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))
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 <8, then
1/n

t + — f(t)] — f(t)

n

o

[ft (t+n) — .f 411 dri

1/n

l[f (t +ri)—f(t)11 dri Consequently, the sequence of almost periodic functions On (t)=

n[f(t +1/n) —f(t)] converges uniformly on J to f(t). Now we only
need to use Property 3.

2 Bochner's criterion
The main results of this section are also valid for almost
periodic functions with values in an arbitrary metric space X. But
for simplicity we shall assume that X is a Banach space. We shall
use the following notation:
X denotes a complex Banach space; x, y, z, ... are elements of X,
and iix11 is the norm of x EX. C(X) denotes the Banach space of
continuous bounded functions f: J --2Y with the norm
iifit)ilc(x) = sup
teJ

and 0(X) is the subspace of C(X) consisting of almost periodic
functions. Let us note that the spaces C(X) and O(X) are invariant
under translations, that is, C(X) (0(X)) contains together with
f = f(s) the function f (s)= f(s +t) for all t E J.
1. Bochner's theorem. Let f:J -+X be a continuous function. For f
to be almost periodic it is necessary and sufficient that the family
of functions H = = {fit + h)}, —co < h Proof. (a) Necessity. We assume that f is an almost periodic function (see § 1, Definition 3). We denote by {r} the set of all rational
points on J and let If hnl= {fit + hn )} be an arbitrary sequence of
functions from H. By using Property 1 and applying the diagonal
process, we can select from the sequence {Pt + hn )} a subsequence
(we denote it again by {f(t + hn)}) which converges for any r E {r}.
We prove that the sequence {fit + hn )} converges in C(X). We take
an arbitrary E >0 and let 1= l be the corresponding length. Let

Bochner' s criterion

5

8 = 8(s) be chosen in accordance with Property 2. We subdivide the
segment [0, 1] into p segments ilk (k = 1, 2, .. . , p) of length not
greater than 8, and in each Ai( we choose a rational point rk. Suppose
that n = nE is chosen so that

lif(rk + hn) — f(rk + hm)ii < 6

(4)

for n, m._.-. nE and k = 1, 2, . . . , p. For every to e J we find a r = ro such
that

0.---to +r--.5/

Suppose that the number t'o = to + r falls in the interval 41(0 and that
rko E Ak o is the rational point chosen earlier. Then by our choice of
8 we have
lifle o + hn) — firk o + hn)li< 6,
ilf(t 10+ hm) — firk o + h„,)il
(5)

It follows from (4) and (5) that

lif(to + hn) — f(to+ h. )1I

=ii.f.(to+ hn) — Pt'o + hn)II±ii.f(t1 o + hn) — Erk o + h,, )II
+11f(rko+hn)—f(rko + LAI ±iif(ko+ hm) — Re 0 + hm)li
+Mt'o+ h.) — fit° + LAI <5E.
Since to E j was chosen arbitrarily, the last inequality implies that
the sequence {f(t + hn)} converges in C(X), that is, the set H is
compact in C (X).
(b) Sufficiency. We assume that the family {f(t + h )}, —oo < h is compact in C(X) and prove that f(t) is almost periodic (in the
sense of Definition 3, § 1). First of all we show that f is a bounded
function. For if this were not the case, then we could find a sequence
of numbers hn for which Ilf(hn)11-*co. But then neither the sequence
If (t + hn)} nor any subsequence of it would be convergent at t = 0.
From the boundedness of f it follows that the family of functions
{P } = { f(t + h)}, —co < h By a criterion of Hausdorff, for all E >0 there are numbers
h l , h2, . . . , h„ such that for all h € J. there is a k = k(h) such that
sup Ilf(t+h)—f(t + hk)ii< E.
teJ

From (6) we have
sup lif(t + h - hk) - f(01 <
teJ

E,

(6)


6

Almost periodic functions in metric spaces

that is, the numbers h — h k (h) (k =1, 2, ... , p) are E-almost periods
for f(t). Now we only need to prove that the set of numbers h — hk
is relatively dense. We set

L= max Ihki.
1--.1c--çp

Then

h—L----5h—h k -h+L,

and since h is arbitrary this inequality implies that every interval
of length 2L contains an 6-almost period for f.
2. Now we are going to deduce further properties of almost periodic
functions that are obtained more simply from Bochner's criterion
than from our definition.

Property 6. The sum f(t)+g(t) of two almost periodic functions is
almost periodic. The product of an almost periodic function f(t) and
a numerical almost periodic function 0 (t) is almost periodic.
Proof. Let hn } be an arbitrary sequence of real numbers. Firstly
we extract from it a subsequence {h' n } such that the sequence of
functions If ( t + h'n )} converges, and then a subsequence { h"n } of h'n}
for which the subsequence of functions {g(t + h"n )} is convergent.
Then, clearly, the subsequence { f(t + h"n )+g(t + h"n )} is convergent.
Similarly, the product can be proved to be an almost periodic
{

{

function.

Let X 1 , X2,. . . , Xn be Banach spaces, and let X = nkXk be their
cartesian product, that is, the Banach space with elements x =
(x 1 , x2,. . . , Xn) and the norm
11x11 = kil l lixkil.
It follows easily from Bochner's criterion that if fi (t), f2(t), . . . , fn (t)
are almost periodic functions from J into X 1 , X2,. . . , Xn, then the
function f(t)= (Mt), f2(t), . . . , fn (t)) is an almost periodic function
from J into X. The next property is easily deduced from this remark.

Property 7. Let fi(t), f2(t), . . . , fn(t) be almost periodic functions from
J into Banach spaces X 1 , X2, . . . , X n, respectively. Then for every
s > 0, all the functions Mt), f2(t), . . . , fn (t) have a common relatively
dense set of E -almost periods.


Bochner's criterion
Proof. Suppose that
fn(t)), that is,
f2(t),
.

.

.

T

7

is an E-almost period for f(t)= (Mt),

,

ilf(t+T) — Pt)lix ==k1
i lifk(t+7)—fk(t)liXk
=
for all t € J. Obviously, for this

lifk(t +7) — fk(t)ii< E
as we required to prove.

T

<6

we have

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

3. The next property gives a condition for the compactness of a set
of functions from 0(X), and is known as Lyusternik's theorem.
Lyusternik's theorem. A set M c 0(X) is compact if and only if the

following three conditions are satisfied:
(1) For every fixed to E J the set
Eto = {x eX: x = f(to)JEM}C X

is compact.
(2) The set M is equicontinuous, that is, for every £ > 0 there is a
8 = 8(e) such that lif(e)— f(r)ii(3) The set M is equi-almost periodic, that is, for every 6 > 0 there
is an 1 = lE such that every interval (a, a + 1)c f contains a common
e-almost period for all f € M.
Proof. (a) Sufficiency. The proof is exactly the same as that of the
necessity for the conditions in Bochner's theorem.
(b) Necessity. By the criterion of Hausdorff, for every e >0 M
contains a finite e-net: fi , f2, ... , fn. Therefore, for all f € M there is
a ko, 1 ---_ ko ---- n, such that
sup ilf(t)—fko(t)II < B.
tei

(7)

For any to e J, from (7) we obtain

Ilf(to)—fko(to)11< 8,
and so the finite set of elements fi(to), f2(t0), ... , fn(to) forms a finite
e-net for the set Etc,. Consequently, Etc, is compact in X, that is,
condition (1) of Lyusternik's theorem holds. Condition (2) follows
from the uniform continuity of each fk (t) (k = 1, 2, ... , n) on J and
from (7). Finally, condition (3) follows from (7) and Property 7.
Remark. For numerical almost periodic functions, condition (1) of
Lyusternik's theorem can be restated as follows: the set Eto is
bounded.


Almost periodic functions in metric spaces

8

3.

The connection with stable dynamical systems

Suppose that we are given a 1-parameter group of homeomorphisms of a metric space X, S(t) : X -› X (t E J). If for any x E X
the corresponding trajectory x t = S(t)x is a continuous function J -+ X
we shall call S(t) a dynamical system or flow.
A flow S(t) is called two -sidedly stable or equicontinuous if the
transformations S(t) (t EJ) are equicontinuous on every compact
set from X.
The next property is obtained from Bochner's criterion.

Property 8. Every compact trajectory of a two -sidedly stable flow is
an almost periodic function.
Proof. We set f(t)= S(t)x. Since a trajectory is compact, we can
extract from any sequence If(tn )1 a fundamental subsequence {Min».
The transformations S(t) are equicontinuous on the set f, and so
sup P(.f(t + t i .), Pt + en ))

B

teJ

whenever p(f(e.), fit' n))----5 8, that is, Bochner's criterion holds.
The converse holds in a certain sense: with each almost periodic
function f :J - - > X can be associated a compact trajectory of a twosidedly stable dynamical system. For if we consider in C(X) a system
of translates, then the trajectory ft = f(s +t) is compact. Since the

distance between two elements of C(X) is invariant under a translation, we have an isometric and so two-sidedly stable flow. It is worth
noting that the difference between isometry and two-sided stability
is essentially insignificant; if a two-sidedly stable flow is defined on
a compact space X, then it can be made isometric by choosing the
following metric

d(xl, x2) = sup p(S(t)xi, S(t)x2).
teJ

It is easy to see that the metric d is invariant under translation and
topologically equivalent to the original metric p.
Letfj Xbe an almost periodic function. We denote by k = Alf)
the closure of the trajectory f = f(s + t) in C(X), and are going to
show that ge is minimal in the sense that any trajectory is everywhere
dense in it. Suppose that f = f(s) is any element from .9e. Then for
some sequence Itni l c J we have
-

A A

sup p(f(s + tin ), i" (s)).--s. 11 m.
seJ


9

Stable dynamical systems

Therefore,
A

sup p(f(s), f(s
S

eJ

A

that is, f(s — t tn)-+ f(s) uniformly with respect to s El. The closure of
the trajectory f t contains f, and so it coincides with Ye.
4 Recurrence
The minimal property of an almost periodic function proved
in the last section is in fact a very simple property of abstract
trajectories.

1. Let X be a Hausdorff topological space.
We shall call a 1-parameter semigroup of continuous operators
S(t) : X - X (t 0) simply a semigroup, and shall use the symbols x t,
x(t) to denote the semitrajectory S(t)x (x EX t 0). A function x(t)
is called a trajectory of a semigroup S(t) if x(t + r) (t
is a
semitrajectory for every T E l. A set Xo c X is called invariant if
through each of its points passes at least one trajectory that is entirely
contained in Xo. An example of a closed invariant set is the closure
of a trajectory.
A set X0 c X is called minimal if it is closed, invariant, and does
not contain proper closed invariant subsets.
,

Birkhoff's theorem. If a semigroup has a compact semitrajectory,

then there exists a compact minimal set.
Proof. Let X1 denote the closure of a compact semitrajectory.

Obviously, the set nt_,0s(t)x,. is compact and invariant. We order
the compact invariant sets by inclusion and apply Zorn's lemma,
thus proving the existence of a minimal compact invariant set.
The trajectories that belong to a compact minimal set are conventionally called recurrent (in the sense of Birkhoff); an example of a
recurrent trajectory is an almost periodic trajectory.
2. Suppose that we are given two semigroups defined on X and Y,
respectively. Then there is an obvious semigroup on the cartesian
product X x Y (the `semigroup product').
Two trajectories x(t), y(t) are called compatibly recurrent if the
trajectory {x (t), y(t)} is recurrent in X x Y. Clearly, compatible recurrence implies the recurrence of each component, but the converse
does not hold.


10

Almost periodic functions in metric spaces

We say that a trajectory is absolutely recurrent if it is compatibly
recurrent with any recurrent trajectory. In Chapter 7 we prove that
an almost periodic trajectory is absolutely recurrent.
5 A theorem of A. A. Markov
We consider a semigroup S(t) (t---- 0) on a complete metric
space X, and call S(t) Lyapunov stable if the transformations S(t)
(t..?- 0) are equicontinuous on every compact set from X.
Markov's theorem. The restriction of a Lyapunov stable semigroup
to a compact invariant subset is a two -sidedly stable group. In
particular, every continuous compact trajectory is almost periodic.

Proof. Let X be a compact invariant subset. We introduce on X the

equivalent metric
d(x i , x2) = sup p(xi t, x2 t ),
to

which has the property d(x i t, x2t )= d(xi, x2) for t_--- 0. Let Z = X x X.
We define a metric on Z by the relation
d(z i , z2) --,-- d(xi, x2) + d(Yi, y2),
where z1= {xi, Yi} and z2 = Ix2, Y21. Since X is invariant, through
every point z = z(0) e Z at least one trajectory z(t) passes. Let A c Z
be the set of elements z = {x, y} such that there is at least one trajectory
z(t) = {x(t), y(t)} with
d(x(t), y(t)) -= d(x, y)

(t e j).

The set A is closed and invariant in Z. We are going to prove that
A =Z. Suppose this is not the case, that is, there is a zo o A.
Let zo(t) = zo t be some trajectory and z 1 be a limit point of the form
z i = lim zo(tm).
t„,-.--co
Since the distance d(zo, A) >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-