Almost periodic solutions of differential equations in Banach spaces

250 Pages·2000·1.05 MB·English

by  Hino Y.

ALMOST PERIODIC SOLUTIONS OF
DIFFERENTIAL EQUATIONS IN
BANACH SPACES
Yoshiyuki Hino
Department of Mathematics and Informatics
Chiba University, Chiba, Japan
Toshiki Naito
Department of Mathematics
University of Electro-Communications, Tokyo, Japan
Nguyen Van Minh
Department of Mathematics
Hanoi University of Science, Hanoi, Vietnam
Jong Son Shin
Department of Mathematics
Korea University, Tokyo, Japan
Preface
Almost periodic solutions of differential equations have been studied since the very
beginning of this century. The theory of almost periodic solutions has been de-
veloped in connection with problems of differential equations, dynamical systems,
stabilitytheoryanditsapplicationstocontroltheoryandotherareasofmathemat-
ics.TheclassicalbooksbyC.Corduneanu[50],A.M.Fink[67],T.Yoshizawa[231],
L. Amerio and G. Prouse [7], B.M. Levitan and V.V. Zhikov [137] gave a very nice
presentationofmethodsaswellasresultsinthearea.Inrecentyears,therehasbeen
an increasing interest in extending certain classical results to differential equations
in Banach spaces. In this book we will make an attempt to gather systematically
certain recent results in this direction.
Weoutlinebrieflythecontentsofourbook.Themainresultspresentedhereare
concernedwithconditionsfortheexistenceofperiodicandalmostperiodicsolutions
and its connection with stability theory. In the qualitative theory of differential
equations there are two classical results which serve as models for many works in
the area. Namely,
Theorem A A periodic inhomogeneous linear equation has a unique
periodic solution (with the same period) if 1 is not an eigenvalue of its
monodromy operator.
Theorem B A periodic inhomogeneous linear equation has a periodic
solution (with the same period) if and only if it has a bounded solution.
In our book, a main part will be devoted to discuss the question as how to ex-
tend these results to the case of almost periodic solutions of (linear and nonlinear)
equations in Banach spaces. To this end, in the first chapter we present introduc-
tions to the theory of semigroups of linear operators (Section 1), its applications
to evolution equations (Section 2) and the harmonic analysis of bounded functions
on the real line (Section 3). In Chapter 2 we present the results concerned with
autonomous as well as periodic evolution equations, extending Theorems A and
B to the infinite dimensional case. In contrast to the finite dimensional case, in
general one cannot treat periodic evolution equations as autonomous ones. This is
due to the fact that in the infinite dimensional case there is no Floquet represen-
tation, though one can prove many similar assertions to the autonomous case (see
e.g. [78], [90], [131]). Sections 1, 2 of this chapter are devoted to the investigation
I
II Preface
bymeansofevolutionsemigroupsintranslationinvariantsubspacesofBUC(R,X)
(ofboundeduniformlycontinuousX-valuedfunctionsontherealline).Anewtech-
niqueofspectraldecompositionispresentedinSection3.Section4presentsvarious
resultsextendingTheoremBtoperiodicsolutionsofabstractfunctionaldifferential
equations. In Section 5 we prove analogues of results in Sections 1, 2, 3 for dis-
crete systems and discuss an alternative method to extend Theorems A and B to
periodic and almost periodic solutions of differential equations. In Sections 6 and 7
we extend the method used in the previous ones to semilinear and fully nonlinear
equations. The conditions are given in terms of the dissipativeness of the equations
under consideration.
In Chapter 3 we present the existence of almost periodic solutions of almost pe-
riodic evolution equations by using stability properties of nonautonomous dynam-
ical systems. Sections 1 and 2 of this chapter extend the concept of skew product
flow of processes to a more general concept which is called skew product flow of
quasi-processesandinvestigatetheexistenceofalmostperiodicintegralsforalmost
periodic quasi-processes. For abstract functional differential equations with infinite
delay,therearethreekindsofdefinitionsofstabilities.InSections3and4,weprove
someequivalenceofthesedefinitionsofstabilitiesandshowthatthesestabilitiesfit
in with quasi-processes. By using results in Section 2, we discuss the existence of
almost periodic solutions for abstract almost periodic evolution equations in Sec-
tion 5. Concrete applications for functional partial differential equations are given
in Section 6.
We wish to thank Professors T.A. Burton and J. Kato for their kind interest,
encouragement, and especially for reading the manuscript and making valuable
comments on the contents as well as on the presentation of this book. It is also our
pleasuretoacknowledgeourindebtednesstoProfessorS.Murakamiforhisinterest,
encouragementandremarkstoimproveseveralresultsaswellastheirpresentation.
The main part of the book was written during the third author (N.V. Minh)’s visit
to the University of Electro-Communications (Tokyo) supported by a fellowship of
the Japan Society for the Promotion of Science. He wishes to thank the University
for its warm hospitality and the Society for the generous support.
Tokyo 2000 Yoshiyuki Hino
Toshiki Naito
Nguyen Van Minh
Jong Son Shin
Contents
1 PRELIMINARIES 7
1.1. STRONGLY CONTINUOUS SEMIGROUPS . . . . . . . . . . . . . 7
1.1.1. Definition and Basic Properties . . . . . . . . . . . . . . . . . 7
1.1.2. CompactSemigroupsandAnalyticStronglyContinuousSemi-
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3. Spectral Mapping Theorems. . . . . . . . . . . . . . . . . . . 11
1.2. EVOLUTION EQUATIONS . . . . . . . . . . . . . . . . . . . . . . 15
1.2.1. Well-Posed Evolution Equations . . . . . . . . . . . . . . . . 15
1.2.2. Functional Differential Equations with Finite Delay . . . . . 18
1.2.3. Equations with Infinite Delay . . . . . . . . . . . . . . . . . . 20
1.3. SPECTRAL THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.1. Spectrum of a Bounded Function . . . . . . . . . . . . . . . . 24
1.3.2. Almost Periodic Functions. . . . . . . . . . . . . . . . . . . . 26
1.3.3. Sprectrum of an Almost Periodic Function. . . . . . . . . . . 27
1.3.4. A Spectral Criterion for Almost Periodicity of a Function . . 28
2 SPECTRAL CRITERIA 31
2.1. EVOLUTION SEMIGROUPS & PERIODIC EQUATIONS . . . . . 31
2.1.1. Evolution Semigroups . . . . . . . . . . . . . . . . . . . . . . 31
2.1.2. Almost Periodic Solutions and Applications . . . . . . . . . . 35
2.2. SUMS OF COMMUTING OPERATORS . . . . . . . . . . . . . . . 45
2.2.1. Differential Operator d/dt−A and Notions of Admissibility . 48
2.2.2. Admissibility for Abstract Ordinary Differential Equations . 53
2.2.3. Higher Order Differential Equations . . . . . . . . . . . . . . 55
2.2.4. Abstract Functional Differential Equations . . . . . . . . . . 62
2.2.5. Examples and Applications . . . . . . . . . . . . . . . . . . . 66
2.3. DECOMPOSITION THEOREM . . . . . . . . . . . . . . . . . . . . 77
2.3.1. Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . 79
2.3.2. Spectral Criteria For Almost Periodic Solutions . . . . . . . . 85
2.3.3. When Does Boundedness Yield Uniform Continuity ? . . . . 89
2.3.4. Periodic Solutions of Partial Functional Differential Equations 91
2.3.5. Almost Periodic Solutions of Partial Functional Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
III
IV CONTENTS
2.4. FIXED POINT THEOREMS AND FREDHOLM OPERATORS . . 109
2.4.1. Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . 109
2.4.2. Decomposition of Solution Operators . . . . . . . . . . . . . . 110
2.4.3. Periodic Solutions and Fixed Point Theorems . . . . . . . . . 113
2.4.4. Existence of Periodic Solutions: Bounded Perturbations . . . 116
2.4.5. Existence of Periodic Solutions : Compact Perturbations . . . 120
2.4.6. Uniqueness of Periodic Solutions I . . . . . . . . . . . . . . . 125
2.4.7. Uniqueness of Periodic Solutions II . . . . . . . . . . . . . . 127
2.4.8. An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.4.9. Periodic Solutions in Equations with Infinite Delay . . . . . . 130
2.5. DISCRETE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . 132
2.5.1. Spectrum of Bounded Sequences and Decomposition . . . . . 133
2.5.2. Almost Periodic Solutions of Discrete Systems . . . . . . . . 137
2.5.3. Applications to Evolution Equations . . . . . . . . . . . . . . 139
2.6. SEMILINEAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . 143
2.6.1. Evolution Semigroups and Semilinear Evolution Equations . 143
2.6.2. Bounded and Periodic Solutions to Abstract Functional Dif-
ferential Equations with Finite Delay. . . . . . . . . . . . . . 151
2.7. NONLINEAR EVOLUTION EQUATIONS . . . . . . . . . . . . . . 153
2.7.1. Nonlinear Evolution Semigroups in AP(∆) . . . . . . . . . . 153
2.7.2. Almost Periodic Solutions of Dissipative Equations . . . . . . 157
2.7.3. An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.8. NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3 STABILITY METHODS 163
3.1. SKEW PRODUCT FLOWS . . . . . . . . . . . . . . . . . . . . . . . 163
3.2. EXISTENCE THEOREMS . . . . . . . . . . . . . . . . . . . . . . . 168
3.2.1. Asymptotic Almost Periodicity and Almost Periodicity. . . . 168
3.2.2. Uniform Asymptotic Stability and Existence of Almost Peri-
odic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.2.3. Separation Condition and Existence of Almost Periodic Inte-
grals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.2.4. Relationship between the Uniform Asymptotic Stability and
the Separation Condition . . . . . . . . . . . . . . . . . . . . 175
3.2.5. Existence of an Almost Periodic Integral of Almost Quasi-
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.3. PROCESSES AND QUASI-PROCESSES . . . . . . . . . . . . . . . 176
3.3.1. Abstract Functional Differential Equations with Infinite Delay 176
3.3.2. Processes and Quasi-Processes Generated by Abstract Func-
tional Differential Equations with Infinite Delay. . . . . . . . 180
3.3.3. StabilityPropertiesforAbstractFunctionalDifferentialEqua-
tions with Infinite Delay . . . . . . . . . . . . . . . . . . . . . 185
3.4. BC-STABILITIES & ρ-STABILITIES . . . . . . . . . . . . . . . . . 190
3.4.1. BC-Stabilities in Abstract Functional Differential Equations
with Infinite Delay . . . . . . . . . . . . . . . . . . . . . . . . 190
CONTENTS V
3.4.2. EquivalentRelationshipbetweenBC-UniformAsymptoticSta-
bility and ρ-Uniform Asymptotic Stability . . . . . . . . . . . 192
3.4.3. Equivalent Relationship Between BC-Total Stability and ρ-
Total Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.4.4. Equivalent Relationships of Stabilities for Linear Abstract
Functional Differential Equations with Infinite Delay . . . . . 198
3.5. EXISTENCE OF ALMOST PERIODIC SOLUTIONS . . . . . . . . 202
3.5.1. Almost Periodic Abstract Functional Differential Equations
with Infinite Delay . . . . . . . . . . . . . . . . . . . . . . . . 202
3.5.2. Existence Theorems of Almost Periodic Solutions for Nonlin-
ear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
3.5.3. Existence Theorems of Almost Periodic Solutions for Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
3.6. APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
3.6.1. Damped Wave Equation . . . . . . . . . . . . . . . . . . . . . 207
3.6.2. Integrodifferential Equation with Duffusion . . . . . . . . . . 210
3.6.3. Partial Functional Differential Equation . . . . . . . . . . . . 214
3.7. NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4 APPENDICES 221
4.1. FREDHOLM OPERATORS . . . . . . . . . . . . . . . . . . . . . . 221
4.2. MEASURES OF NONCOMPACTNESS . . . . . . . . . . . . . . . . 224
4.3. SUMS OF COMMUTING OPERATORS . . . . . . . . . . . . . . . 231
4.4. LIPSCHITZ OPERATORS . . . . . . . . . . . . . . . . . . . . . . . 232
Index 249
CHAPTER 1
C -SEMIGROUPS, WELL POSED EVOLUTION
0
EQUATIONS, SPECTRAL THEORY AND
ALMOST PERIODICITY OF FUNCTIONS
1.1. STRONGLY CONTINUOUS SEMIGROUPS OF LINEAR OP-
ERATORS
In this section we collect some well-known facts from the theory of strongly contin-
uous semigroups of operators on a Banach space for the reader’s convenience. We
will focus the reader’s attention on several important classes of semigroups such as
analyticandcompactsemigroupswhichwillbediscussedlaterinthenextchapters.
Amongthebasicpropertiesofstronglycontinuoussemigroupswewillputemphasis
on the spectral mapping theorem. Since the materials of this section as well as of
thechapterinthewholecanbefoundinanystandardbookcoveringthearea,here
we aim at freshening up the reader’s memory rather than giving a logically self
contained account of the theory.
Throughout the book we will denote by X a complex Banach space. The set
of all real numbers and the set of nonnegative real numbers will be denoted by
R and R+, respectively. BC(R,X), BUC(R,X) stand for the spaces of bounded,
continuous functions and bounded, uniformly continuous functions, respectively.
1.1.1. Definition and Basic Properties
Definition 1.1 Afamily(T(t)) ofboundedlinearoperatorsactingonaBanach
t≥0
space X is a strongly continuous semigroup of bounded linear operators, or briefly,
a C -semigroup if the following three properties are satisfied:
0
i) T(0)=I, the identity operator on X;
ii) T(t)T(s)=T(t+s) for all t,s≥0;
iii) lim kT(t)x−xk=0 for all x∈X.
t↓0
The infinitesimal generator of (T(t)) , or briefly, the generator, is the linear op-
t≥0
erator A with domain D(A) defined by
1
D(A) = {x∈X:lim (T(t)x−x) exists},
t↓0 t
1
Ax = lim (T(t)x−x), x∈D(A).
t↓0 t
The generator is always a closed, densely defined operator.
7
8 CHAPTER 1. PRELIMINARIES
Theorem 1.1 Let (T(t)) be a C -semigroup. Then there exist constants ω ≥ 0
t≥0 0
and M ≥1 such that
kT(t)k≤Meωt, ∀t≥0.
Proof. For the proof see e.g. [179, p. 4].
Corollary 1.1 If (T(t)) is a C -semigroup, then the mapping (x,t)7→T(t)x is
t≥0 0
a continuous function from X×R+ →X.
Proof. For any x,y ∈X and t≤s∈R+ :=[0,∞),
kT(t)x−T(s)yk ≤ kT(t)x−T(s)xk+kT(s)x−T(s)yk
≤ Meωskx−yk+kT(t)kkT(s−t)x−xk
≤ Meωskx−yk+MeωtkT(s−t)x−xk. (1.1)
Hence, for fixed x,t (t ≤s) if (y,s) → (x,t), then kT(t)x−T(s)yk→ 0. Similarly,
for s≤t
kT(t)x−T(s)yk ≤ kT(t)x−T(s)xk+kT(s)x−T(s)yk
≤ Meωskx−yk+kT(s)kkT(t−s)x−xk
≤ Meωskx−yk+MeωskT(t−s)x−xk. (1.2)
Hence, if (y,s)→(x,t), then kT(t)x−T(s)yk→0.
OtherbasicpropertiesofaC -semigroupanditsgeneratorarelistedinthefollowing:
0
Theorem 1.2 Let A be the generator of a C -semigroup (T(t)) on X. Then
0 t≥0
i) For x∈X,
1 Z t+h
lim T(s)xds=T(t)x.
h→0h t
Rt
ii) For x∈X, T(s)xds∈D(A) and
0
(cid:18)Z t (cid:19)
A T(s)xds =T(t)x−x.
0
iii) For x∈D(A),T(t)x∈D(A) and
d
T(t)x=AT(t)x=T(t)Ax.
dt
iv) For x∈D(A),
Z t Z t
T(t)x−T(s)x= T(τ)Axdτ = AT(τ)xdτ.
s s
CHAPTER 1. PRELIMINARIES 9
Proof. For the proof see e.g. [179, p. 5].
We continue with some useful fact about semigroups that will be used through-
out this book. The first of these is the Hille-Yosida theorem, which characterizes
the generators of C -semigroups among the class of all linear operators.
0
Theorem 1.3 Let A be a linear operator on a Banach space X, and let ω ∈R and
M ≥1 be constants. Then the following assertions are equivalent:
i) A is the generator of a C -semigroup (T(t)) satisfying kT(t)k≤Meωt for
0 t≥0
all t≥0;
ii) A is closed, densely defined, the half-line (ω,∞) is contained in the resolvent
set ρ(A) of A, and we have the estimates
M
kR(λ,A)nk≤ , ∀λ>ω, n=1,2,... (1.3)
(λ−ω)n
Here, R(λ,A) := (λ − A)−1 denotes the resolvent of A at λ. If one of the
equivalent assertions of the theorem holds, then actually {Reλ>ω}⊂ρ(A) and
M
kR(λ,A)nk≤ , ∀Reλ>ω, n=1,2,... (1.4)
(Reλ−ω)n
Moreover, for Reλ>ω the resolvent is given explicitly by
Z ∞
R(λ,A)x= e−λtT(t)xdt, ∀x∈X. (1.5)
0
We shall mostly need the implication (i)⇒(ii), which is the easy part of the
theorem. In fact, one checks directly from the definitions that
Z ∞
R x:= e−λtT(t)xdt
λ
0
definesatwo-sidedinverseforλ−A.Theestimate(1.4)andtheidentity(1.5)follow
trivially from this.
A useful consequence of (1.3) is that
lim kλR(λ,A)x−xk=0, ∀x∈X. (1.6)
λ→∞
This is proved as follows. Fix x∈D(A) and µ∈ρ(A), and let y ∈X be such that
x = R(µ,A)y. By (1.3) we have kR(λ,A)k = O(λ−1) as λ → ∞. Therefore, the
resolvent identity
R(λ,A)−R(µ,A)=(µ−λ)R(λ,A)R(µ,A) (1.7)
implies that
lim kλR(λ,A)x−xk= lim kR(λ,A)(µR(µ,A)y−y)k=0.
λ→∞ λ→∞
Thisproves(1.6)forelementsx∈D(A).SinceD(A)isdenseinX andtheoperators
λR(λ,A) are uniformly bounded as λ→∞ by (1.3), (1.6) holds for all x∈X.
10 CHAPTER 1. PRELIMINARIES
1.1.2. Compact Semigroups and Analytic Strongly Continuous Semi-
groups
Definition 1.2 A C -semigroup (T(t)) is called compact for t > t if for every
0 t≥0 0
t > t , T(t) is a compact operator. (T(t)) is called compact if it is compact for
0 t≥0
t>0.
If a C -semigroup (T(t)) is compact for t > t , then it is continuous in the
0 t≥0 0
uniform operator topology for t>t .
0
Theorem 1.4 Let A be the generator of a C -semigroup (T(t)) . Then (T(t))
0 t≥0 t≥0
is a compact semigroup if and only if T(t) is continuous in the uniform operator
topology for t>0 and R(λ;A) is compact for λ∈ρ(A).
Proof. For the proof see e.g. [179, p. 49].
In this book we distinguish the notion of analytic C -semigroups from that of
0
analytic semigroups in general. To this end we recall several notions. Let A be a
linear operator D(A)⊂X→X with not necessarily dense domain.
Definition 1.3 A is said to be sectorial if there are constants ω ∈ R,θ ∈
(π/2,π),M >0 such that the following conditions are satisfied:

i) ρ(A)⊃S ={λ∈C:λ6=ω,|arg(λ−ω)|<θ},
 θ,ω
ii) kR(λ,A)k≤M/|λ−ω| ∀λ∈S .
θ,ω
If we assume in addtion that ρ(A) 6= (cid:11), then A is closed. Thus, D(A), endowed
with the graph norm
kxk :=kxk+kAxk,
D(A)
is a Banach space. For a sectorial operator A, from the definition, we can define a
linear bounded operator etA by means of the Dunford integral
1 Z
etA := etλR(λ,A)dλ,t>0, (1.8)
2πi
ω+γr,η
where r >0,η ∈(π/2,θ) and γ is the curve
r,η
{λ∈C:|argλ|=η,|λ|≥rk}∪{λ∈C:|argλ|≤η,|λ|=r},
oriented counterclockwise. In addition, set e0Ax=x, ∀x∈X.
Theorem 1.5 Under the above notation, for a sectorial operator A the following
assertions hold true:
i) etAx∈D(Ak) for every t>0,x∈X,k ∈N. If x∈D(Ak), then
AketAx=etAAkx, ∀t≥0;