separable. Therefore, without loss of generality we may assume that X itself is separable, and so is isomorphic to a subspace Y of the space of all functions continuous on the interval [0, 11.6 We take a sequence of finite-dimensional linear operators En : Y -> Y with the property

Emy---> m,. y

(y e Y)

(11)

(for this we can use any basis in the space of continuous functions). The function fm(t) --- E rj(t) (m =1, 2, . . .) has values in a finitedimensional space, and so for it the concepts of almost periodicity and weak almost periodicity are equivalent; this is obtained from the equality (Y * , fm(t)) = (y, Etnf(t)) = (Em * Y, f(t)), where Ern * denotes the adjoint operator of En. By (11), fm(t) -> f(t) for every t e J, and in view of the next lemma this convergence is uniform with respect to t e J. Lemma 1. The strong convergence of bounded linear operators is uniform on every compact set K c Y. Proof. Let Am: Y -* Y be a sequence of bounded operators for which Amy -> Ay for every y e Y. Then by the Banach-Steinhaus theorem the norms IlAm Il are bounded by some number 1. 5 6

Thus, the boundedness of fj -0 X and condition (1) are necessary and sufficient for the weak almost periodicity of the function. See L. A. Lyusternik 8c V. I. Sobolev (footnote 2), p. 256; English translation, p. 126.

72

Weakly almost periodic functions

Let lyi l (i = 1, 2, ... , p) be a finite (s/41)-net for K, that is, for all y K we can find a y; (j = 1, ... p) such that (12) el4l. IlY Furthermore, it is obvious that there is an N = N (s) such that (13) IIAyj — AyII =- e/ 2 for n >N and for all j = 1, 2, ... , p. From (12) and (13) we obtain = IRA — An)(Y yi)+Ayi An )(yyj)IH

for all y E K

AnyjIl

llAyjAnyjll

---21(614l)+812=e This proves the lemma, and also Theorem 1.

2. By using this general criterion for almost periodicity we obtain other special criteria which also have important applications. We begin with the following lemma. Lemma 2. Suppose that f(t) is a weakly almost periodic function and that for a given sequence { sn}

lim* f(t +sn )= g(t) uniformly.

(14)

n-).00

Then if lif(t)II and lig(t)11 are almost periodic functions, there is a subsequence {s'} c {sn} such that

Yallif(t + s'n)ii=lig(t)II uniformly.

(15)

is assumed to be almost periodic, there is a subsequence { s'n } fisn l such that

Proof. Since

,Iniipti-s , n1=0(t)

(16)

uniformly, and OW is almost periodic. It follows from (14) and (16) that 8 i[f(t

(17)

From (14) it follows that lim* g(t— s n )= f(t) uniformly.

n->00 8

If xo =

xn, then lixoll ,ç.limn..01ixnii (see Lyusternik & Sobolev

(footnote 2), p. 217, English translation, p. 123).

(18)

73

Criteria for almost periodicity

Since we have also assumed that lig(t)ilis almost periodic, then lim lig(t —s")1 = '(t) ({s"} n-)00

{ s'n })

(19)

uniformly. Now (18) and (19) imply that (20)

from which it follows that Ilf(t+en)ii=0(t+s"n), and consequently, by (17) and (19) 0(0 = lirn i[gt + s"n)I1= lim n—>o0tP(t +s"n)=110)11.

(21)

Now the conclusion (15) follows from (16), (18) and (21). Let Sf denote the set of all sequences s = { sn } for which f(t + sn) uniformly. Theorem 2. Suppose the following conditions are fulfilled: (a) Xis weakly complete; x and xn —> X; 114 (C) ilfs(t)11 is an almost periodic function for all s e Sf. Then f is almost periodic. (b) xn

Proof. It is enough to prove that the set Rf is compact. Let us assume

otherwise. Then there is a p > 0 and a sequence 1 = { ln } such that (22) ilf(ii) — f(4)11=P (j k). By Bochner's criterion we can extract from 1 a subsequence s = Isnle Sf such that lim* f(t + sn ) = fs (t)

n—>c0

(23)

uniformly. Hence f(t) is weakly almost periodic. Since by hypothesis Ilf(t)II is almost periodic, we can choose from s a subsequence s' = {s'} for which fit + s'n )II converges uniformly. Then by using the preceding lemma we obtain Y21,

+ ea! =

(24)

Next by using condition (b), from (23) and (24) we find that for all tej Ilf(t + s'n )— fs (t)I1= 0.

(25)

74

Weakly almost periodic functions

But relation (25) with t = 0 contradicts (22), and so the theorem is proved. There is an important class of Banach spaces for which conditions (a) and (b) of Theorem 2 are automatically fulfilled.

Definition. A Banach space X is called uniformly convex if for 0< o- 2 there exists a function w(o) with 0< (o-) 1, such that for all x l , x2 E X satisfying

lixill = 1 , 11X211

L

(26)

lxi — x211

we have

(27)

+x2)11- 1— w(o-).

Condition (27) can also be stated as follows: for all x i , x2 e X satisfying

11x2 — xiii

0. max (ilxiii, ilx211),

we have

+x2)11

— (a)) max

i1x21i).

(28)

Examples of uniformly convex spaces are the spaces V and ..VP with l< p 0 for some subsequence fx,il c fxk l. Therefore

+

= ( 1— (0 (a)) max (ilx

By taking the weak limit we obtain the impossible inequality 11x11.-.

(1 - ( 0- ) )1 1x The following result of Kadets [62] should also be noted: in any separable Banach space we can introduce an equivalent norm with the property (b).

3. It is possible to give quite simple conditions for almost periodicity in the case of a Hilbert space.

Criteria for almost periodicity

75

Let X be a separable Hilbert space and en} be an orthonormal basis in X. Then {

00

E ekOk(t), Ok(t)= (f(t), PO= k=1

ek),

Ilf(t)1 1 2 =k=1 i l o k( t) 12.

(29)

Theorem 3. For a bounded function f :J ÷ X to be almost periodic it is necessary and sufficient that: (1) the scalar functions ck k (t) are almost periodic; (2) the series (29) is uniformly convergent. -

Proof. Suppose that f(t) is almost periodic. Then since every functional on X has the form (x, h) (h is a fixed element of X), the functions Ok(t) are almost periodic. Next we consider the projection operators En, :X ---> X with Ein x =ET=ickek for x = 5= 1 ckek. By Lemma 1 the series EZ=iekrbk(t) converges uniformly with respect to t ŒJ, that is, for any s > 0 we can find an N = N(s) such that for any t cf.

I

oo

2

E lcbk(t)1 2 --_e. k-.N ek(h(t) 11 = k=N Therefore, the series (29) is uniformly convergent. Conversely, if the Ok(t) are almost periodic and the series (29) is uniformly convergent, then the series f(t) =5=1 ekch(t) is also uniformly convergent, that is, f is almost periodic.

4. We end this section by giving an example of a weakly but not strongly almost periodic function with values in a Hilbert space. Let {O k (t)} be a sequence of numerical almost periodic functions with the properties: (1) the Ok(t) are uniformly bounded, more precisely,

sup t „i 10k (t)i-__ 1; (2) the supports of Then the function

Oi (t)

00

f(t)= E ekOk(t) k=1

and

01 (t ) (i j)

are disjoint.

76

Weakly almost periodic functions

is weakly but not strongly almost periodic. For by properties (1) and (2) .0 sup E 10k(t)1 2 = 1.

k=1 Therefore f(t) is a weakly almost periodic function (see footnote 5 tEl

on p. 71). Then, also by properties (1) and (2), for an arbitrary natural number N 00

sup E IgSk (t)1 2 = I. tEl k=N

Therefore, the series Ecic°-- 1 10k(t)1 2 cannot converge uniformly, and so it follows from Theorem 3 that f(t) is not strongly almost periodic.

Comments and references to the literature §§ 1 and 2. The concept of a weakly almost periodic function and all the properties mentioned in these sections are due to Amerio. It is interesting to note that for Banach spaces that are not weakly complete, it is not known whether every weakly almost periodic function has a countable spectrum.

§ 3. All the results in this section are also due to Amerio. The proof of the Important Theorem 1 that we have given belongs to Zhikov. For other proofs of this theorem see Amerio & Prouse [2] and Levitan [77]. Theorem 3 is due to Amerio. We make an observation about the example at the end of § 3. In all it is simpler to construct a sequence of periodic functions with the same periods that satisfies conditions (1) and (2) of Theorem 3. Thus, there exists a weakly periodic function with values in a Hilbert space that is not strongly periodic.

A theorem concerning the integral and certain questions of harmonic analysis

6

The Bohl—Bohr—Amerio theorem 1. We are going to study the important question: if f(t) is almost periodic, when is the indefinite integral u(t) = f4 f(t) dt almost periodic? In the periodic case there is a simple criterion for the periodicity of the integral, namely, the mean value of f(t) is zero. But if f(t) is almost periodic, when the mean value is equated to zero the spectrum can be condensed at zero, and so the indefinite integral can turn out to be unbounded. We prove that for numerical functions, the almost periodicity of the indefinite integral follows from its boundedness. Clearly, it is enough to examine the case of real functions. Let u(t) be a bounded solution of the differential equation u'(t) = fit), and let 1

m = inf u(t), M = sup u(t). teJ

tel

We consider all possible sequences {tin } c j for which there is uniform convergence

fit + tm ) i >), i(t

(1)

and denote the set of limit functions i(t) by 9r( f ). From the equality t (2) u(t+tm )= u(tm )+f f(s+tm )ds o it is clear that the sequence {u (t + tm )} is compact in the sense of local convergence on A Let 4(t) be a limit point of this sequence; then obviously, û'(t) = f(t). We prove rii, = inf 12(t)= m, /1"/ = sup 4(t)= M. tel

tel

(3)

78

Theorem on the integral; harmonic analysis

Since ti(t) is a local limit of some subsequence of {u(t + tni )} we have A M. To be specific we assume that M <M. m

1 fit). Let v(t) i - tin ) --> The sequence {-tin } is 'returning', that is, (t be a local limit point of {ii(t - t m )}; then v'(t) = f(t), that is v(t)u(t) C. Then we have inf (C +u)ñ inf u, tel

tel

‘ sup (C+u)-_./11

sup u, tel

tEJ

which is impossible, and so the equalities (3) are proved. Clearly, if we add a non-zero constant to the solution 4(t), then the equalities (3) are violated. Thus, with each f(t) from the class Z(f) we can associate a unique preferred solution of the equation u' = f. From here, in turn, we obtain the following observations. Firstly, if {tm } satisfies (1), then the corresponding sequence {u(t + tin )} has a unique limit point in the sense of local convergence. In fact, assuming non-uniqueness we obtain two solutions of one and the same equation of the form u' = f, and both solutions must be subject to (3). Secondly, the convergence u(t + tin ) -> 12(t ) is actually uniform on the whole line. For if this were not the case, then we could find a sequence {s in } cJ such that A

A

A

lu(sm

>O.

(4)

Then by going over to subsequences, if necessary, we can assume that {f(s m + tm +t)} and {f(s m + t)} are fundamental in the sense of uniform convergence. Here it is important that the limits of these sequences must coincide. But then by (4) the local limits of {u(sm + tm + t)} and {12 (Sm + t)} are different, which is impossible. The general case of abstract functions turns out to be more complicated. To appreciate this it is enough to observe that the boundedness of the indefinite integral does not by any means imply almost periodicity (an illustrative example is given in § 2); in fact, we can only assert that boundedness implies weak almost periodicity. For this, let y E X*, then A

(y, u(t))= f (y, f) ds o

and use the result we have proved for numerical functions.

The Bohl-Bohl-Amerio theorem

79

Theorem 1 in Chapter 5 shows that almost periodicity is a consequence of weak almost periodicity and compactness, and so we need to study the question of the compactness of the indefinite integral. It turns out that the problem of compactness is closely connected with geometrical properties of the domain of values, that is, the Banach space X. The most general solution of this problem will be presented in § 2, but for the time being we restrict ourselves to the important special case of a uniformly convex space. Let X be a uniformly convex space and f:J->X be an almost periodic function with a bounded indefinite integral u(t) = 14 Rs) ds. We set M = sup Ma tEl

From preceding results for numerical functions we obtain that if a sequence ft,2 1 satisfies condition (1), then the sequence (y, u(t + tin )) (y EX*) is fundamental in the sense of uniform convergence. The I space X is weakly complete, therefore (y, u(t + tin )) —> (y, t2(t)). Since taking a weak limit does not increase norms, we have

AI = sup 1112(t)1k sup 11u(t + tm)ii tEl

tej

= sup 110A = M. tEl But the sequence {-tm } is 'returning', that is, we have the convergence J (y, et(t - tin )) —> (y, u(t)). Therefore, ki‘ = M. We recall that a space X is called uniformly convex if from the inequality

iixi - x211 = P max {OIL 11x211 } it follows that

ii(xl + x2)/ 2 1 .-- (1 - OW) max {11x111, 11x211}

(OW > 0) .

Now we are going to show that u(t) is compact. If we assume otherwise, then we would have a sequence {OE/ such that

11u(tm) - u(tn)11 =pM >0 (m 0 n, p >0). Therefore, by taking subsequences if necessary, we can assume the convergence:

f(t+tk)-14(t),

(y,

u(t+tk)) 4 (y, /2(0).

80

Theorem on the integral; harmonic analysis

Furthermore, for any fixed t E J we have

tm) - u(t+tn)ii = u(tm) - u(tn)+f ifis +tm) - f(s+tn)1 ds o rt PM j ilf(s + tm) f(s + trz)11 ds =Pm,n(t)M.

o

Since pm,n (t)->p as m, n -*op (m n), for any t el we have

114(t)ii= bin

iiu(t + ttn )+u(t+tn )li 2

mOn--00

lirn (1 - (pm,(t))M (1- (p))M < M. mn This contradicts the equality A';/ = M that we established earlier, and so the compactness of u(t) is proved. It is not difficult to check that our arguments in both numerical and abstract cases are only slightly changed if f (t) is almost periodic in the sense of Stepanov rather than in the sense of Bohr, that is, it is an element of /0 1 (X). Thus, the following result holds

Theorem 1 (The Bohl-Bohr-Amerio theorem). If X is a uniformly convex space, then from the boundedness of an indefinite integral of an almost periodic function f(t) (and even of a Stepanov almost periodic function) follows its almost periodicity.

2. There are more general classes of functions for which the theorem on the integral holds. They are characterised by one or another property of being 'returning'. Let f :J -> X be a continuous function. A sequence of real numbers tn, -> 00 is called returning (more precisely, f-returning) if io. f(t). f(t + tm )-->

Definition 1. A function f(t) is stable in the sense of Poisson if it has at least one returning sequence. Definition 2. A continuous function f : J - > X is called recurrent if for any e, N >0 the set L(e, N, f)= {7

E j:

sup Ilf(r+ t)-f(t)li< el tIN

is relatively dense on the real line.

The Bohl-Bohr-Amerio theorem

81

Obviously, the class of almost periodic functions belongs to the class of recurrent functions, which belongs to the class of functions stable in the sense of Poisson.

Proposition 1. If the indefinite integral of a Poisson stable (respectively, recurrent) function is compact, then it is Poisson stable (respectively, recurrent). Proof. First we consider the Poisson stable case. It is clear from (2) that the sequence {u(t + 40} is compact in the sense of local convergence. Let v(t) be a limit point of this sequence, that is, v(t) = lim u(t + s„,), where {sm } is a subsequence of Itni l. Clearly, v(t) = u(t)+ c, where c is a constant element from X. We denote by Tg the local limit of the subsequence {g(t + s772 )}, assuming that it exists. Since Tu = u + c we have

T2u = u +2c, T3u = u +3c, . . . , Tnu = u + nc.

(5)

Because sup tEJ II Tgij--. -. sup tEj Ile, for a sufficiently large n the equality (5) is impossible for c 0. Thus we obtain v(t).- u(t), which clearly means that the sequence {4,2 } is u-returning. The following property holds for recurrent functions: for any e, N >0 we can find a 8 = 8(e, N) such that L(8, N, f) c L(s, N, u). It is not difficult to prove this property by contradiction, using the conservation of returning sequences that has been proved already. In the next section the requirement that the integral is compact will be weakened significantly.

2. Further theorems concerning the integral 1. Let co denote the Banach space of numerical sequences K = 161 (i = 1, 2, . . .) that converge to 0 with the norm ilicil = max {161}. We consider the function f : J -> cc, defined by f(t) ={(1/n) cos (t/n)}. It is easily seen that f(t) is weakly almost periodic and compact, and that the indefinite integral u(t) =14 f(s) ds has the form {sin (tin )}. Since every functional on co has the form E eini, where E Ind 0 would be uniform on it. But On (u (t)) = sin (t1 n) -> 0 non-uniformly with respect to t E j.

82

Theorem on the integral; harmonic analysis

Thus, co contains an almost periodic function whose indefinite integral is bounded but not compact. It might seem from the simplicity of the preceding construction that similar counterexamples are possible for other Banach spaces. Fortunately, this expectation is not justified. The remarkable fact is that if a space X has no subspace isomorphic to c o, then no similar counterexample is possible. We say that a space Xn does not contain c o if it has no subspace isomorphic to c o. A reflexive or even a weakly complete space does not contain co. For what follows we need an explicit characterisation of spaces that do not contain co. Suppose that we have a series Enœ)=1 en (en EX); we denote by A the set of all partial sums obtained from any permutation of the terms of this series. We call the series unconditionally bounded if the set A is bounded. If in co we take the sequence of unit vectors el = {1, 0, 0, .1, e2 ={0, 1, 0, • • • 1, • . . , then we obtain a divergent unconditionally bounded series E en . In a certain sense the converse assertion also holds. Proposition

2.

If there exists a divergent unconditionally bounded

series, then X contains co. Proof. Let a be any element in A, that is, an arbitrary partial sum.

We set A. (a) = sup

IIE

where the ai are real numbers and the summation is taken over all elements ei in a. From Abel's identity we easily find that def

sup A (a)

= Act

aEzi

2 suP a€4

If we discard the first n terms of our series, then A o can decrease, but it does not tend to zero as n -> co because the series is divergent. By taking this into account (and by discarding if necessary finitely many terms of the series) we find a sequence of non-overlapping partial sums an for which A (an ) = 3A0/4. We set gn = E a e7, where the e7 are the elements in the sum an, and the a7 are chosen so that

Urn II = (an ).

Further theorems concerning the integral

83

We introduce a new series E gn. From our construction we immediately obtain the following important properties: (6) IIE eigill --_ ito max 11611, iigill ---. 3,10/4, (7) where lei l is a sequence of real numbers that are non-zero only for finitely many indices i. For any functional y E X* the series E y(gn ) is absolutely convergent. From the estimate (6) we obtain CO

(8)

E lu(gn)I -- AollYII.

n=1

We consider the element K = E eigi. Let ek = max {II}. We choose a functional y € X* for which y(gk)= 1, y= 1 Aga. Then we have Ilull Ilic11.--- Y(K)=1«kl — E IY(gi)1Ia.1 iok 2

_- max Ilaill (1 — E IY(gi)I)• i Ok

From the inequalities (7) and (6) we obtain the estimate iok

Hence, finally, Aollic II _...- i max 1141. (9) Now we consider the linear manifold of all finite sums of the form K = E aigi. The estimates (6) and (9) show that on this manifold the original norm and the norm IIK ili = max {lai II are equivalent, and hence the closure of this manifold is isomorphic to c o, as we required. We make another observation about the divergent unconditionally bounded series Ecnip=i en. We prove that we can find a series of the form ET=ie, that is not weakly convergent to an element in X. Assume that this is not so. Then it is easy to see that the set A must be weakly compact. We choose a finite sum K = E aiei, where 'a i l -._ 1. We rearrange the terms in this sum so that the coefficients ai are non-increasing. Then by using Abel's identity we see that K belongs to the 'doubled' convex hull of the set A. We now consider the series Y n = 1 gn introduced earlier. From its construction we obtain I-, w

E gn = E aiei, n=1

1=1

84

Theorem on the integral; harmonic analysis

where Icri l--._ 1. Hence the series must converge weakly to an element in X. But this is impossible, since under the isomorphism, gn is mapped into the nth unit vector in co. 2. Let f(t) be an almost periodic function and u(t) =14 f(s) ds. It is clear that for each a E j the difference u(t + a)— u(t) =jtt-l-a fi(s)■ ds is an almost periodic function. Therefore the problem concerning the almost periodicity of the integral is a particular case of the following problem on differences. Let G be an arbitrary group with the group operation written multiplicatively. Let u : G -*X be a function on the group. We assume that for each y e G the difference

u(ty)—u(t)=g y (t)

(10)

is an almost periodic function of t E G, and pose the question, is u(t) almost periodic? The problem can be extended in a sense by taking the function g,(t) in a wider class, say the class of N-almost periodic functions or the class of recurrent functions on the group G, and asking whether u(t) belongs to the appropriate class. We solve the problem on differences under the assumption that one of the following two conditions holds: (a) the function u(t) is bounded and the space X does not contain co; (b) u(t) is weakly compact. Let e be the unit element of G. We set, by definition,

U(e)= {7: ilu(7) — u(e)il< El, 17,(3)= fr: ljg,(7)—gy (e)110 there is a 8 >0 and a finite collection of elements yl , ... , yr such that

i=1

V(8).

Proof. Relation (10) determines can assume that u(e)= 0.

u(t) up to a constant; therefore we

Suppose that (11) does not hold. Then (-1.-) V,(8) n G\U (e 0) 0 0 ,

i=1

for some S o > 0, any 3 >0 and any collection y i , . . . , y,. e G.

(12)

Further theorems concerning the integral

85

We show that we can form a sequence of elements tn e G for which e o (n

iiu(tn)ii

1),

(13)

s 0 /2n

iiii(tntik • • • th) — u(tn)— u (tik •

where the

. . , ik are

2),

(14)

integers such that

1 --i i X is Poisson stable, that is, it has a 'returning' sequence Itni l. Then Itm l is also 'returning' for u(t). 1

Another proof of Theorem 2 in the case when condition (b) holds is given in Chapter 8, § 5.

Further theorems concerning the integral

87

Proof. From the equality (2) it is clear that it is enough to prove I OC > that liu (01-> O. The convergence f(t + tn,) f(t) means that tn., E V(8) for m -.--. m(s, 8). Then from Lemma 1 we obtain that tn., E U(s) as we required.

Information from harmonic analysis Let Y be a collection of measurable complex valued functions OW (t e J) with the norm 110112 = SII0 MI dt, and let Y denote the collection of Fourier transforms c4 (A ) = Si 4)(t) exp (-iAt) dt. We denote by Co the subset of Sê consisting of infinitely differentiable functions with compact support, and the support of (PA ) E Cô by supp i4. We fix a c1)4 0(A ) E CIS° such that iff)0(A ) = 1 for IA I --. 1 and supp 4 0 c [-2,]. For every a>0 we set kW = (f, o(A/a) and Ka (t) = ack(at). We recall that C = C(X) stands for . . the set of all bounded continuous functions J * X, and C = C(X) for the subspace of C(X) consisting of almost periodic functions. In the space C(X), 1?esides norm convergence, a local convergence can be defined: fn —> °c f if the sequence Ifn l is bounded in C(X) and f(t)-f(t) uniformly on 3.

-

each finite interval. In particular, the set of trigonometric polynomials is locally dense in C(X).

A point A E j is called regular for the function f € C(X) if we can find a neighbourhood of it such that 1.

f *4) = f f(t - s)ck(s) ds

I

0

for every c4 E Co with support in this neighbourhood. The set of all regular points is obviously open; its complement is called the spectrum of f and is denoted by u = cr ( f). We can deduce directly from the definition a number of important properties of the spectrum. First of all, the spectrum is not empty provided that f0 O. For if we assume that cr(t)= 0, then we can decompose any element ik‘ e Ciô into a sum of a finite number of elements with sufficiently small supports. Then we obtain that f * 4. ----. O. But since the following proposition holds, we then have f =O. bo,

Proposition 3. For f E C (X) we set f n = f *Kn. Then fn —> f as n -> co.

88

Theorem on the integral; harmonic analysis

j f. if f is uniformly continuous, then f n ---> Proof. We consider the case of uniform continuity; the changes

necessary for the general case will be self-evident. Since IT Kn (0 dt = itn (0) = 1, we have f(t)— fn(t)= f [f(t)— f(s)].Kn(t — s) ds I

=

–fI

[f(t) – f(t – u)]./Cn(u) du

= –J [f(t)– f(t – ul n)]0 0(u) du. I

We choose a number uo = uo(E ) for whichlm--n o ictoo(u)Idu--. -. e. Then by uniform continuity we have ilf(t)–f(t–u/n)II-E (n .---N(e), t eh. But then for any t ef we have Ilf(t) – fn(t)11- 6 110016 ± 26 'Vic, as we required. Now we consider a function f € C(X) with a compact spectrum. Suppose that (4(A) E CT is such that (4 ( A ) =1 for A € o(f). It is not difficult to see that the spectrum of f – f *0 is empty, that is, f(t)= f *0 = f f(s)4)(t – s) ds.

(16)

I

The function cfi (being the inverse Fourier transform of the function irS of compact support) belongs to the class 2 together with all its derivatives. Therefore, f belongs to C(X) together with all its derivatives. We note another completely obvious fact. Suppose that the spectrum of f E C(X) lies outside an interval [a, b]. Then we can find a sequence of trigonometric polynomials fn such that loc fn — )f, cr(fn)c na, b].

To prove this we must first choose an arbirtrary sequence of I OC polynimials gn for which gn —--> f, and then 'cut off' the part of the spectrum lying in the interval [a, b]. This means the following. The set u( f) obviously lies outside some interval [a – s, b + el. Therefore we can set ..,.n fn = ..o – im.on * di where the function tif is such that tfi(A) = 1 for A E [a, b] and IPA ) = 0 for A,g [a – e, b + e]. 1-

,

89

Information from harmonic analysis

2. For what follows we shall need the following lemma. Lemma 2. Suppose that a function f e C(X) is such that the mean T+a

1 lirn -

T-.«) 2 T f_T±, f(t)

dt =0

exists and is attained uniformly with respect to a E J. We set f e = f* Ke and f2 = f — fe. Then —flic -> 0 as s -O. Proof. We integrate by parts the expression f, = II f(t — s)Ech(Es) ds: s f,(t)= s 2 f f f(t — u) du (1); (es) ds

11f

=

J.

J0

17/

s

--e-ff(t—u) du Tic WO d'ri

.1 77 0

=1

77/ E

- E— f

f(t — u) du nc A() dn

1771 - - Ve 71 0

71/E

+J17,_,I.1ii fo

f(t — u) du

(77) dn

_ = le +Ie. The term fe is estimated easily since

_

WIG. ---VEllflIc sup TIE/

For the estimation of f, we observe that from the uniformness of the mean it follows that the convergence

.' f f(t—u) du -+ 0 11 Jo is uniform with respect to t E.1, ini = •I s. Therefore, lilac -+0 as we required to prove. The function f in Lemma 2 has the property that its spectrum is separated from 0. The following important theorem holds for functions whose spectrum is separated from 0. Theorem 3. If the spectrum of f E C(X) is separated from 0, then the indefinite integral V, f(s) ds is an element of C (X). Here if f E 6(X), then 14 f(s) ds E C° (X). Proof. Suppose that the set cr(f) lies outside the interval [—a, a]. We choose a twice continuously differentiable function 14 ( A ) such that 4r4(A. ) = 1/iÀ for IA I _--- a. We are going to show that the inverse transform 0 (t) belongs to the class Y.

90

Theorem on the integral; harmonic analysis

We integrate OW = (1/2r) fi c4(A) exp (iAt) dik twice by parts; then we obtain 10(01.„.5 1 f Icii"(A)exp (W ) dA 277. ii i it2 (17)

— 2 ,711.2fi lik(A)IdA. In addition we have

ow. 1 fa ((A) exp (iikt) a ±-1 f c° sin At d'A. ir J a A 2ar J-a

Therefore, 1

a

0 (t) = yi-r f a CPA) exp (iAt) clik + 1 r sin

7 i at

n dri

71

(t >0)

a

Ck (t) =i; 1 A La c4( ) exp (iAt) clA

_ 1 r sin ri eln (t V '.

loc

Then we have Lvn = gn, where gn ---* g. Since the spectrum of the function v(t) is contained in the interval [—a +A 0, a +A 0] we can suppose that the spectrum of the polynomial Vn is contained in [-2a +A 0, 2a + Ad• We prove that vn (t)= f F (s)gn (t — s) ds I

(19)

94

Theorem on the integral; harmonic analysis

Let g(t) =Ek ak exp (iAkt). Since (4(4) = 1, we have

Ak

e [-2a +A o, 2a +A o] and so

fj F (s)gn (t — s) ds =E exp (iA kt)f F (s) exp (—iA ks)akit ds k

I

=E 0 (4)R,, ak exp (iA kt) =E (A — iikkrlak exp (iikkt)= v(t) , k

which proves (19). By taking the limit in (19) we obtain v(t)= f u(s)0(t—s)ds I

=

fJ

F(s)g(t — s) ds,

from which it follows that v(t) e 6(X). But the latter means that A o i A (u), and so the lemma is proved. The next result is obtained from this lemma and Theorem 4. Theorem 5. Suppose that the intersection of the spectrum of the operator A and the imaginary axis is a rarified set. Then every bounded solution is weakly almost periodic. If, in addition, one of conditions (a) and (b) of Theorem 4 holds, then every bounded uniformly continuous solution is almost periodic. Proof. The second part of the theorem is obtained directly from Lemma 3 and Theorem 4. To prove the first part we need to verify the weak uniform continuity of a bounded solution. Let u(t) be a bounded solution. Clearly, it is enough to verify the uniform continuity of the function (y, u(t)), where y ranges over a set dense in X. But if we choose y from the domain of the adjoint operator A*, then we obtain

(y, u') = — (u, A * Y) + (Y, f),

from which it follows that (y, u(t)) is uniformly continuous. Let us note that in the general case (when the operator A is not bounded), a bounded solution is not necessarily uniformly continuous. Example. We consider the equation m-1

k=0

A k (ox ( k ) = fit),

(20)

Harmonic analysis of bounded solutions

95

o

Here f E C(X), and the Ak (t) (k = O,. .. , m —1) are continuous periodic (with a common period) functions J -+ Horn (X, X). We assume that the operator A k (t) (t E j, k = 0, ... , m — 1) is compact and that the space X does not contain co. We are going to prove that every bounded solution x(t) is almost periodic together with its derivatives of orders up to m. We set u(t)= lx(t), x'(t), . . . , x (m -1) (t)} and write (26) in the form of a first order equation: u' +A (t)u =g. As will be shown below (see Proposition 4), the vector-function u(t) is bounded. The operator A(t) has the property that its mth power is compact. Hence our equation reduces to one of the form v'+ Qu = A where Q does not depend on t and Qm is a compact operator. Since the spectrum of Q is rarefied, the matter is reduced to Theorem 5. Proposition 4. If x(t) is a bounded solution of the equation, then the derivatives x (k) (k =1, . . . , in) are also bounded. Proof. We begin by deducing a certain inequality, first for real and then for abstract functions. Let g(t) be a real twice continuously differentiable function on [—T, T]. Then

sup let)12 = 4 sup ig(t)I sup

10-5 T

10-5 T

Iti.-g-T

+ inf let)12 . itIT

(21)

To prove this we choose a point t1 E [- T, T] for which ig i (ti)1 2 = supitiT let)1 2. We take to to be the nearest point to t1 for which leto)I 2 = infit*T le(t)12. It is clear that g'(t) does not change sign on [to, td. Since d(g 1)2/dt = 2g"g', by the generalised mean value theorem we have tl ig1 (02- 1001 2 = 2 f g"g' dt to

t =2g"(e)f g' dt to

= 2g"(e)(g(ti)-g(t0)), after which the inequality (21) is obvious. Now let x(t) be an abstract function [—T, T]-+ X, where X is a real space, and let y EX* be a real functional with unit norm. We

96

Theorem on the integral; harmonic analysis

apply (21) to g(t) = (y, x(t)). Then we obtain for any t e [— T, T] l(Y, xV))1 2 ---- 4 sup Ry, OD sup (u, x"(t))1 Itl-T

Itl 0 tEl

(respectively, lim p(i(t), x (t))> 0) t-0-00

(*)

for every trajectory x(t)#1(t). When every trajectory is separated (semiseparated) the semigroup and flow are called distal (semidistal). The simplest example of a distal flow is a flow (IC, t); another example is any compact equicontinuous flow. But there are examples of compact distal flows that are not equicontinuous (see 'Bohr's example' at the end of this section). A semigroup S(t) is called an extension of a minimal flow (M, t) if there is a continuous mapping j: X -+ M such that

j(S(t)x)= (j(x))* (x EX, t...-.0).

The separation properties

99

Since every semitrajectory is dense in M we have j(X) = M. The inverse image j -1 (h) is called a fibre over h E M and is denoted by Xh. Obviously, the space is partitioned into fibres, and under the action of S(t) fibres go into fibres. We say that a trajectory i(t) is separated (semiseparated) in a fibre if the inequality (*) holds for every trajectory x(t)#1(t) such that j(x(t))= j(i(t)). A fibre is called distal (semidistal) if every trajectory passing through it is separated (semiseparated) in the fibre. We give an example of an extension which is important for what follows, and also examples of separated trajectories. 2. We consider in a Banach space B a non-autonomous evolutionary equation u; = F(t)u,

(1)

where F is in general a non-linear unbounded operator (the notation u; is used for partial differentiation au/at). For the time being the nature of the operator F is not essential because our analysis will be based only on the existence and properties of a solving operator. We assume that the dependence of F on t is almost periodic, that is, F(t) is an almost periodic function with values in some metric space R. Therefore we can consider (at least formally) a family k = k(F) of 'limit' operator-functions h = F(s) of the form F(s) = lim„,_>co F(s + t,„,). We denote a shift on the space ge by h t ; we also denote the initial operator-function F(s) by ho and identify it with (1). For every h E k we introduce the corresponding 'limit' equation A

u; = F(t)u.

(1h)

Suppose (for the time being, formally) that S h (t)p (t_.-- 0, Sh(0)P = p) denotes a solution of equation (1 h ) corresponding to an initial condition p E B. We set X=Bxk and define a transformation S(t): X -* X (t---- 0) by S(t)x = S(t){P, h} = ISh(t)p, WI. It can be verified in a straight-forward way that the operators S(t) (t -- 0) commute. If the following continuity condition holds, then these transformations form a continuous semigroup.

Continuity condition. For any initial value p e B and any h E H, the equation (1h) has a unique strongly continuous solution u(t) = 5 h(t)P

100

(t

Lyapunov stability and almost periodicity 0, u(0) = p), and the mapping

Sh (t):B x 9e-*./3 is continuous for every t

(2) 0.

We call the semigroup S(t) a basic semigroup; it is an extension of a dynamical system (9e, t), and the corresponding homomorphism is the projection j: X -> ge. In this case a trajectory is a pair OM, h t }, where 4(t) is the solution of (1 h ) which is defined on the whole line. Now we turn to a more detailed study of the trajectories of a basic semigroup. Let OM (t e J) be a fixed solution of the original equation (1). We say that it is separated (semiseparated) if inf110 (t) -p(t)11> ( lim 110 (t)- p tej

t-y—oo

>0)

for every solution p(t) # 0 (t ). It is clear that we can associate with a separated (semiseparated) solution fi(t) a trajectory {Ø(t), ho t }, which is separated (semiseparated) in a fibre. The semiseparation property holds in the important case when a solution OM is uniformly stable in the sense of Lyapunov (we omit the simple verification of this fact). If 0(t) is compact as well, then we can consider the limits /3(t + tm) -*bp, (t). It is important that Ø(t) is also a uniformly Lyapunov stable solution of (1h). Next we observe that for a basic semigroup (and its invariant subsets), we only need to verify the separation property for trajectories from one fibre. Indeed, the separation of the trajectories {p i (t), h i t }, {p 2(t), h2t } from different fibres (that is, h 1 h2) is ensured automatically by the distal property of the system (k, t). Hence we obtain the following result.

Proposition 1. Let OM be a compact uniformly Lyapunov stable solution. Then the closure of the corresponding trajectory {Ø(t), ho t } is a semidistal set. We end this section by giving a classical example of Bohr which is important in a number of situations.

Example 1. Let a(t) be a real almost periodic function whose indefinite integral g(t)= Sot a(s) ds cannot be represented as g(t)= co t + OM, where c o is a constant and OM is an almost periodic function. We consider the non-autonomous equation p ; = ia(t)p and the corresponding dynamical system on X = R 2 x (Rm denotes the

The separation properties

101

m-dimensional Euclidean space). It is obvious that this system is distal, since solutions of the linear equation p't = icl(t)p satisfy the identity il P (t)ii =11p (0)11. On the other hand, the non-trivial solutions are not almost periodic; this follows from Bohr's theorem of the argument. Thus, we have a distal but not uniformly continuous flow. By refining the properties of a(t) we can obtain sharper negative results, for instance, a minimal distal but not strictly ergodic l flow on a three-dimensional torus. Bohr's example will be used significantly in Chapter 8, § 7, where, in particular, we give a 'concrete' construction of a(t).

2 A lemma about separation 1. We denote by X x the set of all mappings X -*X endowed with the topology of pointwise convergence (the Tikhonov topology). The space X x is a semigroup with resi3ect to the composition uv (by definition, (uv )(x) = u(v (x ))). It follows directly from the definition of pointwise convergence that the composition uv is continuous in u for any v e X x, and continuous in v if u is a continuous mapping. From here we obtain the following simple fact.

Proposition 2. Let A c X x be a semigroup of continuous operators. Then the closure A is also a semigroup. If A is commutative, then the elements of A commute with those of A. Proof. Let uc, -> uo and v o -> vo . From the continuity of the composition uv with respect to an argument u it follows that u c,vo e A. But then ua vo -> uovo E A, proving the first part of the proposition. Next if uvc, = vu (u, vc, EA), then by taking the limit we obtain uv o = v ou (u EA, v o EA). We observe that the commutativity of the semigroup A does not by any means imply that of A. Proposition 2 is proved. Now we suppose that a dynamical system S(t) acts on the space X; we denote the closure of the set {S(t)} (t E J) in X x by T = T(X), and call it the enveloping Ellis semigroup. There is defined on the space T a natural dynamical system 7r t : 77-tti = /i t = S(t)24 = uS(t). For the concept of an Ellis semigroup to be meaningful we must assume, in addition, that all the trajectories of the system S(t) are compact. In this case T is a compact Hausdorff space.

1

A flow is called strictly ergodic if it has a unique invariant measure.

102

Lyapunov stability and almost periodicity

2. Suppose that a continuous semigroup S(t) defined on X is an extension of a minimal system (M, t). Lemma 1 (The separation lemma). The following assertions hold: (1) a compact trajectory that is semiseparated in a fibre is recurrent; (2) two compact trajectories x i (t) and x 2(t) from a single fibre, each of which is semiseparated in the fibre, are jointly recurrent and mutually separated, that is, inft. j p(xi(t), x2(t))> O. Proof. First of all we assume that there is defined on X a group and not a semigroup. Since we are concerned with the properties of compact trajectories, we can assume from the outset that X is

compact. We denote the fibre over an element h E M by F(h), and the closure of the family {SW, t 0} in X x by T - ; T- is a semigroup with respect to composition that is invariant under the transformations ir t (t ...._ 0).

The set r--1, 0 7rt T - is non-empty and invariant under the group of transformations /r e (t e j). By Birkhoff's theorem, there exists a compact minimal set V c T - . The set V is also a semigroup with respect to composition. Indeed, it follows from minimality that V = fir tvol = {S(t)v o}, where vo is an element from V. Then it is enough to refer to Proposition 2. We introduce a family of continuous mappings cf)„ : V -> X defined by

4)„(v)= v(x) (vE V, x eX). It follows easily from the minimality of V that ckx (V) is a minimal set in X for every x E X. Hence it follows that the trajectory x ot is recurrent provided that we can prove that Xo E Ox0( V). We set ho =i(xo). The group 71-t acting on V is an extension of the system (M, t). To find the corresponding homomorphism 1: V -* M we must fix an element x E X and set ck , I . , 1 : V --- > A ---> 114 .

If 1 -1 (h0) is the fibre over ho, then the transformations v c 1 -1 (h0) have the property v(F(h0))OE T(h o). We denote by Vh 0 the set of restrictions of these transformations to 1 (120), and by 3 the restriction operation. Since 8 is continuous, Vho is a semigroup of transforma-

A lemma about separation

103

tions F(h0)-*F(h0). Next we use the following important fact from the theory of topological semigroups. Proposition 3. Let V be a compact Hausdorff space with the structure of a semigroup, where the operation a -> ag is continuous for any 0 E V. Then V contains an idempotent element. We apply this proposition to the semigroup Vho. Suppose that w is an idempotent element, that is, co 2 = co, and let co = 8(v). We are going to prove that v (xo ) = xo. Since v e T - there is a generalised sequence S(tA ) (tA -.5 0) for which S(tA )

xx

>V.

By setting x 1 = v (xo) we have v( x0 ) = lim S(tA )xo = v 2(xo) = v (x i ) = lim S(tA )x i . The relation lim S(tA )xo = lim S(tA )xi does not contradict the semiseparation of the trajectory xot only in the case when v (xo) = xi = xo. Thus, v(x0) = xo, and therefore xo belongs to the minimal set q5( V). This proves assertion (1) of Lemma 1. To prove assertion (2) we consider the natural dynamical system on the space Z ={{x, y} EX xX:j(x)=j(y)}.

(3)

Since the trajectory {xi', x2t } is semiseparated in a fibre it is recurrent (by assertion (1)). Since the metric p is continuous on Z, the function P (xi', x2t ) is recurrent (see Chapter 1, § 6). Therefore inf p(xi t, x2t ) = t•--OD inf p(xi t, x2 t ), tEj

which proves Lemma 1 in the case of a group of transformations. To prove Lemma 1 in the case of a semigroup we need quite a little more. Let 0(X) be a collection of continuous functions J -*X with the topology of uniform convergence on every finite interval, and let y be the metric corresponding to this convergence. There is defined on the product 0(X) x M an obvious dynamical system: if g E {Rs ), h} then g t = {f(t + s), W}. We consider two elements g1 = {x i (s), kJ and g2 = Ix2(s), hob where x i (s) and x2(s) are two trajectories of the original semigroup. It follows from Proposition 2 in Chapter 1 that the trajectories gi t and g2t are compact. Let Go be the smallest compact invariant subset

104

Lyapunov stability and almost periodicity

containing these two trajectories. The corresponding fibre F(ho) consists of elements of the form g = Ix (s), hob where x(s) is a trajectory of the original semigroup and j(x (0)) = ho. From the assumption that the trajectories xi (t) (i = 1, 2) are semiseparated in a fibre, it follows immediately that the trajectories git are semiseparated in a fibre, that is, inf y(x(t+s), x i (t+s))>0 t.,..0

(x(s)0x i (s)).

Therefore, Lemma 1 for a semigroup follows from Lemma 1 for dynamical systems, and so this lemma is proved. Now we prove Proposition 3. Let 12 be the class of all non-empty subsets A Œ V such that AA c A. Then 12 0 0 since V E Q. We order 12 by inclusion, then by Zorn's lemma it contains a minimal element B. If co E B, then Bai is compact, and (Bco)(Bco)OE BBa) Œ Bai; consequently, Bai E f2 and Bai Œ B. Since B is minimal, Ba) = B. Therefore there exists an element p e B such that pc° = co. Let L = {a e B: aw = co}; then p e L. Since a multiplication of an element on the right is continuous in V, the set L is closed and therefore compact. If k, 1 e L, then l(kw) = lai = co, that is, LL c L. Thus, L e (2 and L E B, that is, L = B. Hence co E L, that is, (o 2 = co, and Proposition 3 is proved. 3. Let (X, t) be a flow with compact trajectories which is an extension of a minimal flow (M, t), T = T (X) be an Ellis semigroup, and ho be a fixed element from M. We consider the set of those transformations u e T for which u(Xho) c Xho, and let Tho be the set of restrictions of these transformations to Xho.

Lemma 2. If the fibre Xho is distal, then Tho is a group. Proof. We choose an arbitrary element 13 E Tho and are going to prove that (3 -1 E Tho. The set A = Thof3 is a compact semigroup, and so it

contains an idempotent co. By using the same arguments as in Lemma 1, we see that to (x ) = x (x e Xh0). Thus, a i g = e (the unit element of Tho) for some a l e Tho. Similarly, we can find an a 2 e Tho such that a 1 a 2 = e. Hence 13 = e13 = a2a1f3 = a 2, that is, a l = 0 -1 , thus proving Lemma 2. We consider the special case of Lemma 2 when the space M is a point. We obtain that the Ellis semigroup of a distal flow is a group (Ellis's theorem).

A lemma about separation

105

We introduce in the set of fibres the Hausdorff metric P(Xhi, Xh 2) = sup d(X hi, x2) + sup d(x i , Xh2). x2eXh2

xleXhi

Then the following proposition holds.

Proposition 4. If a fibre Xho is distal, then the mapping h -* Xh is continuous at ho. Proof. Let hm -*ho and xo E X. We require to prove that we can find elements xm E Xhin for which lim,n _.co xin = xo. Let um e T be any element such that um(X) = Xhni. (it follows from the minimality of M such that such a um exists). The set {u m } has a limit point a E T. Since

a

by Lemma 2 a (Xho) = Xho, that is, a (x i ) = xo for x i e Xh i . But then xo = a (x i) = lim um (xi), which proves our proposition. Note that the continuity of the mapping h -* Xh at each point means that the mapping j: X -*M is open. E Tho ,

3 Corollaries of the separation lemma Below we suppose that X is a complete metric space with an invariantly acting semigroup S(t) (t---- 0). It is assumed that every trajectory of this semigroup is compact. 1. The first group of corollaries is obtained by applying Lemma 1 in the special case when M is a point.

Corollary 1. If X is compact and the semigroup S(t) is semidistal, then S(t) is a distal flow. Proof. Because the distal property of S(t) is obtained at once from Lemma 1, we prove that S(t) is a group. Since all the trajectories are recurrent, X is split into minimal sets. If X is minimal, then . . S(t)X =X for every t -- 0. Hence it follows that S(t)X = X for every t .-- 0, and consequently, the inverse transformations S -1 (t) are continuous, as we required to prove. Remark 1. If the space X is not compact, then in general, our semidistal semigroup is not a flow since in general the operators S 1 (t) (t ---- 0) are not continuous. The continuity of these operators is easily ensured by the following additional condition: for every

compact set K ΠX the set U S(t)K is compact.

t_,..-o

(4)

106

Lyapunov stability and almost periodicity

The following condition for a trajectory to be absolutely recurrent holds.

Corollary 2. A compact trajectory x ot is absolutely recurrent if and only if it is semiseparated in X0 ={x ot }, that is, it is semiseparated in its closure.

The necessity of the condition is obvious. Indeed, if a trajectory xot is absolutely recurrent, then the pair {xot, x t} is recurrent for any trajectory x t from Xo. If x t *xot, then the minimal set {xot, x t } must be separated from the diagonal set in X0 x X0, which means that the trajectories xot and x t are mutually separated, that is, Proof.

infteiP(xo(t), x(t))> O. To prove the sufficiency we consider an arbitrary recurrent trajectory yot (of some semigroup on Y). Let

Ze = {x, Yot l, Ye = {Yot }, and j: 2 --> Y be the projection onto the second component. Consider the fibre j -1 (y0). The trajectories passing through it have the form Ix% yot l• Hence it is clear that the trajectory {xot, y t } is semiseparated in the fibre, and so by Lemma 1 it is recurrent, as we required to prove.

In particular, an almost periodic trajectory is absolutely recurrent. 2. We say that an extension j: X -*M is positively stable if for every e > 0 and every compact set K c X there is a 8 = 8(e; K) such that p(S(t)xi, S(t)x 2)-s (t 0) whenever p (xi, x2) .-.. 8 and j(xi) =i(x2), xl, x 2 E K. -

It is useful to note that from the positive stability of an extension it follows at once that every fibre is semidistal. Therefore, from Lemma 1 we have

Corollary 3. A semigroup S(t) is a distal and two-sidedly stable extension. When X is compact or condition (4) holds, then S(t) is a flow. In the proof we need to prove only the two-sided stability property. For the proof we consider on the set Z defined by (3) the continuous function p(z) = p(x, y). Since all the trajectories z t are recurrent, the function p (z t ) is recurrent. Therefore suP P(x t, y t )= suP P(x t, y t ) tej

ti:i

(i(x)=AY)),

which gives the two-sided stability of the extension.

Corollaries of the separation lemma

107

Corollaries of the separation lemma (continued) 1. We return to the properties of compact solutions of equation (1), and recall that we mean solutions defined on the whole time axis. 4

Corollary 4. Let p i (t) and p 2(t) be semiseparated compact solutions. Then they are jointly recurrent and mutually separated, that is inf 1119 1 (t)— p 2(t)II> O. teJ

(5)

Corollary 5. Let p° (t) be a uniformly Lyapunov stable solution and _et be the closure of the corresponding trajectory. Then the restriction of the basic semigroup to )2' is a minimal distal flow. For the proof we need to take into account Proposition 1 and Corollary 1. Next, let V( p, q) be a non-negative continuous function on B x B which is zero only on the diagonal. We say that equation (1) is V-monotonic if for every two solutions Pi(t), P2(t) of (1) the scalar function V(pi(t), P2(t)) is non-increasing on that part of the time axis where both solutions are defined.

Corollary 6. If (1) is V-monotonic for any pair of compact solutions p i (t) and p2(t) (t e J), then the 'identical' invariance V(pi(t), P2(t)) --=----constant holds. Indeed, the solutions pi(t), P2(t) are jointly recurrent, and therefore the function g(t) = V(pi(t), P2(t)) is recurrent. Since (1) is Vmonotonic g(t) is non-increasing. But then it is recurrent only when it is identically constant. The properties of mutual separation and identical invariance play an especially important role in what follows. We show that these properties are peculiar to equations with almost periodic coefficients and do not hold, for instance, in the case of bounded coefficients. In Lemma 1 the only condition on the system (M, t) is minimality. This means that the corresponding corollaries (in particular, the separation property (5) and identical invariance) are valid for equations with recurrent coefficients. But it is impossible to weaken recurrence to boundedness, as is shown by the scalar equation P ; + a Mp(t) = 0, where a(t) is a continuous function such that a(t)= 0 for t---5 0 and a(t)= 1 for t---- 1. Here all the solutions are bounded and semiseparated but not mutually separated. Our monotonicity condition holds ( V= lp — q I) but identical invariance does not.

108

Lyapunov stability and almost periodicity

Let us also note that the two-sided uniqueness theorem holds in the class of semiseparated solutions; this assertion is not true without the recurrence condition, as is not difficult to demonstrate by an example. 2. We denote by U(t, 7) (t 7) an operator B B which maps an initial value p = p(r) E B into the value of the solution of (1) at the point t. Suppose that uniform positive stability holds in the sense that the operators U(t, 7) (t 7) are equicontinuous on each compact set in

B. In our earlier notation we have U(t, 7) = Shor(t -7). Since the set fhorl is dense in Ye and the reflection (2) is continuous, the family of operators Sh (t): B B (t 0, h E

(6)

is equicontinuous on compact sets, that is, the semigroup S(t) on B x Ye is a positively stable extension of the flow (Ye, t). Hence, if X is the collection of all compact trajectories in B X c9e, then it is a distal two-sidedly stable extension (see Corollary 3). We discuss especially the case when B is finite-dimensional. Then the set X is closed in B X k. For suppose that xm = {pm, LIE X, x,n (t) = {pm(t), hm t l are the corresponding compact trajectories, and x,n ->{0, h }. By hypothesis the operators (6) are equicontinuous on compact sets; therefore, these operators are uniformly bounded on compact sets provided that X 0 0. From here and from the recurrence property of the solutions pm (t) we have

sup teLm P.(t)II = SUP iiP m (011 = sup IIShm (t)pm (0)11< co.

By taking the limit we see that the solution (t) is bounded, that is, X is closed. Then condition (4) follows easily from the uniform boundedness of the operators (6). Thus, the following assertion holds. Corollary 7. Let B be finite-dimensional and suppose that the uniform positive stability condition holds. Then the set of compact trajectories is closed in B x 9r, and the restriction of the basic semigroup to this set is a distal flow which is a two -sidedly stable extension of the flow (,t).

Almost periodic trajectories

109

5 A theorem about almost periodic trajectories 1. In this section we are going to discuss the simplest principles that enable us to judge whether almost periodic trajectories or solutions exist. Let (X, t) be a compact flow which is an extension of a flow (e, t), j : X -> A' be the corresponding homomorphism, and F(h) be the fibre over an element h.

Theorem 1 (Favard [106]). If each fibre F(h) consists of a single element, then all the trajectories are almost periodic. To prove the theorem it is enough to note that in this case the homomorphism j: X ---> A' is an isomorphism, that is, the inverse mapping r i : .9e-> X is continuous. Theorem 2 (Levitan [74]). If a fibre F(ho) consists of a single element, then the corresponding trajectory is an N -almost periodic function. For a proof we consider the 'winding' {hot } and give it the topology of a subspace of ge. The correspondence h ot --> x(t), where j(x(0)) = ho, defines a function on the winding. From the fact that every fibre F(h ot ) consists of one point and the set X =Li h , g e F (h) is closed, it follows that our function is continuous on the winding. But then x(t) is N-almost periodic.

Remark 2. In applications, it is usual to preassign on a space X (chosen in a suitable way) not a flow but an invariantly acting semigroup. Clearly, Theorem 1 still holds in this case; but in Theorem 2 we need to require that every fibre F(hot ) (t E J) consists of a single element. The next theorem is a fairly significant generalisation of Favard's theorem.

Theorem 3 (Amerio [1]). If a flow (X, t) is distal and some fibre 1 (h0) is finite, then all the trajectories are almost periodic. Proof. We show that all the fibres are finite and consist of one and the same number m of elements. For suppose that F(ho) = {x i ,. . . , x.}, {Y1, Y2, ... , yk } c F (h 1 ), and k > m. We choose a sequence {t„} Cf for which lim,,_,co hi tn = h and lim,,_›co yi(tn)=zi exist (1--.. i ---.. k). Then zi E F(h), and zi 0 zi (i j) because the fibre ['(h 1 ) is distal; this contradicts the inequality k > m. Let A(h) = infxoyEr(h) d(x t, y t ); similar arguments show that A(h) -= A> O.

110

Lyapunov stability and almost periodicity

Suppose that h is an arbitrary element in .9e. It is not difficult to see that there is a neighbourhood 12 = 12(h) of h such that the set F(n)=UhEit T(h) is partitioned into m parts 171 , 172, ... , 11m and that the following conditions are satisfied: (1) d(11i, 11i )-- A/2 (i 0 j); (2) if h ef2, then 1-7 (h)= {0i (h)}, where (h ) are continuous on 12. By selecting a finite covering of the space A' by neighbourhoods 12(h) we obtain that X is an m-sheeted covering of .9e. Therefore, for any e E (0, A/2) we can find a 8 = 8(e) so that one of the following inequalities holds: (7) P (x 1, x2) = 6 P (x 1, x2) = A/2, provided that x i E r(hi), X2 E F(h2) and d(hi, h2) ---- 8(6). Now we are going to prove that the flow (X, t) is uniformly continuous. Suppose that p(xl, x2) --- Et, (hi, h2) --- 8 (6), i(ri) = hi, and p (x i t, x2t) --._E for any t E j. Indeed, i(x2)= h2. We shall show that g(t) = if we assume that g(ti )> e (taking for definiteness t 1 > 0), then since P(hi t, h2t )=P(hi, h2)= 8/6) by (7) we have g(ti)-?-- A/4. Since g(0) --._ e ---. A/4 and g(t 1)---114, we can find a to e (0, t 1) for which g(to) = p (x I to, x2 to) =i/3. But because p(hi t°, h2t°)---.8(e) we obtain a contradiction to (7), and the theorem is proved. 2 Levitan's theorem, in contrast to those of Favard and Amerio, operates with a fixed fibre. This gives rise to the important question of the possibility of generalising Levitan's theorem in the spirit of Amerio's result. In other words, are the trajectories of a distal fibre N-almost periodic functions if the fibre is finite? Unfortunately, the answer to this question turns out to be negative; a counterexample is given at the end of this section. Another question, which arises in connection with Amerio's theorem, is whether the finiteness condition can be lifted. This question is resolved by the following theorem. ,

Theorem 4 (Zhikov [50]). If a flow (X, t) is distal and if some fibre has dimension 0, then all the trajectories are almost periodic. A proof is given in § 6. Theorems 1 4 form a system of 'elementary principles' to which in the final analysis all theorems about almost periodic solutions reduce. Later on we shall develop different methods that will enable -

2

It may seem that the last step relies on the connectedness of the real line. However, we do not use it here, and the proof we have given is suitable without changes for discrete dynamical systems (cascades).

111

Almost periodic trajectories

us to find for the basic semigroup compact invariant sets with appropriate properties; but now we dwell only on a simple application of Amerio's theorem.

Proposition 5. Let 73(t) be a compact uniformly Lyapunov stable solution of (1), and assume that Ø(t) is asymptotically stable. Then this solution is almost periodic. Proof. By Corollary 4 the set ± = {73 (0, hot } is distal and minimal. The fibre 1 (120) contains the point 0(0) ; it follows immediately from the asymptotic stability of /3 (t) and the distal character or 1 (h0) that /3(0) is an isolated point in 1 (h0). But then it is easy to obtain from considerations of minimality and distalness that every point of the fibre 1 (h0) is isolated, that is, T(h o) is finite. Now we only need to use Theorem 3.

2. Example 2. Now we are going to construct a counterexample to show that it is impossible to generalise Theorem 2 in the spirit of Amerio's theorem. The construction is based on the following auxiliary proposition. Let f(t) be (in general) a complex valued almost periodic function satisfying the condition 'POI >0 (t e f),

inf if(t)I = 0. tEl

A

In addition, we assume that every function f(t)E 9e( f) has not more than one zero.

Proposition 6. If the continuous root If(t)I"2 is an N -almost periodic function, then it is almost periodic. Proof. We can write the N-almost periodic function If(t)I 1/2 in the 1/2 . v(hot where hot is the trajectory of some minimal form If(t)1 isometric flow (G, t), and v is a continuous function on {hot} c G. The function v 2, being uniformly continuous on {hot}, can be extended to a continuous function f(h) on G (so that f(t) = f(hot )). We consider a sufficiently small neighbourhood A of h that ,

)

If ( h)I. - - 4 > 0 (h € /1).

(8)

(This is possible because f( h0 ) = f(0) 0 0.) Since y is continuous on {hot} we can find a neighbourhood A 1 c A

such that iu(hot ) — u(ho)l =A/2

112

Lyapunov stability and almost periodicity

for hot E A l . It follows from (8) that the values of v (hot ) for ho t E Ai are the same as those of a branch of [ f (h)] 1/2 (h E Ai). In other words, v (hot ) is extended to a function v (h) continuous on Al. Now we prove that v (hot ) can be extended to a function v (h) continuous on G. Assuming otherwise we obtain a sequence {tm } and two subsequences {t' m }, {t"m } such that urn h o rn = h, lim[f(t',„)]1/2 0 lim [Et",n )1 112, f(h) 0 O. The function f(h t ) vanishes at not more than one point; to be specific suppose that f (h t ) 0 0 for t.---- 0. We can find a number t1 > 0 such that li tl = h l E A i . Since u(h) is continuous on A1, 'fin [ ph + t , m11 112 =iim[f(ti+ t,, m)} 112 . It is now clear from continuity considerations that lim [f (tim )] 1/2 = lim [f (t"„,)] 1/2 because I f(t + tni )l---- e > 0 on 0 .-.. t .-_. t i . This proves Proposition 6. Now we construct the counterexample. Let 9e be the twodimensional torus with the minimal shift. We denote the trajectory {h t } by fh, and fix three points ho, h 1 h2 E k such that ,

,

h1, h2) Jho, AI 0J1/2. We consider a continuous complex-valued function f(h) on .9e with the properties: (1) f(hr) = f (h2) = 0, f(h )0 0 for h 0 h l and h 0 h2; (2) the index of the vector field f(h) at h l is equal to 1 and at h2 to —1. Let u i (t) = [f(h ot )] 1/2 and u2(t) = —u l (t) be two continuous roots. We denote by X the closure of the trajectory x i = {ui(t + s), hot } in the space O(R 2) x k. The fibre Rh o) consists of not more than two elements x 1 = fu l (s), hol and x2 = {u2(s), ho} (it will be clear later on that the case of a single element is excluded). It is important to note that the fibre Rh o) is distal, and hence the system (X, t) is minimal (by Lemma 1). If, for instance, x t is an N-almost periodic trajectory, then u 1 (s) is an N-almost periodic function. Then (by Proposition 6) u 1 (s) is an almost periodic function. This means that the trajectories of the system (X, t) are almost periodic and each fibre consists of two elements. In this situation (as in general in the situation of Amerio's theorem), X is a finite-sheeted covering of the space 9e. Therefore, we can find a neighbourood A of h l not containing h2

Almost periodic trajectories

113

such that

(h)= {0 1 (h), 02(h), h}, where the çb = (f) 1 (h, s) are continuous functions A -> (R 2). But then the numerical functions Oi (h, 0) = I f(h)1 112 are continuous on A, which is impossible by conditions (1), (2). This completes the construction of the example. In this example a distal fibre consists of two elements, but the corresponding trajectories are not N-almost periodic. The number of non-distal fibres in the example is two (with accuracy up to a shift in X). Therefore, arguments that the 'majority' of fibres must be distal do not help us. A more detailed analysis shows that the constructed flow (X, t) is minimal, strictly ergodic and has a mixed spectrum.

6 Proof of the theorem about a zero dimensional fibre -

We split the proof of Theorem 4 into two separate lemmas.

Lemma 3. If the extension X -> .9e is stable and some fibre Rh o) is zero-dimensional, then all the trajectories of the system (X, t) are almost periodic. Proof. Since we are speaking of a separate trajectory, we may assume that the system (X, t) is minimal. We set Z ={(x, y)EXxX:j(x)=j(y)}.

(9)

We define on the set Z an invariant function p(z)=-- p(xi, x2) = suptEJP(xi t, x2 t ). From the stability of the extension it follows that the metric p is equivalent to the original metric on every fibre. We fix a point I E f(h 0). The numerical set A = p (, x): x E r(h0)} does not entirely contain any interval [0, 6] because p is a metric on a zero-dimensional fibre. Then since functions of the form p(xi t, x2 t ) are recurrent and the system (X, t) is minimal, we obtain A = {p(z): z Z}. We prove that any trajectory x t is almost periodic. By assuming that Bochner's criterion does not hold we have two sequences {tin }, {Tm }e j such that x t+tm. ->loc i t, P(x ttn+Tnt, m)5 >O. Here it is clear that the convergence j(x j(i t ) is uniform on J, so that the trajectory h t = j(x t ) is almost periodic. Now we choose a number p=( x t±tm, t‘ . AA with O +00. By taking subsequences if necessary, we may assume the convergence: x t+tni+Orn --ioc yi t and i t+61n --* Ioc Y2 t. Clearly, , \3 1(Y1) =.l(Y2). But from (10) we obtain sup p(Yi t, Y2 t )=P(YI, Y2) = A t_..5_0

= sup P (Y 1

t,

tEj

Y

2t

) = P (Y1, Y2),

which contradicts the choice of A. This proves Lemma 3. Lemma 4. Suppose that the extension X -> M is distal. Then if some fibre is zero-dimensional, then this extension is stable. For the proof we need the concept of an equitransitive system (Gottschalk & Hedlund [38]). A system (X, t) is called equitransitive at a point x if for any neighbourhood U of x we can find a neighbourhood V of x and a compact set K C J such that IV C KU, where

IV = U S (t)17, KU= U S (t) U. teJ

tEK

Note that a transitive (that is, minimal) system is equitransitive.

For if U is an open neighbourhood, then JU is open and invariant, that is, JU = X. But then, in view of compactness, it is enough to choose a finite covering. The main step in proving Lemma 4 is the proof that the system (Z, t) is equitransitive at every point. The stability of the extension X > M follows fairly simply from equitransitivity. We give this reduction. For a given open neighbourhood U of the diagonal in Z we choose an open neighbourhood W of the diagonal such that Wc U. By equitransitivity, for every y E Z\ U. there exists an open neighbourhood f2, and a compact set IC, c J such that Jay c Ky (X1141. Since the set X\U is compact, we can find a finite set L c X \U such that X\U =U vEL f2y. On setting K =UyeL K y we obtain J (X1U) c K (X\W). The set V = r),EK S(t)W is non-empty (it contains the diagonal), and open since the operators St, where t E K, are equicon-

3 Since limm_..3 (1imn_.03 h t-

"^"— = iimm_.00

(ht±ttn+0,i).

Theorem on a zero-dimensional fibre

115

tinuous. Hence the relation we have obtained, rewritten in the form IV C U, gives the stability of the extension X -> M.

Proof of Lemma 4. first of all let us note that from the zero-

dimensionality of some fibre of a distal extension it follows trivially that every fibre is zero-dimensional. For the natural extension Z > M we denote the fibre F(h)x I' (h) by E(h). For any set A e M we put E (A)=Uhn E E(h). Let A be an open neighbourhood of h = j(z) in M. Since the fibre E(h) is zerodimensional, for a sufficiently small diameter diam (A) we can find open sets P and Q such that z e P, E(A)= Pu Q, F n Q. = 0 . (11) Now we assume that the system (Z, t) is not equitransitive at z. Then the relation JP c KP is not fulfilled for some neighbourhood P (of the form described above) and any compact set K c.f. In particular, if we take K = [-m, m], then we can find points zn, E P and tm Ef for which zmt P for t E [tm -- M, tm + In] = A m . Since 0 d m, to be definite we can assume that Arn e (0, co). We consider a maximal interval [am, 13m ] such that A m, [am, 13m ], z m t P for t e [a,n, gm ]. (12) From the condition for maximality we obtain zmam E P. Without loss of generality we may assume that zniam --> zo. Then from (11) we obtain immediately that zo e P but zot P for t---. O. In the recurrent case the closure of the trajectory coincides with the closure of a semitrajectory. Therefore, ZOE P, {zot} n P = 0 . (13) We write ho = j(zo) and Zo = {z ot }. Since ho e A we have h0 = limn, hm, where hm E A. Because the mapping j: Zo ->M is open (see Proposition 4), zo = lim ni , z,„ for some sequence of points z„, e Zo n E(hm ). But from (13) and (11) it follows that zm e Q and so zo e Q. This contradicts the fact that p(P, 0) > 0 and the lemma is proved. It remains to examine the numerical module corresponding to the almost periodic solution in Theorem 4. Let T? be the module corresponding to the almost periodic function h ot, and let Erat be its rational hull, that is, 911rat = {A/ m}, where A E E and m is an integer. By using Kronecker's theorem it is not difficult to see that the module corresponding to the almost periodic solution in Theorem 4 belongs to Erat • -

Lyapunov stability and almost periodicity

116

Statement of the principle of the stationary point

7

Let the space B = II' be finite-dimensional. We assume that the uniform positive stability condition holds for (1), and that this equation has at least one solution bounded for t _.-. O. Let X be the collection of all the compact trajectories of the basic semigroup. Then X 0 0 and is closed in B x k. We denote the fibre Xi,. over ho by P, in other words, P is a set of initial values {p(0)} corresponding to bounded solutions of (1). The set P has the following important property.

Proposition 7. The set P is a homogeneous space relative to a compact topological transformation group; moreover, every element of this group of transformations is the restriction to P of a continuous mapping lim > P. -

Proof. We denote by T and T + the closure of the sets {S(t)} and {S(t): t--- 0} in X x, respectively. By Ellis's theorem, T is a group with respect to composition, and so the natural dynamical system on T is minimal. Hence we obtain the important relationship T = T + . We consider the set of elements u e T for which u(F(h0))c 1(h). Let E = Tho stand for the set of restrictions of these transformations to F(ho) = P; Tho is a group with respect to composition and is compact in the topology of pointwise convergence (see Lemma 2). From the condition for uniform positive stability and the equality T = T + it follows immediately that every u e Tho is the restriction to P of some continuous transformation II' > P. The set Tho is a topological group, since by the equicontinuity (on each compact set in P) of the transformations u E Tho, the operation of composition is continuous in the collection of variables. This proves the proposition. Now we introduce some more terminology. Let A be a subgroup of the group E. By an A-orbit of p E P we mean the set Ap = {u(p): u E A}. A set Po c P is called A-invariant if it contains the A-orbit of each of its points. It is easy to see that P is partitioned into disjoint A-orbits, and that the group A acts transitively on each A-orbit. 4 Since the trajectories of the system (X, t) are recurrent, the set X splits into minimal subsets. The fibre over an element ho corresponding to some minimal set is called a minimal fibre. It is sufficiently -

4

That is, for any two elements PI) u E A such that u(p1)= p2,

P2

belonging to an A-orbit, there is a

Principle of the stationary point

117

clear that the partitioning of P into minimal fibres is the same as that into E-orbits. Thus, a minimal fibre is a homogeneous space with respect to a compact transitive transformation group. It can turn out that the group E = Tho is not connected. Let G denote the component of the unit element. We say that a point p o e P is stationary if u (p o) = p o for every u E G. It is easily seen that if po is stationary, then the minimal fibre containing it consists of stationary points. Indeed, the minimal fibre coincides with the E-orbit of p o. Let Pi= u (Po), where u E E. Since u -l Gu = G (the component of the unit element is always a normal divisor), we have G 1 = {p i}. Let T2 stand for the numerical module corresponding to the righthand side of (1), and Era, denote its rational hull. Principle of the stationary point. For an almost periodic solution with a module of Fourier exponents belonging to V rat to correspond to an initial value po it is necessary and sufficient that po is a stationary point. Proof. Assume that po is a stationary point. By Theorem 4 it is enough to prove that the corresponding minimal fibre is zerodimensional. Let no° denote the restriction of the group E = Tho to the minimal fibre containing p o . There is a natural homomorphism of Tho to Tf,g (the restriction homomorphism). Clearly, under this homomorphism, the component of the unit element is mapped into the component of the unit element. Since the action of the group G on a minimal fibre is trivial, the component the unit element of the group Tgg is trivial, that is, the group ng is zero-dimensional. But the zero-dimensionality of a group implies the zero-dimensionality of the homogeneous space, that is, of the minimal fibre. The proof of the sufficiency does not present any difficulty. Thus, at least formally, the problem about almost periodic solutions is reduced to one about a common fixed point of some compact topological transformation group. If the group G is non-commutative (in general this is the case), then the problem about a common fixed point is very difficult. A facilitation of this is the fact that the transformations forming G act on the entire space II' and G is connected. 8 Realisation of the principle of the stationary point when the dimension m T, where T is a Lie group, such that T(uv ) = T(vu). Since dim 1-(r).-. dim r = 1, r(r) is one-dimensional and consequently it is a commutative Lie group. This contradiction proves Proposition 8 completely. 3. Proof of Theorem 6. By Proposition 7 every element u E E is generated by a continuous mapping cfi: R'n —> P. From (14) and the equality T = T± it follows that 012 c 12; in other words, u has at least one fixed point. We consider the collection A of all closed sets L c R 3 such that

(a) u(L) = L for every u e G (G-invariance); (b) every u e G has at least one fixed point belonging to L; (c) if L contains a topological sphere, then it also contains its interior. Since P n12 E A the set A is non-empty. We order A by inclusion, and by Zorn's lemma obtain a minimal element P EA. Let 6 be the restriction of G to the set 13. The group G° is connected; we shall prove that it is commutative. By assuming otherwise 0 we obtain that G acts non-commutatively on some orbit G(p 0) c P. Since by Proposition 5 G(p 0) is a sphere, in view of the minimality of P and the property (c), the set P is a three-dimensional ball A. Next, the set of all those p e ./3 for which the group G, is non-commutative is obviously open. Therefore the interior of the ball A contains a G-orbit which is a sphere, and we obtain a second closed ball A 1 c A. Since the boundary of this ball is G-invariant, then A 1 itself is G-invariant (this is easily seen by considering the transformations u E G as the limits of solving operators), which contradicts the minimality of P. Thus, the group 6 is commutative and connected. In G (just as in every connected compact commutative group) we can find an element g such that

Lyapunov stability and almost periodicity

120

. {gn} = G (n =0, ±1,. . .). 5 By condition (b) the transformation g has a fixed point po, but then every transformation u E G leaves po fixed. this proves Theorem 6.

4. Proof of Theorem 5. For m =2 the only G-orbits are a circle and a point. Let Q denote the set of all G-stationary points. We consider the collection A of all compact sets L c P such that (a) u(L)= L for any u E G, (b) if L contains a circle, then it also contains its interior. If Q 0 P, then A is non-empty since the closed disc corresponding

to a non-trivial G-orbit is an element of A. Consider a 'minimal' element /3 E A. It follows immediately from minimality that 0 consists of G-stationary points. Thus, we have proved that Q 0 0. We shall show that either Q = {Po} or Q = P.

(15)

In the first case the module of the almost periodic solution po(t) belongs to 2J1 because the corresponding minimal fibre or E-orbit consists of a single point. In the second case every bounded solution is almost periodic and has a module in Erat. Suppose that there is a non-trivial orbit G(p0). We consider an arbitrary point p E P lying outside this orbit. Since the orbits G(p) depend continuously on p (in the Hausdorff metric) and since the set P is linearly connected (as the continuous image of the plane R2), to p corresponds a G-orbit enveloping the orbit G(p0). Hence it follows that there exists a unique stationary point and the G-orbits of the other points are circles surrounding it. This proves (15). It can happen that to a non-trivial G-orbit there corresponds an almost periodic solution /3 1 (t); the module of such a solution does not belong to 9.nrat • Let El denote the smallest module containing the modules E(F) and gn(p i (t)). We shall prove that every bounded solution is almost periodic and its module belongs to J?i . With this aim, in (1) we make the substitution p(t)= pi (t)+ q (t) and obtain 4 = Fi(q, t)= F(q + pi (t), t) F(p i (t), t). This equation has two almost periodic solutions with modules belonging to E l (namely, —

5

Proof. Let a(t) be an everywhere dense 1-parameter subgroup, and x„, the characters of the group 6, x,,,(a(t))= exp (iAmt). We choose a to E f such that exp (iA nito) 0 1 for every m. Then we can take g = a(to). Indeed, if G 1 = { gn} 0 6, then we can find a non-trivial character of 6 that is trivial on G. This is excluded by the choice of to.

The principle under monotonicity conditions

121

q -=- 0 and q = (po(t) - Pi(t)), and consequently it is enough to refer to a property of the form (15).

Realisation of the principle of the stationary point under monotonicity conditions Suppose that the condition for V-monotonicity holds and that equation (1) has a solution bounded on t ---- O. First we assume that V has the form V(p, q)= W(p -q), where W is a homogeneous convex function on R m, or more briefly, W is a norm on R. But even under these conditions the existence of at least one almost periodic solution is unknown in the case m-.- 5. To ensure the existence of at least one almost periodic solution, independently of the dimension we impose the additional condition that the norm W is strictly convex. 9

Theorem 7. If the W-norm is strictly convex, then equation (1) has at least one almost periodic solution with a module in E. When m.--.. 4, the strict convexity condition is not required; here the module of an almost periodic solution belongs to Prat. Proof. By Corollary 6, IIP1(t) - P2(t)liw = iiP1(0) - P2(0)Ilw holds identically for any /3 1 (0), p 2(0) e P; hence we obtain that E is a group of W-isometric operators. Every element of the group G is generated by some mapping 0: fr -> P (Proposition 7). From the monotonicity condition and the equality T = T ± it follows that 0 is a W-nonexpansion operator. We denote the set of all 'generating' mappings by Ô. In particular, the unit element in G is generated by the retract co: Rill -> P with co(p)= p for p e P. 1. We prove the first assertion in Theorem 5. We shall show that the set P is convex and the transformations u E E are affine on P. Let PI, P2 e P, P = (Pi ± P2)/2 and q = co(p). Since

IIci—pillw-11p—pillw=1-11pi—p26 and

11q— P liw = lip — P211w = iliPi — P211 w, it follows immediately from the strict convexity of the W-norm that q = co(p)= p. In other words, the set P is convex. Similarly, it can be established that the transformations u E E are affine on P. Now

122

Lyapunov stability and almost periodicity

we shall prove that the transformations u E E have a common fixed point po. It will then follow that the minimal fibre E(po) consists of a single point, and so our problem reduces to Favard's theorem. Let il, be an invariant measure on the compact group E. We consider a mapping 0: E -0 Fr, where 0(u) = u (p 1 ) (p l is a fixed point in E). It is easy to see that p o = fE 0(u) dp, is an E-stationary point. 2. Now we prove the theorem for the case m -.4. We assume that a W-norm is defined on Finz ; all geometrical notions (distance, diameter, isometry) relate to the W-norm. We denote by S(p, r) the W-ball with centre at the point p and radius r. We consider the set A of all convex compact sets A c Jr' such that 6- (A)c A, and order A by inclusion. The set A is non-empty. Indeed, let L be an arbitrary compact G-invariant set, for instance, L = G(p), and set L = npeL S(p, d), where d = diam L. Then it is not difficult to see that L is convex, L c L, diam L = diam L, and d(L)c L. Let A be a 'minimal' element of the set A. We assume that dim A = 4, since if dim A -. .. 3 it is not difficult to repeat the arguments of Theorem 3. We study the set A in greater detail. Let M denote the boundary of A. We set do = diam A and P =A n P. The inclusion P c A is an important consequence of `minimality'. To prove this we take an arbitrary point p o E P and note that it follows directly from `minimality' that diam G(po) = do. Therefore, we can find points p i., q l E G(p0) for which lip 1 — qiii w = do. Since the group G acts transitively and isometrically on G(p0) we can find a qo e M such that liqo — Paw = do = diam A, that is, p o E M, which proves the inclusion P c M. Two cases are logically possible: (1) P = M, (2) 13 a.A. The first of these cannot be realised since it would mean the existence of a retract c o: A -0 aii. When P o M we reason in exactly the same way as in Theorem 6, with the set A playing the role of Q. The connected orbits have the same structure since P belongs to a part of ail homeomorphic to a three-dimensional ball. This proves Theorem 7. Theorem 7 can be given a more general form without essential changes in the proof. We call V(p, q) (strictly) convex if it is (strictly) convex with respect to the first variable for a fixed value of the second. Then for m ----. 4 the condition of V-monotonicity with a convex V is sufficient, while for m >4 V is strictly convex.

Comments and references

123

Comments and references to the literature The monograph of Nemytskii & Stepanov [95], Chapter 5, contains a very full account of the general theory of dynamical systems (including 'topological dynamics'). The book by Gottschalk •St Hedlund [38] is also interesting.

§ 1. The concept of a distal system is due to Ellis [119] who evidently used the important work on flows on a multi-dimensional torus by Furstenberg [108]. An approach to a non-autonomous equation from the point of view of dynamical systems was given by Millionshchikov [88] and also by Shcherbakov [114]. Later (and apparently independently) similar ideas were developed by American mathematicians (Miller, Sell, and others). § 2. The separation lemma and also the majority of its applications is given in the article by Zhikov [50]. It must be mentioned that the method of proving this lemma is a generalisation of a method of Ellis [119]. Proposition 2 was proved by Wallace [105]. Proposition 4 (in the case when all the fibres are distal) was proved by Furstenberg [109] and then generalised by Bronshtein [31]. §§ 3 and 4. The criterion for absolute recurrence is a generalisation of a similar criterion of Flor [110]. The remaining material in §§ 4 and 5 is due to Zhikov. We mention that Bronshtein & Chernyi [32]

have obtained certain further results. § 5. Example 2 is taken from the article of Zhikov [50]. §§ 7 -9. The formulation and application of the principle of the stationary point were given by Zhikov [50]. It appears that a study

of higher dimensions by topological methods is interesting. We also note that the concept of V-monotonicity in a general form has been studied by Cheresiz [113].

Favard theory

8

1

Introduction

In this chapter we consider the problem of almost periodic solutions of a general linear equation in a Banach space

Lu --=:. eii +A (t)u = f(t).

(1)

In the simplest case, when A(t)=--- 0, we obtain the problem about the integral we studied in Chapter 6, which will serve as a natural motivation for the construction of a more general theory. The Banach space B is assumed to be separable and the operators A(t) unbounded. We do not need more precise information concerning the nature of these operators since we shall rely only on the properties of a solving Cauchy operator. In other words, the conditions on the operator A (t) (and also on the way it depends on t) are implicit in the conditions on solving operators. We assume that A(t) and f(t) are almost periodic functions with values in certain metric spaces. This enables us (at least formally) to define the set k of all 'limiting' pairs h = {24(s),.Af(s)} of the form ;1(s ) = lim„,,,,,„ A (s + tn., ), .i(s) = limr„,,,, f(s + tni ). We denote the initial pair {A(s), f(s)} by ho and identify it with the original equation (1). For any h = 14(s), i(s)} we introduce the corresponding 'limiting' equation

iu ---- u' + ii(t)u =/(t).

(1h)

We rely on the following condition.

Condition 1 (Condition of 'weak' continuity). For any initial value p e B and any h e 9e, equation (1 h ) has a unique strongly continuous

Introduction

125

solution u(t)= Sh(t)p, and the map

Sh (t):Bx 9e- B is continuous for each t.-.- 0 if the space B is equipped with the weak topology on each bounded set. In other words, Condition 1 means that for any equation (1 h ) the Cauchy problem is uniquely right-solvable for positive t for any initial value /4(0) e B, and that this solution depends continuously both on the initial condition and on the variation of the 'coefficients' 4(t), i(t) at the limits of the set 9e. We define a transformation S(t): X -> X(t 0) on the space X= by the formula S(t)x = S (t)fp, h1 = {S h (t), 11 } . It is not difficultBxk to verify that the operators S(t) (t_--- 0) commute. In what follows the space B will be endowed with the weak topology on bounded sets. The corresponding topology on X=B XX' will, for simplicity, also be called weak. From Condition 1 we obtain that S(t) (t -.- 0) is a semigroup of continuous transformations. It is also clear that if we give Condition 1 a two-sided character (by allowing t 0 whenever L(u) = 0, u(t) O, sup teillU(t)11 ‹ (x). Note that non-trivial solutions of the homogeneous equation can vanish since we do not assume that the left uniqueness theorem holds; if we were to make this assumption, then semi-isolation away from zero would reduce to the inequality inf o iiu(t)ii > O.

Theorem 3. Let the space B be uniformly convex. Then if the condition of semiseparation holds for the original equation (1), then it has at least one weakly N -almost periodic solution. This solution is weakly almost periodic if each equation (1 h ) satisfies the condition of semiseparation. Proof. We introduce the quantity

il(h) = inf sup Ilu(t)ii, tEl

where the inf is taken over all bounded solutions of (1 h ). By taking a minimising sequence and extracting from it a subsequence weakly convergent for each t E L2 we see that the inf is attained on some solution uh (t), that is, sup iluh(t)ii = A (h).

(3)

tEl

2

The possibility of taking the weak limit (uniform on each finite segment) is ensured by Proposition 1.

128

Favard theory

We are going to show that ,u(h)--mi p.. Assume that kt(h 1 ) h2. By taking a subsequence if necessary, we can assume the weak convergence uhi (t + trn ) — v(t). But v(t) is a solution of (1 h2) and sup liv (Oil .--. sup Iluhi (t + tm)11 tEl

tEl

--.. sup liuhl (t + tm)li = tk(h1)0). We shall prove that the unique solution of (1) satisfies the minimax condition (3). Indeed, suppose that uo(t) and u l (t) are two such solutions. If uo(t)*ui(t), then by the condition of semiseparation we can find a to e j such that tint0f iiuo(t) - u i (t)II = 8p, >0. We choose a subsequence trn -> -00 for which hottn -> ho . Without loss of generality we may assume that uo(t+t.) — u i (t+t,n ) — v i (t)

(t E I).

Then for any t e f we have

liivo(t)+ vi(t)ii =1 m-,..0 lim iluo(t +t„,)+ul (t +tm )II --(1--0(8))14 I1 with the norm -

IluilB2 = f liu (s )1I2 Q (s ) ds, I

where the weight function Q(s) is continuous and positive, and Q(s) — 1/s as s -> co. Let C be a Hermitian completely continuous operator H >H with a complete system of eigenvectors {e m } corresponding to the positive eigenvalues {Ani }. For each u(s) E B the translation u(t + s) is a continuous function J >B; this function is, obviously, a 'generalised' solution of the equation u't — u's =0. We now fix a weakly almost periodic but not compact function /2(s):J->H (an example is given at the end of Chapter 4). It is easy to verify that to it there corresponds a weakly almost periodic but not compact function '6(t+ s):J B. The function es (t + s) satisfies the equation u' t — u', + Cu = f(t + s), where f(t) = Ca (t); this equation is considered in the space B and has the form u't +Au = g(t), where A = —d/ds + C and g(t)= f(t + s). Thus, the equation has a weakly almost periodic solution; we are going to show that the homogeneous equation u't +Au = 0 has no bounded -

-

solutions.

Indeed, let u(t, s):J. > B be a non-trivial solution of the homogeneous equation. We set u = E um em and assume that um (t, s)$ 0. Then -

a a Urn — — um + Amtim = 0. at as

—

If now we set vm (t, s)= um (t, s) exp (—A mt), then vm (t, s) satisfies the relation

a at

a as

—v m --v m = 0, and so vm must have the form vm = vm (t + s). On the other hand (since 11B is a bounded solution and An, >0), the norm livm (t + 5 u J livm (t + s)IIQ(s) ds decreases exponentially as t -> +co. But this is impossible (here we must use the condition Q(s)— 1/s). This, then, is the required example. Clearly, the operator A can be chosen to be bounded; for this we must replace B by a subspace corresponding to a fixed width of the spectrum.

Favard theory

130

Certain auxiliary questions

3

From now on, unless stated otherwise there will be no restrictions on the space B (apart from separability). 1. Construction of a minimal set. We consider the set of all weakly compact convex sets K c B such that K x k contains at least one trajectory of the basic semigroup 5(t),3 and order this set by inclusion. Every minimal element of this set is called a Favard minimal set. Let K be a Favard minimal set. The set of all trajectories belonging to K x Ye is denoted by F, and the fibre over h E k by Fh. It is obvious that the fibre Fh is a convex weakly compact set. Let p be the metric corresponding to the topology of weak convergence on K. We shall prove that for any elements p (0), q (0) E Fh inf p(p(t), q(t))= inf p(p (t), q(t))= 0. t,so tej

(4)

With this aim we embed our space B in a Hilbert space H so that every weakly convergent sequence in B is strongly convergent in H.4 Then, in particular, the weak topology on K is equivalent to the strong topology on H. Now we show that sup liu(t)111/ = sup ilu Willi--- . iL tEl

(5)

t .--50

for any trajectory fts (t), h t } c F. Suppose, for instance, that there are trajectories xi(t) = fu i (t), h i t} and x 2(t) ={/42(t), h2t } for which a = sup, o ilui(t)16 < supteJ 11/42(t)iix. We consider the set K1 = 1p e K: 1119 6 ---. al. Then the set Ki x .9e contains the semitrajectory {x i (t)} (t --. 0) but does not coincide with K x ye, which is impossible. Now the property (4) follows from (5). For suppose, for instance, that inft-,_o IlPi(t) — P2(t)IIH >0. Then for the half-sum (Pi(t) + p2(t))/2 we have (by the parallelogram equality in H) supto ii(Pi(t)+ P2(t))/216 < p, which contradicts (5).

Proposition 2. Suppose that equation (1) has a (weakly) compact solution on t.--0. Suppose that the non-trivial (weakly) compact solutions of the homogeneous equations Lu = 0 are (weakly) semi3

For this it is sufficient that the set K

X

9e contains at least one

semitrajectory. 4

For this it is enough to realise B as a subspace of C(0, 1) and take H to be 22(0, 1).

Certain auxiliary questions

131

separated from zero. Then (1) has at least one (weakly) almost periodic solution. Proof. In the 'weak' case each fibre Fh is trivial, as follows at once from (4). In the 'strong' case we must notice that a compact Favard minimal set exists, and then property (4) holds in the sense of the norm. The matter is then reduced to Theorem 1. It also follows from property (4) that the set F contains a unique Birkhoff minimal subset;

we denote it by

P.

2. Some properties of extreme points. It is known that if P is a weakly compact set in B, then every extreme point of its closed convex hull belongs to P and so is an extreme point of P (see Dunford & Schwartz, [40], p. 440). Let K be a Favard minimal set, and i denote the natural projection

B x 9e + B. -

We call a point x = fp, Ille F extreme if the component p is an extreme point of the fibre Fh. From the right uniqueness theorem it follows that the point x(t)= {p(t), h t } is also extreme for t .---- 0. Hence it follows that the set of extreme points form an everywhere dense subset of P. Indeed, if we assume otherwise, then (by taking into account that in a minimal set every semitrajectory is everywhere dense) Ê does not contain any extreme points at all. But it follows from Favard minimality that K is the convex closure of each of the following weakly compact sets i(F)= U

11€2?

Fh) i(P) = U Ph. liege

Hence it follows that every extreme point of K is an extreme point of some fibre Fh and belongs to its subset Ph. This contradiction proves our assertion. The next subsection is devoted to a general study of the important property (4). 3. On completely non distal extensions. Let X be a metric compact space on which a semigroup S(t) (t..? 0) acts invariantly. Let M be a compact set that is minimal with respect to the action of a dynamical system on it. We assume that the semigroup S(t) is an extension of the dynamical system on M, that is, there is a continuous mapping j : X -> M such that -

-

j(s(t)x)= h t (h = j(x), x €X, t-.-- 0).

132

Favard theory

We say that S(t) is a completely non-distal extension if inf p(xi(t), x2(0) = o

(6)

tel

whenever i(x1(0)) =i(x2(0)). The main problem regarding completely non-distal extensions is the following: is there at least one trivial (that is, one element) fibre Xh? Unfortunately, the solution to this problem in such a general form seems to be negative. Here we give an affirmative solution of the problem under a certain additional condition. We call a fibre Xh o positively stable if the transformations S(t) (t _ .- 0) are equicontinuous on it. -

Lemma 1. Let Xh o be a positively stable fibre. Then the trivial fibres form a set of the second category in M. Proof. For convenience of notation we denote the fibre Xh (h E M) by T (h). We consider the function g(h)= diam (F(h)) (the diameter of 1"(h)). Since this function is semicontinuous on M it has a nonempty set 12 of points of continuity (12 is a set of the second Baire category). We show that g(h) = 0 for h E Q. By assuming otherwise we obtain g(h) ...-- k o > 0 for h E A , where A is an open set in M. Since M is minimal there is a finite set Iti l c r such that M = U A. From

this we conclude easily that g(h)--k 1 >0 (hEM). (7) Let x i , X2,. . . , xi?, be an arbitrary finite collection of elements in the single fibre F(h). Then inf diam {x i (t)} =0. (8) t_,..._o

For the proof let Z denote the product of m copies of X. Then if (8) does not hold, the closure of the semitrajectory {x i (t), . . . , x,„ (t)} (t-.- 0) is separated from the diagonal in Z. But then the minimal subset contained in this closure is also separated from the diagonal. As is easily seen, this means that the original set X contains at least two different minimal sets, which is impossible in view of property (6). It follows from the positive stability of a fibre ['(h0) that for any 6 > 0 we can find a finite set {x i (0)} c Rho) such that {x i (t)} is an E-net of F(h ot ) for any t...--- O. But then from (8) we find that inf diam ['(hot )--- e, t--0

which contradicts (7). This proves Lemma 1.

Certain auxiliary questions

133

4. The finite dimensional case. Let the space B be finite-

dimensional.

Corollary 1. There is a set 12 c k of the second Baire category such that for h e 12 equation (1h ) has at least one N-almost periodic solution. Proof. We consider the set F. If we can prove that every fibre is positively stable (it is sufficient to do this for one fibre), then the problem is reduced to Lemma 1 and Theorem 2. But the stability of each fibre is an immediate consequence of finite-dimensionality. Indeed, consider the hyperplane Q defined by

Q = fu (0)E B: Lu =f, sup liu (t)ii < col.

tEl The transformations Sho(t) (t-.- 0) are uniformly bounded on Q, and being affine, they are equicontinuous. In § 7 we give an example which shows that the set 12 does not in general coincide with k (this example is important in a number

of other connections). We also remark that the question of the validity of Corollary 1, for instance, for a Hilbert space, is open. 5. On an equivalent norm. It is known (see Kadets [62]) that a separable Banach space B can be given an equivalent norm with the property: if xn, -÷ x and lixn, ii -+ 11X11, then I Ix - xin II -+ O.

Lemma 2. Let K be a metric compact set and a function 0 :K -> B be weakly continuous. Then the points of strong continuity form a set of the second category in K Proof. We assume that an equivalent norm with the above property has been introduced on B. Now lick II is lower semicontinuous, and so every point of continuity of it is, obviously, a point of strong continuity of ck, as we required. 6. Remarks on semigroups. Let 1 be a compact space with a semigroup S(t) acting invariantly and minimally. We need the following elementary fact: if A is an open set in _et then we can find a finite set {t} €J for which / = U S(4)A. For the proof we set ti = S(t)i.1 for t ---- 0 and ti = {x EX: S t)X E LI} for t 0.

t-.... to

(9)

We regard the projection i : P -* B as a weakly continuous mapping of the compact set P into B. By Lemma 2, there is at least one point of strong continuity. This means that for any positive E -'-- 8 there is an open neighbourhood d = 1 (e) such that (10) iii(x1) - i(x2)11= e/2 (x i , x2 E A ). We consider the solution p(t) = (p i (t)+p 2(t))/2 and let X be the closure of the semitrajectory {p (t), ho t } (t --- 0) in F. As was mentioned in § 3.2, the extreme points form an everywhere dense set in P. We choose an extreme point x = {q(0), h} belonging to the open set d and show that x it X. By assuming otherwise we obtain P(tm)= -1(Pi(tm)+P2(tm.)) — q(0), hot- -> h for some sequence tm, -> -00. Without loss of generality we may assume that p i(tm ) — q i and p2(tm) — q2 weakly. Since q(0) is an extreme point of the fibre Fh, we have q i = q2 = q. But then, for large m, the points {pi(t.), ho tm} and {P2(tm), ho t-} belong to d, which is incompatible with (9) and (10). Thus, x1C. But then the Birkhoff minimal set in X also does not contain x, and consequently, does not coincide with P. This (as we noted in § 3.1) contradicts property (4), and so we have proved that the fibres F(h o t ) are trivial. It remains

Weak almost periodicity (the general case)

135

to apply Theorem 2. The second part of Theorem 4 is proved similarly. This completes the proof. Now we make several remarks. 1. At the expense of some complication of the preceding proof we can establish that not only 1(h0) but also Fho is trivial. This means that the Favard minimal set does not contain any solutions of (1) apart from the unique weakly N-almost periodic solution. A similar result in the minimax method is: every ball of minimal radius containing at least one solution of (1), actually contains no solutions other than the unique weakly N-almost periodic solution. 2. The condition of semiseparation can be stated in the following generalised form: if u(t) is a bounded solution of the homogeneous equation (1), then we can find a sequence of intervals Tn c J whose lengths increase to infinity, such that inft e Iln O. 3. We say that a bounded solution of equation (4) is uniformly positively stable if there is a constant l> 0 such that T. II— WI>

liu Mil --- nu NA

(11)

(t = to)

whenever Lu =0, u E C. It is clear that in this case the condition of semiseparation holds. 4. If the operator-function A(t) does not depend on t or is periodic in t, then for the generalised condition of semiseparation to hold it is sufficient that one of the following manifolds is closed:

N = {14(0) e B : Lu = 0, u E C}, N ± = fu (0)E B : Lu =0, u E Cl, N - = fu (0) e B : Lu =0, u € C -1. The proof follows directly from the Banach-Steinhaus theorem.

5 Problems of compactness and almost periodicity The problem of compactness consists in finding a compact solution of (1) if a weakly compact solution is known to exist. In a narrower formulation the problem could be, for instance, the compactness of the weakly almost periodic solutions in Theorems 3 and 4. The compactness problem is non-trivial even for the simplest equation u' = f(t) (this is clear from the discussion in Chapter 6). To clarify our approach to problems of compactness, we consider what is in a certain sense the 'simplest' situation. Let T(t) (t -..-- 0) be a strongly continuous semigroup of linear operators B + B. Since linear operators are automatically weakly continuous, it is natural -

Favard theory

136

to endow B with the weak topology. Suppose that the semigroup T(t) is uniformly bounded (that is, sup,0 il T(t)ii< (x)) and that X c B is a weakly compact minimal5 subset. We claim that the set X is compact in the strong sense, and consequently, by Markov's theorem it consists of almost periodic trajectories. The idea of the proof is as follows: according to Lemma 2 we must find a point of strong continuity of the identity mapping X -> B, and then use the minimality and uniform boundedness to 'carry over' the strong continuity to each point of X. The approach to the compactness problem developed below is not based on the linearity of the equations but only on the weak continuity of the solving operators Sh ( t) (t.--- 0, h E ge), which follows from linearity. Consequently, it makes sense to give the main results in a form that relates to general dynamical systems.

1. General lemmas on compactness. Let M be a compact metric space with a minimal dynamical system. We consider the weak topology on B; the corresponding topology on B x M is also called weak. Let a semigroup of continuous transformations Sh(t) (t_.-- 0) be given on B x M that is an extension of the dynamical system on M, that is, having the form S(t){p, h} = {S h (t)p, W}. For x = { p, h} we set i(x) = pi, j(x) = h. Let d denote the metric on M. We need a stronger form of the continuity condition.

Condition 3. For any positive e, to and K there is a 3 = 3(E, to, K) such that

iiS h i (to)p - Sk(to)Pli --ç- E whenever d(h i, h2) ,-5 E and iii911--- K

Lemma 3. Let 0 = 0(p, q) be a metric on B that is strongly uniformly continuous on each bounded set. Suppose that the following conditions hold: (1) the operators Sh(t) (t...-. 0, h E M) satisfy Condition 3; (2) the operators S h(t) (t -?- 0, h E M) are 0-equicontinuous on each bounded set; (3) on a weakly compact minimal set dt'c BxM the weak topology is subordinate to the topology ckfined by the metric p(x, y) =

i(Y))+ d(i(x), .i(Y)). 5

Here and in what follows minimality is understood in the Birkhoff sense.

Compactness and almost periodicity

137

Then the weak topology on t is equivalent to that defined by p. By considering the special case of Lemma 3 when 0(p, q) =11p — we obtain the following assertion.

iI

Lemma 4 (Compactness lemma). Suppose that the following conditions hold: (1) there exists at least one weakly compact trajectory; (2) the transformations Sh(t) (t 0, h EM) satisfy Condition 3: (3) the transformations Sh (t) (t 0, h M) are strongly equicontinuous on bounded sets. Then there exists a compact trajectory; moreover, every weakly recurrent trajectory is compact. We begin by proving Lemma 4, which is simpler in form. Since the semigroup S(t) has a weakly compact trajectory, there is a minimal weakly compact set c B X M. We denote the fibre over h E M by F(h). Let K = supxEkiii(x )11. By using the fact that the operators Sh (t) hE M) are strongly equicontinuous on sets bounded in B, we can choose an a = a (s) so small that

sup

IISh(t)pi - Sh(t)P2Il

E/2,

(12)

whenever 11pia(6), 11Pill, IIP211---K By reasoning as in the proof of Theorem 4 we find an open set d = A(E) such that (10) holds. As we mentioned in § 3.4, there is a finite set L = {th } cj + such that = UteL S(t)A. We put Ak = S(4). Then we use Condition 3 and choose a 8 = 8(s) small enough that sup liShi (t)p — S k(t)Pii s teL

(13)

whenever d(h i, 8(s), p E 0,4 Since a translation on M is continuous, we can find a = such that d(h i t, 3(e) if t e L and d(h i , h2) 13. Let M =U Ms be a decomposition of M into a finite number of sets Ms with diameters diam (m-8 )--5 0. Then it = U it, where ±.9 = UheM s r(h )• We set G sk = 5ts n Ak, and prove that for arbitrary elements x i, X2 E G sk

ow

i(X2)11 - -- E.

(14)

Since the sets G sk form a finite covering of the space dt, (14) means that i(it) is strongly compact. If x i , x2 e G sk, then there is a TEL such that x i = S(r)y i and x2 = S (r)Y 2, where yl, y2 E L.

Favard theory

138

Putting p l = i(Yi), P2 = i(Y2)) hl = i(Y1), and h2=i(Y2) we obtain d(h i , h2) -5 8(e) and iiPi — P2ii = E/2. Now, by (12) and (13) we have Ili (xi) — i(x2)J1 = liShi(T)P 1 — Sk(T)P2ii -- iiShi(T)Pi — Sh i (T)P211+11Sh i (r)P2 — Sh2(7)P2ii -5 E.

This proves Lemma 4. Lemma 3 is proved in a completely similar way; we merely make a few observations. Obviously, Lemma 3 would be proved if we could show that the set X endowed with the metric p is compact. By arguing as in Lemma 4, instead of the estimate (14) we have 0(i(x1), i(x2))= E (xi, x2 e Gsk). The continuity of 0(i(x), i(x)) on X does not follow immediately from here because, in general, the sets G s k are not open. 6 However, it follows from this estimate that X is compact in the metric p, that is, this metric is topologically equivalent to the weak metric on

2. Some consequences. Lemmas 3 and 4 have a whole range of applications.

Definition. We say that the condition of uniform positive stability holds if (11) holds for any solution u(t) (t ---- to) of the homogeneous equation Lu = O. The property of uniform positive stability means that the operators S1 (t) (t_.. 0, T EJ) are equicontinuous on sets bounded in B. Since the set {hot } is dense in Ye and the mapping Sh (t): B X Ye + B is continuous, the operators Sh(t) (t --- 0, h E .9e) are also equicontinuous. In particular, the inequality (11) holds for solutions of Lu = O. -

-

Theorem 5. Suppose that Conditions 1-3 and the condition of uniform positive stability hold. Then equation (1) has at least one almost periodic solution and every weakly recurrent solution is compact. For the proof we must use Lemma 4 together with Theorem 4. We consider the special case A (t) -=- O.

Corollary 2. If the indefinite integral of an almost periodic function is weakly compact, then it is almost periodic. 6

In general, the sets Ak are not open.

139

Compactness and almost periodicity

Now we discuss in more detail the case when the operator A does not depend on t. The condition of uniform positive stability means that A generates a uniformly bounded semigroup of operators, that is s up t_.---4D

Ilex p (— tA) I1 < co.

Corollary 3. Every weakly recurrent

solution of the

non-

homogeneous equation (1) is almost periodic. Proof. From Lemma 4 it follows that a weakly recurrent solution

is compact. On the other hand, by Theorem 5 there exists an almost periodic solution. By taking their difference we obtain a compact solution of the homogeneous equation; by Markov's theorem it is almost periodic, as we required. Corollary 4 (criterion for a point spectrum). An operator A has a point spectrum on the imaginary axis if and only if the homogeneous equation Lu -- u' + Au = 0 has at least one weakly compact solution weakly separated from zero. Proof. First we prove the sufficiency. By Birkhoff's theorem, the

homogeneous equation has a non-trivial weakly recurrent solution, which by Corollary 3 is almost periodic. But then the Fourier coefficients are the eigenvectors of A. To prove the necessity we note that if Ax = idlox (Ao EJ), then the solution u(t) = x exp (—iA ot) is weakly separated from zero. We consider some examples which use the compactness theorem in this section together with the method of harmonic analysis (see Chapter 6, §§ 4, 5). Example 1. Suppose that the operator A generates a uniformly bounded semigroup and that the resolvent of A is completely continuous. We consider the perturbed equation u'+Au +A i u =g(t), where A 1 is a compact operator and g E C; (13). We shall prove that every weakly compact solution is almost periodic. The operator A +A 1 also has a completely continuous resolvent; therefore the spectrum of A +A 1 is rarified. Then by Theorem 5 of Chapter 6, a weakly compact solution u(t) is weakly almost periodic. Furthermore, the function A l u(t) is compact and, consequently, almost periodic. We set gi (t)= g(t)— A i u(t); then we have u' +Au = gl . By applying Theorem 5 to this equation we obtain that the solution u(t) is compact.

Favard theory

140

Example 2. In the Hilbert space H we consider the hyperbolic equation u"+ Q2u + Q i u =f, where Q is a positive definite operator with a completely continuous inverse, and Q i is completvly continuous relative to Q (this means that Q 1 Q -1 is completely continuous). Let W denote the Hilbert space of pairs z = {x, y} with the norm liZ IlW2 = 11Qx112 +11w112. By setting z = lu, u'l we obtain z' +Az +A iz =g, where Az = { — y, 12 2 x}, Alz = {0, () i x}, and g = {0, f } . The operator A generates in W a group of unitary operators, and A 1 is completely continuous; thus, the matter is reduced to Example 1, that is, every solution bounded in the 'energy' norm of W is almost periodic in this norm. 6

Weakening of the stability conditions Stability type conditions played a part in all the results in

the previous section. None the less, there is a class of linear problems for which stability conditions do not hold, but assertions of the type of the point spectrum criterion are still valid. We begin by stating a condition that replaces stability. Let A(t) -.a A and T(t)= exp (—tA) (t ---- 0). Instead of requiring the operators T(t) (t .--. 0) to be uniformly bounded we need a more general condition to be fulfilled: the function v(t) = T*(t)v(0) is bounded on t ---- 0 for a set of initial values D = {v(0)} everywhere dense in B*. As before, we denote by (6, p) the value of a functional 6 E B* at p E B. Under these conditions the following assertion holds. Theorem 6. Every compact (weakly recurrent) solution of the equation Lu =f is (weakly) almost periodic; in particular, the criterion for a point spectrum holds. Proof. Since B is separable, the set D contains a countable total

subset 161; here we can assume that iiedi = 1 . We set 6i(t) = T*(t)6, (i. .... 0) and define on B X B the metric (P(P, q) = ci sup l(6(7) i=1

1-.-.-1:31

'

P

2i

—q)1

.

It is not difficult to see that on each weakly compact set in B, (10 is not weaker than the metric of weak convergence. It is important that the operators Sh (t) (t-.- 0, h e k) are 1P-non-expanding. Indeed, the

Weakening of the stability conditions

141

difference u(t) = Sh(t)p — Sh(t)q is a solution of the homogeneous equation, and therefore for t ..- 0 we have c° sup Rei(T), u(t))I 0(Sh(t)p, Sh(t)q)= E 2i i =1 1-?--0

co

l(ei(7 + t), p — q)I 2i

= E sup i=1 i-_,---o

--- 0(p, q).

First we study the more difficult case of a weakly recurrent solution. Let p(t) be such a solution. Instead of p(t) We consider a solution q(t) lying in the set F (see § 3.1). By Lemma 3 the metric p(x, y)= (i(x), i(y))+ d(j(x), j(y)) is equivalent to the weak metric on ,t. But then any solutions from the same fibre .th must be weakly semiseparated, as follows at once from (15). Hence (with due regard for the fundamental property (4)), we conclude that each fibre Fh is trivial, that is, q(t) is a weakly almost periodic function. The difference u(t) = p(t)— q(t) is a weakly recurrent7 solution of the homogeneous equation. On the closure of the trajectory {u(t)} in B the metric (10 is equivalent to the weak metric (again by Lemma 3). By Markov's theorem u(t) is weakly almost periodic, and so Theorem 6 is proved for weakly recurrent solutions. The case of compact solutions is simpler. We first clarify the character of a compact solution of the homogeneous equation. It can be seen immediately that the metric 4) defines strong convergence on the closure of a compact trajectory. Therefore, a compact trajectory is almost periodic. The non-homogeneous equation is settled by referring to Proposition 2, and the theorem is proved completely. Example 3. We consider the equation [pil [A.11 ...

_

Ailpi •

=0,

Ami • • • Amm _Pm where the Ay are commuting bounded normal operators in a Hilbert space; let H denote the product of m copies of this space. By relying on Theorem 6 we show that every compact (weakly recurrent) solution is (weakly) almost periodic. The operators Ai; are functions of a self-adjoint operator C, which splits into cyclic components, and P'm

7

Here we have used the fact that an almost periodic function is simultaneously recurrent with any recurrent function, see Chapter 7, § 3.

Favard theory

142

so we may assume that C is a cyclic operator. We realise C in the form of the operator of multiplication by A in the space Y2(-1, 1, dg ). Let

A = (AO,

exp (—tA) = B(A, t), u = (pi,. . . , pm ).

We distinguish two cases. The first is when the bounded solutions of the equation u' +Au =0 are dense in H. In this case the norm IlB(A, 011 is bounded in t EJ for almost all A e [-1, 1] (in the sense of the measure g ); then the norm IIB*(A, 011 also has this property. Hence, in turn, it is easy to deduce that the set of bounded solutions of u'+A*u =0 is dense in H. In other words, Theorem 6 applies. Next suppose that the closure of the bounded solutions is H1 0 H; let H2 denote the orthogonal complement of I/1 , and P1, P2 be the corresponding projection operators. It is easy to see that P1 and P2 are defined by operators of multiplication by measurable matrices. Since u = Piu = ui, we find that u l ' +PiAu i = u i ' +A l u i = 0 in H1, and we can apply the preceding arguments to this equation.

7

On solvability in the Besicovitch class

In this section we extend the original problem and study the solvability of (1) not in the class of Bohr almost periodic functions but in the wider class of Besicovitch almost periodic functions. 8 It turns out that in this wider setting the problem can be solved fairly simply and without stability or separation type conditions. We also give a construction of a linear equation that is solvable in the class of bounded Besicovitch almost periodic functions but not in the Bohr class. 1. Let g be a normalised invariant measure on 9r. We introduce the Lebesgue space B1 consisting of g -measurable mappings 0 : with the norm ii(MIBI2 = fx ii0 (h )112 CIA.

Definition. A measurable mapping 0 : k-->B is called an invariant section if 0 (ht) = Sh(t)0(h) for any t E [0, 03 ) =J ± almost everywhere on k. Let us explain this definition. It is obvious that 0 (h t ) and Sh(t)0(h) are measurable on kxr and are equal almost everywhere. By s A measurable function f :J -* B is said to be Besicovitch almost periodic if for any E >0 there is a trigonometric polynomial l'(t) such that lim , 1,.,

f T IWO — P.(0112

T-co Z 1 —T

dt ---5 E.

Solvability in the Besicovitch class

143

Fubini's theorem there is a set F C k of full measure such that cfi(h t )= Sh(t)0(h) for almost all t E J + , and after a suitable adjustment, for all t Er. We set F1= Ut.0 rt and bo = U0 Fi t . Then F0 is an invariant set, it has full measure, and 0 (V) = Sh (t)0(h) for every h € ro and t E J. Consequently, c(ht) is a solution of equation (1h)

defined on the entire real line. Theorem 7. Under Conditions 1 and 2 there is at least one (essentially) bounded invariant section. Proof. We consider the family of operators H (t): B 1 ---> B 1 (t-.- 0)

defined by H(t)0(h )=Sh -t(t)cgh ). It is easy to see that these operators commute. What is important for us is that they have the following property: an element 0 E B 1 is an invariant section if it is a common fixed point for the operators H(t) (t-.- 0). We consider the set F introduced in § 3.1, and denote by Z the set of all measurable mappings 0 : 9e ---> B for which cfi(h )E Fh. The set Z is convex (since the fibres Fh are convex), invariant under 11(t) and weakly compact in B 1 . By the Tikhonov—Markov theorem the operators H(t) have a fixed point, as we required. This theorem implies that for each h in a set A c 9e of full measure, equation (1 h ) has a bounded Besicovitch almost periodic solution. Indeed, let 0 be an invariant section. We choose a continuous function O s : 9 e ---> B for which U. — 0.118, --. s. Then by the ergodic theorem we have fT , 2T 'Irif 1 L T II° (V)

dt =110 _&11B12 — (fie

82

("112

for almost all h E H. The function cfi E (h t ) is almost periodic, and so cf) (h t ) is Besicovitch almost periodic. The situation regarding the existence of a non-trivial section for homogeneous equations is slightly more complicated. Definition. We say that A € J belongs to the point spectrum of the homogeneous problems if there is a weakly compact solution of a homogeneous equation of the form (1 h ) such that the expression 1fT 7 exp (—iAt)u(t) dt ,

0

(T ---> +a))

has a non-zero weak limiting point.

144

Favard theory

Let mA (u) denote an arbitrary limiting point of this expression: Theorem 8. If A is a point in the point spectrum, then the equation Lu +iAu =0 has a non-trivial section. Proof. Without loss of generality we may assume that A =0, that is, that some equation Lu = 0 has a weakly compact solution u(t) with a non-zero mean mo(u) = a. Let a e B* with Re a (a) 0 0. We denote by K the closed convex hull of u(t). For each h E k we consider the homogeneous equation (1 h ); let Ki, be the collection of values {u(0)} of those solutions of this equation that lie entirely in K. We put g*(h)= max Re a (p), g * (h)= min Re a(p). pexh pErch The functions g*(h) and g(h) are semicontinuous on A', and at least one of them is non-trivial. Indeed, if g*(h)= g(h) almost

everywhere on k, then both these functions are continuous almost everywhere, and so they are Riemann integrable. Then, by the Kronecker—Weyl theorem9 the means m o(g*(h t )) and m o(g* (h t )) exist for any h e and are equal to zero, but this contradicts the condition mo(Re a (u)) 0 0. To be specific, assume that g*(h) is non-trivial. Let Z stand for the set of measurable mappings ck : k --> B for which cgh )E Kh and Re a (0(h)) = g*(h ); it is easy to see that Z is a nonempty, closed, convex set in B 1 that is separated from zero. For the remainder we must proceed as in Theorem 7. Note that in the periodic case (here k is a circle) an invariant section is always continuous (more precisely, equivalent to a continuous one); in the almost periodic case this is not so, as is clear from the examples given below. 2. We give a construction of a scalar equation u' +a(t)u = f(t) (a and f are almost periodic functions with an integer two-term basis), which has bounded solutions but not almost periodic ones. For sets 12 and A is introduced in § 3.4) we have f2 o .9e and A = 9r. Other properties will be mentioned during the course of the construction. For convenience in reading we give separately a slightly modified version of the classical example of Bohr.

co

Bohr's example. Let k be the two-dimensional torus, realised as a square of side 2 1T, and let A 1 , A2 be linearly independent numbers. 9

See Levitan [73], p. 109.

Solvability in the Besicovitch class

145

We put h = {x i, x2 } , h t = {xi+ al, x2+ tA.2}, ho= {0, 0 } . We choose numbers tom = i mi A i + /L1 2 (l,n1 and /2m are integers) so that M 2/3 < 2com < 2m 2"3. W wee set a(h)= E 04,22 sin (/ m 1 x i +/x 2), a(t)= a(ho t )= E 07, com t. 4 2 sin Obviously, a(t) is almost periodic. Let n (t) =Ç a(s) ds. Then (t) = E (on, (1 — cos (o m t) I 7

=E con, sin2 (com t/2).

By using the inequality Isin xi Ix 1/2 for 1x1--... 1 we have -ri(t).--. E con, sin2 (com t12)

(t

2/4)

42/4)

m_,E iti3/2

r

(i/m2) ds

Jiti3/2-1 5 2

t2 4(1t1 312 -1)'

that is, limt _.. n(t) = +co. The numbers l mican be taken to be odd; then a(h) is antiperiodic (in the first argument) with period 7T, that is, a(x l +z-, x2) = –a(xi, x2). Now we come to our construction. We use the notation of Bohr's example. In addition, for 0(h) (h € k) we set by definition

at

n)=A1—+A. 2 . axi

ax2

Let a(t) be the function in Bohr's example, and z(t)= exp Hot a(s) ds); z(t) satisfies the equation z'+a(t)z = 0 and decreases with its derivative faster than any power. We take an interval 4 cf of length 2s small enough that the corresponding portion d = ItA i , tA21= {hot} (t E 4) lies strictly inside the square {0 .--_ x 1 .--.2 ,77-, 0.---. x 2 --. 27}. By using a partition of unity for J we can write z(t) in the form c° 1 -,,, n where the un (t) are concentrated on 4 and are uniformly bounded together with their derivatives We put zm = un i(t). ET—. (1/n 2 )un(t + En). It is clear that z,n (t)= z(t) on some interval [ Tm, Tm] and that Tm z(t)=

–

146

Favard theory

Next, it is not difficult to realise the construction of periodic functions on (h) with the following properties: (1) On (h ) and a4i(h)/at are continuous everywhere apart from d where they have a discontinuity of the first kind with jumps u(t), un '(t), respectively; (2) t/i(h ) and a/i(h)/at are uniformly bounded. Then we set ml

4).(h)= ,En--.0(h"), c°

I

_con It

is easy to see that

at

-00 n

at Ch')

—

for lt, {ho t }.

(15)

We define fm (h) and f(h) by

a at a f(h)=-0(h)+ a(h)0(h). at By construction, fm(h ) is continuous at each point ht {hot}. On the compact part of {hot}, where It' ----. T., 0. and ackmlat have jumps equal to z,, and zn.,', respectively. Since z'(t)+ a (t)z (t) = z in i(t)+ a (hot )z„,(t) = 0 for I ti ----. Tm, the function fm (h) does not have disconfm (h)=-0m (h)+ a(h)(1)„t (h),

tinuities on this interval. Since the convergence fn, -+ f is uniform on k, f(h) is continuous on Thus, for the equation ts' - - P a(ht )u = f(h t ) (16) the function o(h) is an invariant section with discontinuities (which are not removable) only on {hot }. This means that the solution ck (ht), where ht {hot}, is N-almost periodic but not almost periodic. We show that (16) does not have almost periodic solutions. Suppose this were not so, then for h 1 = {7r, 0 } the equation u' + a(hi t )u = f(hit) would have an almost periodic solution u(t). The difference q5(h i t)_ u ( t ) is a non-trivial bounded solution of the homogeneous equation. But since a(h) is antiperiodic with period Ir, we have a(hi t )= –a(t), and so the homogeneous equation does not have a non-trivial bounded solution.

Solvability in the Besicovitch class

147

Now we prove the stronger assertion: Ch ) is the unique (essentially) bounded invariant section. By assuming non-uniqueness we have a non-trivial bounded section for the homogeneous equations; we denote it by a (h). Returning to Bohr's example we set g(h)= —E Wm cos (l m' xi+ / 2.x2). The series E 0),„2 converges and so g(h) is square integrable over k. The relation g(h t )— g(h)= f(t, a (h ) ds (for almost all h € Ye) is readily verified by considering partial sums for a(h) and g(h). The functions a(h) and exp (—g(h)) are non-trivial sections for one and the same scalar equation u'+ a(h t )u = 0; consequently, their ratio a (h) exp (g(h)) is invariant under translation on 9e. Since translations are strictly ergodic, this ratio is constant almost everywhere on Ye. We find that exp (—g(h)) is essentially bounded. Let g = g+ —g be the decomposition into positive and negative parts. From exp (—g) 1—g it follows that g - is bounded, while the antiperiodicity of g implies that g + is bounded. Thus, we have found that exp (—g) and exp (g) are bounded. Then there is a point h2 e 9e for which u(t)= exp (—g(h 2t )) and 1/u(t) are bounded. This means that the equation u' + a(h2t )u =0 has a bounded solution that is separated from zero. But then u; + a (hot )u = 0 also must have such a solution, which contradicts the original properties of Bohr's example. This completes the construction of the required example. We can also give for the homogeneous equation an example of an invariant section that is not equivalent to a continuous one. For this we must consider the equation u; + ia(Ou =0, where a(h) is the function in Bohr's example. The required section is defined by 0(h) = exp (—ig(h)).

Comments and references to the literature § 2. The minimax method was given by Favard [106] in the finitedimensional case and in the case of a uniformly convex space by Amerio (see Amerio & Prouse [2]). Both Favard and Amerio assumed the condition of (two-sided) separation and not of semiseparation, and they studied almost periodic solutions and not N-almost periodic ones. The general formulation of the `minimax' Theorem 3 and also the example of a non-compact solution is due to the authors. For finite-dimensional spaces the condition of semiseparation implies that of separation (see Chapter 7, § 4). This is not so in the infinite-dimensional case. As an example we can take the semigroup of right translations in Y2(0, co).

148

Favard theory

§§ 3 and 4. Lemma 2 (in a more general form) is in Gel'fand's article [36]. The other results are taken from the works of Zhikov. Note that the problem of completely non-distal extensions and related questions of the existence of N-almost periodic solutions (see § 3, Corollary 1) are open.

§ 5. Questions of compactness occupy a central place in the investigations of Amerio, who started from the well-known works of Sobolev [102] on the homogeneous wave equation. He succeeded in studying the non-homogeneous wave equation, and in proving the theorem on the integral (see Chapter 6, § 1) and certain abstract results about compactness within the framework of the minimax method (all these results are contained, for example, in Theorem 5). Our approach to questions of compactness differs from Amerio's. The important argument about a point of strong continuity is given in an article by Kadets [63], where there is a proof of Corollary 2. (This argument was used independently by Zhikov [50], p. 184, in connection with the criterion for a point spectrum.) For a unitary group this criterion can be proved on the basis of a spectral resolution (see Lax & Phillips [69], p. 139). § 6. Example 3 is in the fundamental article of Bochner & von Neumann [29], who discuss the case of compact solutions. The Bochner—von Neumann method is based on the rather complicated techniques of generalised harmonic analysis. The original formulation of Theorem 6 (with a derivation of the results of Bochner & von Neumann) is given in an article of Zhikov [47]. K. V. Valikov (private communication) and Perov and Ta Kuang Khai [97] have made some simplifications to the original proof. § 7. The problem of constructing a linear equation solvable in the class of bounded functions but not in the class of almost periodic functions was discussed in the original article of Favard [106]. But Favard only succeeded in showing (on the basis of Bohr's example) that the separation property does not carry over automatically to the limiting equations.

9

The method of monotonic operators

1 General properties of monotonic operators 1. Let V be a separable reflexive Banach space, V* be its dual, and (y, x) denote the value of y E V*at x E V. Definition. An operator A: V -> V* is called monotonic if Re (Ax i -Ax2, x i - x2) ?.-- 0

(x i , x2 E V).

In what follows, to simplify the notation we assume that V is real, so that the symbol Re in the definition can be dropped. An operator A is called bounded if it carries bounded sets in V into bounded sets in V*. Definition. An operator A: V -> V* is called semicontinuous if the scalar function t -> (A(u + tv), w) is continuous for any u, v, w E V. Lemma 1. Let A: V -> V* be a monotonic, semicontinuous operator. An element u

E

V satisfies the equation Au = f if and only if

(Av - f, v - u)--.0

(1)

for any v e V. Proof. Let Au =f. Then we have (Av -f, v - u)= (Av - Au, v - u) ---0. Conversely, if in (1) we take v = u + tw (t > 0, tV E V), then we obtain

(A(u +tw)-f, tw)?---0. After cancelling by t and letting t tend to 0 we obtain (Au -f, w)_-.-. 0 for any w e V. Hence it is clear that Au =f

150

The method of monotonic operators

Remark 1. We have assumed that the operator A is defined on the whole of V, but it is clear that Lemma 1 is still valid when the domain D(A) of A is a linear manifold dense in V; it is merely necessary to assume, in addition, that u belongs to D(A). The next proposition is a consequence of Lemma 1.

Proposition 1. Fora monotonic semicontinuous operator A: V-+ V* the set of solutions of the equation Au =f is closed and convex. For a proof it is enough to note that if two elements u li u2 satisfy (1), then (u i + u2)/2 also satisfies (1).

Proposition 2. A monotonic semicontinuous bounded operator A: V -> V* gives a continuous mapping if Vis endowed with the strong topology and V* with the weak.' Proof. Let un -> u and fn = A (un ). The sequence ffn l is bounded, and consequently it has a weak limit point, which we denote by y. We can take the limit in the inequality (A v -fn, y - un ) 0, and as a result obtain Au = y, as we required.

Theorem 1. Suppose that a monotonic semicontinuous bounded operator A: V -> V* is such that (Au, u) 0 (Iluil = ro)

(2)

Then there is auE V with lull ro, such that Au = 0. Proof. Let Vn be an expanding sequence of finite-dimensional subspaces of V whose union is dense in V, and let 3n denote the restriction operator of v E V * to Vn. We consider the continuous finite-dimensional operators

=

: Vn -> Vn * . Since (Au, u) 0 (11u11= ro, u e Va ), we obtain from the finitedimensional result (see Proposition 3 below) fin E Vn with Ilunil ro such that Au n = 0. We have the inequality (Anv, v - un ) = (Av, v - u n ) 0, (3) where v e Vm m n. The bounded sequence {u n } contains a weak limit point which we denote by u. We take the limit as n -> co in (3) for a fixed v e Vff, ; as a result we obtain (A v, v - u) 0 (v E V,n ). Then ,

This property is called demicontinuity.

151

General properties

by continuity (using Proposition 2) this inequality turns out to hold for any y E V. It remains to apply Lemma 1. Theorem 2. Suppose that a monotonic semicontinuous bounded operator A: V -> V* has the coercive property

lim

(Au, u)

'lull

Ilull->00

= + 00.

Then for any fE V* the equation Au =f has at least one solution. For a proof it is sufficient to recognise that Au -f satisfies (2) for a suitable choice of ro.

Proposition 3. Let P:R m

> R m.

be a continuous mapping of the Euclidean space Irn into itself such that (p(e), 6) 0 for any e in the sphere llI = ro. Then there is a 6, ro, such that p(e) = O. Proof. If I() o 0 on the ball K = ileil ro}, then we can consider the operator -

P(f) ( ) il . C > which in this case is continuous. Then from the classical fixed point theorem of Bohl-Brouwer it follows that that there is a e such that

- roP(6)

But

6 = ii-P(e)ii then lei = ro, and

(4) by multiplying both sides of (4) scalarly by

P() we obtain (P(6),

= - roiiP(e )ll < 0,

which is impossible.2

2. So far we have assumed that the monotonic operator is defined on the whole of the space V. For an application to evolution problems we require to study the problem of the solvability of an equation of the form

Lu

Au + Au = f,

(5)

where A is, as before, a monotonic semicontinuous bounded operator V-+ V*, and A is an unbounded linear operator D(A).-> V* (here 2

To use Proposition 3 in the proof of Theorem 1 we need to equip the space Vn with a scalar product. This enables us to identify Vn * with Vi,.

152

The method of monotonic operators

D (A) c V and D (A) is dense in V) subject to the following con-

ditions: (1) A is closed and monotonic, that is, (Au, u) 0 for all u (2) there is a sequence of expanding subspaces Vn c D(A) whose union is dense in V and which are such that A: Vn -> V* is bounded for each n; (3) if 6, is the restriction operator of the functional y E V * to Vt.') then there is a constant 1 such that E Va ). mull liOnAUll Theorem 3. Suppose that, in addition to conditions (1)-(3), the operator A is coercive. Then (5) has at least one solution for any f e V*. Proof. The operator Q :D(A)-> V*, where Qu Au +Au -f, is monotonic. There is a number ro (depending on in such that (Qu, u) 0 (u E D (A), Iluil = r 0) . The operator Qn = ang : Vn Vn * satisfies the conditions of Theorem 1. Let Qu n=0, liun 11 ,-5 ro. Since the operators &A are uniformly bounded, the sequence 3nAun, and

so the sequence Aug, is bounded. Hence, since A is a closed operator, there is at least one u E d(A) for which un, —u and Aun, Au. Then it is necessary to proceed as in Theorem 1 (using Remark 1). This proves Theorem 3. 3. Now we consider the question of the fixed points of non-expansive

operators. Let H be a Hilbert space. An operator T :H -> H is called nonexpansive if II Txi- Tx211=lixi - x211. Here there is an important connection with monotonicity since the operator A = I - T is monotonic. In fact (Axi -Ax2, xi -x2) = (xi - x2, xi - x2) - (Txi - Tx2, X1 - X2) - Tx21111xi - x211.- 0.

Theorem 4. A semigroup U(t) (t

0) of non-linear non-expansive

operators has a common fixed point if and only if it is bounded, that

is,

sup II U(t)xli < co

(x E H).

Similarly, a non-expansive operator T has a fixed point if and only if the semigroup of powers T" (m = 1, 2, .. .) is bounded.

General properties

153

Proof. The necessity is obvious; we prove the sufficiency. First we study the single operator case. We fix an xo E H and set Km = {Tnxo,n ....- m } . Then we choose a positive number d > diam Ko and denote by the ball in H of radius d and with centre at x. We define Ko = ITnxo , n--.- 01,

nm

=

n

xErcin

sd(x),

a=

u

m9

Sd ( x )

nm.

It is obvious that 12m is convex, 12m c 12m , i, and the set 12 is bounded. Since T is a non-expansive operator we obtain

7-'12m c nm+i,

Tf2 c t2.

(6)

We introduce a family of operators TE defined by

Tex = E (Tx

–

Tx0)+ Tx 0 (0 < 6 < 1).

Since xo E 120 and the sets 12m are convex, taking (6) into account we obtain T6f2 c 12. For e < 1 the operator T, satisfies the contraction mapping principle; we denote its fixed point by xs. The boundedness of the set { xs } follows from (6) and the boundedness of 12; without loss of generality we may assume that xs —x (weakly) as E —> 1. Setting As = I Te we have A sxs = O. Taking the limit in (As y, y – xs ) -- 0 (y e H) we conclude that Ax = 0 and Tx = x. Thus, the operator T has a closed convex set of fixed points. Now we turn to the semigroup U(t). The operator T = U(to) satisfies the boundedness condition; let Ntc, be the closed convex set of its fixed points. From the commutativity of the operators U(t) an invariant set, that is, U(t)Nk, c (t_- - 0) it follows that N Nk, (t.-.-- 0). Let N be a minimal (relative to inclusion) closed convex invariant set. It can be proved by the preceding argument that the operator T = U(t0) has at least one fixed point in N, and so it follows that N consists of the common fixed points of the operators U(to). –

2 Solvability of the Cauchy problem for an evolution equation Let E be a separable reflexive embedded space; this means that there is a Hilbert space H such that the embedding EclIcE* is dense and continuous, and that the bilinear form (y, x) (y e E*, x e E) coincides with the scalar product on H whenever x, y E H. We shall use the following notation for norms: 11x11 is an E-norm, WI* is an E*-norm, and lx 1 is an H-norm.

154

The method of monotonic operators

Our aim is a study of the evolution operator d/dt +A, but first we shall be concerned with constructing the differentiation operator d/dt. 1. Let [0, T] be a finite interval, S be the set of all trigonometric polynomials with coefficients in E,p>l, and p-l+q-i.... i We define D to be the closure of the set S with respect to the norm „T

=(

IIUIID

Jo

IlUIIP

‘ 1Ip „ T d9 +(j o

‘

1/q

Ilu'11,7 di)

= Ilull21(0,T;E)+11u'll2q(0,T;E*)• The following important proposition holds. Proposition 4. Each element in D is an H-continuous function on [0, T], so that it is meaningful to talk about the initial condition ul t =0 = u(0)E H. Moreover, for each uo E H there is at least one u(t) e D such that u (0) = UO. Proof. Suppose that u E S. Since lui = l lull, by the theorem about the mean there is a point to = to(u) for which

T

100)1 =

1/p

Tif IWO?) dt)---5 T-liitiiip. o

Therefore, from the formula lu(t)12 = lu 40)12 + 2it To (u i, u) dî, and by using the obvious inequality ‘ llq 1/p T Jo Rui, oidt '-- (f iluiii*q

do

Jo

(

'Fur do

--_iluiii)2 we obtain sup lu(t)1 2 ---5 (T -1 + 2)liuilD2.

0,-5t---5T

It follows from here that the elements in D are H-continuous functions. Then, by an uncomplicated construction, for any uo e H we can find an extension u(t)E D, and Proposition 4 is proved. We put V = ..e (0, T ; E), V* = ..cfq (0, T ; E*). We define the duality between V and V* by the usual formula: the value (y, u) of y E V * at u E V is given by (y, u)= for (v, u) dt. We consider the operator A= d/dt with the domain {u E D: u(0) =0 } , and shall prove that it satisfies conditions (1)-(3) in § 1.

Cauchy problem for an evolution equation

155

The operator A is monotonic since T

2(Lu, u)= 2 f (u', u) dt

o =1u(T)12 —iu(0)12 = lu(T)1 2 --- O.

Because it is obvious that A is a closed operator, we only need to give the construction of the subspaces Vn. For V„, we take the collection of all trigonometric polynomials n

ti

(0 = E bk sin (krta) (bk E E). Ic---1

It is well known that every element z E V is the sum of its Fourier sine series, and if zn, denotes the nth partial sum of this series, then liznilv ----. ilizliv. For u E V, we have T

T

(Au, z) = f (u 1 , z) dt = f (u' , zn ) dt.

o

o

Therefore, MAW = sup (Au, z)---. sup (Au, zn) iizIlv=i = /113„Aull,

iiznii---5/

that is, condition (3) is fulfilled. 2. We impose the following conditions on an operator A: (1) A :E > E* is monotonic and semicontinuous; (2) there are constants ci >0 (i =1, 2, 3, 4) and a p >1 such that -

(Ax, x).---. ciiixr - c2)

(7)

l iAx 11* -- c 311 x 1 ' + C4.

(8)

The first of these estimates means that A is coercive, and the second that it is bounded. Now we must interpret A as an operator from V into V*. With this aim we note that A :E > E* sends strongly convergent sequences into weakly convergent ones. Then, for a function with values in a separable space (for instance, E and E*), strong and weak measurability coincide. Hence it follows that the function Au(t) is E* measurable provided u(t) is E-measurable. It is clear from (8) that -

-

156

The method of monotonic operators

A acts from V into V* since T J liAtill* q

T

dt.--.5 c3 f o

= C3

jJo

luuii(P-1)q dt + c4 T

T

Muir

dt + c4 T.

It is not difficult also to verify that A: V -> V* is monotonic and semicontinuous. In what follows we shall be interested not only in the operator A but also an operator B: V * V* of the form Bu = A(e + u), where e is a fixed element in V. We prove the coercive estimate -

T

(Bu, u)= f (A(e + u), u) dt Jo

--- kiliulivP —k2 with positive constants k1 and k2 depending on we have T

ivy and T. Indeed,

T

I co(11(e + ti), u) dt = Jof (A(6-1-u),e+u)dt

—

T .

.1 0

(A( e + u), e) dt.

We estimate the first term on the right from below using (7) and the second from above with the help of (8). As a result we obtain: T

(Bu, u),-.- c 1 f lie + ur dt—c 2 T—c 3 o

X f lie + ully -i llelldt —C4 j

rT

T

o

0

'lei' dt,

T

Ile + ur dt

(Bu, u) ?---- cl f 1 r

x ( .1

0

—

c3

o

\ 1/ q

T

Ile +ullP dt)

, r

T

‘

1/p

(i IlellP dt) 0

(Bu, u)= c iiie + uliv P — C3116 + uliv P/ q ileliv — C 5.

— C5, (9)

Since pl q E* with the norm

1/q VIIX =sup

f0 iv(t+s)ii*ci ds) .

Let Xeo„,„ denote the subspace of X consisting of those f for which the family of translates {f( t +7)} (7 E I) is compactin ...r( - T, T; E*), and X denote the subspace of those f E X for which the family of translates {fit +7) } (7. EJ) is compact in X. Obviously, X c Xcompc X.

1. The next lemma establishes the boundedness (in the H-norm) of solutions on the half-line [to , co).

Lemma 2. Let f E X. Then any solution u(t) (t

to) of the equation

Lu =f has the estimate sup lu(t)I t .?-to

max {1u(to)

where the constant 1 does not depend on the solution.

(10)

158

The method of monotonic operators

Proof. We scalar multiply the equality u'+Au=f by u(t) and

integrate from s to t (s t

to):

ilu(t)I 2 —ilu (8)1 2 + is (Au, u) dn = fs (f, u) dn. By applying the coercive estimate (7) we obtain

ilu(t)1 2 —11u (s)I 2 + c2(t — s)+

i.t Ilur dn

1/q C2(t

)

fs

IlfIl*q dn)

1/p

t

(Is lull

d'n)

•

We estimate the last term on the right by using the inequality abl p -1 sPaP + q-1 6' bq ; as a result we obtain

10)12 + ci Ilur cin =lu(s)1 2 +co{(t—s) t

12226

1/q

+(fs IlfIl*q dn) }•

(11)

Below we shall refer to the boundedness of a number of quantities; we shall have in mind boundedness by constants not depending on solutions. We assume that the half-line [to, CO) contains an interval =[t 1 - 1, tu] such that V =Iwo= max t€4 lu(t)I. Then from (11) follows the boundedness of .1.4 Ilur dn, and consequently, of L lur dn. Then we can find a t2 in A such that the norm lu(t2)lis bounded. Now if in (11) we take s = t2 and t = t i , then we conclude that I' is bounded by some constant 11 . Hence, as is easily seen, we obtain the inequality max It/ (t )0l }. sup lu(t)Imax { t0•-•-çt---ç-t0+1 ttn

By estimating maxto -to +ilu(t)I in terms of Itc(to)1 with the help of (11) we obtain the required inequality. This proves the lemma. From (11) we obtain Sup

sup Ilu i(t + s)Ii* q ds Iiu (t s )11 ds, tto-f-T0 t-40T

C(Iti(to)I, T).

(12)

We obtain some other inequalities. Let f1 f2 e X, and u i (t), u2(t) (t to) be solutions of the equations Lu = fi Lu = f2, respectively. ,

,

159

Evolution equation on the entire line

We have 2 I1 141(t) -142(012 4 14 1(t0) - U2(t0)1 t t (Au i — Au 2, u 1 —u2) dt = f (fi —f2, ui— u 2) dt.

+f

to to By using monotonicity and then (12) we find that Itil(t)— u2(0 2

ii i(to)

—

u2(t0)1 2

, 1 /q t +2(i t .to lifi —f211*q dt) I/41— u21lP dt) (fto I

,

lui(t) — u2(t)I 2 .- Iui(to) — u2(t0)1 2 1/q

t 4.

(j.

to

lifi —f211*g dti)

C(lui(0)1, iu2(0)l, iinix, iifilix). (13)

2. In this subsection we digress from parabolic evolution equations and prove certain abstract propositions about compactness. Suppose that we have a triple of continuously embedded separable Banach spaces B1 c B C B2, where the embedding B 1 c B is compact and the space B2* is dense in B*. For certain p -- 1 and P 1 > 1 we set 17 = {u E .TP(0, T; BO: u E C(0, T; B), lu In

If E

.."(O, T; B2)},

--= IlullY.P(0,T;Bo+ Ilullc(0,T;B)+ Ilu'll2Pi(0,T;Bo•

Lemma 3. The embedding II -+ ce (0, T; B) is compact. As a preliminary we prove the following assertion.

Proposition 5. Suppose that the embedding B 1 c B is compact, and that there is a family {u a (t)} (0 ..-- t --.T) for which we postulate (a) boundedness in .r(0, T; B 1 ) and C(0, T; B); (b) weak equicontinuity in B. Then this family is compact in ..ce (0, T; B). Proof. Without loss of generality we may assume that there is a sequence of finite-dimensional operators In,: B + B such that /„,,x -+ Ix =x (x €B). Indeed, we can replace the space B by the universal space C(0, 1) while preserving the conditions of Proposition 5, and such a sequence exists in C(0, 1). Since the strong convergence of linear operators is always uniform on compact sets, -

iiinix —x1I-Eilxli B, (x E B1, In

160

The method of monotonic operators

By putting u arn. = /m u°, we obtain

(J

T

1/p

—

o iiiiau:z rdt,)

f0

T

- - S (‘ j

\ 1/p

ia1 lti113? dt)

'--5. si,

that is, the set {_Lk, (t)} is an si-net for fua l. But the compactness of the set trinua l follows easily from conditions (a) and (b), and Proposition 5 is proved. To prove Lemma 3 we verify that the unit ball in 17 satisfies condition (b). But for g E B2 * we have

I( U (t), g )I tg+ 4

1(u' , g)1 dt litiliBiliglIB2* dt

/ f 0t +4

-

--11g11B2*

U.

\

Ilur dt)

1/p

A"

-11g11132.41" . From this (b) follows if we take into account the fact that the ball that is being considered is bounded in C(0, T; B), and the assumption that the space B2 * is dense in B*.

3. Now we explain when the equation Lu = f has a solution bounded on the whole line, and when this solution is compact. With this aim we assume, in addition, that the embedding E c H is compact (the compactness condition'). '

Lemma 4. For any f e X the equation Lu =f has at least one solution that is H -bounded on the whole line. If, additionally, f E Xcomp, then a bounded solution is compact. Proof. Let un(t) (t ---.- —n) be a solution with the initial condition II It=- n = 0. In view of the estimates (10) and (12) and Lemma 3, the sequence {u(t)} is compact in ...WP ( T, T; 111. Without loss of generality we may assume that lurn (t)—un (t)i -- 0 almost everywhere on J. But since lum(t) — un(t)i --- lum.(to) — un(to)i (t .---' to), the sequence {u(t)} is in fact fundamental in the sense of local convergence of J. If u(t) (t EJ) is a limit function, then it must be a solution of our equation. For let z(t) (t...-- to) be a solution with the initial condition z(to )= u(to). Since lu(t)—z(t) --jun(to)—z(t 0)1- 0, we have u(t)= z(t) (t.- -.- to). —

Evolution equation on the entire line

161

We turn to the question of compactness. To be specific we consider a solution u(t) defined on the half-line t.--- to . For an arbitrary sequence of numbers tm -*00 the sequence {u (t + tin )} is fundamental in ...e (— T, T; H). Without loss of generality we may assume the convergence u(t + tm ) -> û(t) almost everywhere on J, and that f(t + t,n )-> Af(t) in the sense of ..Wq (— T, T; E*). But then from (13) we obtain (by analogy with the preceding) that t2(t) must be a solution of the limiting equation Lu —J and that the convergence u(t + 4, ) 14 il(t) is uniform on intervals. This proves the lemma.

4 Almost periodic solutions of the evolution equation 1. Suppose that the free term f(t) is in i., that is, f(t) is an almost periodic (in the sense of Stepanov) function J > E. We are interested in solutions that are Bohr almost periodic functions j > H, and denote the class of such solutions by C (H). Let E be the module of Fourier exponents of f (t). -

-

Theorem 6. If the compactness condition holds, then the equation Lu —f has at least one solution u e 6(H) with a module of Fourier exponents belonging to E. Note that when f(t) -= I, the almost periodic solution in Theorem 6 also does not depend on t, we obtain the solvability of the stationary equation Au =I. We only need to emphasise that Theorem 6 is based essentially on the compactness condition in contrast to a stationary problem and evolution problems on a finite interval. At the end of this section we discuss the situation when the compactness condition is not assumed. Proof of Theorem 6. Let 9e denote the closure of the family A := fA(S + t)* {f(s + t)} (t E J) in X, and for fA(s)e 9( we set P It follows from the monotonicity of the operator A that if u 1(t) and u2 (t) (t _--. to) are solutions of the equation Lu —f, then the norm lui(t) — u2(01 is non-increasing on [to, 00). But then, as was shown in Chapter 7, § 4, any two solutions compact on the entire line must satisfy the invariance identity A

-- const. (t en. itti(t) — u2(t)1=-

(14)

Now we give another proof of this identity that does not depend on the special results in Chapter 7. Let /4 1 (0 and u2 (t) (t E J) be fixed compact solutions of an equation of the form Lu = J We consider in the space H x k the natural A

The method of monotonic operators

162

semigroup connected with the equation Lu =1.3 We select a compact invariant subset ZcHx 9e containing the fixed trajectories {u i (t), i t} and {u2(t), i t } (Z can be the union of the closures of these trajectories). Let P denote the fibre over fek; thus, PcH is a collection of initial values {u(0)} of those solutions of Lu =f(f E Ye is fixed) such that the corresponding trajectory {u(t),P} belongs A to Z. For an element f we fix a 'returning' sequence tm -- -co: x fit +tm )-> f(t). A

A A

For each u(0) E P we fix (in view of possible non-uniqueness) a compact solution u(t) such that {a (t), it } c Z. We introduce operators Tm :P -> H, where by definition Tmu(0) = u(tm ). It is obvious that the sequence of operators Tm is compact in the topology of pointwise convergence; let T be some limit point. By the definition of pointwise convergence, for any v 1 (0), v 2(0) E P we have lim PI = Tv 1(0) = m-.00

VI(G) )

P2 = TV2(0) = m..00 lim v2(G),

where le m l is some subsequence of {tm }. Since the solutions v ,(t) and v2(t) are compact and uniformly continuous on the entire line, we can assume the local convergence IOC io. v i (t + ttm )--> p i (t), v 2(t +ti m ) —--> p 2(t). From the invariance of the set Z it follows at once that p i (0), p 2(0) E P. Since the norm I vi(t) - v2(t)I is a non-increasing function of t EL = const. lim Ivi(t +e m )- v 2 (t +ent)117)1(0 - P2(t)1=m-›co In other words, we have proved (14) for any p 1 (0), p 2 (0)E TP. Consequently, it remains to prove that TP = P. Since T is the limit of non-contraction operators Tm, it is itself non-contracting, that is, I TP1 - TP2I = IP 1 - P2I. Hence it is enough to refer to Proposition 1 in Chapter 1. Now we prove that the half-sum of compact solutions is also a solution. We put v (to) = (ui(to)+ u2(to))/2 and a =1141(0) - u2(0 )I. Suppose that v(t) (t ---- to) is a solution with the initial condition vit= 40 = v(to). From the inequalities lui (t)- v(t)i ui(to) - v(to)l =a/2 (i .1, 2 ; t _.--. to), 3

The important 'continuity condition' (see Chapter 7, § 1) follows easily from (13).

Almost periodic solutions

163

the identity lui(t) — u2(t)I-- a, and the strict convexity of the norm in a Hilbert space, we immediately obtain v(t) = (ui(t)+ u2(t))/2

(t = to).

Then by letting to -4 —00 we see that the half-sum (ui(t)+ u2(t))/2 is a solution. Thus, the set of compact solutions of an equation of the form Lu =f is convex. This makes the situation close enough to the linear case (see Chapter 8, § 2). An almost periodic solution 0) can be selected by the minimax method: sup lei(t)i = inf sup iu(t)i, tEl

teJ

where inf is taken over all the compact solutions of the equation Lu =f. This proves Theorem 6. 2. Now we turn to a discussion of this theorem. The proof of Theorem 6 (after Lemma 4) essentially is not based on special properties of a parabolic evolution equation. Therefore it is not difficult to give Theorem 6 a completely general form. We consider in a Banach space B a general non-linear equation u'= F(u, t).

(15)

We assume that the right-hand side is almost periodic, and the 'continuity condition' (for the details see Chapter 7). Let p = p(p, q) be a function that is symmetric and continuous on B x B, and vanishes only on the diagonal. Suppose that p satisfies the strict triangle inequality, that is, p (p i , q)+ p(q, p 2 ) = P (P 1, P2) implies that q belongs to the interval [Pi, /92].

Theorem 7. Suppose that the solving operators of the equations u' = P(u, t) are p-non -expansive. Then if equation (15) has a compact solution for t---. 0, then it has at least one almost periodic solution. Now we consider the parabolic evolution equation without the condition that the embedding E c H is compact. Lemma 4 is a key result; without the compactness condition one cannot succeed in proving the existence of a solution bounded on the whole line, not referring to compactness. Let ,u, be a normalised invariant measure on k. It turns out that for almost all h =/e .9e the corresponding equation Lu = .f has a solution bounded on the entire line; moreover, these solutions are bounded by a common constant and are Besicovitch almost periodic.

164

The method of monotonic operators

To prove this let H1 be the space of A-measurable mappings 0 : 9 e -- H with the norm*? = S ge 10 (h)12 d,u. Let S h(t) (t --- 0, h e 90 be the solving operators for the equation Lu =1 We introduce a family of operators //(t): HI -* H1 (t -...-- 0) by

/7(t)0(h )= Sh -0(h -t ). It is easy to see that these operators commute, that is, they form a semigroup. Next, they are Hi -non-expansive. For since the operators S h (t) : H -+H are non-expansive we have

111 (001 - H(t)0211 2 = Liol(h-t)-02(h-t)12d,i,

= f 101(h)_02(h)12 diu =101 - 02112 . ge

Consider the estimate (10); from it we obtain immediately the inequality

1/7(t)0(h )12 -.10 (it -t )I2 + 12. Therefore, I11(t)01 12 101 12 +12, that is, the semigroup /7(t) is bounded. If tfr(h) is a common fixed point, then as in the linear case (see Chapter 8, § 7), for almost all h e 9r the solutions u(t)= tfr(h t ) are Besicovitch almost periodic, and the following limit exists: (16) lim -1 T-->co 2T f- TT 10 (11!)1 2 dt =1011 2. The solutions u(t)= t/j(ht) are bounded by a common constant; indeed, (16) and the inequality IWO --1/4(to)1 ---. 1 (t --. to) implies that supteilu(t)1 2 - 12 +101 2. Finally, we mention a simple result that does not require the compactness condition. We retain the estimates (7) and (8) for the operator A but replace the monotonicity condition by the stronger

condition: (Ax - Ay, x - y) --- yix - y1 2 (x, y

E

E)

with the constant y >0. We prove that the equation Lu =f has a unique almost periodic solution. For two solutions 11 1 (0 and u2(t) (t -.-- to) we have

1 d lu i-—- u212 +1Aui -Au21= 0, 2 dt

d

lui - u212 + Ylui - u 2 12 -- 0, — dt lui(t) — u2(t)l

-

exP( y(t — to))lui(to) — u2(to)I. —

Almost periodic solutions

165

Therefore, if U(t, to) (t to) is a solving operator u(t o)-4 u(t), then I Mt, to)P — U(t, to)q exp( — y(t — t o))1/9 — q. We consider a family of operators 17 (t ): 6(H) -4 6(H), where

11(t)0(s) = U(t +s, s)0(s — t). The operators 17(t) commute, and for t > 0 they are contractions; consequently, they have a common fixed point, which is obviously an almost periodic solution.

Comments and references to the literature §§ 1 and 2. The material in these sections is mainly classical for that subject area of non-linear analysis conventionally called 'the method of monotonic operators'. A valuable feature of this method is its application to non-linear elliptic and parabolic equations. Although these applications are very important in their own right and served as the source of the method of monotonic operators, we have not touched on them (as regards this question, see the extensive literature, for example, the book by Lions [79 1). The fixed point theorem (proved by Zhikov) is a generalisation of the standard theorem (usually attributed to Browder): a non-expansive operator T : H -4 H has afixed point if it leaves a bounded convex set invariant. § 3. Lemma 3 and Proposition 5 are versions of fairly well known compactness lemmas' (Lions, [79], Russian p. 70). '

§ 4. The results here are extracted from Zhikov's article [52]. It should be noted that we have restricted ourselves only to the results about almost periodic functions that are proved most easily. A more complete theory embracing, in particular, non-linear equations of hyperbolic type, is presented in the article of Zhikov & Levitan [57].

10

Linear equations in a Banach space (questions of admissibility and dichotomy)

Notation We use the notation: C = C(B) is the space of continuous functions J --* B with the sup norm; C± = C ±(B) and C - = C - (B) are analogous spaces for the half-lines j ± = [0, co) and j - = (-00, 0 1 , respectively; Ar = Ar (B) and ..ce = ...cr (--co, oo; B) (p __--- 1) are the spaces of measurable functions u J--.13 with the norms

1 ilp sup u(t ( i. Il + sr ds)1 tEl

.0

i/p and ( . . 'lull' dt) , —oo

0

respectively; the constants c o, c 1 ,. .. , /0, 1f, . . . are assumed positive.

1

Preliminary results

1. Our basic object is a linear operator L ---- d/dt +A(t), where A(t) (t E j) is generally an unbounded operator in a Banach space B. We assume the following condition of right-solvability: for any initial value u(to)E B, the equation Lu = 0 has a unique strongly continuous solution u(t) (t = to). We assume also that the solving operators U(t, to) (t -.-- to) are strongly continuous with respect to t..--- to and satisfy the estimate

(1) IIU(t, ton ----- bo exP (co(t — to)) (t =" to). The solution u(t) (t__-- to) of the non-homogeneous equation Lu = f (fE M l (B)) is determined from the usual formula t

u (t) = U(t, to)u (to) + f U (t, s)f(s) ds. to

If u(t) (t en satisfies this equality for any t._-. to E l, we speak of the solution on the entire line, and write Lu =f (t e n. We obtain the

167

Preliminary results following estimate immediately from (1):

t0 +1

colliu(to)ii+

to

lif(s)ii ds}(

to

to + 1).

t

(2)

We mention some properties of the solutions. Let Lu =f (t > to) and let 40 (t €J) be a smooth scalar function with support on (to, oo). We set v = cAu (t P. Then Lv = + 4)' u (t e J). In what follows we always take to be a smooth mapping [0, 1]; when cA needs to be prescribed more exactly, we give its support (supp 0) and the set on which OW = 1. Now we define L as an unbounded operator in the space C = C(B). For this it is enough to specify its domain D = 1.4 E C : Lu E C}. Similarly for the spaces Ar (B) and (—co, 00; B). From (2) it follows that in all cases we have the embedding D C. It is perfectly clear that these operators are closed. An operator L is called regular if the equation Lu =f is uniquely solvable in C for any f E C. From Banach's theorem we obtain the existence of a bounded inverse L -1 : C C and the estimate {

iluilc

kilLuilc.

(3)

L is called correct if (3) holds for any u with Lu e C, and strongly correct if I lu Ilc k IlLullm2for any u E C with Lu E M2. Since C c M2, strong correctness implies correctness. A pair (W, F) of Banach function spaces is called admissible for an operator L if Lu = f has at least one solution u E W for any f e F. From Banach's open mapping theorem we obtain that for any fE F there is a solution u E W such that

(4) k Ilf IIF with a constant k not depending on f € F. If (C, C) is an admissible pair, then L is called weakly regular.

liuliw

It turns out that the concept of regularity is closely linked with an important concept in stability theory: the exponential splitting or the exponential dichotomy. Suppose that we have a variable decomposition B = Ni (to)+N2(to) into a direct sum of subspaces, and let Pi(to) and P2(t0) be the corresponding projection operators. Also let u(t) (t % to) be a solution of Lu = 0. The operator L has the exponential dichotomy on the whole line J if

(d1) U(t, to)Ni(to) c Ni (t) (t (d2)

sup to IIPi(to)li
0 that 11 -1 exp (c iA)-/ i exp (-c iA)= a > 0. Let u(to) E N(to) with ilui(to)li = 1. For t = to +LI we have

ilui(t)11=li exP ( -- cd),

1u2(t)11=11-1 exP (ciA),

which together with (1) gives

(t0)11= liul(to) + u2(to)ii exp ( - cod)iiui(t)+ u2(t)11 exp ( - co4){11142(t)ii ilui(t)111 exp ( -coA) =y>0, that is, liPi(to)ii Now note that from the exponential dichotomy on J it follows that L is regular; the proof consists of a straightforward construction of the Green's function. For this, by using condition (d4), we introduce the bounded operators 12(4 to) = U(t, t0)P2(t0) for any t, to e J. Further, we set

G(t, s)=

U(t, s)/31 (s)

s)

for t

for t

s, s.

(5)

Conditions (d2)-(d4) imply the estimate

(6) 11G(t, s)II / exp (-c i lt Therefore, the function u(t)= j G(t, s)f(s) ds is bounded. Then it is easy to verify the identity U(t, to)G(to , s)= G(t, s)-x to (s)U(t, s), where x(s) is the characteristic function of the interval (to, t). By applying this identity to f(s) and integrating, we see that u(t) is a solution of the non-homogeneous equation. The uniqueness of the solution in C follows from conditions (d2)-(d4). Next we formulate the concept of exponential dichotomy on the half-line J . Let N2 be a complemented subspace in B, N 1 be complementary to N2, and u(t) (t to) be a solution of the

Preliminary results

169

homogeneous equation. We say that the exponential dichotomy holds on J if (d1 - ) for any u (0) E N2 there exists a unique extension to j - such that

to)

/111u(to)ii exP (ei(t (t-to--0); a solution u(t) (to ---5 t - --5 0) satisfies the estimate

ilu(t)ii (d2 )

Plu (t) /Au (to)ilexP (—ci(t — to)) (to ---5 t -- 0) provided that u(0) E N1. We construct a Green's function. For to --- 0 we set by definition -

N2(t0) = {u (to) E B : u (0) E N2}) Ni(to) = Iti (to) E B : u (0) E N1},

where in the first we take u(t) to be the solution in condition (dC), and in the second u(t) is arbitrary. It follows from condition (d1 -) that N1(t0)riN2(t0)= {O}. We show that these subspaces are mutually complementary. Suppose that x (to) E B and x(t) (t --.. to) is the corresponding solution of the homogeneous equation. We set y(0) = Pi(0)x(0) and z(0) = P2(0)x(0). By assumption, for z(0) there is a unique extension z(t) to j with the estimate in (d1- ); therefore z (to) E N2(to). Since y (to) = x (to) z (to) E Ni (to), we have proved that the subspaces are mutually complementary. The subsequent arguments are similar to the case of the whole line. We introduce the operator-function 12(t, s) = U (t, s)P2 (t, s) (t, s E r). The Green's function G(t, s) (t, s E J - ) is given by (5), and satisfies the exponential estimate. Therefore, the function u(t)=1_?c,,, G(t, s)f(s) ds is a solution in C - , that is, the pair (C -, C - ) is admissible. —

2. Now we study the concept of correctness in more detail. Let L be a correct operator and u(t) (a---ç.t --..b) a solution of Lu =f Suppose that T >0 is such that the interval [to — 2 T, to + 2T] lies in (a, b). We choose a smooth function 0 : 1 .- [0, 1] for which

OM = 1 for t E [ to — T, to+ T], supp 0 c [to - 2T, to+211, 10'1--- 2/T. For y = Ou we have Lv =f0 +0'u (t E j). Therefore, from (2) we obtain 2k

sup Ilu (t)II ---5 k sup WWII+ .,. _, sup Ilu(t)II. 1

It—toi,...T

It—toi _.,2T

[a, b]

(7)

170

Linear equations in a Banach space

Before using this estimate let us note that the notions of local convergence, local closure, etc. can be naturally introduced in the spaces C, M", and others. For instance, we say that fm i0f in AI' if the sequence {fm } is bounded in M", andliO(fm — / )lI mp 0 for any smooth with compact support. It turns out that the range of a correct operator is locally closed in C. For suppose that Lei n-, = fr., where um, fm E C, and that fin -4 loc f If we apply (7) to the difference u m — un we find that the sequence {um } is locally fundamental in C. The limit function is a solution of the equation Lu =f. Similarly it can be proved that the range of a strongly correct operator is locally closed in M 2(B).

2 The connection between regularity and the exponential dichotomy on the whole line In this section our aim is to prove the following assertion. Theorem 1. Regularity is equivalent to the exponential dichotomy on J. In one direction this assertion has already been proved in § 1; it remains to prove that regularity implies exponential dichotomy on J. Our proof is constructed so that regularity is really only used in the last stage, until then our arguments are based only on correctness. Suppose that u(t) (t to) is a solution of Lu =0. We call it uniformly stable to the right if (8) liu(t)11. ( t s to) and uniformly exponentially stable to the right if /1 exP (—ci(t —s)) s to). (9) In a similar way we speak of uniform stability and exponential uniform stability to the left if a solution u (t) is defined for t to. Lemma 1. If L is correct, then uniform stability (to the right or left) implies uniform exponential stability (to the right or left), and the constants l and c 1 depend only on 1 in (8) and k in (3). Proof. To be definite we consider stability to the right. We show that we can find a number T = T(1, k) such that (10) 2"; s to). iiu(s + liu(s)11/2 For suppose there is a solution z(t) with the estimate (8) and an interval A = [a, bic [to, co) such that iz(a)11= 1 and liz(b)11---- From

Regularity and exponential dichotomy

171

(8) we obtain immediately (11) 4(01= 1, Ilz(t)11--- 1/2 1 (t E A). Let 0 be such that supp 4) c A and OM = 1 for t E [a +s, b — e], and let

f= 0 (t)z(t)Ilz MIL' , t u(t) = z(t) f co o (s)liz(s)11 - ' ds. By using (11) it is easy to calculate that iluli c ...--. 2-1 1 -2 (4 —2E). Since liflic = 1 and Lu =1' from (3) we obtain that A --.21c1 2 , which proves (10). Then this estimate and (8) imply (11) if we take / 1 = 2 1 and c l = T-1 ln 2. Thus, Lemma 1 is proved. We introduce the following manifolds: Ni (to) = {u (to) e B : Lu = 0, sup lits(t)11< 001,

(12)

N2(to) = 114 (to) E B :Lu =0, sup Ilu(t)11< col.

(13)

t- - to t to --..

First of all we note that under our conditions the solution of the homogeneous equation can have an infinite number of extensions to the whole line. In (13) it is required that such an extension is bounded on (-00, to); correctness implies the uniqueness of this extension. We prove the inequalities (t % to, ti (to) E N 140)))

(14)

liti (Oil -- liiti (t 011 (t to, U (to) E N2(to)). To this end let 0 be such that:

(15)

liti(t)11--- Iliti(to)li

supp 0 c {to, 00 ), OW = 1 (t---- to+ 1), 10'(t)i ---- 2. (16) If u (to) e Ni(to), then for v = Ou we have Lv = (Yu =f (t En. Therefore, from (3) and (1) we obtain

sup ilu (t)II -_ II v li c .--.. k lifli c --. 2k exp collu (to)ii,

tt 0+1

which together with (1) gives (14). Next, if u(to) E N2 (t 0), then by extending the solution to the whole line and setting v = (1— (1))u, we obtain (15) by a similar method. The estimates (14) and (15) indicate that the manifolds Ni (to) and N2(t0) are closed; by Lemma 1, these estimates imply the corresponding exponential estimates. Consequently, it remains to prove that the subspaces Ni(to) and N2 (t0) are mutually complementary.

Linear equations in a Banach space

172

Let x (to) be arbitrary and x(t) (t-- -- to) be a solution of the homogeneous equation. We put f= O'x(t) (t En, where 0 satisfies the conditions (16). By the regularity property, the equation Lu =f has a unique solution u E C ; the function v (t)= 0 (t)x (t) (t eJ) also satisfies this equation. We set z(t)= v(t)-u(t) and y(t)= x(t)z(t). Then obviously Lz = 0 (t EJ) and Ly = 0 (t...-- to). It is clear from the definition of 0 that

sup ilY(t)11< co, t?--t0+1 that is, z(to) e N2(t0) and y(to) e Ni(to). Thus we have obtained the decomposition x(t0)= y(t0)+ z(t0), which proves that N1 (t0) and N2(t0) are mutually complementary. Theorem 1 is proved. We consider a family of operators L a satisfying the inequality (1) uniformly with respect to a. We say that the La are uniformly regular if the inverse operators La-1 : C --> C are uniformly bounded. Furthersup Ilz(t)ii ) C (B). If f E M i (B), then we set of a bounded inverse u (t) = f_c°,,o G(t, s)f(s) ds. In view of (6) we have ilti(t)11= i f OE) exp (- c i ls plifis +Oil ds.

By dividing the axis of integration into unit intervals we find that 1 00 E exp (-CM) i lu (t)II= 21 sup (f Ilf(s + t)11 ds)m=0 tel

= killfIlmi.

I)

(17)

Theorems on regularity In many cases it is desirable to deduce regularity from simpler properties, such as correctness or weak regularity. We prefer to start from correctness since firstly it arises in a number of problems on stability, and secondly, weak regularity can be reduced to correctness by duality arguments. Below we distinguish two basic situations in which correctness implies regularity. 3

Theorems on regularity

173

1. We say that the compactness condition holds if the operators U(t, to) (t > to) are compact. The simplest examples show that the compactness condition is not enough, we need certain conditions on the dependence of A(t) on t e J. A function 6 j -> R (R is a metric space) is called Poisson stable (has the P-property) if e(t + 4,)-* e(t) (t e J) for some sequence tm -> —00. If the operator A(t) were bounded, then the P-property would mean the Poisson stability of the function A(t) :J > Horn (B, B). In general, the P-property must be formulated in terms of a solving operator, namely, there exists a sequence tm -> —co such that -

U(t + tm, s + tm )x -> U(t, s)x (t -.- s, x E B).

Theorem 2. If the compactness condition holds, then a correct operator having the P-property is regular. Proof. We prove that the exponential dichotomy on the half-line J holds. To this end we consider the subspace N2 = N2(0) defined by (13). The compactness condition implies that this subspace is finite-dimensional; suppose N1 is a complement of it. Since property (d 1- ) is already ensured by the estimate (15) and Lemma 1, we prove property (d2 -). As a preliminary we prove uniform stability to the right, that is, the inequality

ilu(t)ii= iiiu(s)11

(t 0 ---. s ---. t ---. 0, u(0)

E

NO.

(18)

By assuming otherwise we find a sequence of solutions um with sup Ilum II = 1 and Mum(1m)II -> 0 (am .... t--._ 0). It follows from the compactness condition that the set {u m (t)} is compact for any t--- 0. Hence, if z(t) is a limit point of the sequence u m (t), then z(t)E C - , Lz =0, and z (0) e Ni, that is, z -= 0 and it follows that um (t) ->ioc0. But then the norm ilum (t)II reaches its maximum value (unity) at a point sm such that sm -> —co and am — s m .- —co. We apply the estimate (7) to Urn (t), setting (a, b)= (am , 0), to = sm , and T = min {1 5 mi, lam —5.1 } . As a result we obtain a contradiction which proves (18), and together with it the exponential dichotomy on J, and consequently, the admissibility of the pair (C - , C - ). But then, for every f E C - the equation Lu = f has a solution with the estimate liulic- ---- kilif lic - . For each f e C we put fm(t) = f(t — t m ), and suppose that vm (t) (t E j -) is a solution of Lv =fm with the above estimate. Then we set um (t)= vm (t + tm).

174

Linear equations in a Banach space

Thus, we have

Lu m = f (t---. — t m ),

sup lium(t)II ---Ç. kiiif ilo

t-,-.—trn

The difference zm,n = um — un satisfies the homogeneous equation on the half-line ( — co, bm,n), where bm,a, = min (—tm, —ta ). From (7) we easily obtain sup t .,..i b„,,„ lizm (t)ii -> 0, that is, in any case the sequence {um } is locally fundamental. By taking the limit we obtain a bounded solution of Lu =f This proves Theorem 2.

2. As well as the operator L we consider the dual operator L* formally given by L* = —d/dt +A*(t). Regarding L* (since it is referred to in general) we assume the condition of left-solvability and an estimate of the form (1). As before, we denote the value of y e B* at X €B by (y, x). An informal determination that the operators L and L* are duals is contained in Green's formula. Namely, it is assumed that if Lu = f (t E J) and L*v = g (t e J) for f E M 2(B) and g E M 2(B * ), then

(y, ti)I t4 = j

t2

.

ti

[(g, u)— (v, f)] dt.

Theorem 3. If L is strongly correct and L* is correct, then they are both regular. Proof. We define L and L* as unbounded operators in the spaces B 1 = .T2(-00, co; B) and B 1* = Y 2( 00, 00; B*). It follows from inequalities of the form (2) that WWII -> 0 and Ilv(t)ii -000 as t -0 co for u E D(L) and y e D(L*). Therefore, Green's formula becomes -

t (v, Lu) dt = f (L*v, u) dt I

(u e D(L),

yE

D (L*)).

Hence it follows that the operator adjoint to L (relative to the natural duality between B 1 and B 1 *) is some extension of L*. To prove the coincidence of these operators it is enough to establish that L* has a non-empty resolvent set. With this in mind we consider the operator L* AI, where A >0 is sufficiently large. Since the solutions of the homogeneous equation L*v — Av =0 decrease exponentially as t -4 —co, this operator is regular. Consequently, it is sufficient to prove the invertibility in B i* of a regular operator. To this end we take a function f e B i * with compact support and put v = fr G(t, s)f(s) ds, —

175

Theorems on regularity

where G is the Green's function of a regular operator. The solution y decreases exponentially and satisfies the estimate t

iiv(t)ii - -- /If . exp (—ci(t—s))lif( Oil ds exp (c i (t — s))iif(s)lids}

t

= /{01(t)+ 02(t)}. Since oaf +clip, = 'If'', after multiplying by tfri and integrating we have cif J

lOt12 dt = f oillfil dt

--

J

1/2 (fJ

11/412 dt)

1/2 (f lifil2 dt) J

,

that is, liv 6, --- //2ciiitlia,, which (in view of closedness) gives invertibility in B 1*. Now we assume that L is not regular. Then we can find a function fo e C with compact support such that Lu = fo has no solutions in C. We prove that the range of L (as of an operator in B 1 ) is not dense in B 1. Assuming otherwise, we find a sequence urn E B 1 such that Lu n., = fn., * fo. Since um E C and lifm — f0lim2= Ilfm —follBi, from strong correctness we obtain Mum — uolic -O, that is, the equation Lu =fo is solvable in C. Hence, the range of L is not dense in B 1 . Since L* is adjoint to L, the orthogonal complement of the range of L consists of the zeros of L*. These are the elements of C (B*) that contradict the correctness of L*. This proves that L is regular. From what has been proved above the regular operator L is invertible in B 1 . By a theorem of Phillips (Yosida [61], p. 273), a conjugate operator L* is invertible in B 1.. This, together with correctness, gives regularity, and Theorem 3 is proved. -

3. It would be useful to establish the duality of the concepts of correctness and weak regularity. In this direction we have only been able to obtain the following result. Lemma 2. If one of L or L* is weakly regular, then the other is correct. Proof. We prove that L* is correct if L is weakly regular. Let n(t) be an odd continuous scalar function such that n (t) = 1 for t. .- 1. We fix an E > 0 sufficiently small that the pair (C, C) remains admissible -

176

Linear equations in a Banach space

for the family of operators d/dt +A(t + s) - en MI with a common constant k in an equality of the form (4)• 1 Assuming that L* is not correct, we can find sequences fy m l, {gm }c C(B*) for which L * y,,,, = gm, IIYmII = 1 and ligm lic O. We select a tm EJ such that liYm(tm)lid an_ put vm (t)= ym (t + tm ), gm (t)= g(t + tm ), and L n, = dt + A(t + tm ). From Ilvm (0)11-1- and an estimate of the form (2) we obtain immediately (19) Then putting v;,,,(t)= vm (t) exp (-Ef

= m (t) exp

71(S) dS),

a (s) ds),

and bearing in mind that v ins (t).- 0 as t -> 00 and Lm *v,„' envEm = by Green's formula we have -

(f, v;n ) dt = (u, R:n ) dt,

(20)

where f is an arbitrary element in C, and u is a solution of L mu eu =f satisfying an estimate of the form (4). We consider the expression Am (f) = fj (f, em ) dt as a functional over C. Since IlAm ii = SjI!vII dt, then as follows from (19), iiAmii =0 >O. On the other hand, from (20) we have liAm II

kf

dt--> O.

In a similar way, from the admissibility of (C, C) for L* we can obtain the correctness of L. 4 Examples 1. Suppose that the space B is finite-dimensional, and A(t):J.-.> Hom (B, B) is a bounded, continuous, Poisson-stable function (for example, an almost periodic function). Then the operator L is regular only if it is correct or weakly regular. To prove this we must apply Theorem 2 and Lemma 2. Weak regularity (just as regularity and correctness) is preserved under small (with respect to the norm of Hom (C, C)) perturbations.

177

Examples

2. Suppose that the solving operator U(t, s) exists for any t, s e J and satisfies the estimate s)li ii exP ( —cult — s i). For any f € C the formula

u(t)= sgn ti. U(t, s) f(s) ds o

gives a bounded solution of Lu = f. The operator L is weakly regular but not regular. If B is one-dimensional, then an example of such an operator is dl dt ri(t), where 71(t) is a continuous odd function such that 77(t)= 1 for t 1. 3. Let Uo(t)= exp (—tA0) (t 0) be a semigroup of isometric operators. Let us assume that, for t >0, U0(t) is not unitary (B = H is a Hilbert

space). We prove that the operator L = d/dt +A o / is correct. Indeed, suppose that Lu =f (u, f e C). We find a point to for which Ilu(to)li Ili/silo Then from the usual formula t0 +1 U(1)u(to)= u(to + 1)—i.

to

U(t o + 1— s)f(s) ds

we have iellullc ilullc ±elk +1 11f(s)11 ds, which actually gives strong correctness. However, the operator L is non-regular. For the solutions of the homogeneous equation Lu = 0 increase exponentially as t + co. Therefore, the exponential dichotomy on J is possible if and only if these solutions can be extended to the whole line (see the definition of the exponential dichotomy on J). The latter is incompatible with the assumption that U0(t) is not unitary. 4. Let H+ and H_ be orthogonal mutually complementary subspaces in H, and P+ and P_ the projection operators. We consider the group of unitary operators Uo(t) satisfying the conditions

Uo(t)H+

II+ (t 0), U o(t)H_c H_ (t

0).

(21)

In addition, we assume that H+ , H_ are not invariant under the whole group. Let A o be the infinitesimal generator of this group. We set Rx = P.fx — P_x = x ± — x_, and A = Ao +R. We show that (C, C) is admissible for L = d/dt —A, but L is not regular. Since R is bounded, A generates a strongly continuous group U(t).

178

Linear equations in a Banach space

From (21) we easily obtain U(t)x ± = exp ( t)U0(t)x + (t--0), —

U(t)x_ =exp (t)U0(t)x_ (t ---5- 0).

(22) (23)

Now we note that the restriction of the semigroup U(t) (t ....-- 0) to H, is a semigroup with a generating operator A + . In view of the exponential estimate (22), L + = d/dt —A + is regular (considered relative to H±). Similarly, the restriction of the semigroup U(t) (t ---Ç. 0) to H_ leads to a regular operator L_. Now let f e C and f = f+ + f_. Then u = L +-1f+ +L_ -if_ is a bounded solution of Lu =f, that is, (C, C) is admissible for L. However, the operator L is not regular. For if we were to assume regularity, then by Theorem 1 we would have the exponential dichotomy on the whole line J. It follows from (22) and (23) that the exponential dichotomy must be given by subspaces H+ and H_. But then these subspaces are invariant under U(t) (t El); hence we easily obtain that they are also invariant under U0(t) (t EJ), which contradicts our assumption. To realise the several conditions we can take H . y2(_ 00, 00 ) , H-± = y2(0 , 00 ), H = 2( -00, 0) , 2 and U0(t) to be the group of right translations.

5. Instead of R in the preceding example we can take ER, where E is small. Then from the elementary theory of perturbations of semigroups we obtain the representation U(1) = U0(1)(I +K), where K is an operator whose norm is small. Therefore, I +K = exp (F), where F is bounded. The operator U0(1) (as any unitary operator) admits the representation U0(1) = exp ( W), where W is bounded. Then, U(1) = exp ( W) exp (F). For this situation Massera & Schdffer [87] (p. 354) have indicated a constructive method of producing a periodic equation Lu 7--- u' + A(t)u = 0 (A(t) is bounded) for which the monodromy operator is U(1). L is weakly regular but not regular; this follows from the readily seen reducibility to the operator in Example 4. 6. Suppose that A(t)= A is bounded, B is arbitrary, and that one of the following conditions holds: (1) (C ± , 6) is admissible; (2) L is correct. We prove regularity.

Examples

179

For definiteness we assume that (C +, 6) is admissible. Then for each f e ei there is a solution u E Cy + such that (24) ilu lic-+ = klIfilc. It suffices to prove that the spectrum of A does not intersect the imaginary axis. We assume otherwise and let —iA o be a boundary point of the spectrum lying on the imaginary axis. By a theorem about boundary points2 there is a sequence an E B such that Aan, + ii1.0a. = b. 00, 10,11= 1. -

(25)

Putting v An, = exp (iAt)a n and 4= IA —A 01 (A is real) we obtain Lv A, = exp (iAt)(Aan + WO =.n. From (25) it follows that Ilfdlc ----0(4), where 1im 4 .00(4) = 0. Let /4 be a solution of Lu = f,A, satisfying the estimate (24). We set znA = vnA —u. We define the weak topology on the dual space B** of B*; it is a fact that the unit ball in B** is weakly compact. For z E C + we denote by niA (z) a weak limit point of '--, foT z(t) exp (—iAt) dt (T-0+00). Clearly, if Lz =0, then mA (z) is an eigenvector of the operator A** with the eigenvalue —iA; this is easily proved by taking into account the weak continuity of A**. Now we note the following. Since niA (v An) = a. we have iinh( znA )ii = iinh(v .,A ) — m(u)ll = 1— WA) =i, provided that IA —A 01= 4 is small enough. This means that —IA is an eigenvalue of A**. Since the spectra of A** and A coincide, we obtain a contradiction to the assumption that —a. ° is a boundary point of the spectrum.

7. We study the regularity of an operator with coefficients that have small oscillations. A typical example is d/dt +A(et), where e is small (here and below A(t) is assumed to be bounded). We define a 'measure of the oscillation' of an operator-function A(t) by

T

itC(T) = sup je IIA(t0+ t) — A(t0)11 dt. toe J —T 2

See, for example, Daletskii 8r Krein [39], Russian p. 44, English translation, p. 28.

180

Linear equations in a Banach space

Lemma 3. Let the operators 1,0 = dIdt +A(t o) (toe f ) be uniformly regular. Then if 12,(T) is sufficiently small for a sufficiently large T, then L = d/dt +A(t) is regular.

Proof. From (17) it follows that the norms of the operators Lto- ':11//'.- C are not greater than some constant kJ.. Let Lu = f, where f e Af' and u e C. We take a point to E I for which iiu(to)ii = Miu lic, and rewrite the equation Lu = f in the form Lu = (A(t 0)— A(t))u +f. We selected a 0 for which 0 (to) = 1, supp 0 e [—T + to, T + to], and 14"(t)i .--.. 2/ T. For y = Ou we have

Ltov =0(t)(A(t 0)—A(t))u +Of +0'u =g(t). Therefore, lliuilc =iivlic = kiiigiimi= ki,u(T)Ilulic + kilfilmi + k i(2/ T)Ilull m i. Assuming that 1/ T and A (T) are sufficiently small, we hence obtain that ilutic =ikil film', which gives, in particular, the strong correctness of L.

Because these arguments can be applied to the dual operator L*, by Theorem 3 the operator L is regular. Corollary 1. Suppose that A(t) is a uniformly continuous function in Hom (B, B) and the operators Lk, are uniformly regular. Then the operator d/dt +A(st) is regular for small e.

Corollary 2. Suppose that the function A(t) (t..?.- 0) is compact in Hom (B, B) and 'stationary at infinity', that is, T liM j. MAU + to) —A(taldt = 0 (T>0). t0-' 00

-T

Suppose that the spectrum of each of the 'limiting' operators A (A = lim tn _>„;) A(tn )) does not intersect the imaginary axis. Then the exponential dichotomy on r = [0, +00) holds for L = d/dt + A (t). For a proof we must choose to >0 large enough that the operators L, (s...--t o.) are uniformly regular (we can do this in view of the compactness) and that the operator-function

{AM for t -.- to, A (to) for t ---- to, satisfies the conditions of Lemma 3. Then the operator L l has the exponential dichotomy on j (by Theorem 1), and consequently, L has it on J. Ai(t)=

Comments and references

181

Comments and references to the literature Questions of admissibility and dichotomy closely connected with the general theory of stability are treated in the books of Massera & Schaffer [87], Daletskii & Krein [39], and Krasnosel'skii, Burd & Kolesov [66]. The first two of these contain extensive bibliographies and historical notes. § 2. A connection between regularity and the exponential dichotomy in the finite-dimensional case was essentially known to Perron [98] (Theorem 1 is explicitly in the article of Maizel' [84]). But there was no similar result for equations in a Banach space. Massera & Schaffer deduced the exponential dichotomy from regularity by assuming a certain 'closedness condition' (regarding this, see also Daletskii & Krein [39], Russian p. 249, English tr. p. 203). The observation that 'conditions of closedness' are redundant was made by Zhikov [55]. § 3. We mention the interesting article of Mukhamadiev [94] who studied a property fairly close to correctness. We consider the operator L = d/dt + A(t), where A(t) is an almost periodic matrixfunction, and assume that equations of the form Lu u' + ii(t)u = 0 do not have bounded solutions. Then L is regular. This result of Mukhamadiev can be reduced to Theorem 2 by proving that L is correct. Indeed, if L is not correct, then there are un, fn e C such that Lu n= f, liuniic = 1, IlfnlIc -) 0. We choose points tn El for which Ilun(tn)ii By taking the limit in Iun (t + tn )1 we obtain a non-trivial bounded solution of one of the limiting equations. § 4. The results in the example in § 4.6 were inspired by a question posed by Daletskii & Krein [39] (Russian p. 44, English translation p. 27). The results in § 4.7 were obtained jointly by Zhikov and Valikov, while the other results in § 4 are due to Zhikov [55]. The reader can find further discussions in the article of Zhikov & Tyurin [58].

11

The averaging principle on the whole line for parabolic equations

1 Bogolyubov's lemma

We consider the linear operators L,,,, = di dt + A(cot), where A(t) is a bounded continuous function f -0 Horn (X, X), X being some Banach space. We are interested in the properties of the operator L„, as co -0 co. Suppose that A(t) is such that the mean

1 j+.T a

urn — A(t) dt = A 7, ,,,, 2T -T+a

(uniformly with respect to a eh

exists in the sense of the operator norm. We introduce the so-called averaged operator L = d/dt +A. Bogolyubov's lemma. If the averaged operator L is regular, then the operators L„, are uniformly regular for co -?.- coo. Proof. We put Â(t) = A (t)— A. The function A(t) has a zero uniform mean; by Lemma 2 and Theorem 3 in Chapter 6 there is a family of functions 71,(t) :f -0 Horn (X, X) such that t lim sup WO–AE (011=0, sup 1 f 218 (s) ds11 0, x e E). 1 It is obvious that the adjoint operator A* : E --> E* is also strongly elliptic. We can regard A as being an unbounded operator in H if we define its domain by fx e H: Ax e HI. This will always be understood when we speak about the spectrum of A. We call an operator d/dt + A(t) parabolic if A(t) is a continuous bounded function in Hom (E, E *) and the coercive inequality holds uniformly with respect to t e J. We characterise a parabolic operator by the constants c l and c2 in the coercive inequality and by the constant 1 = suptET 1 A (t)11E.E*. The solvability of the Cauchy problem for a parabolic operator is readily obtained from the results of Chapter 9. For consider the operator

exp (--At)/(d/dt+A(t)) exp (At)I = (d/dt +A(t)+A./).

If A -- -- c2, then the operators A(t)+ AI are monotonic. Therefore, Theorem 5 of Chapter 9 with p = q = 2 holds. But then Theorem 5 holds for the original operator L. Now we write out the estimates we shall need later. By scalar multiplying the equation u'+A(t)u =f by u(t) and integrating we obtain (by analogy with (11) in Chapter 9) t t iu(t)1 2 - iu(to)1 2 + ci*(f 11u112 ds + f

f

to

to+1

t.

to

11f11* 2 ds)

(t -?-: to),

(1)

t 0 +1

Olu 112 ± Ilu i li*2) ds :--- c 1 * ( iu (t0)12 ± I.

11f11*2 ds).

(2)

Here the constant c l * depends only on the constants c l , c2 and 1. Let us note that if we introduce a parameter co in the operator A, then it turns out that (1) and (2) are uniform with respect to co. 1

It is worth emphasising that in applications, usually beforehand we have only the embedding E c H and a bilinear form a(u, y) continuous on E. Then E * is determined as the closure of H in the norm

iiWil*= sup 1(y, x)i, 11.1k1 and the operator A : E * E* is defined by (Au, v)= a(u, y). -

Some properties of parabolic operators

185

2. The operator L* = —d/dt +A*(t) is by virtue of its properties, analogous to L, except that the Cauchy problem is solvable from the left. We show that L and L* are duals, that is, they are connected by Green's formula. Let Lu =f and L*v = g (a ---5. t .-.-. b), where f, g e Y 2(a, b ; E*). From results in § 2 in Chapter 9 it follows that d(u, v)/dt = (u', v)+ (u, v')

almost everywhere on [a, b]. Therefore, d(u, v)/dt = (—Au +f, v)+ (u, A* v — g) = (f, v)— (u, g); b (it, 2))1 6 = i

: {(Lu, v)—(L*v, u)} dt,

as we required. We define the concepts of regularity, correctness, and weak regularity relative to the space C = C (H).

Corollary 1. Suppose that the embedding E c H is compact and A(t) is a Poisson-stable function in Horn (E, E*) (for example, an almost periodic function). Then the operator L is regular whenever it is correct or weakly regular. Proof. We apply Theorem 2 of Chapter 10. Since the compactness condition follows from the compactness of the embedding E c H, we must verify the P-property. Let tm -0 —00 and A (t + tm) .- io, A(t) (t e J). We set Lm = d/dt +A(t + t,n ). Suppose that u(t) and um (t) (t .-.- 0) are solutions of Lu = 0 and Lmu =0 with a common initial condition um (0) = u(0) =0. We must prove that lu(t)— um (t)! -0 0 for t _-.- 0. From the relation L (u — urn) = (A (t) — A (t + tm ))un, and the estimates (1) and (2), for 0 ---5. t ---5 T we have T lu (t) — Um(t)I 2 •-- c 1 * SUP 11A(t

+ tm) — AMME-.E* .1. Ilum112 ds,

0----t----T

0

which proves the P-property. Weak regularity is dealt with by using Lemma 2 of Chapter 10. 3. In what follows the space E* is subject to the following condition: there is a sequence of linear operators Pm : E* -0 E* such that E*

Pm E* c E, Prny --> y

(y e E*).

(3)

186

The averaging principle for parabolic equations

We introduce the following spaces of functions: X = M 2(E*), Xeon,„ is the subspace of X consisting of those f(s) E X for which the family of translates f(t+s) (te 1,0-s---51) is compact in ,Y2(0, 1; E*); V = M2(E) n C(H) with the norm

„1 1/2 1 14 11V = sup t O J ( 110+8)112 dS)l ±st elf lii(t)l; W is

the space of those

u e V for which u' eX with the norm given

by lu 11w = liuliv -Filu'llx. In these spaces the norm is translation-invariant. This enables us in an intrinsic way to distinguish subspaces * 1°7, li7 consisting of almost periodic elements. For instance, f € X is almost periodic if {f(t + s)} is compact in X. In terms of the above spaces, the estimates (1) and (2) have the form

(4)

Mu liV= iits iiTv= ciliu awn ± Val.

3 The linear problem about averaging 1. We assume that the embedding E c H is compact, and that A(t) is a compact function in Horn (E, E*) with a uniform mean A. Theorem 1. Suppose that the averaged operatorf, is regular, that is, the spectrum of A does not intersect the imaginary axis. Then the operators L„, (w . . coo) are uniformly regular. Additionally, the following assertions hold: (1) The operator L,„ (w.--- coo) realises a homeomorphism W -0 X, and the inverses are uniformly bounded; in particular - -

(5)

iitilly = kliiiwuilx. (2) If f

e Xcomp and f is the uniform mean in E*, then

lill4 alo 1.-lf(tot) — A-lfilv Proof.

(6)

O.

(1) Let C = C(H). First we prove the estimate Mulic =kilLoullx (u E C, Lu

(7)

EX, to

Assuming that this estimate does not hold, we find

vn, gn and con suchta

L,„„vn = g, iivnilc =2, lIgnilx -00

as con -+00.

The linear problem about averaging

187

We choose a tn € 1 for which Ivn(tn)1--.- 1 and put un (t)= v n (t+

ta), ii: n(t) = A(CO n (t ± ta )),

fn(t)=g+.

Then un ' + Aun + Anti = fn. The sequence {un } is bounded in W (the estimate (4)). Therefore, { un } and fun 'l are compact in the sense of weak convergence in SP2(—T, T; E) and SE2(—T, T; E*), respectively. By going over to subsequences if necessary, we can assume the weak convergence Un -0 z, un ' -0 z'. We show that fa = 0 (t e j). To this end we consider the identity

L

(un ',

e) dt + fI (Au, e) dt = —.{J (Antin, e) dt + fI (fa, e) dt, (8)

e(t)

where is a fixed smooth function with compact support. We denote the first term on the right by An and prove that An -0 (it is obvious that the second term tends to zero). We compare An = f ( 14n, An* I

A ,n,z = f (un,

0

) dt with

Pm/L*6 dt.

(9)

I

Since the convergence Piny -0 y (see (3)) is uniform on compact sets in E*, and since A*(t) is a compact function in Hom (E, E*), the limit rnlim ->oo IIAn*e —PmAn*eL= 0

exists uniformly in n and t. Hence (and from the boundedness of Un in V) it follows that 1imm _>,,3 A nm = An uniformly with respect to n. Therefore, it is enough to prove that limn_.., A nm = 0 (m is fixed). We set Bn(t) = fOt ji n(S) ds. Since 1 J. tn wn -Ftwa t Bn(t)= f (A(con(s + tn))— A) ds =— (On tnwn o

(A(s)

from the uniformity of the mean A we obtain ioc

O. IlBn(t)E.*-

By integrating (9) by parts we obtain A,7 = --f

(un', PmBn *e) dt

—

f

(un, PmBn*e) dt.

—

A) ds,

188

The averaging principle for parabolic equations

Since IIP„,13n * (t)iiE-.E —010c 0, it is now obvious that A 1,7 -0 0 as n -0 00. Thus we can take the limit in the identity (8); as a result we obtain Lz = 0 (t ef). We show that z (t) o 0. Since IzIc 2, this contradicts the regularity of L. By Lemma 3 of Chapter 9, the sequence {un } is compact in 2(_ T, T; H). Since we already have weak convergence, un (t) -0 z(t) in .2'2(– T, T; H). Recall that lun (0)I 1. Allowing that z(t)=- 0, from (1) we have lun(0)12 = lun(t)I 2 +c1

ds

0).

-T

The integral on the right-hand side tends to zero (by assumption), therefore iun(t)1=1 provided that n is large enough ; but then iun(t)li 0 in '2(– T, 0) is impossible. This proves (7). 2. The estimate (7) means, in particular, that the operators L„, are strongly correct, since there is a natural embedding M 2(H) M 2(E*) = X. To prove regularity we use Theorem 3 of Chapter 10. For this it is enough to note that the dual operator L* is regular together with L, and that the preceding arguments could be repeated for the operators L*, that is, the L o,*(co coo) are strongly correct. Then the operators L co (co coo) are regular. 3. Now we regard Lo, as an operator W -0 X. From the estimates (7) and (4) we obtain that (10) IPuMw kiaiwu lix. In Chapter 10 we proved assertions of the following type: the range of a correct operator is dense in C, and the range of a strongly correct operator, in M2 . Similarly it can be proved that the range of an operator with the estimate (10) is locally closed in X. Since the range of L Û, already contains C(H) (regularity), it must coincide with X. This proves assertion (1).

4. The operator A E --0 E* is a homeomorphism. In fact, the preceding arguments could be carried out not for L„, but for L, that is, : W -0 X is a homeomorphism. In particular, if f y e E*, then it is clear that x = E-lf does not depend on t, that is, Ax = y, x e E. Now we prove assertion (2). We put Obviously, Las = f (cot) – = g (tot).

The linear problem about averaging

189

Let iteemp stand for the subspace of elements of Xe , having a uniform mean in E*. Since the estimate (5) is already proved, it is sufficient to prove (6) for a set dense in Jteernp; we obtain such a dense set in the following way. For any f E it omp we consider the relation Pmf(t + s) › f(t + s). Since we have convergence in Y2(0, 1; E*) for each t E J (see property (3)), in view of compactness this convergence is uniform with respect to t. Thus, f is approximated by an element in Aleen,p(E). Then by using Steklov averaging in a similar sense we obtain an approximation by an element in C (E) with a uniform mean. Furthermore, by Lemma 2 and Theorem 3 of Chapter 6, we have a best possible approximation by an element of the form 1+ g(t), where FEE, g(t) e C (E) and n(t) = t g(s) ds E -

C (E).

After this we make the substitution u = z + co -i n (wt) in the equation _Lou =g(wt) to give Loz w = co -1A(cot)g(cot).

But then Ilz iv -› 0, and so from (5) it follows that Ile 11 1 v -) O. Theorem 1 is completely proved. Remark 1. If A(t) is an almost periodic function in Horn (E, E*), then in Theorem 1 we can take X., 1°7, IA/ instead of X, V, W. The proof is straightforward and we omit it. Remark 2. We return to Bogolyubov's lemma and discuss when similar arguments can be applied to a parabolic operator. As we have already noted, an averaged regular operator 1: realises a homeomorphism W -› X, which is obviously preserved under perturbations that are small with respect to the norm of Horn (W, X). Let us assume that Â(t) = A(t) — Ã is a completely continuous operator E › E* for every t € J. Then Â(t) can be represented as a sum of a 'small' perturbation and a finite-dimensional operator-function. This enables us to repeat the proof of Bogolyubov's lemma and to find a transformation T for which IT = E+ 8, where 8 is a small perturbation. It can be shown that the converse also holds: if there exists a transformation T with the indicated property, then A(t) —24- is a completely continuous operator E › E*. -

-

4 A non-linear equation By using Theorem 1 the construction of the non-linear theory

is completely trivial; the only point requiring some attention is that in studying the properties of conditional stability it is not possible

190

The averaging principle for parabolic equations

to use the usual apparatus of differential inequalities. We resolve this minor deficiency by some very simple additional analysis of a linear approximation in special spaces. We introduce the spaces XA VA (A 0) consisting of those f € X, u V for which the following are finite: Ilexp (A It pf(t)11x, 11exP (A itpu(t)11v; these are used as norms in XA VA respectively. We also introduce the spaces X +, V +, XA+, VA, which consist of functions on J + and are the analogues of X, V, X,„ VA. Suppose that the parabolic operator L has an inverse L -1 :X > V; we denote its norm by 'co. -

Lemma 1. There is a A 1 >0 such that for A e [0, A l] the operators

exist and their norms are not greater than 21c0. Proof. Let f e XA and u = L -1 f. We put for It1 on(t) = {exp (A1t1) exp (An) for It' n. Then v„ = Onu satisfies Lvn = erfij + On 'u. Therefore, L -1 : XA •

VA

+ A 110nu Ilx}. 11vn 11v ka14411x +110n Since 114nulix liebnuliv, for kA we obtain 110nully ---lkolle/411x. The required inequality is obtained by taking the limit, and Lemma 1 is proved. Next, under the conditions of Theorem 1 the operators La, have the uniform exponential dichotomy on J (see Chapter 10, § 2). Let Nwl(to) and N°2(to) be the subspaces giving this dichotomy, p°1(t 0 ) and P'°2(tØ) be the corresponding projection operators, and c l the constant in conditions (d 3) and (d4) in Chapter 10. We put Ao = min {A 1 , cd, and extend an arbitrary f E X A+0 by zero to the whole line; let u' = Lco-lf. We set by definition Tw +f = uw (t)- U (t, 0)13'1 (0)u (0) (t 0). The fact that we obtain a bounded family of operators T'°+ :X 0 -> VL follows from Lemma 1 and the inequality (11) 11U w (t, 0)a Il v, o+ lia (a eNwi(0)), which in turn is easily derived from condition (d 3) and the a priori estimate (2). Suppose that the norms of and Tw, are not greater than a number k. We consider the non-linear equation u' + F(u, (00=0. (12)

191

A non-linear equation

Let there be a point 1 E E for which f(t)= -F (i, t) is an element of Xcomp with zero uniform mean, and assume that the following conditions hold: (1) The operators F(x, t): E . E* are Fréchet differentiable at i. (2) The operator L = d/dt +A(t), where A (t) = Fx '(i, t), satisfies the conditions of Theorem 1. (3) The operator F(u(t), t) acts from V into X, and is continuously differentiable in some V-neighbourhood of i. Remark. Since there are natural embeddings E c V and E* c X, then condition (1) follows from condition (3), at least when F(x, t)=--(the converse is not true). F(x) Under conditions (1)-(3) we have -

Theorem 2. For co __.- w o equation (12) has a unique solution e E V such that Ile -M -> 0. This solution is conditionally exponentially stable uniformly with respect to the initial value to E J and to co = too. In particular, for any a E Nw1( to) there is a unique solution y'0 (t) (t -- to) for which 13'1(t0) (yw (t 0)- uw (to)) = a, and IV -14w Ilv;--loicil whenever la l--- Po. The constants lo and po do not depend on to and (0.2 Proof. For convenience we take i = 0; we rewrite (12) as

Lu = -F(u, (00 + A(tot)u = The operator Ow satisfies the Lipschitz condition (13) IlO w (ill) - V(u2)IIx,, -(1/2k)Ilu 1 - u2lIvA (A = 0), whenever Will v and Mud v--- -- ro . For A =0 this follows from the identity 1

o 0:(u2+ s(ui - u2))(ui - u 2) ds,

since PIP :(v)IIHoni(v,x) -- 0 as Ilvilv -- 0 (condition (3)). For A >0 we need to multiply this identity by exp (A ItI) and use the fact that the derivative O.' commutes with multiplication by a scalar function. We prove the existence of a solution u' e V. To this end, we consider in the space V the equation u = L„,1 { 0") (u)} = Qw (u). For the operator Qw 2

the Lipschitz constant is -A- provided that I lu II v ---. ro.

We have stated only part of the property is formulated and proved similarly.

of conditional stability; the rest

192

The averaging principle for parabolic equations

Since the first approximation u'i = L 0,-1 {f(cot)} satisfies the condition liu ''''ll v -> 0 (see (6)), we obtain the required solution by iterations. Now we establish conditional stability, assuming for convenience that to = O. To prove the existence of solutions y'°(t) (t- . 0) we consider the equation y = LI' (40)a + T'±{11' (y)}. By making the substitution z = y u`" we obtain z = W (t , 0)a + Tw +fir' (it') + z) Ow (uw)} = D' (z). Let us estimate the Lipschitz constant of Da' in VA . . Since lielly -> 0, it follows from (13) that this constant is not greater than 1 provided liz liv, o+ --- r i . Therefore the iteration process converges if the first approximation z a'l =13' (e°) = 13``) (0) = LI' (t,O)a is sufficiently small in VA., that is, if lai ---. po (see (11)). Here the fixed point (as is always the case in the contraction mapping principle) satisfies the estimate -

—

—

llz iliT 0 + = lie — z °IVA 0 + 2 1IZ wi — Z livx 0+ 21Iz c°1 11V).0+ --- 101a l• This proves Theorem 2. Unfortunately, the actual range of application of Theorem 4 is quite narrow; the situation is that the differentiability condition (condition 3) is delicate. It is not difficult to give the cases (relative to non-linear parabolic equations) when this condition automatically does not hold. For this it is useful to bear in mind the following observations. 1°. The operator F :B1 -> B2 (assumed to be independent of t and such that F(0) = 0) generates a differentiable mapping Y2 (0, 1; B1) -4 2(0,1; B2 ) only when it is linear. Hence we obtain the following. Y Suppose that F :E -> E* is a non-linear operator. Then we can establish its differentiability as an operator V .- X only at the expense of requiring that the term Ilu ic occurs in the expression for lu iv. 2°. The superposition operator G(0(x)) gives a differentiable mapping .cf2(f2) .--> 22 (12) only when G is a linear function. 3 We denote by H(f2) (s = 0, 1, . . .) the Sobolev space of functions u(x) with the norm A

J

E lu (i)(x) 2 dx,

f2 i ..-s

where u (i) denotes the derivative of order j and the summation is over all derivatives. We consider the Dirichlet problem for a second3

Here and in what follows 12 is a finite domain in the Euclidean space of variables x = (x 1 ,. . . , xm).

A non-linear equation

193

order equation. Then H = Ho =Y2(12), E = A i (Pi is the closure in H1 of the set of infinitely smooth functions with compact supports), and typical non-linearities are given by expressions of the form a aG G(ux), — (u), G(u). ax ax We consider the differentiability of these operators. The first cannot be differentiable (not being linear) even as an operator E .- E*; this follows easily from 2°. The second can define a differentiable mapping E .- E*, but not V .- X. A rigorous proof is fairly lengthy, but it is easy to construct examples of non-differentiable operators with an infinitely smooth G with compact support. There remain the operators aG(u)lax and G(u). We discuss the operator aG(u)lax in detail. Suppose that IG'(u )1 --._ /{1 + lu I A } (A ..-. 0). Then it can be proved that: 3°. The operator aG(u)lax :E -> E* exists and is continuously differentiable for m .---. 3 if A --. 2/(m -2), and for m = 1, 2 if A H. By means of a projection, equation (14) can be written as

u'+F(u)=Pf=g. Suppose that the following conditions are fulfilled: (a) If m =2, then f E M 2comp(E * ); .1 is its uniform mean. (b) If m --_ 3, then f E M 2compVe2(f2)), 1 is its uniform mean. It is known that the stationary equation F(u)= g has at least one solution 6 E E.

194

The averaging principle for parabolic equations

(c) The operator A = F1 u (6) (A :E -> E* is strongly elliptic; see property 3°) does not have a spectrum on the imaginary axis. From Theorem 2 with due regard for the important property 4°

we obtain Proposition 1. For m = 2 equation (14) has a unique solution le for which 1 sup ilu — 1 1 H + sup iiuw (t + s)— II 1 ds O. tEl

tEl

JO

This solution is conditionally exponentially stable in H. The case m =3 requires a modification of the energy method since property 4° does not hold for A = 1 and m =3. It is assumed that fl is sufficiently smooth; then it is known that 1.1 e H2(0).

Proposition 2. For m which

3 equation (14) has a unique solution is' for

. sup jiiu(t sup Ile — fiiiiii(n)+ teJ

tEl

0

€1 11112(.0)

ds -> O.

This solution is conditionally exponentially stable in E. The proof of Proposition 2 imitates the main arguments of the energy method, but instead of a triple E, H, E* we must take the triple H2, E, H. Let us note that for the equation u' +Au = g we have the estimate t0

(011E2

to

+1

(IIU 111/2(0)2 1114 11H2 ) ds t0

c

11612 d.s1

(01E2 L

(to t

to + 1),

t0

which is completely analogous to the estimates (1) and (2) (see Ladyzhenskaya [68], Russian p. 127). By proceeding as in Theorem 1, from condition (c) we find that L = d/ dt +A has an inverse L -1 : X -> V, where the spaces X and V differ from those introduced earlier by replacing the triple E, H, E* by H2(f2), E, H. It remains to show that the operator F acts from V into X and is continuously differentiable. Since the basic classes of functions are preserved under the projection operator P, it suffices to solve this problem for the operator l a(u)2/ax = uux = 0(u).

The Navier—Stokes equation

195

We set Q = [0,1] x i? and 'lull, = lul dx) 1/ P. By Holder's inequality and Sobolev's embedding theorem we have

L

dt,

Ivux 12 dx dt j.

lu 114

k11/2 k (f1 ),

f,Ivux12

lluXll4

k iits162(n);

dx dt ki sup Iv iii(n)

iluil 2H2(n) dt, (15)

It follows from (15) that 0 acts from V into X; it also follows that = vxh + vh

and that 0'.(v) is continuous with respect to u e V. For this it suffices to consider the relations 0(u + h)— 0(u)— u xh — h xu = hhx, 01 .(z)h — 0' u (v)h = (zx — v x )h +(z — v)hx, and to apply an inequality of the form (15) to them. Here we can assume that Proposition 2 is proved. The case of a zero mean (1= 0) is especially interesting. Here =0 and A= —P4 =-4 , and consequently condition (c) holds

automatically. 6

The problem on the whole space The method developed in § 3 is directly applicable only to

boundary-value problems in a finite domain (the requirement that the embedding E c H is compact). A careful analysis shows that the problem in the whole space, in a reasonable formulation, can be studied by close methods. The 'reasonable formulation' requirement concerns above all the choice of the function spaces in which the operator L„, is studied; these spaces must firstly be similar to the corresponding spaces for boundary-value problems (the spaces V, X, W), and secondly have a norm invariant under translations not only with respect to t but also x. In brief, for the whole space we need to establish the complete analogue of Theorem 1 (with precise estimates for Lc° that are uniform with respect to co). However, this question will not be dealt with fully here. Instead we consider the more transparent averaging problem for second-order equations in the H61der classes. We give the main attention to the new details that arise, which are generally typical for problems in the whole

196

The averaging principle for parabolic equations

space. These details are connected with certain complications in the regularity problem. We consider the uniformly parabolic operator L = cl/ dt + A(t), where -

A(t)u =

a (ai;(t, x)u,+ ai (t, x)u) + bi (t, x)u, + ao(t, x)u. axi

—

Let D= Jxffm, C =C (D) be the space of continuous functions u(z)=u(t, x):D -> 17' with the sup norm, and C" 2(D) be the set of u(t, x) E C (D) for which the following norm is finite: iitt licc..-/ 2(D) = iiu ilc + +

sup

lu(t,x)-u(t,x')I

(t,x),(c.'),D

Ix - xr

its(t,x) - u(t i ,x)1

sup

It — t f la/2

(t,x),(e,x)eD

•

Suppose that the following conditions hold: (1) aii, bi, ai, a, aaiilax, ab/ax E CY' 7/2 (D) (y E (0, 1)). (2) The uniform means eiti, 56 di, and d o exist and are attained uniformly with respect to x E R m. It then follows that the coefficients of the averaged operator have the same smoothness properties as in the unaveraged case, that is,

_

ei , bi, aciii/ax, abi/ ax E Cy (Rm). •

By analogy with the finite domain case, we introduce the space V consisting of functions u(z) (z e D) with the norm

I

l

liti V = (i. 11,1(4 2 + 1 —ai

2

1/2 CIZ) ,

I aX 1 Q where Q is the unit cube in D and the sup is taken over the whole unit cube. By a solution of an equation of the form Lu =f (Je C (D)) we mean a u (z) E V satisfying the identity jr, u(z )L* 0(z) dz = f f(z)0(z) dz, where L* is the formally adjoint operator and 0 e Cô° (D). The result that u(z) and the derivative u(z) belong to the class C = C (D) is obtained from the following important estimates:

Ilul l c 1 1 11tilly + lifIld

Iltilly --- 1111u Ilc + liflIc}

(The maximum principle; see Ladyzhenskaya, Solonnikov & Urartseva [67], Russian p. 225, English tr. p. 181) (The energy estimate [67], Russian p. 170, English tr. p. 139)

The problem on the whole space

197

IlUX liCe6a/2 1 11114 11C +

(A Nash type estimate [67], Russian p. 246, English tr. p. 219) We emphasise that these estimates remain uniform if we introduce a parameter co in the operator A(t). Theorem 3. Suppose that the averaged operator is regular, that is, Lu =1 has a unique solution u E C for any f E C. Then for co (00 the operators L„, are uniformly regular with the estimate (16) k11L.ulic (co wo, a E (0, 1)). Then if/ =1(x) is the uniform mean of f(t,x), attained uniformly with respect to x e Rm, and = f( cot, x), A =] then

x

xiic +iiu w - eiiic

(17)

0.

We preface the proof with several auxiliary propositions.

Proposition 3. If L is weakly regular, then L*u = 0 has no solutions in C. Proof. Let L*g =0 with g e C. Since L is weakly regular, Lu = g for some u E C. Consider a smooth function (/)(z) with compact support such that cP(z) = 1 for 1z1-._ 1. Then u8 (z) = 0(ez)u satisfies Lu e = .rb(ez)g +ne (z). Since u, ux E C (D), from the readily obtained explicit expression for n it follows that

177 8 1--- cols IcYz(sz)1+ 6 2 10"zz(ez)1}. Now by Green's formula we have

0

= fD

(gLu e - ueL* g) dz =

gne dz.

g2cfi (ez) dz + D

(18)

D

Because SD ni dz = o (1), by taking the limit as e -0 0 we see that SD g 2 dz co. Suppose that fo e Co but fo it R (L). Then we can find a functional g e C*(D) orthogonal to a subspace of 92(L) and such that (g, fo) 0. The restriction of g to Co(D) defines a non-trivial finite measure tz. Therefore, we have ID Lcf) d,u, = 0 (0 e Cô (D)). This means that the measure it is a generalised solution of L* u = 0. From the hypoellipticity properties of a parabolic operator (see Eidel'man [118], pp. 213-14, English tr., pp. 218-20), it follows that L*u =0 has a nontrivial sufficiently smooth solution u(z) belonging to 21(D). This solution lies in C(D) since sup lu(z)1 ---c f lu(z)l dz, Q(1) Q(2)

where Q(1) and Q(2) are arbitrary concentric cubes with sides of length 1 and 2, respectively (see, for example, Eiderman [118], pp. 215-16, English tr. pp. 221-23). This proves Proposition 4. Now we introduce all possible operators Eh whose coefficients are obtained from f, by a translation followed by a local passage to the limit with respect to x e Rm.

Proposition 5. The operators Lk are weakly regular. Proof. Let Eh be obtained from f.,.. by a limiting translation with respect to fxkl c Rm. Let f E C (D), fi = f(t, x — x k ), and Lu k = fk . We put v k = uk (t, x - x k ). It follows from Nash's estimate that fvk l is locally compact in C(D). By taking the limit we obtain that Ehv = f, v e C(D) (as yet we have not established that the solution is unique). Proposition 5 is proved.

Proposition 6. The operators Lk* are correct. Proof. For simplicity of notation we assume that we are speaking of the operator E. Assuming that it is not correct we obtain sequences {un } , UnlcC(D) such that Escur, =f, IlunlIc = 1 and IlfnlIc -*0. We select a zn E D such that lu (zn )1 ...- 4 and put vn = un (z +z). Since {v} is compact in C(D), by taking the limit we obtain a non-trivial locay solution of an equation of the form Eh*u =0, which contradicts Propositions 5 and 3. From Propositions 3-6 it follows immediately that the operators fh and Eh * are regular; in other words, the conditions of Theorem 3 are in fact symmetric in L and L*.

The problem on the whole space

199

Now we briefly outline a proof of Theorem 3. First we must show that the L„, (co .--. coo) are uniformly correct. Assuming otherwise we have a sequence w a --> co and sequences lual, {fn } c C such that Lu n = fn, Ilunlic = 1, Ilfnlic --* 0. We take a point za such that lua (zn )1 = ta )--- 1- and put va (z ) = tia(z +z). It is important that {v a } is locally compact in C (a Nash type estimate); we may assume that v a --> io. 6. Now we must realise a passage to the limit similar to that in Theorem 1. A small feature is that as well as the main averaging procedure, in this case we have a passage to the limit with respect to x. Therefore, /5 is a solution not necessarily of Lu = 0 but of some equation rhu = 0 • 4 But this contradicts the regularity of L. The uniform correctness of L.* (co> coo) is proved in exactly the same way. But then the regularity of L a, follows directly from Proposition 4. The proof of the remaining assertions of the theorem should not present difficulty. In conclusion we mention a most commonly used sufficient condition for the regularity of L: ão(x)0 and every doh(x) * O. This condition is fulfilled if, for example, 'do ---. 0, ao € 6(R m) and do * 0.

Comments and references to the literature Bogolyubov's lemma is proved in his book [8]. A very extensive bibliography on the averaging method is given in the monograph of Mitropol'skii [90]. The contents of Chapter 11 are mainly from Zhikov's article [56]. The results in § 6 were obtained by Zhikov jointly with L. Tsend and M. Otel'baev (unpublished). A technically different approach to averaging in parabolic problems which has been developed by Simonenko [101] must be mentioned.

4

At this point it is important that the mean values of the coefficients are uniform with respect to x e R.

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Additional references Sell, G. R. (1973). Almost periodic solutions of linear partial differential equations, Journal of Mathematical Analysis and Applications, 42, 302 12. -

Fink, A. M. (1974). Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377, Berlin-New York, Springer-Verlag. Brom, J. (1977). The theory of almost periodic functions in constructive mathematics, Pacific Journal of Mathematics, 70, 67 81. Shubin, M. A. (1978). Almost periodic functions and partial differential operators, Uspekhi Matematicheskykh Nauk, 32 (2), 3 47 (Russian -

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Mathematical Surveys, 33 (2), 1 52). Kozlov, S. M. (1978). Homogenization of differential operators with almost periodic rapidly oscillating coefficients, Matematicheskii Sbornik, 107, 199 217 (Mathematics of the USSR Sbomik, 35,481 98). Zhikov, V. V. (1979). A pointwise stabilization criterion for second order parabolic equations with almost periodic coefficients, Matematicheskii Sbornik, 38, Sbornik, 110, 309 18 ((1980) Mathematics of the USSR 279-92). -

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Zhikov, V. V., Kozlov, S. M. & Oleinik, O. A. (1982). Homogenization of parabolic operators with almost periodic coefficients, Matematicheskii Sbornik, 117, 69-85. Kozlov, S. M., Oleinik, O. A. & Zhikov, V. V. (1981). Sur l'homogénéisation d'opérateurs différentiels paraboliques a coefficients presque périodiques, Comptes Rendus des Séances de l'Académie de Sciences, Paris, 293, Series 1, no. 4,245-8. Zhikov, V. V. & Sirazhudinov, M. M. (1981). Homogenization of nondivergent second order elliptic and parabolic operators, and the stabilization of the solution of the Cauchy problem, Matematicheskii Sbomik, 116, 166-86.

Index

admissible pair, 167 almost automorphic function, 63 almost period E-almost period, 1 E,N-almost period, 53 almost periodic function, 1, 60, 71, 73, 75, 79, 84, 99, 100, 111, 115, 124, 144 almost periodic solution, 109, 113, 115, 117, 121, 126,.138, 146, 147, 148, 163, 164, 165 Amerio, L., 36, 76, 96, 147 Amerio, L. & Prouse, G., 76, 96, 147 approximation theorem, 17, 36 for N-almost periodic functions, 60 for Stepanov almost periodic functions, 36

basic semigroup, 100 basis finite, 26 integer, 26 rational, 26, 69 Baskakov, A. G., 63, 97 Besicovitch almost periodic function, 142 Besicovitch almost periodic solution, 143, 163 Birkhoff minimal set, 9, 131 Bochner, S., 13, 33, 36, 52, 62, 96, 148 Bochner, S. & von Neumann, J., 148 Bochner's criterion, 4, 6, 42, 70, 73 Bochner-Fejer composite kernel, 28 Bochner-Fejer kernel of order n, 28 Bochner-Fejer polynomial, 29, 36, 69, 70 Bochner-Fejer sum, 25 Bochner-Fejer summation method for weakly almost periodic functions, 69

Bogolyubov, N. N., 15, 36, 199 Bogolyubov, N. N. & Krylov, N. M., 36 Bogolyubov's lemma, 182, 189, 199 Bohl, P., 12, 96 Bohr, H., 12, 52, 96 Bohr, H. & Neugebauer, 0., 97 Bohr's example, 100, 144 Bohr transformation, 23 of weakly almost periodic functions, 69 Boles Basit, R., 63, 97 Boles Basit, R. & Tsend, L., 97 bounded almost periodic function, 70, 75 bounded solutions, 77, 92, 93, 94, 118, 120, 157, 160, 174, 177, 178 Brodskii, M. S. & Mil'man, D. P., 13 Bronshtein, I. U., 97, 123 Bronshtein, I. U. slir Chernyi, B. F. 0., 123 Cheresiz, V. M., 123 coercive estimate, 156 coercive inequality, 184 coercive property, 151 compactness condition, 160, 1.73 compactness lemma, 137 compactness of a set of almost periodic functions, 7 compact solution, 107, 140 condition for almost periodicity, 70, 91, 92, 97 condition of right solvability, 166 condition for uniform positive stability, 138 condition of weak continuity, 124 conditional exponential stability, 191 conditionally periodic function, 12 contraction mapping principle, 153, 192

Index convolution, 16 criterion for almost periodicity, 70, 91 criterion for N-almost periodicity, 58 criterion for a point spectrum, 139 Daletskii, Yu, L. & Krein, M. G., 179, 181 demicontinuity, 150

derivative of an almost periodic function, 3 deviation from an almost periodic function, 31 duality, 154 Dunford, N. & Schwartz, J. T., 13 dynamical system (flow), 8 Ellis, R., 123 Ellis semigroup, 101, 104 equi-almost periodic set, 7 equitransitive system, 114, 140 Esclangon, E., 12 Esclangon's lemma, 97 exponential dichotomy, 167, 168, 169 extension completely non-distal, 132 distal, 98 of a minimal flow, 98 positively stable, 106 extreme point, 131 evolution equation, 99, 157, 160, 161, 164 Favard, J., 52, 97, 147, 148 Favard minimal set, 130, 131, 134, 135 Fejer sum, 25 Fejer kernel, 28 fibre, 99 distal, 99, 104, 105 minimal, 116 positively stable, 132 semidistal, 99 Flor, P., 123 flow, 9 compact, 98 distal, 98, 105 equicontinuous, 8 semidistal, 98 strictly ergodic, 101 two-sidely stable, 8

Fourier coefficients of almost periodic function, 24 of N-almost periodic function, 60 of Stepanov almost periodic function, 35

minimal property of, 32 Fourier exponent, 24, 32, 40, 41, 50 Fourier-Bohr exponent, 42, 43

209 Fourier series of almost periodic function, 24 of N-almost periodic function, 59, 60 Fourier-Stieltjes integral, 16, 36, 56 Fourier-Stieltjes transform, 14 Fourier transform, 87 Furstenberg, H., 123 Gel'fand, I. M., 148 Gorin, E. A., 52 Gottschalk, W. A. & Hedlund, G. A., 114, 123

harmonic analysis of weakly almost periodic functions, 68 homogeneous space, 118 hyperbolic equation, 140, 148, 165 identical invariance, 107 indefinite integral, 77, 81, 138 invariant section, 142, 143 Jessen, B., 52 Jessen, B. (Sr Tornehave, H., 52 Kadets, M. I., 74, 96, 133, 148 Krasnosel'skii, M. A., Burd, B. Sh. & Kolesov, Yu, S., 181 Ladyzhenskaya, O. A., 194 Ladyzhenskaya, O. A., Solonnikov, V. A. Sr Uranseva, N. N., 196 Lax, P. D. & Phillips, R. S., 148 Levin, B. Ya., 62 Levin, B. Ya, & Levitan, B. M., 62 Levitan, B. M., 36, 52, 62, 76, 97, 144 limit periodic function, 46 linearly independent set, 26 Lions, J.-L., 165 local convergence, 87 Loomis, L. H., 97 Lyapunov stable semigroup, 10 Lyapunov solution, 100 Lyapunov trajectory, 13 Maizel', A. D., 181 Marchenko, V. A., 62 Massera, L. L. & Schaffer, I. I., 178, 181 mean value, 22 mean-value theorem, 22

measure of oscillation of an operatorfunction, 179 Millionshchikov, V. M., 123 minimax method, 127, 135, 147 Mitropolskii, Yu. A., 199

210

Index

module, 59 rational hull ?rat, 115 Montgomery, D. & Samelson, H., 118 Muckenhoupt, C. F., 13 Mukhamadiev, E., 181

Sell, G. R., 123 semigroup

continuous, 11 distal, 98 Lyapunov stable, 10 semidistal, 98 semiseparated solution, 100 semiseparation condition, 127, 134, 147 semitrajectory, 9 separated solution, 100 separation lemma, 102, 105

N-almost periodic function, 54, 57, 58, 63, 84, 111, 112, 146 N-almost periodic solution, 127 Navier-Stokes equation, 193 Nemytskii, V. V. & Stepanov, V. V., 13, 123

sequence

operator averaged, 182 coercive, 151, 152, 184 correct, 167, 169 evolution, 154

set

monodromy, 178 monotonic, 149 non-expansive, 152 parabolic, 183, 189, 190, 196 regular, 167 semicontinuous, 149 strongly correct, 167 strongly elliptic, 184 uniformly regular, 172 weakly regular, 167 orbit, 116 Parsevars relation, 31, 32, 36 Pelczynski, A., 96 Perov, A. I. & Ta Kuang Khai, 148 Pérron, 0., 181

point of almost periodicity, 91 of non-almost periodicity, 91 regular, 87 stationary, 117 point spectrum of homogeneous problems, 143 P-property, 173

Poisson stable function, 80, 86, 97, 173, 185 Pontryagin, L. S., 118

positive definite function, 17, 19, 55 principle of the stationary point, 116, 117, 121 quasiperiodic function, 12

f-increasing, 42 f-normal, 42 f-returning, 80 returning, 57, 78, 80, 81 weakly convergent, 64 equi-almost periodic, 7 A-invariant, 116 invariant, 9 minimal, 9, 130 relatively dense, 1, 86 Shcherbakov, B. A., 96, 97, 123 Simonenko, N. B., 199 space co, 81 C(X), 4 6(X), 4 H8 (0), 193 Y, 1, 87 2'' (X), 33 21) (-00, co; B), 166 lte(X), 34 Ye(X), 34 .et, 157 Xcomp, 157 komp 190 X, 190 not containing c o, 82 ,

reflexive embedded, 183 weakly complete, 64 spectrum, 24, 87, 143 spectral condition for almost periodicity, 91 stationary equation, 161 stationary subgroup, 118 Stepanov, V. V., 33, 36 Stepanov almost periodic function, 33, 34, 36, 80

strictly convex norm, 121 rarified subset, 92 rational hull Erat, 115 recurrent function, 12, 80, 84 Reich, A., 63 resolvent, 93

theorem Amerio, 109, 111, 112 Birkhoff, 9, 102, 125, 139 Bochner-Khinchin, 17, 19, 36

Index theorem-continued Bogolyubov, 13, 41, 55, 62, 97 Bohl-Bohr-Amerio, 77, 80 Ellis, 104, 116 Favard, 62, 109 Kronecker, 37, 39, 46, 63, 66, 71, 115 Kronecker-Weyl, 51, 52, 144 Levitan, 109 Lyusternik, 7, 13 Markov, 10, 13, 136, 141 Zhikov, 110 theorem of the argument, 49, 101

trajectory absolutely recurrent, 10, 106 compact, 8, 20, 106, 137 compatibly recurrent, 9 of a group, 8 minimal closure of, 8 recurrent, 9, 102, 115 of a semigroup, 9 semiseparated, 100, 103, 106

separated, 100 weakly recurrent, 137 unconditionally bounded series, 82 uniform exponential dichotomy, 167 uniform positive stability, 108, 135 uniformly convex space, 74 uniformly exponentially stable to the right, 170 uniformly stable to the right, 170

211 uniqueness theorem for almost periodic functions, 24 for N-almost periodic functions, 62 two sided, 108 Valikov, K. V., 148 Veech, W. A., 63 V-monotonic equation, 107, 121, 123

Wallace, A. D., 123 weak convergence, 64 weak limit, 64 weakly almost periodic function, 65, 66, 68, 69, 70, 71, 72, 75, 81, 126

weakly almost periodic solution, 127, 129, 131, 134, 140

weakly compact solution, 125, 130, 143 weakly N-almost periodic solution, 126, 127, 128, 134, 135

weakly recurrent solution, 139, 140, 141

Wiener, N., 36 Wintner, A., 52 Wolff, F., 97 Zhikov, V. V. , 62, 76, 97, 123, 148, 165, 181, 199 Zhikov, V. V. & Levitan, B. M., 165 Zhikov, V. V . & Tyurin, V. M., 181 Zhikov, V. V. & Valikov, K. V., 181