On Evolution Equations in Banach Spaces and Commuting

ON EVOLUTION EQUATIONS IN BANACH SPACES AND COMMUTING

SEMIGROUPS

A dissertation presented to

the faculty of

the College of Arts and Sciences of Ohio University

In partial fulfillment

of the requirements for the degree

Doctor of Philosophy

Saud M. A. Alsulami

June 2005 This dissertation entitled

ON EVOLUTION EQUATIONS IN BANACH SPACES AND COMMUTING

SEMIGROUPS

by

SAUD M. A. ALSULAMI

has been approved

for the Department of Mathematics

and the College of Arts and Sciences by

Quoc-Phong Vu

Professor of Mathematics

Leslie A. Flemming Dean, College of Arts and Sciences ALSULAMI, SAUD M. A. Ph.D. June 2005. Mathematics

On Evolution Equations In Banach Spaces And Commuting Semigroups (102 pp.)

Director of Dissertation: Quoc-Phong Vu

This dissertation is concerned with several questions about the qualitative behavior of mild solutions of differential equations with multi-dimensional time on Banach spaces. For the differential equations

⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t) where A and B are linear (in general, unbounded) operators defined on a Banach space E, we give a definition of the mild solution of (∗). In order for Eq.(∗)tohave a (mild) solution, we introduce the condition(Ds − A)f2 =(Dt − B)f1 (where Ds and Dt are the partial differential operators with respect to s and t, respectively) which is understood in a generalized (mild) sense. We extend the notion of ad- missible subspaces, for closed translation-invariant subspaces M of BUC(R2,E) with respect to Eq.(∗), and we give characterization of the admissibility in terms

M M of solvability of the operator equations AX − XDs = C and BX − XDt = CT .

As a tool to investigate almost periodic functions of several variables which is important for the study of asymptotic behavior of Eq.(∗), we give answer to a question raised by Basit in 1971, which is an extension of the classical Bohl-Bohr theorem (to almost periodic functions with two or more variables). We also extend a theorem due to Loomis (who obtained it for functions with values in a finite dimensional space) to functions with values in Banach space E, under conditions introduced by Kadets. As an application, we obtain a result on the almost peri- odicity of a double integral of an almost periodic function defined on the plane.

Applying to Eq.(∗), we show that the boundedness implies the almost periodicity for solutions of (∗) in the finite dimensional case. For the infinite dimensional case, and under the conditions that A and B are bounded linear operators, we reduce the question of the almost periodicity (resp, almost automorphicity) of the differential equations (∗) for given almost periodic (resp, almost automorphic) functions f1 and f2 to the question of the almost periodicity(resp, almost auto- morphicity) of the homogeneous system (i.e., f1 = f2 = 0). Finally, in chapter 6, we introduce the notion of C-admissible subspaces and obtain various conditions of C-admissibilities, generalizing well known results of Schuler-Vu and others. Approved:

Quoc-Phong Vu

Professor of Mathematics DEDICATION

I would like to dedicate this work to:

My parents,

My grandmother,

My brothers and sisters,

My wife,

and My kids. Acknowledgments

In the Name of ALLAH[God], the Most Gracious and the Most Merciful.

All praises and thanks are due to ALLAH, the Lord of all that exists. May the peace and blessings of ALLAH be upon Muhammad, the messenger of ALLAH, and his family and companions and all who follow them in righteousness until the day of judgment. I would like to express my sincere gratitude to Prof. Quoc-

Phong Vu, my advisor, for his guidance and his friendship. Also, I would like to thank the members of my dissertation committee: Prof. A. Gulisashvili, Prof. H.

Pasic and Prof. N. Pavel for taking their time to go through my manuscript .

There is no doubt that none of this work would have been possible if it was not for the guidance of ALLAH and then for the love, patience and sacrifice of my beloved parents, grandmother and grandfather (who died while I am working on my dissertation) whose sacrifice and prayers have been a major source of strength for me all throughout my life. May ALLAH reward them with the best of reward and grant them mercy, success and happiness as long as they live and when they meet Him on the day of judgment. Also, special thanks go to each one of my brothers and my sisters whose spiritual support greatly enhanced my academic accomplishment. I am also greatly indebted to my beloved wife Um Omar for her patience and sacrifice. A special word of appreciation goes to my beloved kids.

Moreover, I would like to thank all of my relatives and all of my friends. Also,

I would like to thank King Abdul Aziz University for the scholarship and Ohio

University for giving my the chance to continue my higher education.

Last but not least, I thank those who help, encourage or pray for me during my stay in the United States. 7

Contents

Abstract 3

Dedication 5

Acknowledgments 6

Introduction 9

1 Preliminaries 18

1.1 Almost periodic and almost automorphic functions from R2 into a

Banachspace...... 18

1.2 Thespectrumofvector-valuedfunctions...... 22

1.3Semigroupsofoperators...... 24

1.4 The operator equation AX − XB = C ...... 30

2 On mild solutions of differential equations 35

2.1 On mild solutions of differential equations with multi-time . . . . 35

3 Regular admissibility of subspaces 50

3.1 Operator equations AX − XDs = C and BX − XDt = CT .... 50

3.2 Admissibility of difference equations ...... 56

4 On the integral of almost periodic functions 64

4.1OnBasit’squestion...... 64

4.2 Loomis’ theorem for vector-valued almost periodic functions . . 69 8

4.3 An alternative form of the theorem on the integral of almost peri-

odicfunctions ...... 72

5 Almost periodic solutions of differential equations 74

5.1Thefinitedimensionalcase ...... 74

5.2 The infinite dimensional case ...... 79

6 C-admissibility and analytic C-semigroups 87 6.1 C-admissibility ...... 87

6.2 C-semigroups and analytic C-semigroups...... 90

6.3 Operator equation AX −XB = CD,withA generating an analytic

C-semigroup...... 92

References 96 9

Introduction

Consider the differential equation

u(t)=Au(t)+f(t),t∈ R (1) where A is a linear operator on a Banach space E and f is a (continuous) function from R to E.IfA is a bounded operator, then the solution of (1) is defined by the following formula

 t u(t)=eA(t−s)u(s)+ eA(t−v)f(v)dv ∀t ≥ s (2) s where eA is defined by ∞  Ak eA = . (3) k=0 k!

InthecasethatA is unbounded, but is generator of a C0-semigroup T (t), then (2) is replaced by

 t u(t)=T (t − s)u(s)+ T (t − v)f(v)dv ∀t ≥ s (4) s

In general, the formula (4) is used as a definition of mild solution of equation (1).

See for example: [5], [20] and [36] . The question of asymptotic behavior of (1) is intensively studied by many authors during the last decades, see e.g. [5], [16] and

[54].

The primary objective of this dissertation is to initiate a systematic study of the questions of solvability and asymptotic behavior of differential equations with multi-dimensional time (or, for short, equations with multi-time). We are moti- vated by series of papers by Perov and others. See for examples: [37], [38], [39], 10

[40], [41] and [42] and the references therein. The theory of differential equations with multi-time has applications in different fields. We mention for example the multi-time Hamilton-Jacobi equations which arise from the study of problems in mathematical economics, see for example [7] and [31] and the references therein.

The importance of the study of almost periodic solutions of differential equations rely on the applications in mathematical physics and astronomy (where are some objects has the almost periodic behavior). For sake of simplicity we restrict our- selves to the case of time dimension two, but we note that the results remain true for arbitrary finite time dimension.

The first question we are concerned about in this dissertation is to find a general formula for mild solutions of the following differential equations ⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t) where A and B are commuting closed linear operators on a Banach space E.

In chapter 2, we obtain the following formula for mild solutions of Eq. (∗): in order to be a mild solution of (∗)onR × R, u(s, t) must satisfy simultaneously

 t u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − s0) T2(t − w) f2(s0,w) dw t0  s + T1(s − v) f1(v,t) dv (s ≥ s0,t≥ t0) s0 and

 s u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − v) T2(t − t0) f1(v,t0) dv s0 11

 t + T2(t − w) f2(s, w) dw (s ≥ s0,t≥ t0) t0 Note that mild solutions of ( ∗ )onR+ × R+ are given by

 t u(s, t)=T1(s) T2(t) u(0, 0) + T1(s) T2(t − w) f2(0,w) dw 0  s + T1(s − v) f1(v,t) dv (s ≥ 0,t≥ 0) 0 which must simultaneously satisfy

 s u(s, t)=T1(s) T2(t) u(0, 0) + T1(s − v) T2(t) f1(v,0) dv 0  t + T2(t − w) f2(s, w) dw (s ≥ 0,t≥ 0) 0

We show that in order to the system (∗) to have a solution, the following condition must be satisfied: (Ds − A)f2 =(Dt − B)f1 where Ds and Dt are the partial differentiations with respect to s and t, respectively.

In chapter 3, we study the admissibility of subspaces with respect to Eq. (∗).

Let M be a closed subspace of BUC(R,E), the space of bounded uniformly con- tinuous functions from R to E. Recall that M is said to be regularly admissible with respect to equation (1) if for every function f ∈M, (1) has a unique mild so- lution in M. The question of regular admissibility of a subspace M in BUC(R,E) plays an important role in the study of asymptotic behavior of solutions of dif- ferential equations. A classical approach to the question of regular admissibility of a subspace M is to use the so called Green’s function. The following classical result about the regular admissibility of BUC(R,E) is due to Daleckii and Krein.

Theorem 0.1 (Daleckii-Krein [16]) Suppose that A is a bounded operator on a

Banach space E. Then the following are equivalent: 12

(i) The space BUC(R,E) is regularly admissible with respect to the equation

(1).

(ii) σ(A) ∩ iR = ∅.

(iii) The semigroup eAt is exponentially dichotomic, i.e, there is a bounded pro-

jection P on E, which is commuting with A, and positive constants M and

ω, such that

a) etAP ≤Me−ωt for all t ≥ 0,

b) etA(I − P )≤Meωt for all t<0.

In any case, the mild solution of (1) is given by

 ∞ u(t)= GA(t − s)f(s)ds, −∞ where GA(t) is Green’s function defined by the following formula: ⎧ ⎨⎪ tA e P for t ≥ 0, GA(t)=⎪ ⎩ −etA(I − P ) for t<0.

An alternative way to study the admissibility is to use a method introduced by

Schuler and Vu, in [54], which connects the regular admissibility of the space M with the solvability of the operator equation of the form

AX − XDM = C. (5)

where D is the differentiation operator, DM is its restriction to the space M, C is a bounded operator from M to E and X : M→E is the unknown bounded operator. 13

Suppose M is regularly admissible w.r.t. equation (1). If we define X : M → E by Xf := u(0) where u is the unique mild solution of (1) corresponding to f,then it turns out that X is the unique bounded solution of the operator equation

M AX − XDM = −δ0 , (6)

M where δ0 ≡ δ0|M and δ0 : BUC(R,E) → E is the Dirac operator such that

δ0f := f(0). Conversely, if X is a bounded solution of equation (6), then u(t):=

XS(t)f,whereS(t) is the translation operator on M, is a mild solution of (1). Thus, to investigate the regular admissibility of a subspace M, we can use the results on operator equation in the form of (5).

In section 3.1, we extend the above results by characterizing the regular admis- sibility of a subspace M with respect to the differential equations

∂u(s, t) = Au(s, t)+f1(s, t) ∂s ∂u(s, t) = Bu(s, t)+f2(s, t) ∂t by the solvability of the operator equations

M M AX − XDs = −δ0

M M BX − XDt = −δ0 T

In section 3.2, we apply the same method to show the admissibility with respect to the following difference equation

xn+1 = Axn + yn ∀n ∈ Z 14 where A is a linear operator on E and l∞(Z,E) is the Banach space of all bounded sequences x := (xn)n∈Z with the sup-norm.

In chapter 4, we deal with special cases of (∗) which generalize some known results.

Let us recall the following classical theorem of Bohl and Bohr concerning inte- grals of almost periodic functions.

t Theorem 0.2 Assume f(t) is almost periodic and F (t)= 0 f(s) ds.Then,F is almost periodic if and only if it is bounded on R.

The Bohl-Bohr Theorem is not true, in general, for almost periodic functions with values in a Banach space, see e.g. [30]. Thus, the question is under what additional conditions, on the function F or the space E, does the boundedness of

F (t) imply its almost periodicity.

Kadets discovered that the Bohl-Bohr Theorem for the vector-valued functions is valid under either one of the following conditions:

(C1) The Banach space does not contain a subspace isomorphic to c0 [28].

(C2) f has weakly relatively compact range [27].

In 1971, Basit (see [8]) showed that for a complex valued almost periodic func- tion f(x, y) defined on the plane, the boundedness of the integral F (x, y)= x 0 f(t, y) dt does not imply its almost periodicity. In section 4.1, we impose a condition on the function f(x, y) to ensure the almost periodicity of F (x, y). Note that this, when we relate to Eq. (∗), is the case when A = B =0 and f2 =0.

We also generalize this result to vector-valued functions. 15

Another related result is due to Loomis ([32]) who considered the following equations in the finite dimensional case:

∂u(s, t) = f1(s, t) ∂s ∂u(s, t) = f2(s, t) ∂t and proved that if f1(s, t)andf2(s, t) are scalar-valued almost periodic functions on the plane and u is bounded, then u(s, t) is a scalar-valued almost periodic function. In section 4.2, we extend Loomis’s theorem to vector-valued functions under the above mentioned conditions of Kadets. Namely, we show that if f1(s, t) and f2(s, t) are vector-valued almost periodic functions on the plane and u(s, t) is bounded, then, under one of the conditions of Kadets, u(s, t) is a vector-valued almost periodic function on the plane.

In section 4.3, we apply the above results to the following related question : assuming that f(s, t)((s, t) ∈ R2)isa(scalarorvector-valued)almostperiodic t s function, when is the double integral F (s, t)= 0 0 f(x, y) dx dy almost periodic?

In section 5.1, by using Fink’s techniques [22], we show that the boundedness of solutions of Eq. (∗) implies the almost periodicity in the finite dimensional case ( the converse is well-known). This result generalize a well-known fact for the differential equation (1) (with one-dimensional time) .

In section 5.2, by using another equivalent definition of the almost periodic

(resp, almost automorphic ) functions introduced by S. Bochner in 1962, see [14], and his related methods of iterated double limits (cf. Sell, [47], where similar methods are used for partial differential equations) we reduce the question of 16 almost periodicity of the inhomogeneous system to that of the corresponding homogeneous system ⎫ ∂u(s,t) ⎪ ∂s = Au(s, t) ⎬ ∗∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t) (under the additional assumptions that A and B are bounded linear operators).

Moreover, we prove a nice result for the differential equation (with one- dimensional time), namely:

If f(t)(t ∈ R) is almost periodic (resp, almost automorphic), A is a bounded operator and (1) has a unique bounded solution, then u(t) is almost periodic (resp, almost automorphic).

In [51], Phong Vu proved the following theorem:

Theorem 0.3 Let A be a generator of an analytic semigroup in a Banach space

E. Assume that B isaclosedlinearoperatorinaBanachspaceF such that

−1 Σω,θ ⊂ ρ(B) and λ(B − λ)  is uniformly bounded when λ belongs to the sector { ∈ C | − | }∪{ }  − −1 Σω,θ (where Σω,θ = λ : arg(ω λ) <θ ω and supλ∈C\Σω,θ λ(A λ) < ∞). Then, the operator equation

AX − XB = D has a unique solution which is expressed by  1 X = (A − λ)−1D(B − λ)−1dλ. 2πi Γ In chapter 6, we generalize such theorem to the case where A is a generator of an analytic C-semigroup . 17

The result in section 3.2 with more investigation of stability and almost peri- odicity of the solutions of difference equations on Banach spaces are presented in

[1]. While the results of chapters 2, 4 and 5 are presented in [2]. 18

1 Preliminaries

In this preliminary Chapter, we give basic definitions and results which are needed for the next chapters.

1.1 Almost periodic and almost automorphic functions

from R2 into a Banach space

In this dissertation, we use the following notations: we denote the space of all con- tinuous functions from R2 to a Banach space E by C(R2,E). Note that C(R2,E) is a Banach space equipped with the supremum norm:

fC(R2,E) := sup f(s, t). (s,t)∈R2

We will denote the Banach space of all bounded uniformly continuous functions

(equipped with the supremum norm) by BUC(R2,E).

Let f be a function in L1(R2,E). The Fourier transform fˆ of f is defined by

 ∞  ∞ fˆ(s, t)= e−isve−itwf(v,w) dv dw −∞ −∞

Moreover, let f ∈ L1(R2,E)andu ∈ BUC(R2,E). Then, the convolution of f and u, f ∗ u, is defined by

 ∞  ∞ (f ∗ u)(s, t)= f(s − v,t − w)u(v,w) dv dw −∞ −∞

Next we recall the definition of almost-periodic functions.AsetΛ⊂ R2 is said to be relatively dense in R2 if there exists a number >0 such that the 19

2 2 intersection B(t, ) ∩ Λ = ∅ for all t ∈ R ,whereB(t, )= {t1 ∈ R : |t1 − t| < }

2 (t, t1 ∈ R ).

Definition 1.1 A continuous function f(s, t):R2 → E is called almost periodic if for every ε>0 there exists a relative dense set Λ ⊂ R2 such that

sup f(t1 + τ1,t2 + τ2) − f(t1,t2) <ε,∀(τ1,τ2) ∈ Λ. 2 (t1,t2)∈R

We will denote the space of E-valued almost periodic functions on R2 by

AP(R2,E).

Note that H. Bohr was the first one, in 1925, who used the relatively dense sets to define the complex valued almost periodic functions defined on the whole line .

In 1926, S. Bochner established his first equivalent definition for the complex valued almost periodic functions defined on the whole line. The following def- inition is an analog of his definition for vector-valued almost periodic functions defined on the plane :

Definition 1.2 The function f(t)(t ∈ R2) is almost periodic if and only if any

2 sequence {λn}⊂R contains a subsequence {λn} for which

lim f(λn + t)=g(t) n→∞ exists uniformly in R2.

Remark 1.3 J. von Neumann noted that this definition is meaningful for func- tions defined on groups. 20

For convenience, let us denote a sequence {νn} by ν. Correspondingly , if limn→∞ f(νn + t)=g(t) then we write T νf = g .

In 1962, S. Bochner [14] gave his second equivalent definition of complex valued almost periodic functions defined on the whole line.

The following definition is an analog of his definition that we adapt for vector- valued almost periodic functions defined on the plane:

Definition 1.4 f(t)(t ∈ R2) is almost periodic if and only if for any two se-

quences β ⊂ R2 and ν ⊂ R2, there exist subsequences β of β and ν of ν such that

T β+ νf = T βT νf (7)

Note that equation (7) is equivalent to

2 lim f(t + βn + νn) = lim lim f(t + βn + νm),t∈ R n→∞ n→∞ m→∞

There are some other equivalent definitions of the almost periodicity by Markoff

[33] and Seifert [46] . We refer the reader to an interesting paper of Fink [21], in which equivalent definitions of almost periodicity and their analysis are presented.

Next, we give a definition of almost automorphic functions.

Definition 1.5 Afunctionf(t),t ∈ R2, is called almost automorphic if for any

sequence β ⊂ R2, there exists a subsequence β such that

T− βT βf = f where if β = {βn},then− β = {−βn}. 21

Proposition 1.6 (a) If f(t) ∈ AP(R2,E),thenf(t) ∈ BUC(R2,E).

(b) Let f(t) ∈ AP(R2,E) and assume that g is continuous function from the range of f in E into a Banach space Y . Then, the composed function g ◦ f := g(f(t)),t∈ R2, is almost periodic from R2 into Y .

∞ 2 (c) Let {fn(t)}n=1 be a sequence of almost periodic functions from R into E

2 such that limn→∞ fn(t)=f(t) uniformly on R .Then,f(t) is an almost periodic function.

For the proof and related facts, we refer the reader to [56, Ch.9]. 22

1.2 The spectrum of vector-valued functions

Let f ∈ BUC(R2,E). A point λ ∈ R2 is called a regular point of f if there is a neighborhood U of λ such that for any ϕ ∈ L1(R2)withsuppϕˆ ⊂Uwe have

ϕ ∗ f ≡ 0.

The spectrum, Sp(f), of f is defined as the complement in R2 of the set of regular points of f.

For a point λ ∈ R2, we say it is an almost periodic regular point of f if there is a neighborhood U of λ such that for any ϕ ∈ L1(R2)withsuppϕˆ ⊂U we have

ϕ ∗ f ∈ AP(E).

We also define the almost periodic spectrum of f, denoted by Spap(f), as the complement in R2 of the set of almost periodic regular points of f.

The proof of the following lemma can be found in [29] and [43]:

Lemma 1.7 Let f ∈ BUC(R2,E) and φ ∈ L1(R2,E).Then

(i) sp(f) is closed.

(ii) sp(f)=∅ if and only if f =0.

(iii) sp(ft)=sp(f),whereft is the translation of f.

(iv) sp(f + g) ⊂ sp(f) ∪ sp(g).

(v) sp(f ∗ φ) ⊂ sp(f) ∩ suppφˆ,whereφˆ is the Fourier transform of φ.

Similar results hold for the almost periodic spectrum.

Loomis, in [32], proved that the countability of the almost periodic spectrum of a scalar-valued function implies the almost periodicity of the function. 23

Here, we will give our proof of the analog theorem for vector-valued functions.

For other proofs and related results, we refer the reader to [11] and [26].

Theorem 1.8 Let x(s, t) ∈ BUC(R2,E) and assume that the almost periodic

2 spectrum Spap(x) is countable. Then, x(s, t) ∈ AP(R ,E) provided one of the following conditions holds :

(a) the Banach space E does not contain a subspace isomorphic to c0.

(b) the range of the vector x(s, t) is weakly precompact.

Proof:

We must show that Spap(x) is empty. For this, it is sufficient to show that the spectrum Spap(x) cannot have isolated points, since any non-empty closed countable set has an isolated point.

Assume the contrary, and let λ be an isolated point of Spap(x). Then,there

1 2 exists a neighborhood U such that U∩¯ Spap(f)={λ} and a function ϕ ∈ L (R ) with suppϕˆ ⊂U, but f1 = ϕ ∗ f/∈ AP(E). From this, Spap(f1) ⊂ Spap(f). Now,

Spap(f1)=Spap(ϕ ∗ f) ⊂ suppϕˆ ∩ Spap(f)

⊂U∩Spap(f) ⊂ U¯

Thus, Spap(f1) ⊂{λ}. WLOG, we may assume that Spap(f1) ⊂{0}.Thisgives

2 2 us that f1(s + t) − f1(t) ∈ AP(E) ∀s ∈ R (t ∈ R ) (see [26, Lemma4.3.3] or

[9, Theorem 4.2.2] ). Therfore, using a result of Basit [8] (which is a natural generalization of the cited theorem of Kadets), we see that when one of the two conditions of the theorem holds, the function f1 is almost periodic, which is a contradiction. Hence, Spap(x) cannot have an isolated point. Thus, Spap(x)=∅ which leads to conclude that x(s, t) ∈ AP(R2,E). 24

1.3 Semigroups of operators

In this section, we collect some basic facts concerning the theory of C0-semigroups of operators on a Banach space E.LetL(E) denote the Banach space of all bounded linear operators on E.IfA is a closed operator on E, then the resolvent set of A is (A)={λ ∈ C :(λI − A) is invertible},theresolvent of A is (the operator-valued function) R(λ, A)=(λI − A)−1,λ∈ (A), and the spectrum of

A, σ(A), is the complement of the resolvent set.

Definition 1.9 The family (T (t))t≥0 of bounded linear operators T (t) on E is called a strongly continuous semigroup (or C0-semigroup) if it satisfies the follow- ing properties.

(i) T (0) = I,

(ii) T (t + s)=T (t)T (s) for all t, s ≥ 0,

(iii) The function T (t)x :[0, ∞) → E is continuous for every x ∈ E.

The generator A : D(A) ⊂ E → E of the semigroup (T (t))t≥0 is defined by

1 D(A)={x ∈ E : lim (T (h)x − x)exists} h→0+ h and 1 Ax := lim (T (h)x − x), (x ∈ D(A)). h→0+ h

It is well-known that A is a linear, closed and densely defined operator which determines the semigroup uniquely. Moreover, it is bounded if and only if the C0- semigroup generated by A is uniformly continuous (i.e.,limt→0+ T (t) − I =0). 25

The next theorem is a summary of well-known properties of a C0-semigroup. For details, we refer the reader to [17], [20], [23], [25], [35] and [36].

Theorem 1.10 Let (T (t))t≥0 be a C0-semigroup generated by A.Then

(i) There exists constants ω ≥ 0 and M ≥ 1 such that

T (t)≤Meωt for t ≥ 0, (8)

(ii) If x ∈ D(A),thenT (t)x ∈ D(A) and

d T (t)x = AT (t)x = T (t)Ax dt ∈ ≥ t ∈ (iii) For every x E and t 0, we have 0 T (s)xds D(A) and

 t T (t)x − x = A T (s)xds 0  t = T (s)Axds if x ∈ D(A). 0

(iv) For all λ ∈ C such that Reλ > ω,(whereω is given in formula (8)), we

have that (λ − A)−1 exists and

 ∞ (λ − A)−1x = e−λsT (s)xds. (9) 0

If the C0-semigroup (T (t))t≥0 is defined for all t ∈ R such that T (t + s)=

T (t)T (s) for all t, s ∈ R, then it is called a C0-group.

It is well known that a C0-semigroup T (t) can be extended to a C0-group if and only if T (t) is invertible for some (and hence for all) t>0. A C0-group is called an isometric group if T (t) =1foreveryt ∈ R. 26

Definition 1.11 Let (T (t))t≥0 be a C0-semigroup generated by the generator A.

Then the growth bound of (T (t))t≥0 is defined by

ωt ω(A):=inf{ω : ∃Mω s.t. T (t)≤Mωe ,t≥ 0}. (10)

It is well known that ω(A) = lim 1 ln T (t). A strongly continuous semigroup t→∞ t T (t) is called exponentially stable if ω(A) < 0. We have some properties about exponentially stable semigroups:

Theorem 1.12 The following are equivalent:

(1) (T (t))t≥0 is an exponentially stable semigroup

(2) lim T (t) =0 t→∞

(3) T (t0) < 1 for some t0 > 0.

(4) r(T (t1)) < 1 for some t1 > 0,wherer(T (t1)) is the spectrum radius of T (t1).

Example 1.13 Let M beaclosedsubspaceofBUC(R,E) which contains with every function all its translations (translation-invariant subspace). Then, the fam- ily of translation operators ϕt : M →Mdefined by (ϕtf)(s):=f(t + s) is an isometric C0 group on M whose generator is the differentiation operator D given by (Df)(t)=f (t) on its domain

D(DM)={f ∈M: f ∈M}.

It is well known and easily verified that σ(DM)=iR,ifM = BUC(R,E). 27

Example 1.14 Let u ∈ BUC(R,E) and Mu be the closed subspace of BUC(R,E) spanned by all translations of u.Then,S(t) is an isometric C0-group on Mu whose

generator is the differentiation operator DMu . It was proved in [52] that

σ(DMu )=sp(u).

Consider the abstract Cauchy problem ⎧ ⎨⎪ u (t)=Au(t),t≥ 0, (ACP1) ⎪ ⎩ u(0) = x ∈ E,

A classical solution of (ACP1) is a function u ∈ C1(R+,E) such that u(t) ∈ D(A) and (ACP1) holds. We have the following results which connect classical solutions of abstract Cauchy problems and C0-semigroups.

Theorem 1.15 Let A be a closed, densely defined operator with non-empty re- solvent set. Then, the following are equivalent:

(1) The operator A generates a C0-semigroup T (t).

(2) The abstract Cauchy problem (ACP1) is well posed, i.e, for every x ∈ D(A),

the abstract Cauchy problem (ACP1) has a unique classical solution and,

for every sequence {xn}⊂D(A),with lim xn =0, one has lim un(t)=0, n→∞ n→∞

where un(t) is the classical solution of (ACP1) corresponding to the initial

value xn.

(3) For every x ∈ D(A), the abstract Cauchy problem (ACP1) has a unique

classical solution defined by u(t):=T (t)x. 28

With the operator A associates a non-homogenous differential equation ⎧ ⎨⎪ u (t)=Au(t)+f(t),t≥ 0, (ACP2) ⎪ ⎩ u(0) = x ∈ E, where f :[0,T) → E is a (Bochner) integrable function.

Definition 1.16 Afunctionu :[0,T) → E is a classical solution of (ACP2) on

[0,T) if u is continuous on [0,T), continuously differentiable on (0,T), u(t) ∈

D(A) for 0

The following is well known fact:

Theorem 1.17 If f ∈ L1(0,T : E), then for every x ∈ E the initial value problem

(ACP2) has at most one solution. If it has a solution, this solution is given by

 t u(t)=T (t)x + T (t − s)f(s) ds. 0

The definition of the mild solution of the initial value problem (ACP2) is given in the following form:

Definition 1.18 Let A be the generator of a C0-semigroup T (t), x ∈ E and f ∈ L1(0,T : E). The function u given by

 t u(t)=T (t)x + T (t − s)f(s) ds, 0 ≤ t ≤ T, 0 is the mild solution of the initial value problem (ACP2) on [0,T].

Note that for (ACP2) to have a classical solution, we have to require more than just the continuity of f. For example, let us recall the following known results: 29

Theorem 1.19 [36, Corollary 4.2.5] Let A be the generator of a C0-semigroup

T (t).Iff(s) is continuously differentiable on [0,T], then the initial value problem

(ACP2) has a solution u on [0,T) for every x ∈ D(A).

Theorem 1.20 [36, Corollary 4.2.6] Let A be the generator of a C0-semigroup

T (t) and f ∈ L1(0,T : E) be continuous on (0,T).Iff(s) ∈ D(A) for 0

(ACP2) has a solution u on [0,T).

One of the objectives of the present dissertation is to extend the preceding theory to the multi-parameter semigroups of bounded linear operators. For sim- plicity of presentation, we give statements in the following chapters for two- parameter semigroups. Note that if T (s, t) is a two-parameter semigroup, then

T (s, t)=T1(s)T2(t)whereT1(s)andT2(t) are two commuting C0-semigroups. For more information see [25]. 30

1.4 The operator equation AX − XB = C

Let A : D(A) ⊂ E → E and B : D(B) ⊂ F → F be closed densely defined, generally unbounded, linear operators on Banach spaces E and F , respectively and let C : F → E be a bounded linear operator.

Definition 1.21 A bounded linear operator X : F → E is called a solution of the operator equation

AX − XB = C (11) if for every f ∈D(B), we have Xf ∈D(A) and AXf − XBf = Cf.

As opposed to the case of bounded A and B, there exist unbounded operators

A and B such that σ(A) ∩ σ(B)=∅, but equation (11) does not have a solution.

Equation (11) has been considered by many authors. It was first studied inten- sively for bounded operators by Daleckii and Krein [16] and Rosenblum [44]. For unbounded operators, the case when A and B are generators of C0-semigroups was considered in [6] and [51] and the general case was considered in [45] and [55].

The following Theorem is a brief summary of known results about the unique solvability of equation (11)

Theorem 1.22 (1) If A and B are bounded operators, the equation (11) has a

unique solution for every bounded C if and only if σ(A) ∩ σ(B)=∅.Inthis

case, the solution is given by  1 X = − (λ − A)−1C(λ − B)−1dλ, (12) 2πi Γ 31

where Γ is a Cauchy contour which separates σ(A) and σ(B) such that σ(B)

is inside of Γ.

Note: If we take Γ as a contour around σ(A), then the solution is the same

integral (now over new Γ) but with positive sign, i.e.  1 X = (λ − A)−1C(λ − B)−1dλ. (13) 2πi Γ

(2) If A and −B are generators of C0-semigroup T (t) and S(t) with growth bound ω(A) and ω(−B) respectively such that ω(A)+ω(−B) < 0, then for

every bounded C,equation(11) has a unique solution, which is given by

 ∞ X = − T (t)CS(t)dt. (14) 0

(3) If A and −B are generators of C0-semigroups with σ(A) ∩ σ(B)=∅ and if

one of them is the generator of an analytic semigroup, then equation (11)

has a unique solution.

(4) If A is the generator of an exponentially dichotomic C0-semigroup T (t) and

−B is the generator of an isometric C0-group S(t), then for every C,equa-

tion (11) has a unique solution given by

 ∞ X = − GA(t)CS(t)dt, (15) −∞

where ⎧ ⎨⎪ T (t)P, t ≥ 0 GA(t)=⎪ ⎩ −T (t)(I − P ),t<0

is the Green function. Here, P denotes the dichotomic projection. 32

(5) If A and B are closed operators with disjoint spectra and if one of them is

bounded, say B, then for every C,equation(11) has a unique solution given

by  1 X = − (λ − A)−1C(λ − B)−1dλ. 2πi Γ

where Γ is a Cauchy contour around σ(B) and disjoint from σ(A).

(6) If for every bounded operator C,equation(11) has a unique solution, then

σ(A) ∩ σ(B)=∅.

(7) If A and B are closed unbounded operators, then the condition σ(A)∩σ(B)=

∅ is, in general, not sufficient for the solvability of (11).

Consider the operator τA,B : X → AX − XB which is defined as follows:

D(τA,B)):={X ∈ L(F, E):XD(B) ⊂D(A),

there exists Y ∈ L(E,F):AXf − XBf = Yf,∀f ∈D(B)}, and

τA,B X = Y.

Lemma 1.23 τA,B is a closed operator on L(F, E).

In general, the domain of τA,B is not dense. This is the case, for instance, if

E = F , A is bounded and B is unbounded, since the domain D(τA,B )doesnot contain any bounded invertible operator X ( and the set of bounded invertible operators is open in L(E)). 33

Theorem 1.24 (Arendt-Raebinger-Sourour)[6].

(i) σ(A) − σ(B) ⊂ σ(τA,B).

(ii) If one of the operator A or B is bounded, then σ(A) − σ(B)=σ(τA,B).

(iii) If A and B generate C0-semigroups one of which is analytic, then σ(A) −

σ(B)=σ(τA,B).

Remark 1.25 (a) Part (i) implies that if for every C ∈L(F, E), equation (11) has a unique bounded solution, then necessarily σ(A) ∩ σ(B)=∅.

(b) The operator τA,B can be regarded as sum of two commuting operators

AX := AX and BX := −BX, and the above result can be stated (and proved in the same way) for sums of commuting operators.

There is a connection between the solvability of the operator equation (11) and the admissibility of some spaces with respect to the following differential equation:

u(t)=Au(t)+f(t),t∈ R (16)

Assume that M is a closed translation-invariant subspace of the space

BUC(R,E) which satisfies the following additional assumption :

∀C ∈L(M,E)and∀f ∈M, the function φ(t):=CS(t)f ∈M. (17)

Definition 1.26 M⊂BUC(R,E) is called a regularly admissible for equa- tion (16), if for every f ∈M, there exists a unique mild solution u ∈Mof equation (16). 34

Schuler and Vu, in [54], has proved the following theorem:

Theorem 1.27 Let M be as above. Assume that A is a generator of C0- semigroup T (t), D is the differentiation operator and δ0 is the Dirac operator

M ( DM and δ0 are respectively the restriction of D and δ0 to the space M ). Then, the following are equivalent:

(i) M is regularly admissible with respect to (16).

(ii) The operator equation :

M AX − XDM = −δ0 has a unique bounded solution.

(iii) For every bounded linear operator C : M−→E, the operator equation

AX − XDM = C has a unique bounded solution. 35

2 On mild solutions of differential equations

2.1 On mild solutions of differential equations with multi-

time

Let A and B be generators of two commuting semigroups T1(s)andT2(t)ona

Banach space E, respectively.

Note that if dim(E) < ∞,thenA and B are commuting matrices and the

As Bt corresponding semigroups are T1(s)=e and T2(t)=e , respectively. Most of our results seem to be new even in this particularly important finite dimensional case.

Consider the equations

⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t)

u(s0,t0)=x

We introduce the following definitions of solutions, which are natural extensions of definitions from equations with one-dimensional time.

Definition 2.1 Afunctionu is called a classical solution of (∗)onR × R if u is continuously differentiable on R × R, u(s, t) ∈ D(A) ∩ D(B) for s, t ∈ R and (∗) is satisfied on R × R. 36

Definition 2.2 Afunctionu is called a classical solution of (∗)on[S0,S]×[T0,T] if u is continuous on [S0,S] × [T0,T], continuously differentiable on (S0,S) ×

(T0,T), u(s, t) ∈ D(A) ∩ D(B) for S0 ≤ s ≤ S, T0 ≤ t ≤ T and (∗)issatisfiedon

[S0,S] × [T0,T].

Definition 2.3 Afunctionu is called a classical solution of (∗)onR+ × R+ if u is continuous on [0, ∞) × [0, ∞), continuously differentiable on (0, ∞) × (0, ∞), u(s, t) ∈ D(A) ∩ D(B) for 0 ≤ s<∞, 0 ≤ t<∞ and (∗)issatisfiedon [0, ∞) × [0, ∞).

Remark 2.4 Let u be a solution of the homogeneous abstract Cauchy problem

(HACP) ⎫ ∂u(s,t) ⎪ ∂s = Au(s, t) ⎬ ∗∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)

u(0, 0) = x

Then, u(s, t)=T1(s)u(0,t) and u(s, t)=T2(t)u(s, 0), which imply that u(s, t)=

+ T1(s)u(0,t)=T1(s)T2(t)u(0, 0) = T1(s)T2(t)x for every s, t ∈ R .Fromthisit follows that the solution of the initial value problem for the homogeneous equation

(i.e. f1 ≡ 0 and f2 ≡ 0)withinitialvalueu(0, 0) = 0 is the trivial solution u ≡ 0.

Proposition 2.5 If T2(t)x ∈ D(A) and T1(s)x ∈ D(B), then the HACP (∗∗) has a unique classical solution. 37

Proof: Since T2(t)x ∈ D(A)andT1(s)x ∈ D(B), then u(s, t)=T1(s)T2(t)x is differentiable w.r.t. s and t on R+ ×R+, respectively. Thus, u satisfies the HACP.

+ + If u1 and u2 are two solutions on R × R ,thenu1 − u2 is also a solution of

HACP with initial value x = 0. By the remark, u1 − u2 ≡ 0. i.e. u1 ≡ u2 which proves the uniqueness of the HACP.

The question of asymptotic behavior ( stability and almost periodicity ) of (∗∗) is investigated by Batty-Vu in [12]. Indeed, they investigated the stability and the almost periodicity of strongly continuous representations of abelian semigroups, but their results include the most important particular case of multi-parameter semigroups.

The following is the variation of parameter formula for differential equations with multi-time. It is analogous to the variation of parameter formula for the classical case, but it has some specific features for the multi dimensional case.

Theorem 2.6 (I) Let u(s, t) be a classical solution of ( ∗ )onR × R, T2(t − w)u(v,w) ∈ D(A) and T1(s − v)u(v,w) ∈ D(B) for every s ≥ v,t ≥ w.Then,

 t u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − s0) T2(t − w) f2(s0,w) dw t0

 s + T1(s − v) f1(v,t) dv (s ≥ s0,t≥ t0)(1) s0 and, simultaneously,

 s u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − v) T2(t − t0) f1(v,t0) dv s0

 t + T2(t − w) f2(s, w) dw (s ≥ s0,t≥ t0)(2) t0 38

+ + (II) Let u(s, t) be a (classical) solution of ( ∗ )onR × R , T2(t − w)u(v,w) ∈

D(A) and T1(s − v)u(v,w) ∈ D(B) for every s ≥ v,t ≥ w.Then,

 t u(s, t)=T1(s) T2(t) u(0, 0) + T1(s) T2(t − w) f2(0,w) dw 0  s + T1(s − v) f1(v,t) dv (3) 0 and, simultaneously,

 s u(s, t)=T1(s) T2(t) u(0, 0) + T1(s − v) T2(t) f1(v,0) dv 0  t + T2(t − w) f2(s, w) dw (4) 0

Proof.

Let u(s, t) be a solution of ( ∗ ). Consider y(v,w)=T1(s − v)T2(t − w)u(v,w), where s, t are fixed and v ∈ (−∞,s],w∈ (−∞,t].

Since T2(t−w)u(v,w) ∈ D(A)andT1(s−v)u(v,w) ∈ D(B) for every s ≥ v,t ≥ w,wehavethaty(v,w) is differentiable and

∂y(v,w) ∂u(v,w) = −AT1(s − v)T2(t − w)u(v,w)+T1(s − v)T2(t − w) ∂v ∂v

Hence

∂y(v,w) = T1(s − v)T2(t − w)f1(v,w)(5) ∂v Similarly 39

∂y(v,w) = T1(s − v)T2(t − w)f2(v,w)(6) ∂w

Integrate (5) with respect to v from s0 to s where −∞

  s ∂y(v,w) s dv = T1(s − v)T2(t − w)f1(v,w) dv = y(s, w) − y(s0,w) s0 ∂v s0

On the other hand,

y(s, w) − y(s0,w)=T2(t − w)u(s, w) − T1(s − s0)T2(t − w)u(s0,w) ,s≥ s0,t≥ w.

Therefore

 s T2(t − w)u(s, w) − T1(s − s0)T2(t − w)u(s0,w)= T1(s − v)T2(t − w)f1(v,w) dv s0

By letting w = t,weobtain

 s u(s, t)=T1(s − s0)u(s0,t)+ T1(s − v)f1(v,t) dv (7) s0 By doing the same process to equation (6), we get:

 t u(s, t)=T2(t − t0)u(s, t0)+ T2(t − w)f2(s, w) dw (8) t0 From this it follows

 t u(s0,t)=T2(t − t0)u(s0,t0)+ T2(t − w)f2(s0,w) dw (9) t0 40

From (7) and (9) we have

 t u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − s0) T2(t − w) f2(s0,w) dw t0  s + T1(s − v) f1(v,t) dv , s ≥ s0,t≥ t0. s0 From (7) we have

 s u(s, to)=T1(s − s0)u(s0,to)+ T1(s − v)f1(v,to) dv s0

By substitute it in (8), we get

 s u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − v) T2(t − t0) f1(v,t0) dv s0  t + T2(t − w) f2(s, w) dw (s ≥ s0,t≥ t0). t0 + + Analogously, if u(s, t) is a solution of ( ∗ )onR ×R , T2(t−w)u(v,w) ∈ D(A) and T1(s − v)u(v,w) ∈ D(B) for every s ≥ v,t ≥ w,then

 t u(s, t)=T1(s) T2(t) u(0, 0) + T1(s) T2(t − w) f2(0,w) dw 0  s + T1(s − v) f1(v,t) dv 0 and

 s u(s, t)=T1(s) T2(t) u(0, 0) + T1(s − v) T2(t) f1(v,0) dv 0  t + T2(t − w) f2(s, w) dw 0 41

Remark 2.7 From the previous theorem it is clear that in order for the ACP to have a solution on R × R (resp, on R+ × R+ ), the following condition must be satisfied:

 t  s T1(s − s0) T2(t − w) f2(s0,w) dw + T1(s − v) f1(v,t) dv (10) t0 s0  s  t = T1(s − v) T2(t − t0) f1(v,t0) dv + T2(t − w) f2(s, w) dw (11) s0 t0 and, respectively,

 t  s T1(s) T2(t − w) f2(0,w) dw + T1(s − v) f1(v,t) dv (12) 0 0

 s  t = T1(s − v) T2(t) f1(v,0) dv + T2(t − w) f2(s, w) dw (13) 0 0

The above theorem justifies the following definitions.

1 Definition 2.8 Let u(s0,t0)=x ∈ E, f1( ·,t) ∈ L ([s0,s],E) and f2(s, ·) ∈

1 L ([t0,t],E),wheres0 ≤ s, t0 ≤ t. Then, the continuous function u(s, t) defined by

 t u(s, t)=T1(s − s0) T2(t − t0) x + T1(s − s0) T2(t − w) f2(s0,w) dw t0

 s + T1(s − v) f1(v,t) dv s0  s = T1(s − s0) T2(t − t0) x + T1(s − v) T2(t − t0) f1(v,t0) dv s0  t + T2(t − w) f2(s, w) dw t0

is called a mild solution on [s0, ∞) × [t0, ∞) of the ACP: 42

∂u(s, t) = Au(s, t)+f1(s, t) ∂s ∂u(s, t) = Bu(s, t)+f2(s, t) ∂t

u(s0,t0)=x

Definition 2.9 Afunctionu(s, t) defined on R × R is called a mild solution on

R × R of the differential equations

∂u(s, t) = Au(s, t)+f1(s, t) ∂s ∂u(s, t) = Bu(s, t)+f2(s, t) ∂t if for every s0 ∈ R and every t0 ∈ R, u(s, t) (t ≥ t0,s≥ s0) is a mild solution on

[s0, ∞) × [t0, ∞) of the ACP.

Equivalently, u(s, t) is a mild solution on R × R if:

 t u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − s0) T2(t − w) f2(s0,w) dw t0

 s + T1(s − v) f1(v,t) dv (∀t ≥ t0, ∀s ≥ so) s0  s = T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − v) T2(t − t0) f1(v,t0) dv s0  t + T2(t − w) f2(s, w) dw (∀t ≥ t0, ∀s ≥ so) t0 43

Remark 2.10 (i) If T2(t)=I ∀t ≥ 0 (i.e. B =0)andf2 ≡ 0, then we have the one parameter case (dealing with t as a constant). Thus, our definition of mild solution coincide of the well-known one of u = Au + f.(cf. [36])

(ii) It is clear then not every mild solution is a classical one.

We can show that the continuity of f1 and f2, in general, is not sufficient to ensure the existence of solutions of ( ∗ )evenforu(0, 0) = x ∈ D(A) ∩ D(B).

Example 2.11 Let A be a generator of C0-semigroup T1(s) and let x ∈ E be such that T1(s)x/∈ D(A) for any s ≥ 0.

Let f1(s, t)=T1(s)x.Then,f1(s, t) is continuous for s ≥ 0.

Let T2(t)=I ∀t ≥ 0, f2 ≡ 0.

Consider ∂u(s, t) = Au(s, t)+T1(s)x (14) ∂s u(0, 0) = 0

We claim that the above equation has a mild solution, but no classical solution even x = u(0, 0) = 0 ∈ D(A).

Indeed, the mild solution of (14) is

 s u(s, t)= T1(s − v)T1(v)xdv = sT1(s)x 0

But sT1(s)x is not differentiable for s>0 and therefore cannot be the solution of (14).

Remark 2.12 : From the above example, in order to prove the existence of so- lution of ( ∗ ) we have to require more than just the continuity of f1 and f2. 44

Assume that A and B are bounded linear operators and f1 and f2 are differ- entiable w.r.t. t and s, respectively. Assume also that ( ∗ ) has a solution u(s, t) which has partial derivatives of second order. In the sequel, we use the same notation A (respectively, B) to denote the operator acting in the function space in the pointwise manner, i.e. (Af)(s, t):=Af(s, t)(and(Bf)(s, t):=Bf(s, t), respectively).

Then Eq. (∗) can be written as (Ds − A)u = f1, (Dt − B)u = f2, which implies that

∂ ∂ ( − A)f2 =( − B)f1 (15) ∂s ∂t

This motivates the following definitions.

Definition 2.13 f1 and f2 are called (classical) solution of the operator equa- tion (15) if f2(s, t) ∈ D(A) , f1(s, t) ∈ D(B) , f1andf2 are differentiable w.r.t. t and s, respectively and (15) is satisfied.

Definition 2.14 f1 and f2 are called mild solution of the operator equation (15)

1 2 if ∀ϕ ∈ L (R ) and suppϕˆ is compact, we have that f˜1 = f1 ∗ ϕ and f˜2 = f2 ∗ ϕ ∂ − ˜ ∂ − ˜ are classical solutions. i.e., ( ∂t B)f1 =(∂s A)f2 is satisfied.

The following proposition relates the differential condition with the integral condition.

Proposition 2.15 Assume that there exists a mild solution u(s, t) of (∗).Then ∂ − ∂ − ( ∂t B)f1 =(∂s A)f2 in the mild sense. 45

Proof: The existence of a mild solution implies, as noted above, that (10) =

1 2 (11) holds. Let ϕ ∈ L (R ). We must show that f˜1 = f1 ∗ ϕ, f˜2 = f2 ∗ ϕ are ∂ − ˜ ∂ − ˜ classical solutions of ( ∂t B)f1 =(∂s A)f2.

From (10) = (11), it follows that (10) = (11) also holds for f1 ∗ ϕ and f2 ∗ ϕ. i.e.,

 t  s T1(s − s0) T2(t − w)(f2 ∗ ϕ)(s0,w) dw + T1(s − v)(f1 ∗ ϕ)(v,t) dv t0 s0  s  t = T1(s − v) T2(t − t0)(f1 ∗ ϕ)(v,t0) dv + T2(t − w)(f2 ∗ ϕ)(s, w) dw s0 t0 Takeu ˜ = u ∗ ϕ.Since˜u is smooth, it can be easily verified that it is classi- cal solution of the corresponding equation (with f˜1, f˜2 as in-homogeneous parts).

Therefore, ∂ ∂ ( − B)f˜1 =( − A)f˜2 ∂t ∂s

Under suitable conditions, we will show that the mild solution is a classical one.

Because of the similarity, we will prove it for the case of R+ × R+.

+ + Theorem 2.16 Let u(s, t) beamildsolutionof(∗ )onR × R , T2(t)x, T2(t − w)f2(0,w), f1(v,t) ∈ D(A), T1(s)x, T1(s − v)f1(v,0), f2(s, w) ∈ D(B) for every s ≥ v,t ≥ w, x ∈ E and u is differentiable. Then, u is a classical solution of (∗).

Proof: Since  t u(s, t)=T1(s) T2(t) x + T1(s) T2(t − w) f2(0,w) dw 0  s + T1(s − v) f1(v,t) dv 0 46

Then

 ∂u(s, t) t = AT1(s) T2(t) x + A T1(s) T2(t − w) f2(0,w) dw ∂s 0  s +f1(s, t)+A T1(s − v) f1(v,t) dv 0 i.e. ∂u = Au + f1 ∂s

and since

 s u(s, t)=T1(s) T2(t) x + T1(s − v) T2(t) f1(v,0) dv 0  t + T2(t − w) f2(s, w) dw 0 Then

 ∂u(s, t) s = BT1(s) T2(t) x + B T1(s − v) T2(t) f1(v,0) dv ∂t 0  t +f2(s, t)+B T2(t − w) f2(s, w) dw 0 i.e. ∂u = Bu + f2 ∂t Thus, (∗) is satisfied and u is a classical solution of (∗).

For x ∈ E, we could define the mild solution of (∗)onR+ ×R+ by the following continuous function:

 t u(s, t)=T1(s) T2(t) x + T1(s) T2(t − w) f2(0,w) dw 0 47

 s + T1(s − v) f1(v,t) dv 0 and in such case, we have the following theorem to ensure the classical solution.

Theorem 2.17 Under the assumptions of the above theorem if T1(s)f2(0,w) ∈

D(B) and f2(s, t) ∈ D(A),thenu is a classical solution provided one of the following conditions satisfied:

(i) The differentiation condition (15) is satisfied.

(ii) The integral condition (12) ≡ (13)is satisfied.

Proof : We will show that u satisfied ( ∗ ).

(i)Sinceu is given by:

 t u(s, t)=T1(s) T2(t) u(0, 0) + T1(s) T2(t − w) f2(0,w) dw 0  s + T1(s − v) f1(v,t) dv 0 Then

 ∂u(s, t) t = AT1(s) T2(t) u(0, 0) + A T1(s) T2(t − w) f2(0,w) dw ∂s 0  s +f1(s, t)+A T1(s − v) f1(v,t) dv 0 i.e. ∂u = Au + f1 ∂s

also 48

 ∂u(s, t) t = BT1(s) T2(t) u(0, 0) + T1(s) f2(0,t)+B T1(s) T2(t − w) f2(0,w) dw ∂t 0

 s ∂f1(v,t) + T1(s − v) dv 0 ∂t We make use of the condition,

 ∂u(s, t) t = BT1(s) T2(t) u(0, 0) + T1(s) f2(0,t)+B T1(s) T2(t − w) f2(0,w) dw ∂t 0

 s ∂f2(v,t) + T1(s − v)[Bf1(v,t)+ − Af2(v,t)] dv 0 ∂v

Then

  ∂u(s, t) t s = BT1(s) T2(t) u(0, 0)+B T1(s) T2(t−w) f2(0,w) dw+ T1(s−v) Bf1(v,t) dv ∂t 0 0

 s  s ∂f2(v,t) +T1(s) f2(0,t)+ T1(s − v) dv − T1(s − v) Af2(v,t)] dv 0 ∂v 0 Then

∂u(s, t) = Bu(s, t) ∂t s ∂ +T1(s) f2(0,t)+ (T1(s − v) f2(v,t)) dv 0 ∂v Thus

∂u(s, t) = Bu(s, t) ∂t

+T1(s) f2(0,t)+f2(s, t)+T1(s) f2(0,t)) 49 i.e.

∂u(s, t) = Bu(s, t)+f2(s, t) ∂t (ii) the proof is similar to the previous theorem. 50

3 Regular admissibility of subspaces

3.1 Operator equations AX −XDs = C and BX −XDt = CT

Let A and B be generators of two commuting semigroups T1(s)andT2(t)ona

Banach space E, respectively.

Consider the following equations:

⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t) Notations : Let M be a closed translation-invariant subspace of

BUC(R × R,E), i.e., if f(s1,t1) ∈M,thenS(s, t)f(s1,t1)=f(s+s1,t+t1) ∈M where S(s, t) is the translation operator. Also, the Dirac operator, δ0, is defined as δ0f(s, t):=f(0, 0) . We assume that M is invariant under any bounded linear operator commuting with S(s, t)wheres, t ∈ R. This condition can be formulated in the following manner.

∀C ∈L(M,E)and∀f ∈M, the function φ(s, t):=CS(s, t)f ∈M (1)

In addition, we assume that M satisfies the following condition: for every f1 ∈Mthere exists a unique f2 ∈Msuch that (Ds − A)f2 =(Dt − B)f1 in the mild sense. Let T be defined by T f1 = f2. By the Closed Graph Theorem, T is a bounded linear operator on M and it is not difficult to see that T commutes with translation operators. 51

Definition 3.1 M⊂BUC(R × R,E) is called regularly admissible if for every f1 ∈Mthere exist a unique mild solution u ∈Mof equation (∗).

Definition 3.2 M⊂BUC(R × R,E) is called admissible if for every f1 ∈M there exist a ( mild) solution u ∈Mof equation (∗).

Let C be a bounded linear operator from M into E. We define solution of the system of operator equations

M AX − XDs = C

M BX − XDt = CT

as a bounded operator X : M→E such that for every f1 ∈M, the following holds

M AXf1 − XDs f1 = Cf1

M BXf1 − XDt f1 = Cf2 where f2 = T f1.

Theorem 3.3 Let M be as above and consider the following properties:

(i) M is admissible.

(ii) The system ⎧ ⎨⎪ M M AX − XDs = −δ0 (S1) ⎩⎪ M M BX − XDt = −δ0 T has a unique bounded solution. 52

(iii) For every bounded linear operator C : M−→E, the system ⎧ ⎨⎪ M AX − XDs = C (S2) ⎩⎪ M BX − XDt = CT has a unique bounded solution.

(iv) M is regularly admissible.

Then, (iv) =⇒ ( (iii) ⇐⇒ (ii) )=⇒ (i).

M M M Where −δ0 is the Dirac operator restricted to M and Ds and Dt are the partial differentiation operators w.r.t. s and t restricted to M, respectively.

Proof :

First, we will show that (ii) and (iii) are equivalent. Then, (iv) =⇒ (ii) =⇒ (i).

(ii)=⇒ (iii). It follows from the uniqueness of the solution of the system (S1) that X = 0 is the only solution of the following system

M AX − XDs =0

M BX − XDt =0

Otherwise, the uniqueness of the solution of (S1) will fail. Therefore, a solution of the system ⎧ ⎨⎪ M AZ − ZDs = C ⎩⎪ M BZ − ZDt = CT is unique if it exists. 53

Let X be the unique bounded solution of the system (S1) and C be a given bounded linear operator. By the assumption (1), we can define operator

Z : M→E byZf = Xf

where f= −CS(s, t)f.

Thus

M M AZf1 − ZDs f1 = −AXCS(s, t)f1 + XDs CS(s, t)f1

M = δ0 CS(s, t)f1 = Cf1, ∀f1 ∈ D(Z) i.e.,

M AZ − ZDs = C

Similarly

M BZ − ZDt = CT i.e. Z satisfies (S2). Moreover, it is a unique bounded solution of system (S2).

(iii)=⇒ (ii). This is trivial.

(iv)=⇒ (ii). Let M be regularly admissible. Let f1,f2 ∈Msuch that

(Ds − A)f2 =(Dt − B)f1 and u is the unique solution of (∗). Let G : M−→M be the bounded operator defined by Gf1 = u where u is the unique solution in

M of the equations (∗)withf1 ∈M. Define the operator X : M→E by

Xf1 =(Gf1)(0, 0). Since, the differential equations are autonomous, it follows that G commutes with S(s, t). In particular, (Gf1)(s, t)=XS(s, t)f1.

M M Therefore, from f1 ∈ D(Ds ∩Dt ) it follows that (Gf1)(s, t) is differentiable and 54

∂ (Gf1)(s, t)=A(Gf1)(s, t)+f1(s, t) ∂s so that ∂ (Gf1)(0, 0) = A(Gf1)(0, 0) + f1(0, 0) ∂s Hence

M M XDs f1 = AXf1 + δ0 f1

Or

M M M AXf1 − XDs f1 = −δ0 f1 ∀f1 ∈ D(Ds )

By a similar argument , we have

∂ (Gf1)(s, t)=B(Gf1)(s, t)+f2(s, t)=B(Gf1)(s, t)+T f1(s, t) ∂t M M which implies BXf1 − XDt f1 = −δ0 T f1 . This means that X is a bounded solution of the system (S1).

On the other hand, if X is a bounded solution of the system (S1), then for every f1 ∈Mthe function u ∈Mdefined by u(s, t)=XS(s, t)f1 is a mild solution of equations (∗) as follows:

∂u(s, t) M M = XD S(s, t)f1 =(AX + δ )S(s, t)f1 = Au(s, t)+f1(s, t) ∂s s 0 and

∂u(s, t) M M = XD S(s, t)f1 =(BX + δ T )S(s, t)f1 = Bu(s, t)+f2(s, t) ∂t t 0 55

Since the solution of (∗)inM is unique for every f1 ∈Mit follows that the solution X of the system (S1) is unique.

(ii) =⇒ (i)

Suppose X is a solution of ⎧ ⎨⎪ M M AX − XDs = −δ0 ⎩⎪ M M BX − XDt = −δ0 T

Let f1 ∈Mand u(s, t):=XS(s, t)f1. By the argument in ((iv)=⇒ (ii)) and the assumptions on M,wehavethatu is a mild solution in M of Eq. (∗).

Corollary 3.4 Assume that M⊂BUC(R × R,E) is a closed translation- invariant subspace satisfying the assumption (1) which is regularly admissible.

If N is another translation-invariant subspace satisfying the assumption (1) and

N⊂M,thenN also is regularly admissible.

Proof : It follows from the fact that if M is regularly admissible and f1 ∈N, then the unique solution u in M of equations (∗)isu(s, t)=XS(s, t)f1 which belongs to N by the assumption (1) . 56

3.2 Admissibility of difference equations

Consider the following equation:

xn+1 = Axn + yn ∀n ∈ Z (2)

Notations : x := (xn)n∈Z , y := (yn)n∈Z and z := (zn)n∈Z (wealsodenote

(x)n ≡ xn).

∞ Let l (Z,E) be the Banach space of all bounded sequences x := (xn)n∈Z with the sup-norm. Consider the shift operator S : l∞(Z,E) → l∞(Z,E)viaS :

(xn)n∈Z → (xn+1)n∈Z (thus,(Sx)n = xn+1). Below, for convenience of notation,

m we also use the notation ϕm = S , i.e. (ϕmx)n =(x)n+m.LetM be a subspace of l∞(Z,E) which is translation invariant, i.e. invariant with respect to the shift operator S. We denote the restriction of S on M by SM and define the Dirac

∞ operator δ0 : l (Z,E) → E by δ0x := x0.

Definition 3.5 We call M⊂l∞(Z,E) a regularly admissible subspace w.r.t. (2) if for every y ∈M, there exists a unique solution x ∈Mof equation (2).

We make the following assumption : M is invariant under any bounded linear operator commuting with S (and hence with all ϕn, n ∈ Z).

k A sequence x := (xn)n∈Z is called a complete trajectory under A if xn+k = A xn for every n ∈ Z and every k ∈ Z+. Below we denote by Γ the unit circle, i.e.

Γ:={λ ∈ C : |λ| =1}.

For a bounded sequence x := (xn)n∈Z in E,itsCarleman transform x˜(z) is a function analytic on C\Γ defined by 57

⎧ ⎪ ∞ n−1 ⎨⎪ n=1 xnz , |z| < 1 x˜(z) = ⎪ ⎩ 0 n−1 − −∞ xnz , |z| > 1. Apointz on Γ is called a regular point of x,ifx˜(z) can be continued analytically into a neighborhood of z. The complement in Γ of the set of regular points is called the spectrum of x and denoted by Sp(x).

The following proposition is from [52].

Proposition 3.6 [52, prop. 5.3] Let A be a linear operator in E and x := (xn)n∈Z be a bounded complete trajectory under A.Then,Sp(x) ⊂ Aσ(A) ∩ Γ.

Lemma 3.7 Let M be admissible w.r.t (2) and assume that the above assumption holds. Let X be the bounded solution operator defined as Xy = x0 where x :=

(xn)n∈Z is the solution of (2) for given y := (yn)n∈Z and S be the shift operator.

Then, for y ∈M, XSy = SXy.

Proof: Given y ∈M, there exist a unique x satisfies (2). Also, it is not difficult to see that (Sx) is the unique solution of the same equation with y replaced by

(Sy):=(yn+1)n∈Z.

Let y := (yn)n∈Z ∈M.Then

SXy =(Sx0)=x1 and

XSy := X(Sy)n = X(y)n+1 =(x)1 = x1, which implies XSy = SXy. 58

Theorem 3.8 Let M be as above. Then, the following are equivalent:

(i) M is regularly admissible.

(ii) The operator equation

M AX − XSM = −δ0 (E1)

has a unique bounded solution.

(iii) For every bounded linear operator C : M−→E

AX − XSM = C (E2)

has a unique bounded solution.

Proof : (i)=⇒ (ii). Let M be admissible and G : M−→Mbe the bounded operator defined by Gy = x where x is the unique solution in M of the equa- tion (2) with given y ∈Mand (Gy)n =(x)n. It is not difficult to see that

G is linear and closed. Thus, by the Closed Graph Theorem, G is a bounded linear operator on M. ItisalsoeasytoseethatG commutes with SM. Define

Xy =(Gy)0 := x0. It can been seen that G commutes with ϕn; in particular,

(Gy)n = Xϕny. Therefore, from y ∈ D(SM) it follows that (Gy)n ∈ D(SM)and

SM(Gy)n =(Gy)n+1 = xn+1 = Axn + yn

= A(Gy)n + yn

In particular, by putting n =0weobtain

SM(Gy)0 = A(Gy)0 + y0 59 i.e.

M SMXy = AXy + δ0 y ∀y ∈ D(SM)

By Lemma 3.7 we have

M XSMy = AXy + δ0 y ∀y ∈ D(SM)

Hence

M AXy − XSMy = −δ0 y ∀y ∈ D(SM)

Thus, X is a bounded solution of the operator equation (E1).

On the other hand, if X is a bounded solution of the operator equation (E1), then for every y ∈ D(SM), the vector x ∈Mdefined by x := (xn)n∈Z where xn = Xϕny ∀n ∈ Z is a solution of equation (2). Indeed, we have

SMxn = SMXϕny = XSMϕny

M =(AX + δ0 )ϕny

M = AXϕny + ϕnδ0 y

= Axn + yn

Thus, for every y ∈Mthe sequence x := (xn)n∈Z,wherexn = Xϕny ∀n ∈ Z, is a solution in M of (2).

Since the solution in M is unique for every y ∈M, it follows that the solution

X of the operator equation (E1) is unique.

(ii)=⇒ (iii). It follows from the uniqueness of the solution of the operator equation (E1) that X = 0 is the only solution of the following operator equation:

AX − XSM =0 60

From this it follows that a solution of the operator equation AZ − ZSM = C is unique, if it exists. Let X be the unique bounded solution of the operator equation

(E1) and C be given bounded linear operator. By the assumption above, we can define operator

Z : M→E by Zy = Xy where (y)n = −Cϕny.

Thus

AZy − ZSMy = −AX(Cϕny)+XSM(Cϕny)=δ0(Cϕny)=Cy, ∀y ∈ D(Z)

i.e., AZ − ZSM = C.

(iii)=⇒ (i). Since for every C there exists a unique solution X of the operator equation (E2), then it follows that σ(A) ∩ σ(SM)=∅ see [6].

But, from part((i)=⇒ (ii)), the sequence x := (xn)n∈Z,where(x)n = Xϕny, is a solution in M of (2). We need to show that the solution is unique. Suppose there exist two solutions v := (vn)n∈Z and w := (wn)n∈Z in M of (2). Consider z := (zn)n∈Z where zn = vn − wn. Then, z is a solution of equation xn+1 = Axn in M. It follows from this and the definition of the complete trajectory that z is a complete trajectory under A . Hence , by [52, prop. 5.3], Sp(z) ⊂ Aσ(A) ∩ Γ. i.e., Sp(z) ⊂ σ(A). Since z ∈M,wehaveSp(z) ⊂ σ(SM). Thus, Sp(z)=∅, which implies that z =0. Therefore, the solution is unique in M and M is regularly admissible.

∞ Moreover, consider the shift operator (forward operator) S1 : l (Z,E) →

∞ l (Z,E)viaS1 :(xn)n∈Z → (xn−1)n∈Z (thus,(S1x)n = xn−1). Clearly,S1 61 is an invertible isometry . In [53], Vu assumed that M is a closed subspace of l∞(Z,E) which satisfies properties :

(1) S1M = M;

n (2) If C : M→E is a bounded linear operator, x ∈Mand yn = CS1 x,then

y ≡ (yn)n∈Z ∈M;

M (3) σ(S1 )={λ ∈ C : |λ| =1}.

M The restriction of S1 on M is denoted by S1 . He also defined the operator

M M δ1 : M→E by δ1 x := x−1 . The space of all almost periodic sequences is an example of a space M satisfying

(1) − (3).

Thus, Theorem 5 in [53] with the previous theorem can be joined in the following theorem:

Theorem 3.9 Let M be as above. Assume that A is a bounded linear operator on a Banach space E such that σ(A)∩Γ=∅. Then, the followings are equivalent:

(1) A is hyperbolic .

(2) For every invertible operator V : F → F such that σ(V ) ⊂ Γ and for every

bounded operator C : F → E, the operator equation

X − AXV = C

has a unique solution. 62

(3) For every isometric operator V : F → F and for every bounded operator

C : F → E, the operator equation

X − AXV = C

has a unique solution.

(4) For every isometric operator U : M→Mand for every bounded operator

C : M→E, the operator equation

X − AXU = C

has a unique solution.

(5) For every bounded operator C : M→E, the operator equation

M X − AXS1 = C

has a unique solution.

(6) The operator equation

M M X − AXS1 = δ1

has a unique solution.

(7) The operator equation

M AX − XSM = −δ0

has a unique solution.

(8) For every bounded linear operator C : M−→E,

AX − XSM = C

has a unique solution. 63

(9) For every sequence y := {yn}n∈Z in M, there exists exactly one sequence

x := {xn}n∈Z in M which is solution of the difference equation (2).

∞ (10) For every sequence y := {yn}n∈Z in l (Z,E), there exists exactly one se-

∞ quence x := {xn}n∈Z in l (Z,E) which is solution of the difference equa-

tion (2). 64

4 On the integral of almost periodic functions

4.1 On Basit’s question

A classical result of Bohl and Bohr for almost periodic functions states that if f(t) R t is a scalar-valued almost periodic function defined on and F (t)= 0 f(s) ds, then F is almost periodic if and only if it is bounded over R. This question plays an important role in the study of asymptotic behavior of differential equation : u(t)=Au(t)+f(t). See e.g. [4], [14], [15] and [24]. For applications to differential equations in Banach space, it is important to have generalizations of the Bohl-Bohr theorem to functions with values in a Banach space. However, a direct extension, without any further condition on the function f or the underlying Banach space

E, is not valid. The following example can be found in ( [3],[30, p.179]).

∞ Example 4.1 Let l be the space of bounded sequences x =(ξ1, ..., ξn, ...) with norm

x =sup|ξn|. n Consider a function f(t) with values in l∞,

f(t)={λn cos λnt} where λn > 0 and λn → 0. We will show that f(t) is an almost periodic function.

Take an arbitrary >0 and let 1 λn <  2 for n>n. Then for arbitrary real τ and n>n

sup |λn cos λn(t + τ) − λn cos λnt| <. (1) t 65

Since a finite sum of periodic functions is almost periodic, there exits a relatively dense set ∆ such that if τ ∈ ∆,then

sup |λn cos λn(t + τ) − λn cos λnt| < (n =1, 2, ..., n)(2) t Thus,(1) and (2) imply that

sup f(t + τ) − f(t) <, t i.e. f(t) is an almost periodic function.

Consider the integral of f(t)  t F (t)= f(s) ds = {sin λnt}. 0 Then F (t) is a bounded function, since

F (t) =sup| sin λnt|≤1. n However, F (t) is not an almost periodic function ( with values in l∞). To see this, consider the linear functionals

an(x)=ξn with norms equal to 1 . We have

|an(F (t + τ)) − an(F (t))|≤F (t + τ) − F (t), which implies that if F (t) were an almost periodic function then the set of numerical-valued almost periodic functions {sin λnt} would be equicontinuous and almost periodic, and hence is compact in C(−∞, ∞). However, this is impossi- ble since the limit of the sequence {sin λnt} must equal zero and zero cannot be the uniform limit of this sequence since for arbitrary n there exists a t such that

π λnt = 2 and hence sin λnt =1. 66

Since boundedness of the integral of a vector-valued almost periodic function does not imply, in general, its almost periodicity, it would be desirable to clarify under what conditions (besides boundedness of the integral) the almost periodicity of the integral would follow. There have been obtained numerous results with conditions either on the space E or on the function F . Below, we briefly recall some of such results.

The first condition was indicated by Bochner in [13] who proved the following theorem.

Theorem 4.2 Let E be a Banach space and f : R → E be almost periodic . t Let F (t)= 0 f(s) ds .Then,F is almost periodic if and only if its range set {F (t):t ∈ R} is totally bounded in E .

Amerio [3] proved that the Bohl-Bohr theorem is valid for vector-valued func- tions with values in a uniformly convex Banach space (in particular, in a Hilbert space).

Levitan [30] proved that if the indefinite integral of a vector-valued almost pe- riodic function is bounded and has uniformly convergent means, then it is almost periodic.

Kadets [27] proved that if the range of the indefinite integral of a vector-valued almost periodic function is weakly relatively compact, then it is almost periodic.

He also proved [28] that if the Banach space does not contain c0, then the bound- edness of the indefinite integral implies its almost periodicity. Let us also mention investigation by Basit [8], Doss [19] and Gunzler [24] of analogous questions for 67 almost periodic functions on locally compact abelian groups. See also [9], [45] and the references therein.

In 1971, Basit (see [8]) showed that for a complex valued almost periodic func- tion f(x, y) defined on the plane, the boundedness of the integral F (x, y)= x 0 f(t, y) dt does not imply its almost periodicity. In this section, we impose a condition on f that ensures the almost periodicity of F .

The proofs of the following lemmas are straightforward and therefore are omit- ted.

Lemma 4.3 If f(s, t) is an almost periodic function in the plane, then f(s, t0) is almost periodic in R for fixed t0 ∈ R.

Lemma 4.4 Let g(s) and h(t) be almost periodic in R.Then,g(s)+h(t) is an almost periodic function in the plane.

Corollary 4.5 If g(s) is an almost periodic function in R,thenG(s, t):=g(s) is an almost periodic function in the plane.

Proof: Let h(t) ≡ 0 in the previous lemma.

We also need the following which is a result of Loomis for scalar-valued functions in [32].

Theorem 4.6 (Loomis’ Theorem )[32, p.366] For the finite dimensional

Euclidean group : If f(x):=f(x1, ..., xn) is bounded scalar-valued function and

∂f ∂xi is almost periodic for every i,thenf is almost periodic. 68

Theorem 4.7 Let f(s, t) be an almost periodic function in the plane, F (x, y)= x 0 f(t, y) dt be bounded and suppose that there exists a scalar-valued almost peri- odic function g(x, y) on the plane such that

∂f ∂g (x, y)= (x, y) ∂y ∂x

Then, F (x, y) is a scalar-valued almost periodic function.

Proof: Let

 x F (x, y)= f(t, y) dt 0 ∂F Note that ∂x (= f(x, y)) is almost periodic on the plane.

Since  ∂F x ∂f = (t, y) dt ∂y ∂y  0 x ∂g = (t, y) dt 0 ∂t = g(x, y) − g(0,y) is almost periodic in the plane, it follows by Loomis’ theorem 4.6 that F is almost periodic in the plane.

Our next objective is to generalize the above theorem to functions with values in a Banach space E. As an intermediate step, we extend, in the next section,

Loomis’ theorem to the vector-valued case. 69

4.2 Loomis’ theorem for vector-valued almost periodic

functions

In this section we continue our consideration of integrals of almost periodic func- tions. Assume that u(s, t) satisfies the following equation (which is a special case of (∗)): ∂u(s, t) = f1(s, t) ∂s ∂u(s, t) = f2(s, t) ∂t

Let f1(s, t)andf2(s, t) be vector-valued almost periodic in the plane. Under what conditions will u(s, t) be a vector-valued almost periodic function in the plane? First, we note the following simple facts, which follow immediately from the definition of almost periodic functions. R τ Lemma 4.8 If f(t) is almost periodic in ,theng(t)= 0 f(t + s) ds is almost periodic for every τ ∈ R.

Lemma 4.9 If f(s, t) is almost periodic in the plane, then for fixed (a, b) ∈ R2 , a 0 f(s + v,t + b) dv is almost periodic in the plane.

Let us also consider the following conditions, which have been introduced by Kadets.

(C1) The Banach space does not contain a subspace isomorphic to c0.

(C2) f has weakly relatively compact range.

We recall the following theorem of Basit, [8]: 70

Theorem 4.10 If the vector-valued function f(t) satisfies the difference equation f(tγ) − f(t)=gγ (t), where for each γ ∈ G (G is a group) the function gγ (t) is almost periodic, then under one of the conditions (C1)-(C2) the function f(t) is almost periodic.

∂u(s,t) ∂u(s,t) Theorem 4.11 Let ∂s and ∂t be vector-valued almost periodic functions with values in a Banach space E. If one of the conditions (C1)-(C2) is satisfied, then u(s, t) is almost periodic.

Proof:

We will show that for each (a, b) ∈ R2, u(s + a, t + b) − u(s, t)isanalmost periodic function.

Let (a, b) ∈ R2. Then, we can express the difference as a sum of line integral.

u(s + a, t + b) − u(s, t)   a ∂u b ∂u = (s + v,t + b) dv + (s, t + w) dw 0 ∂v 0 ∂w a ∂u b ∂u But, by lemma 4.9, both 0 ∂v (s + v,t+ b) dv and 0 ∂w(s, t + w) dw are almost periodic functions in the plane. Thus, their sum is almost periodic.

By the Basit’s theorem (when G = R2), u(s, t) is almost periodic in the plane.

The above theorem can be reformulate in the following form: 71

Theorem 4.12 Consider ∂u(s, t) = f1(s, t) ∂s ∂u(s, t) = f2(s, t) ∂t

If f1 and f2 are vector-valued almost periodic in the plane and one of the conditions

(C1)-(C2) is satisfied, then the solution of the equations is almost periodic.

The following theorem extends theorem 4.7 for vector-valued functions under the conditions of Kadets. By using theorem 4.11, the proof is similar to that of theorem 4.7 and therefore is omitted.

Theorem 4.13 Let f(x, y) be a vector-valued almost periodic function and x F (x, y)= 0 f(t, y) dt be bounded. Assume that there exists a vector-valued almost periodic function g(x, y) on the plane such that

∂f ∂g (x, y)= (x, y) ∂y ∂x

If one of the conditions (C1)-(C2) is satisfied, then F (x, y) is a vector-valued almost periodic function.

Remark 4.14 An almost automorphic version of Loomis’ theorem also hold ( with the same proof). As a consequence, the almost automorphic version of The- orem 4.11 also holds. 72

4.3 An alternative form of the theorem on the integral of

almost periodic functions

Let f(s, t) be an almost periodic function with values in a Banach space E. In this section, we formulate conditions which guarantee the function F (s, t)= t s 0 0 f(x, y) dx dy is almost periodic.

Theorem 4.15 Let f(s, t) be an integrable almost periodic function. Assume that

∂f ∂f1 there exists an almost periodic function f1(s, t) such that ∂s = ∂t and an almost ∂f ∂f2 s t periodic function f2(s, t) such that ∂t = ∂s .IfF (s, t)= 0 0 f(v,w) dw dv is bounded, then F is almost periodic, provided one of the conditions (C1)-(C2) holds.

∂F ∂F Proof : We show that ∂s and ∂t are (vector-valued) almost periodic functions, which implies, by theorem 4.11, that F (s, t) is almost periodic.

∂F Differentiating ∂s with respect to s and with respect to t,wehave

 ∂2F t ∂f 2 = (s, w) dw ∂s 0 ∂s  t ∂f1 = (s, w) dw 0 ∂w

= f1(s, t) − f1(s, 0)

∂2F Thus, ∂s2 is a vector-valued almost periodic function. On the other hand, ∂2F ∂t∂s = f(s, t) is a vector-valued almost periodic function on the plane. Hence, by ∂F ∂F theorem 4.11, ∂s is a vector-valued almost periodic function. Analogously, ∂t is 73 a vector-valued almost periodic function. Therefore, applying theorem 4.11 again, we obtain that F (s, t) is a vector-valued almost periodic function.

By using theorem 4.6 in the above proof, we obtain a scalar-valued version of the above theorem, namely:

Theorem 4.16 Let f(s, t) be an integrable scalar-valued almost periodic function.

∂f ∂f1 Assume that there exists an almost periodic function f1(s, t) such that ∂s = ∂t

∂f ∂f2 and an almost periodic function f2(s, t) such that ∂t = ∂s .Then,F (s, t)= s t 0 0 f(v,w) dw dv is almost periodic if (and only if) F is bounded. 74

5 Almost periodic solutions of differential equa-

tions

5.1 The finite dimensional case

In this section, we consider equations (∗) in a finite-dimensional space E.Since all norms on E are equivalent, the results are valid regardless of what norm is

E equipped with. In particular, we may assume that E is equipped with the

Euclidean norm.

For convenience, from now on, let us denote a sequence {νn} by ν. Correspond- ingly, if limn→∞ f(νn + t)=g(t) then we write T νf = g.

We will need the following equivalent definition of almost periodic functions due to Bochner.

Theorem 5.1 (Bochner)

f is almost periodic if and only if for any two sequences β and ν , there exist β β
ν ν
subsequences of and of such that T β+ νf = T βT νf.

Theorem 5.2 Let

⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t) 2 If f1 and f2 are defined and bounded on R and there exists a bounded solution, then there exists a solution of Eq. (∗) with minimum norm. 75

Proof: Let u0 be a bounded solution. Take K to be any compact set containing the range of u0.

λ =inf{u : u(s, t) ∈ K ∀ (s, t) ∈ R2and u is a solution of (∗)}

2 λn =inf{ sup |u(s, t)| : u(s, t) ∈ K ∀ (s, t) ∈ R and u is a solution of(∗)} B((s,t),n)

Clearly, λn ≤ λn+1 and λ = limn→∞ λn.Letun be a solution such that

1 sup |un(s, t)|≤λn + B((s,t),n) n → Then, there is nk of integers so that unk (s, t) v(s, t) uniformly on all compact R2 ∗ | |≤ subsets of and v is a solution of ( ). Fixed (s, t), for large k : unk (s, t) λnk + 1 . Take the limit, we get : |v(s, t)|≤λ.Thus,v = λ since v(s, t) ∈ K ∀(s, t). nk

Theorem 5.3 Consider ⎧ ⎨⎪ Dsu = Au ⎩⎪ Dtu = Bu If u is a bounded solution on R2,thenu is almost periodic .

Proof: For constant matrices A and B, the solution will be in the following form u(s, t)=eAs eBt c where c is a constant (vector in E). Let M be spanned by Akc, Bkc, k =0, 1, .... Without loss of generality, one can assume that M = E.

Since u is bounded, it follows that eAs and eBt are bounded. But, eAs is a bounded solution of U (s)=AU(s). Thus, it is almost periodic. Similarly, eBt is almost periodic. Since, the almost periodicity on the line implies almost periodicity on the plane, both of them are almost periodic on the plane. Moreover, their multiplication is almost periodic, see [29]. Hence, u is almost periodic on the plane. 76

We also need the following lemma, which is implied by Theorem 5.3 since every almost periodic function generates an equicontinuous flow, hence is distal in Ellis sense. But we give an independent proof.

Lemma 5.4 If u is a non-trivial bounded solution of ⎧ ⎨⎪ Dsu = Au ⎩⎪ Dtu = Bu then there exists γ>0 such that u(s, t)≥γ for all (s, t) ∈ R2.

Proof: Arguing as in the proof of the above theorem, we can assume, without loss of generality, that u(s, t) is a complete trajectory of a bounded two-parameter group esA+tB on E. Since any bounded two-parameter group esA+tB is isometric

sA+tB in the new equivalent norm |x| =sups,t e x, the statement of the lemma immediately follows.

The following is generalization of the well-known theorem for differential equa- tions.

Theorem 5.5 Asolutionof(∗) is almost periodic if (and only if) it is bounded.

Proof: Assume that u0 is a bounded solution of (∗). Then, by theorem 5.2, there is a solution u(f1,f2) of minimum norm. If T α
f1 = g1 and T α
f2 = g2

uniformly, then there exists α ⊂ α such that v = T αu0 uniformly on compact sets and it is a bounded solution of ⎧ ⎨⎪ Dsu = Au + g1 ⎩⎪ Dtu = Bu + g2 77

Thus, every equations of the form ⎧ ⎨⎪ Dsu = Au + T αf1 ⎩⎪ Dtu = Bu + T αf2

have a solution of minimum norm u(T αf1,T αf2). We claim the solution with minimal norm is unique. Consider ⎧ ⎨⎪ Dsu = Au + g1 ⎩⎪ Dtu = Bu + g2

If u1 and u2 are distinct solutions of ⎧ ⎨⎪ Dsu = Au + g1 ⎩⎪ Dtu = Bu + g2

      u1+u2 such that u1 = u2 = u(g1,g2) ,then 2 is a solution of ⎧ ⎨⎪ Dsu = Au + g1 ⎩⎪ Dtu = Bu + g2

u1−u2 and 2 is non-trivial solution of ⎧ ⎨⎪ Dsu = Au ⎩⎪ Dtu = Bu

| u1−u2 |≥ ∀ ∈ R2 Thus, by Lemma 5.4, 2 (s, t) ρ>0 (s, t) . Thus, by the parallelogram law

2 2 u1 + u2 2 u1 − u2 2 |u1(s, t)| + |u2(s, t)| 2 | (s, t)| + | (s, t)| = ≤u(g1,g2) 2 2 2

| u1+u2 |2 ≤ 2 − 2  u1+u2    Then, 2 (s, t) u(g1,g2) ρ and 2 (s, t) < u(g1,g2) contra- dicting the fact that u(g1,g2) has minimum norm. Thus, the solution of the above system is unique. 78

α For any ,wehaveT αu(f1,f2)=u(T αf1,T αf2). Hence, T α+ βu(f1,f2)=

T αT βu(f1,f2) since both minimize the norm over bounded solution of ⎧ ⎪ ⎨ D su = Au + T α+ βf1 ⎪ ⎩ D tu = Bu + T α+ βf2

Thus, u(f1,f2) is almost periodic. But, u0(s, t)=u(f1,f2)(s, t)+w(s, t)where w is a bounded solution of ⎧ ⎨⎪ Dsu = Au ⎩⎪ Dtu = Bu which is almost periodic by Theorem 5.3. This proves the almost periodicity of u0. 79

5.2 The infinite dimensional case

Let E be a Banach space and assume that A and B are bounded linear operators.

We say that u ∈ BUC1(R2,E)ifu is a bounded uniformly continuous function

∂u ∂u such that ∂s and ∂t exist and are bounded uniformly continuous functions.

Consider the following equations:

⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t)

Of concern to us is the question of almost periodicity of solutions u,providedf1 and f2 are almost periodic functions. Denoting, without ambiguity, by the same letter A (respectively, B) the operator on BUC1(R2,E) defined by (Af)(s, t):=

Af(s, t) (resp., (Bf)(s, t):=Bf(s, t)), we can write Eq.(∗) in the following form:

(Ds − A)u = f1

(Dt − B)u = f2 where Ds and Dt are the partial differentiation operators w.r.t. s and t, respec- tively. Moreover, the system can be written as:

L1u = f1

L2u = f2

Consider a general equation of the form

Lu = f 80 where L is an arbitrary linear partial differential operator (with constant coeffi- cients). G. Sell [47] noted that if u is differentiable and bounded, then, by choosing a subsequence if necessary, Lu = f implies T ν(Lu)=LT νu = T νf. Thus, by the continuity of the operators A and B and the above observation, we conclude that if u ∈ BUC1(R2,E), then LT νu = T ν(Lu).

Consider, together with the above non-homogeneous system, the following cor- responding homogeneous system.

⎫ ∂u(s,t) ⎪ ∂s = Au(s, t) ⎬ ∗∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)

In the next theorem, we reduce the question of almost periodicity of the bounded solutions of (∗) to the question of almost periodicity of the bounded solutions of

(∗∗).

Theorem 5.6 Let u ∈ BUC1(R2,E). Assume that vector-valued functions f1(s, t) and f2(s, t) are almost periodic. Assume also that every bounded solu- tion of (∗∗)isalmostperiodic.Ifu ∈ BUC1(R2,E) is a bounded solution of (∗), then u is almost periodic.

The proof of this theorem follows essentially the argument of Bochner [14], with appropriate modifications for partial differential equations. Note that Bochner’s original consideration of scalar functions was subsequently extended to vector- valued functions, with applications to differential equations in mind, by G. Sell

[47], Sibuya [48] and Zaidman [56]. 81

In order to prove Theorem 5.6, let us recall the equivalent definition of the almost periodic functions due to Bochner [14] (Theorem 5.1).

We said that f is almost automorphic if for any sequence β , there exists a β subsequence such that T− βT βf = f.

Replace almost periodicity by almost automorphicity in Theorem 5.6, we obtain

Lemma 5.7 Assume that f1 and f2 are almost automorphic functions and every bounded solution of (∗∗ ) is almost automorphic. If u is a bounded solution of

(∗), then u is almost automorphic .

A Proof of Lemma 5.7 : Let us denote T− βT β by β.So,g is almost
automorphic if and only if for any sequence β there exists a subsequence β such that A βg = g.Sincefi are almost automorphic (i=1,2), then for any sequence
β there exists a subsequence β such that the limits T βfi exist. Apply A β on

Lu = f (1) we get

A βLu = L(A βu)=A βf = f (2)

By subtracting (1) from (2), we have:

L(A βu − u)=0

By the hypothesis, A βu − u is almost automorphic which means there exists an almost automorphic function g such that A βu = u + g. 82

We can again apply the operator A β and since g is almost automorphic, we have:

2 A βu = A βu + A βg

= u + g + g = u +2g

Apply A β successively, we have

n A βu = u + ng or 1 g = (An u − u) n β Clearly, we have A βu≤u.  ≤ 1   Thus, g n 2 u .Sincen is arbitrary, then we conclude that g = 0. i.e.,

A βu = u.Thus,u is almost automorphic.

Lemma 5.8 Let f be a continuous function. Then, f is almost periodic if and

only if for any two sequences β and ν , there exist subsequences β and ν such that:

(i) T βT νf is almost automorphic and

(ii) Cf = f,whereC = T−( β+ ν)T βT ν.

Note that this lemma was proved by G. Sell [47] for the case where the function f : Rm → Rn and m, n ∈ N, but his argument applies in the more general case.

Proof of Lemma 5.8 : If (i)and(ii) hold, then by applying T( β+ ν) to Cf = f and by taking a subsequence if necessary, we get 83

T( β+ ν)T−( β+ ν)T βT νf = T( β+ ν)f

By (i),

T βT νf = T( β+ ν)T−( β+ ν)T βT νf

Hence

T βT νf = T( β+ ν)f

Thus, f is almost periodic.

Conversely, if f is almost periodic then for any two sequences β and ν there β ⊂ β
ν ⊂ ν
exist subsequences and such that the limits T βT νf,T( β+ ν)f and Cf exist. Furthermore, each of these three functions is almost periodic and therefore almost automorphic. Hence (i) holds and

Cf = T−( β+ ν)T βT νf = T−( β+ ν)T( β+ ν)f = f i.e. (ii)holds.

Proof of Theorem 5.6 :

We follow the argument in Lemma 5.7, but instead of using the operator A β we use the operator C := T−( β+ ν)T βT ν.

Since fi (i =1,2) are almost periodic, we apply C to

Lu = f (3) to get:

CLu = L(Cu)=Cf = f (4) 84

By subtracting (3) from (4), we have

L(Cu − u)=0

By the assumptions, Cu = u + g where g is an almost periodic function.

Apply C successively and by using the same argument in Lemma 5.7, we get g = 0. Hence, Cu = u .

Apply T βT ν to Lu = f,wegetL(T βT νu)=T βT νf.

Lemma 5.8 implies that T βT νf is almost automorphic. Lemma 5.7 implies that T βT νu is almost automorphic. Therefore, by Lemma 5.8, u is almost periodic.

The above techniques have been used for ordinary differential operators by

Bochner(see, [14]) and for partial differential operators by Sell(see, [47]) and we use them for the sum of partial differential and bounded operators.

In light of Theorem 5.6, it is natural to ask under what conditions on A, B is a bounded uniformly continuous solution of the homogeneous equation (∗∗) almost periodic. In analogy to the case of equations with one-dimensional time, it is natural to invoke the corresponding version of Loomis theorem on almost periodicity of functions of two variables with countable spectrum, i.e. Theorem

1.8.

Theorem 5.9 Suppose that the joint spectrum of A and B, Sp(A, B),iscount- able. If u ∈ BUC1(R2,E) is a bounded solution of (∗∗), then u is almost periodic provided (C1) [or,(C2)] is satisfied. 85

Proof: In fact, we note Sp(u) ⊂ Sp(A, B), and Spap(u) ⊂ Sp(u),andthenthe statement follows from Theorem 1.8.

Theorem 5.9 and Theorem 5.6 imply the following result.

Corollary 5.10 Suppose that the joint spectrum of A and B, Sp(A, B),iscount- able. If u ∈ BUC1(R2,E) is a bounded solution of (∗), then u is almost periodic provided (C1) [or,(C2)] is satisfied.

The technique used in Theorem 5.6 can also be used to obtain some new results for equations with one-dimensional time.

Theorem 5.11 Let f(t):R → E be an almost periodic (resp, almost automor- phic) and A be a bounded operator. Assume that the differential equation

u(t)=Au(t)+f(t) has unique bounded solution u.Then,u is almost periodic (resp, almost automor- phic).

Proof : We give the proof for the case of almost periodicity. The proof for almost automorphicity is analogous and therefore is omitted.

Let β and ν be arbitrary sequences in R.Sincef is almost periodic, there β ⊂ β
ν ⊂ ν
exist subsequences and such that the limits T β+ νf and

T βT νf are exist.

Furthermore, T β+ νu is the unique bounded solution of

v (t)=Av(t)+T β+ νf(t)(5) 86 and T βT νu is the unique bounded solution of

v (t)=Av(t)+T βT νf(t)(6)

Since f is almost periodic, then (5) = (6) . Thus, T β+ νu = T βT νu.There- fore, u is almost periodic. 87

6 C-admissibility and analytic C-semigroups

6.1 C-admissibility

In this section, we introduce the notion of C-admissible subspaces (for equations with one-dimensional time), which is a generalization of the notion of admissible subspaces, and we extend to this situation the results of Vu [51] and Schuler and

Vu [54].

Consider the differential equation

u(t)=Au(t)+f(t),t∈ R (∗∗∗) where A is a linear closed operator on a Banach space E. Let M be a subspace of BUC(R,E), which is invariant with respect to trans- lations. Let C be a bounded linear operator on E.

Definition 6.1 Afunctionu is called a mild solution of (***) if for every ϕ ∈

L1(R) such that ϕˆ has compact support, v := u ∗ ϕ is a (classical) solution of v = Av + f ∗ ϕ.

Definition 6.2 We say that M is C-admissible if for every f ∈ M there exists a mild solution u ∈ M of Eq. u(t)=Au(t)+Cf(t).

Definition 6.3 We say that M is regularly C-admissible if for every f ∈ M there exists a unique mild solution u ∈ M of Eq. u(t)=Au(t)+Cf(t).

Let D be the differentiation operator on BUC(R,E)andδ0 be the Dirac op- erator defined by δ0f := f(0). Let S(t) be the translation group on BUC(R,E).

M By DM and δ0 we denote the restrictions of D and δ0 on M, respectively. 88

Theorem 6.4 Consider the following properties

(i) M is C-admissible;

M (ii) The operator equation AX − XDM = −Cδ0 has a bounded solution; (iii) M is regularly C-admissible.

Then (iii) =⇒ (ii)=⇒ (1).

Proof. (iii)=⇒ (ii). Suppose M is regularly C-admissible. Define an operator

X : M → E by Xf = u(0), where u is the solution in M of Eq. u(t)=

Au(t)+Cf(t). If u is a classical solution, then by putting t =0intheabove

equation we have u (0) = Au(0)+Cf(0) or AXf −XDM f = −Cδ0f. The general claim follows from this by a standard argument using the approximate unit in

1 M L (R). Thus, the existence of a solution to AX − XDM = −Cδ0 is proved. Note that in this case the solution X also is unique. Indeed, assume there is another solution X1 of this operator equation, such that X1 = X.ThenY = X − X1 satisfies AY = Y DM . Choose a function g ∈ M such that v(t)=YS(t)g is not identically zero. Then v(t)=Y DS(t)g = AY S(t)g = Av(t). This contradicts the fact that M is regularly admissible.

(ii)=⇒ (i). Suppose X is a solution of AXf − XDM f = −Cδ0f.Letf ∈

M and u(t):=XS(t)f. By approximation, one can assume, without loss of generality, that f is smooth so that u is smooth. We have u(t)=XDS(t)f =

(AX + Cδ0)S(t)f = Au(t)+Cf(t)sothatu is a solution in M of Eq. u (t)=

Au(t)+Cf(t). 89

Let Λ be a closed subset of iR and M(Λ) := {f ∈ BUC(R,E):Sp(f) ⊂ Λ}.

Lemma 6.5 Assume M(Λ) is regularly C-admissible, where C has a dense range.

Then, σ(A) ∩ Λ=∅.

Proof. Given λ ∈ Λ, we must show that λ ∈ ρ(A). Let y ∈ E and f(t)=eλty.

Then Sp(f)={λ}, hence f ∈ M. By Theorem 6.4, there exists a unique solution u(t)ofEq. u(t)=Au(t)+Cf(t), which must have form u(t)=XS(t)Cf = · XS(t)eλ Cy. Hence

· λ · u(t)=XDS(t)eλ Cy =(λ)XS(t)e Cy = λu, which implies u(t)=eλtx for some x ∈ E.Fromu(t)=Au(t)+Cf(t)itfollows

(A − λ)x = Cy. Since,

· x = u(0) = Xeλ Cy≤MCy = M(A − λ)x for some M>0, it follows that (A−λ)−1 is defined on a dense set and is bounded, hence λ ∈ ρ(A).

Since f ∈ M(Λ) implies Cf ∈ M(Λ), it follows that if M(Λ) is (regularly) admissible, then it also is (regularly) C-admissible for every bounded operator C on E. It is well known that if Λ is compact, then the condition σ(A) ∩ Λ=∅ implies that M(Λ) is regularly admissible.

Theorem 6.4 reduces the question of C-admissibility to solution of the operator

M equation AX − XDM = −Cδ0 . In section 6.3, we investigate this equation for the case A is the generator of an analytic C-semigroup, generalizing results of

[51]. 90

6.2 C-semigroups and analytic C-semigroups

Again, let E be a Banach space, and B(E) the set of all bounded linear operators from E into itself. The following definitions can be found in [34]

Definition 6.6 Let C be an injective operator in B(E).Afamily{S(t):t ≥ 0} in B(E) is called an exponentially bounded C-semigroup (hereafter abbreviated to

C-semigroup ) on E if

(i) S(s + t)C = S(s)S(t) for s, t ≥ 0 and S(0) = C,

(ii) S( ·)x :[0, ∞) → E is continuous for every x ∈ E,

(iii) there are M ≥ 0 and a ≥ 0 such that S(t)≤Meat for t ≥ 0.

Definition 6.7 The generator A of a C-semigroup {S(t):t ≥ 0} on E is defined by ⎧ ⎨⎪ D(A)={x ∈ E : limt→0+(S(t)x − Cx)/t ∈ R(C)} ⎩⎪ −1 Ax = C limt→0+(S(t)x − Cx)/t for x ∈ D(A) where R(C) denotes the range of C.

Miyadera and Tanaka showed that A is a closed linear operator in E and A =

C−1AC.

iφ Definition 6.8 : Sθ ≡{re : r>0, |φ| <θ}.

Definition 6.9 : The C-semigroup S(t)t≥0 is a uniformly bounded analytic C-

¯ semigroup of angle θ if it extends to a family of bounded operators S(z)z∈Sθ satis- fying 91

(1) The map z → S(z),fromSθ into L(X) ,isanalytic.

(2) S(z)S(ω)=CS(z + ω), ∀z,ω ∈ S¯θ.

(3) S(z) is bounded and strongly continuous on S¯θ.

S(z) is called a strongly uniformly continuous uniformly bounded analytic C- semigroup of angle θ if the continuity in (3) is uniform. But, it is sufficient to have the generator densely defined.

The following theorem due to R. DeLaubenfels [18, Corollary 6.4]:

Theorem 6.10 :SupposeA generates a strongly uniformly continuous uniformly

{ } ¯ bounded analytic C-semigroup, S(z) z∈Sθ ,ofangleθ. Then, there exists a Ba- nach space W and an operator G such that

(1) C extends to a bounded operator on W ,

(2) G generates a uniformly bounded analytic strongly continuous semigroup of

zG zG angle θ on W , such that e C = Ce , ∀z ∈ S¯θ,

zG (3) Ce x = S(z)x, ∀z ∈ S¯θ,x ∈ E,

(4) A = G|E and

(5) [C(W )] → E→ W .

For more information about the subject of analytic (holomorphic) C-semigroup, we refer the reader to [18] and [50]. 92

6.3 Operator equation AX − XB = CD,withA generating

an analytic C-semigroup

Consider AX − XB = D (7)

where A generates an analytic semigroup in a Banach space E if and only if π ⊂ there exist a θ (0<θ< 2 ) and a real number ω such that σ(A) Σω,θ where { ∈ C | − | }∪{ }  − −1 ∞ Σω,θ = λ : arg(ω λ) <θ ω and supλ∈C\Σω,θ λ(A λ) < .There will be no restriction if assume that ω =0.

Let >0 be small enough such that the disk with center at 0 and radius  is contained in ρ(B) and denote by Γ the unbounded contour which is the boundary of the domain

H = {λ ∈ C : |arg(−λ)| <θ}∪{λ ∈ C : |λ| <} oriented in such a way that the domain H remains in the left side of Γ.

The following theorem due to Phong Vu [51, Theorem 15]:

Theorem 6.11 Let A be a generator of an analytic semigroup in a Banach space

E. Assume that B isaclosedlinearoperatorinaBanachspaceF such that

−1 Σω,θ ⊂ ρ(B) and λ(B − λ)  is uniformly bounded when λ belongs to the sector

Σω,θ. Then, equation( 7) has a unique solution which is expressed by  1 X = (A − λ)−1D(B − λ)−1dλ. 2πi Γ 93

We will be interested to generalize it to analytic C-semigroup.

The proof of the following lemma is included in the proof of theorem 1 of [50]:

Lemma 6.12 Let A be the C- complete infinitesimal generator of {S(t):|argt| < } ≤ π ∈ δ δ (where0 <δ 2 ). Let  (0, 2 ).Then, ⎧ ⎪ ⎨⎪ 1 λt − −1 | |≤ − 2πi Γ e (λ A) Cdλ, for argt δ 2, S(t):= ⎪ ⎩C for t =0 | | π ∞ −iφ ∞ iφ where Γ is a curve running, in the sector argλ < 2 + δ,from e to e π − with φ = 2 + δ .

Theorem 6.13 Let A be a generator of an analytic uniformly bounded C- semigroup on a Banach space E. Assume that B is a closed linear operator in a

Banach space F satisfying the conditions on the above theorem. Then, the operator equation

AX − XB = CD (8) has a unique solution for every bounded operator D. The solution is expressed by  1 X = (A − λ)−1CD(B − λ)−1dλ. 2πi Γ

Proof : To show the operator equation has a unique solution, it is enough to show that the following operator equation has the trivial solution the zero operator: AX − XB =0 94

Let X be a solution of the above equation. For λ ∈ ρ(A) ∩ ρ(B), then

AX − XB =0=⇒−CX(λ − B)−1 +(λ − A)−1CX = 0 by use of [50, theorem 1]

From the conditions on A and B and the above equality it follows that the fol- lowing integrals converge and   1 1 etλ(λ − A)−1CXdλ = etλCX(λ − B)−1dλ 2πi Γ 2πi Γ

Since (λ − B)−1 is uniformly bounded on H and C is bounded, then the right hand side integral vanishes. Therefore, by the above Lemma,  1 S(t)X = etλ(λ − A)−1CXdλ =0 2πi Γ

Thus CX = 0, but C is an injective operator. Hence, X =0. 1 tλ − −1 − −1 Note that Xt = 2πi Γ e (λ A) CD(λ B) dλ converges and is a bounded linear operator from F to E.

For every f ∈D(B), we have Xtf ∈D(A)and

AXtf − XtBf = −(λ − A)Xtf + Xt(λ − B)f  1 = etλ[−CD(λ − B)−1 +(λ − A)−1CD]fdλ 2πi Γ Since CD(λ − B)−1 is uniformly bounded as λ →∞, λ ∈ Γ, the first integral vanishes.

Thus  1 tλ −1 AXtf − XtBf = e (λ − A) CDfdλ = S(t)Df 2πi Γ Let t → 0, then AXf − XBf = CDf. 95

From Theorems 6.13 and 6.4 we obtain the following result.

Theorem 6.14 Suppose A is the generator of an analytic C-semigroup and

σ(A) ∩ σ(DM )=∅.Then,M is C-admissible.

Theorem 6.14 has many important partial cases which are obtained when we put M = M(Λ) with specific subsets Λ. Below, we present one such example

(which corresponds to Λ = {2πik/ω : k ∈ Z}).

Corollary 6.15 Suppose A is the generator of an analytic C-semigroup such that

2πik/ω ∈ ρ(A) for all k ∈ Z. Then, for every ω-periodic function f,equation u(t)=Au(t)+Cf(t) has a ω-periodic solution. 96

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