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Almost automorphic solutions of secondorder evolution equationsJames Liu a , Gaston M. N’Guérékata b & Nguyen Van Minh aa Department of Mathematics , James Madison University ,Harrisonburg VA 22801b Department of Mathematics , Morgan State University , 1700 E.Cold Spring Lane, Baltimore, MD 21251, USAc Communicated by R.P. GilbertPublished online: 04 Sep 2006.

To cite this article: James Liu , Gaston M. N’Guérékata & Nguyen Van Minh (2005) Almostautomorphic solutions of second order evolution equations, Applicable Analysis: An InternationalJournal, 84:11, 1173-1184, DOI: 10.1080/00036810410001724372

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Applicable AnalysisVol. 84, No. 11, November 2005, 1173–1184

Almost automorphic solutions of

second order evolution equations

JAMES LIUy�, GASTON M. N’GUEREKATA*z andNGUYEN VAN MINHyx

yDepartment of Mathematics, James Madison University, Harrisonburg VA 22801zDepartment of Mathematics, Morgan State University, 1700 E. Cold Spring Lane,

Baltimore, MD 21251, USA

Communicated by R.P. Gilbert

(Received 19 April 2004; in final form 23 April 2004)

This article is concerned with the existence of almost automorphic mild solutions to secondorder evolution equations of the form €uuðtÞ ¼ AuðtÞ þ fðtÞ ð�Þ, where A generates a stronglycontinuous semigroup and f is an almost automorphic function. Using the notion of uniformspectrum of a function and the method of sums of commuting operators in previous worksfor the case of bounded uniformly continuous solutions, we obtain sufficient conditions forthe existence of almost automorphic mild solutions to ð�Þ in terms of spectrum of A and uniformspectrum of f. Moreover, we study the nonlinear perturbation of this equation and obtain anextension of results by Diagana and N’Guerekata.

Keywords: Analytic semigroup; Almost automorphic solution; Uniform spectrum; Sums ofcommuting operators

1991 Mathematics Subject Classification: Primary: 34G10; Secondary: 43A60

1. Introduction and notations

In this article we deal with the existence of almost automorphic mild solutions to secondorder evolution equations of the form

d 2u

dt2¼ Auþ fðtÞ, ð1Þ

where A is a (unbounded) linear operator which generates a holomorphic semigroup oflinear operators on a Banach space X and f is an almost automorphic function takingvalues in X.

*Corresponding author. Email: gnguerek@jewel.morgan.eduxEmail: nguyenvm@jmu.edu�Email: liujh@jmu.edu

Applicable Analysis ISSN 0003-6811 print: ISSN 1563-504X online � 2005 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/00036810410001724372

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This problem has been of great interest to many mathematicians for decades.Actually, it goes back to the characterization of exponential dichotomy of linearordinary differential equations by O. Perron. The reader can find many extensions ofthe classical result of Perron to the infinite dimensional case in [4,12–15,24] and thereferences therein with results concerned with almost periodic solutions and boundedsolutions. Recently, the interest in finding conditions for the existence of almost auto-morphic solutions has been regained (see e.g. [6,10,16–19,23]). Some extensions ofresults on almost periodic solutions have been made in [13]. In this direction, westudy the conditions for the existence and uniqueness of almost automorphic solutionsto equation (1). The idea of using the method of sums of commuting operators to studythe existence of almost periodic solutions is due to Murakami et al. [13]. This methodworks well with the problem of finding almost automorphic solutions of first orderevolution equations as shown in the recent article [7].

In this article, we show that the method of sums is still useful for second order evolu-tion equations. Our main results are Theorems 3.14 and 3.20. Note that in [25,26] simi-lar results for f 2 BUCðR,XÞ were proved using an operator equation. This methodthat needs the uniform continuity of f is inapplicable to the problem we are con-sidering due to the fact that an almost automorphic function may not be uniformlycontinuous.

Notation

Throughout the article, R, C, and X stand for the sets of all real, all complex numbers,and a complex Banach space, respectively; LðXÞ, BCðR,XÞ, and BUCðR,XÞ denote thespaces of all linear bounded operators on X, all X-valued bounded continuous func-tions, and all X-valued bounded uniformly continuous functions with sup-norm,respectively. The translation group in BCðR,XÞ is denoted by ðSðtÞÞt2R which is stronglycontinuous in BUCðR,XÞ whose infinitesimal generator is the differential operatord / dt. For a linear operator A, we denote by D(A), �(A) and �(A) the domain, spectrumand resolvent set of A, respectively. If Y is a metric space and B is a subset of Y, then �BBdenotes its closure in Y. In this article by the notion of sectorial operators is meantthe one defined in [20]. The notion of closure of an operator is referred to the onedefined in [5].

2. Preliminaries

2.1. Spectral theory of functions

2.1.1. Spectrum of a function in BCðR,XÞ. In the present article, for u 2 BCðR,XÞ,sp(u) stands for the Carleman spectrum, which consists of all � 2 R such that theCarleman–Fourier transform of u, defined by

uuð�Þ :¼

R10 e��tuðtÞ dt ðRe � > 0Þ

�R10 e�tuð�tÞ dt ðRe � < 0Þ,

8<:

has no holomorphic extension to any neighborhoods of i� (see [21, Prop. 0.5, p. 22]).

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Below we list some properties of the spectra of functions which we will need in thesequel.

PROPOSITION 2.1 Let u, un, v 2 BCðR,XÞ such that limn!1 kun � uk ¼ 0, and 2 S.Then

(i) sp(u) is closed,(ii) spðuþ vÞ � spðuÞ [ spðvÞ,(iii) spð � uÞ � spðuÞ \ supp ~ ,(iv) spðu� � uÞ � spðuÞ \ suppð1� ~ Þ,(v) If ~ � 1 on a neighborhood of sp(u) then � u ¼ u,(vi) If spðuÞ \ supp ~ ¼ 6 0 then � u ¼ 0,(vii) If spðunÞ � �, 8n, then spðuÞ � �.

Proof For the proof we refer the reader to [3] and [21, Prop. 0.4, Prop. 0.6,Theorem 0.8, pp. 20–25]. g

2.1.2. Uniform spectrum of a function in BCðR,XÞ. We summarize in this subsectionthe definition and several properties of uniform spectrum that was introduced in [7].Notice that for every � 2 C with <� 6¼ 0 and f 2 BCðR,XÞ the function’fð�Þ : R 3 t � dSðtÞfSðtÞfð�Þ 2 X belongs to Mf � BCðR,XÞ. Moreover, ’f (�) is analyticon CniR.

Definition 2.2 Let f be in BCðR,XÞ. Then,

(i) � 2 R is said to be uniformly regular with respect to f if there exists a neighbor-hood U of i� in C such that the function ’fð�Þ, as a complex function of � with<� 6¼ 0, has an analytic continuation into U.

(ii) The set of � 2 R such that � is not uniformly regular with respect to f 2 BCðR,XÞis called uniform spectrum of f and is denoted by spuð f Þ.

We list below some properties of uniform spectra of functions in BCðR,XÞ.

PROPOSITION 2.3 Let g, f, fn 2 BCðR,XÞ such that fn! f as n!1 and let � � R bea closed subset. Then the following assertions hold:

(i) spuð f Þ ¼ spuð fðhþ �ÞÞ;(ii) spuð�fð�ÞÞ � spuð f Þ, � 2 C;(iii) spð f Þ � spuð f Þ;(iv) spuðBfð�ÞÞ � spuð f Þ, B 2 LðXÞ;(v) spuð fþ gÞ � spuð f Þ [ spuðgÞ;(vi) spuð f Þ � �.

Proof For the proof, see [7]. g

As an immediate consequence of (iii) of the above proposition, we have

COROLLARY 2.4 For any closed subset � � R, the set �uðXÞ :¼ f f 2 BCðR,XÞ:spuð f Þ � �g is a closed subspace of BCðR,XÞ which is invariant under translations.

The notion of uniform spectrum of a function in BCðR,XÞ has some propertiessimilar to those of a function in BUCðR,XÞ as shown in the following.

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LEMMA 2.5 Let � be a closed subset of R and let D�ube the differential operator acting

on �uðXÞ. Then we have

�ðD�uÞ ¼ i�: ð2Þ

Proof For the proof, see [7]. g

2.2. Almost automorphic functions

Definition 2.6 A function f 2 CðR,XÞ is said to be almost automorphic if for anysequence of real numbers ðs 0nÞ, there exists a subsequence ðsnÞ such that

limm!1

limn!1

fðtþ sn � smÞ ¼ fðtÞ ð3Þ

for any t 2 R.

The limit in (3) means

gðtÞ ¼ limn!1

fðtþ snÞ ð4Þ

is well-defined for each t 2 R and

fðtÞ ¼ limn!1

gðt� snÞ ð5Þ

for each t 2 R.

Remark 2.7 Because of pointwise convergence the function g is measurable but notnecessarily continuous. It is also clear from the definition above that constant functionsand continuous almost periodic functions are almost automorphic.

If the limit in (4) is uniform on any compact subset K � R, we say that f is compactalmost automorphic.

THEOREM 2.8 Assume that f, f1, and f2 are almost automorphic and � is any scalar, thenthe following hold true.

(i) �f and f1 þ f2 are almost automorphic,(ii) f�ðtÞ :¼ fðtþ �Þ, t 2 R is almost automorphic,(iii) �ffðtÞ :¼ fð�tÞ, t 2 R is almost automorphic,(iv) The range Rf of f is precompact, so f is bounded.

Proof See [19, Theorems 2.1.3 and 2.1.4] for proofs. g

THEOREM 2.9 If f fng is a sequence of almost automorphic X-valued functions such thatfn � f uniformly on R, then f is almost automorphic.

Proof see [19, Theorem 2.1.10] for the proof. g

Remark 2.10 If we equip AAðXÞ, the space of all almost automorphic functions withthe sup-norm

kfk1 ¼ supt2RkfðtÞk,

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then it turns out to be a Banach space. If we denote KAAðXÞ, the space of all compactalmost automorphic X-valued functions, then we have

APðXÞ � KAAðXÞ � AAðXÞ � BCðR,XÞ

THEOREM 2.11 If f2AAðXÞ and its derivative f 0 exists and is uniformly continuous on R,then f 0 2AAðXÞ.

Proof See [19, Theorem 2.4.1] for a detailed proof. g

THEOREM 2.12 Let us define F : R � X by FðtÞ ¼R t0 fðsÞ ds where f 2 AAðXÞ. Then

F 2 AAðXÞ iff RF ¼ fFðtÞj t 2 Rg is precompact.

Proof See [19, Theorem 2.4.4] for the proof. g

For any closed subset � � R we denote

AA�ðXÞ :¼ fu 2 AAðXÞ: spuðuÞ � �g:

By the basic properties of uniform spectra of functions, AA�ðXÞ is a closed subspaceof BCðR,XÞ.

3. Main results

3.1. Mild solutions of inhomogeneous second order equations

3.1.1. Mild solutions. Let A be any closed linear operator on a Banach space X.We now define the concept of mild solutions on R to equation (1).

Definition 3.1 (See [1, p. 374]) A continuous X-valued function u is called a mildsolution on R of equation (1) ifZ t

0

ðt� sÞuðsÞ ds 2 DðAÞ 8t 2 R ð6Þ

and

uðtÞ ¼ xþ tyþ A

Z t

0

ðt� sÞuðsÞ dsþ

Z t

0

ðt� sÞ fðsÞ ds ðt 2 RÞ ð7Þ

for some fixed x, y 2 X. If u 2 C2ðR,XÞ, uðtÞ 2 DðAÞ, 8t 2 R and equation (1) holdsfor all t 2 R we say that u is a classical solution to equation (1). It is easily seen thatif u 2 CðR,XÞ is a classical solution of equation (1), then it is a mild solution ofequation (1).

3.1.2. Mild solutions and weak solutions

Definition 3.2 A function u 2 BCðR,XÞ is said to be a weak solution to equation (1) ifthere is a sequence fn 2 BCðR,XÞ and a sequence of classical solutions un 2 BCðR,XÞ ofequation (1) with f replaced by fn such that fn! f and un! u in the sup-norm topologyof BCðR,XÞ.

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Definition 3.3 We define an operator L on BCðR,XÞ with domain DðLÞ consisting ofall u 2 BCðR,XÞ such that there exists at least a function f2BCðR,XÞ for which u is amild solution of equation (1); i.e., (6) and (7) hold.

LEMMA 3.4 If A is a closed linear operator, then L is a single-valued closed linearoperator BCðR,XÞ � DðLÞ ! BCðR,XÞ.

Proof First we show that L is a single-valued linear operator. For this it sufficesto show that if uðtÞ � 0 is a mild solution of equation (1), then fðtÞ � 0. In fact, bytaking t ¼ 0 we can see that in equation (7), x ¼ 0. Hence, we have

0 ¼ tyþ

Z t

0

ðt� sÞ fðsÞ ds ðt 2 RÞ: ð8Þ

Differentiating (8) twice, we have fðtÞ ¼ 0 for all t 2 R. Therefore, the operator L is asingle-valued linear operator.

Next, we show its closedness. Let un be in DðLÞ that are mild solutions of equation (1)with f replaced by fn 2 BCðR,XÞ such that un! u 2 BCðR,XÞ and fn! f 2BCðR,XÞ. We have to prove that u is a mild solution to equation (1). Indeed, by ourassumption, we have

unðtÞ ¼ xþ tyþ A

Z t

0

ðt� sÞunðsÞ dsþ

Z t

0

ðt� sÞ fnðsÞ ds ðt 2 RÞ:

Since A is closed and un, fn 2 BCðR,XÞ, for a fixed t 2 R, letting n!1 we haveR t0ðt� sÞ uðsÞ ds 2 DðAÞ, and A

R t0ðt� sÞ unðsÞ ds! A

R t0ðt� sÞ uðsÞ ds. Therefore,

uðtÞ ¼ xþ tyþ A

Z t

0

ðt� sÞ uðsÞ dsþ

Z t

0

ðt� sÞ fðsÞ ds ðt 2 RÞ:

Thus, by definition, u is a mild solution of equation (1). g

Remark 3.5 By a similar argument, we can easily show that the part of L on AA�

(here � is a closed subspace of R) that is denoted by L� is a closed linear operator.

PROPOSITION 3.6 Let A be a closed linear operator. Then every weak solution of equation(1) is a mild solution.

Proof The proof is obvious in view of the above lemma and the remark that classicalsolutions are mild solutions. g

3.2. Operators A

Let � be a closed subset of R. We first consider the operator A� of multiplicationby A and the differential operator d / dt on the function space AA�ðXÞ. By definitionthe operator A� of multiplication by A is defined on DðA�Þ:¼ fg 2 AA�ðXÞ: gðtÞ 2DðAÞ 8t 2 R,Agð�Þ 2 AA�ðXÞg, and Ag :¼ Agð�Þ for all g 2 DðA�Þ.

LEMMA 3.7 Assume that � � R is closed. Then the operator A� of multiplication by Ain AA�ðXÞ is the infinitesimal generator of an analytic C0-semigroup on AA�ðXÞ.

Proof For the proof, see [7]. g

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In the following lemma we assume that � is a closed subset of the real line and thesecond order differential operator D2

� defined in AA�ðXÞ with domain DðD2�Þ consist-

ing of all functions u 2 AA�ðXÞ that are of class C2 such that d2u=dt2 2 AA�ðXÞ. Let us

define ��2:¼ f� 2 Rj � ¼ ��2, for some � 2 �g:

PROPOSITION 3.8 With the above notations the following assertions hold true

�ðD2�Þ ¼ ��2: ð9Þ

Proof We associate with the equation

d2u

dt2¼ �uþ fðtÞ, f 2 AA�ðXÞ ð10Þ

the following first order equation

x01 ¼ x2

x02 ¼ �x1 þ fðtÞ, f 2 AA�ðXÞ:

�ð11Þ

It is easily seen from the theory of ordinary differential equations that the solvability ofthese equations in BCðR,XÞ are equivalent. As shown in [7], if f 2 AA�ðXÞ and

i� \ �ðIð�ÞÞ ¼ �, ð12Þ

where I(�) denotes the operator matrix associated with equation (11), a simple compu-tation shows that �ðIð�ÞÞ consists of all solutions to the equation t2 � � ¼ 0. Thus,

�ðD2�Þ � f� 2 C : � ¼ ��2 for some � 2 �g:

Hence �ðD2�Þ � ��2.

On the other hand, let �2�. Then gð�Þ :¼ xei� 2 AA�ðXÞ. Obviously, D2�g ¼ ��

2g,and thus, ��2 2 �ðD2

�Þ. Therefore, �ðD2�Þ � ��2 and the proposition is proved. g

PROPOSITION 3.9 Let � be a closed subset of the real line and let �ð�Þ denote the setfz 2 Cj z 6¼ 0, jarg zj < � �g, where � is a given small positive number. Then theresolvent Rð�,D2Þ of the operator D2 in AA�ðXÞ satisfies

kRð�,D2Þk �M

j�j, 8� 2 �ð�Þ, ð13Þ

where M is a positive constant depending only on �.

Proof Consider the equation

x 00ðtÞ � �xðtÞ ¼ fðtÞ, t 2 R, ð14Þ

for every given � 2 �ð�Þ and f 2 AA�ðXÞ. As shown in the previous proposition, thereexists a unique function xf 2 AA�ðXÞ that satisfies the above equation. For �2�ð�Þthere are exactly two distinct solutions �1, �2 ¼ ��1 of the equation x2 � � ¼ 0.

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Without loss of generality, we assume that <�1 < 0. It is easy to check that the uniquebounded solution to equation (14) is given by

xfðtÞ ¼1

2�1

Z t

�1

e�1ðt��Þfð�Þ d� þ

Z 1t

e��1ðt��Þfð�Þ d�

� �ð15Þ

that is in AA�ðXÞ. By definition, xf ¼ �Rð�,D2Þ f. Obviously, since � 2 �ð�Þ, we have

kxfk �1

2j�1j�

2

<�1�

M

j�j,

where M is a positive constant depending on �. g

PROPOSITION 3.10 Let A be a closed linear operator such that �ðAÞ \ ��2 ¼ �. Thenfor every f 2 BCðR,XÞ with spð f Þ � �, equation (1) has at most one mild solutionu 2 BCðR,XÞ.

Proof Since the operator L is a closed linear operator, it suffices to show that ifu 2 BCðR,XÞ is a mild solution of equation (1) with f replaced by 0, then spðuÞ ¼ �.This can be checked directly as follows. Assume that u is a solution of equation (7)with f replaced by 0.

Proof Taking Carleman–Laplace transforms of both sides of equation (7) we have

uuð�Þ ¼x

�þ

y

�2þ A

uuð�Þ

�2ð<� 6¼ 0Þ:

Therefore,

ð�2 � AÞ uuð�Þ ¼ �xþ y ð<� 6¼ 0Þ: ð16Þ

If � 2 R and � 62 �ðAÞ, then ð�2 � AÞ is invertible and ð�2 � AÞ�1 is holomorphic in asmall neighborhood of �. Therefore, uuð�Þ has a holomorphic extension to a neighbor-hood of �, so spðuÞ � R \ �ðAÞ. On the other hand, by the assumption thatspðuÞ � ��2 and ��2 \ �ðAÞ ¼ �, we have spðuÞ ¼ �. And hence, u ¼ 0. g

LEMMA 3.11 Let A be the generator of an analytic semigroup. Then the operator A� ofmultiplication by A and the differential operator D2

� on AA�ðXÞ are commuting andsatisfy condition P of Definition 4.2.

Proof Let us denote by T(t) the semigroup generated by A. As shown in [7], the oper-ator A of multiplication by A generates an analytic semigroup T ðtÞ of multiplicationby T(t) on AA�ðXÞ. So, it is sufficient to prove that the semigroup T ðtÞ commuteswith D2, that is T ðtÞDðD2Þ � DðD2Þ and

T ðtÞD2x ¼ D2T ðtÞx, 8x 2 DðD2Þ, t 0:

By the definition of the operator T ðtÞ of multiplication by T(t), the above claim isobvious. The lemma now follows from the above propositions. g

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So, by the spectral properties of sums of commuting operators, we have

COROLLARY 3.12 If A generates an analytic semigroup with �ðAÞ \ ��2 ¼ �, then forevery f 2 AA�ðXÞ there exists a unique u 2 AA�ðXÞ such that

D2� �A�u ¼ f:

Proof Since A� and �D2� commute and satisfy condition P, the sum D2

� �A� isclosable (denote its closure by D2

� �A�). From �ðAÞ \ ��2 ¼ � and Theorem 4.3 inAppendix, it turns out that 0 2 �ðD2

� �A�Þ. Therefore, for every f 2 AA�ðXÞ thereexists a unique u 2 DðD2

� �A�Þ such that

D2� �A�u ¼ f: �

Now our remaining task is just to explain what the above closure means. Moreprecisely, we will relate it with the notion of mild solutions to evolution equations.

LEMMA 3.13 Let u, f2AAðXÞ. If u 2 DðD2� �A�Þ and D

2� �A�u ¼ f, then u is a mild

solution of equation (1).

Proof The lemma follows from the fact that weak solutions are mild solutions. g

As an immediate consequence of the above argument we have:

THEOREM 3.14 Let A be the generator of an analytic semigroup and let � be aclosed subset of R. Then �ðAÞ \��2 ¼ � holds for each f 2 AA�ðXÞ if and only ifthere exists a unique mild solution u 2 AA�ðXÞ to equation (1).

Proof The sufficiency follows from the above argument. The necessity can be shownas follows: For every � 2�, obviously the function h : R 3 t � aei�t is in AA�ðXÞ, wherea 2 X is any given element. By assumption, there is a unique g 2 DðA�Þ such that��gðtÞ � AgðtÞ ¼ hðtÞ for all t 2 R. Following the argument in [13, p. 252] one caneasily show that g(t) is of the form bei�t. Hence, b is the unique solution of the equation��b� Ab ¼ a. That is �� 62 �ðA�Þ, so �� \ �ðA�Þ ¼ �. g

COROLLARY 3.15 Let A be the generator of an analytic semigroup such that�ðAÞ \ ½�spuð f Þ

2¼ �: Then equation (1) has a unique almost automorphic mild solution

w such that spuðwÞ � spuð f Þ.

Proof Set � ¼ spuð f Þ. Then by the above argument we get the theorem. g

Remark 3.16 We notice that all results stated above for almost automorphic solutionshold true for compact almost automorphic solutions if the assumption on the almostautomorphy of f is replaced by the compact almost automorphy of f. Details of theproofs are left to the reader.

3.3. Nonlinear equations

In this subsection we consider nonlinear equations of the form

d2

dt2u ðtÞ ¼ Au ðtÞ þ Fðt, u ðtÞÞ, t 2 R, ð17Þ

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where F : R�X! X is assumed to satisfy the following conditions: Fðt, xÞ is almostautomorphic in t for every fixed x 2 X and F is continuous jointly in ðt, xÞ, Lipschitzin x uniformly in t, that is, there is a positive number � independent of t, x such that

kFðt, xÞ � Fðt, yÞk � �kx� yk, 8x, y 2 X: ð18Þ

We will say that condition H holds for F in AA�ðXÞ if for a closed subset � � R theNemystky operator F , defined by

FgðtÞ :¼ Fðt, gðtÞÞ, 8g 2 AA�ðXÞ, t 2 R

is an operator AA�ðXÞ ! AA�ðXÞ.

LEMMA 3.17 If Fðt, xÞ is almost automorphic in t for every fixed x 2 X, F is continuousjointly in ðt, xÞ and satisfies (18), then the Nemystky operator F is a continuous operatoracting on AAðXÞ, that is, condition H holds for F on AARðXÞ.

Proof For the proof, see [19, Theorem 2.2.6]. g

Remark 3.18 In general, we are still puzzled with conditions on F and � such that con-dition H holds for F on AA�ðXÞ. In the linear case, if Fðt, xÞ ¼ LðtÞx is T-periodic in tand � ¼ f� 2 Rj e i� 2 Gg, where G is a closed subset of the unit circle, then condition Hholds for F and � (see e.g. [4]). In the general nonlinear case, this question is still open.

Definition 3.19 A continuous X-valued function u is called a mild solution on R

of (17) if

Z t

0

ðt� sÞ uðsÞ ds 2 DðAÞ, 8t 2 R ð19Þ

and

u ðtÞ ¼ xþ tyþ A

Z t

0

ðt� sÞ uðsÞ dsþ

Z t

0

ðt� sÞ fðs, uðsÞÞ ðt 2 RÞ ð20Þ

for some fixed x, y 2 X. If u 2 C2ðR,XÞ, uðtÞ 2 DðAÞ, 8t 2 R, and equation (17)holds for all t 2 R we say that u is a classical solution to equation (17). It is easilyseen that if u 2 CðR,XÞ is a classical solution of equation (17), then it is a mild solutionof equation (17).

The following is a main result of this subsection:

THEOREM 3.20 Let A be the generator of an analytic semigroup, and let condition H holdfor F on AA�ðXÞ. Furthermore, assume that �ðAÞ \ ��2 ¼ � and the Lipschitzcoefficient � in (18) is sufficiently small. Then, equation (17) has a unique almost auto-morphic mild solution u 2 AA�ðXÞ.

Proof First we consider the operator L� defined in Definition 3.3 and the remarkthat follows. Under the assumptions, by Theorem 3.14, this operator is invertibleon AA�ðXÞ. Next, we consider the operator L� �F . This operator may be seen as a

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nonlinear perturbation of L�. Since L� is a closed linear operator, if � is sufficientlysmall, and if we consider DðL�Þ � AA�ðXÞ with the graph norm of L�, then by theInverse Function Theorem, the operator L� � F is an invertible Lipschitz operatorfrom DðL�Þ onto AA�ðXÞ. Hence, equation (17) has a unique almost automorphicmild solution. g

References

[1] Arendt, W. and Batty, C.J.K., 1997, Almost periodic solutions of first and second order Cauchyproblems. Journal of Differential Equations, 137, 363–383.

[2] Arendt, W., Rabiger F. and Sourour, A., 1994, Spectral properties of the operators equationsAX þ XB ¼ Y. Quarterly Journal of Mathematics. Oxford (2), 45, 133–149.

[3] Arendt, W., Batty, C.J.K., Hieber, M. and Neubrander, F., 2001, Vector-valued Laplace Transforms andCauchy Problems. Monographs in Mathematics, Vol. 96 (Basel: Birkhauser Verlag).

[4] Batty, C.J.K., Hutter, W. and Rabiger, F., 1999, Almost periodicity of mild solutions of inhomogeneousperiodic Cauchy problems. Journal of Differential Equations, 156, 309–327.

[5] Davies, E.B., 1980, One-parameter Semigroups (London: Academic Press).[6] Diagana, T. and N’Guerekata, G.M., 2003, Some remarks on almost automorphic solutions of

some abstract differential equations. Far East Journal of Mathematical Sciences, 8(3), 313–322.[7] Diagana, T., N’Guerekata, G.M. and Nguyen Van Minh, 2004, Almost automorphic solutions of

evolution equations. Proceedings of the American Mathematical Society, 132(11), 3289–3296.[8] Engel, K.J. and Nagel, R. 1999, One-parameter Semigroups for Linear Evolution Equations

(Berlin: Springer).[9] Goldstein, J.A., 1985, Semigroups of Linear Operators and Applications. Oxford Mathematical

Monographs (Oxford: Oxford University Press).[10] Goldstein, J.A. and N’Guerekata, G.M., 2005, Almost automorphic solutions of semilinear evolution

equations. Proceedings of the American Mathematical Society, 133(8), 2401–2408.[11] Hino, Y. and Murakami, S. Almost automorphic solutions for abstract functional differential equations

(Preprint).[12] Hino, Y., Naito, T., Minh, N.V. and Shin, J.S., 2002, Almost Periodic Solutions of Differential

Equations in Banach Spaces (London–New York: Taylor & Francis).[13] Murakami, S., Naito, T., and Minh, N.V., 2000, Evolution semigroups and sums of commuting

operators: a new approach to the admissibility theory of function spaces. Journal of DifferentialEquations, 164, 240–285.

[14] Naito, T. and Minh, N.V., 1999, Evolution semigroups and spectral criteria for almost periodicsolutions of periodic evolution equations. Journal of Differential Equations, 152, 358–376.

[15] Naito, T., Minh, N.V. and Shin, J.S., 2001, New spectral criteria for almost periodic solutions ofevolution equations. Studia Mathematica, 145, 97–111.

[16] N’Guerekata, G.M., 2004, Existence and uniqueness of almost automorphic mild solutions to somesemilinear abstract differential equations. Semigroup Forum, 69, 80–86.

[17] N’Guerekata, G.M., 1999, Almost automorphic functions and applications to abstract evolutionequations. Contemporary Mathematics, 252, 71–76.

[18] N’Guerekata, G.M., 2000, Almost automorphy, almost periodicity and stability of motions in Banachspaces. Forum Mathematicum, 13, 581–588.

[19] N’Guerekata, G.M., 2001, Almost Automorphic and Almost Periodic Functions in Abstract Spaces(New York–Moscow–London: Kluwer).

[20] Pazy A., 1983, Semigroups of linear operators and applications to partial differential equations.Applied Mathematical Sciences, 44 (Berlin–New York: Springer-Verlag).

[21] Pruss, J., 1993, Evolutionary Integral Equations and Applications (Basel: Birkhauser).[22] Ruess, W.M. and Vu, Q.P., 1995, Asymptotically almost periodic solutions of evolution equations

in Banach spaces. Journal of Differential Equations, 122, 282–301.[23] Shen, W. and Yi, Y., 1998, Almost automorphic and almost periodic dynamics in skew-product

semiflows. Memoirs of the American Mathematical Society, 136.[24] Vu, Q.P. and Schuler, E., 1998, The operator equation AX� XB ¼ C, stability and asymptotic behaviour

of differential equations. Journal of Differential Equations, 145, 394–419.[25] Schweiker, S., 2000, Mild solutions of second-order differential equations on the line. Mathematical

Proceedings of the Cambridge Philosophical Society, 129(1), 129–151.[26] Schuler, E. and Vu Quoc Phong, 2000, The operator equation AX� XD2 ¼ �0 and second

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Vol. 42 (Newport Beach, CA) 1998, Progr. Nonlinear Differential Equations Appl., pp. 352–363(Basel: Birkhauser).

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Appendix: Sums of commuting operators

We recall now the notion of two commuting operators which will be used in the sequel.

Definition 4.1 Let A and B be operators on a Banach space G with nonempty resolventset. We say that A and B commute if one of the following equivalent conditions hold:

(i) Rð�,AÞRð�,BÞ ¼ Rð�,BÞRð�,AÞ for some (all) � 2 �ðAÞ,� 2 �ðBÞ,(ii) x 2 DðAÞ implies Rð�,BÞx 2 DðAÞ and ARð�,BÞx ¼ Rð�,BÞAx for some (all)

� 2 �ðBÞ.

For � 2 ð0,Þ,R > 0 we denote �ð�,RÞ ¼ fz 2 C: jzj R, j arg zj � �g.

Definition 4.2 Let A and B be commuting operators. Then

(i) A is said to be of class �ð� þ =2,RÞ if there are positive constants �,R such that0 < � < =2, and

�ð� þ =2,RÞ � �ðAÞ and sup�2�ð�þ=2,RÞ

k�Rð�,AÞk <1, ð21Þ

(ii) A and B are said to satisfy condition P if there are positive constants �,� 0,R,� 0<�such that A and B are of class �ð� þ =2,RÞ,�ð=2� �0,RÞ, respectively.

If A and B are commuting operators, Aþ B is defined by ðAþ BÞx ¼ Axþ Bx withdomain DðAþ BÞ ¼ DðAÞ \DðBÞ. In this article we will use the following norm, definedby A on the space X, kxkT A

:¼ kRð�,AÞxk, where � 2 �ðAÞ. It is seen that different� 2 �ðAÞ yields equivalent norms. We say that an operator C on X is A-closed if itsgraph is closed with respect to the topology induced by T A on the product X� X.It is easily seen that C is A-closable if xn! 0, xn 2 DðCÞ,Cxn! y with respect to T A

in X implies y ¼ 0. In this case, A-closure of C is denoted by CA.

THEOREM 4.3 Assume that A and B commute. Then the following assertions hold:

(i) If one of the operators is bounded, then

�ðAþ BÞ � �ðAÞ þ �ðBÞ: ð22Þ

(ii) If A and B satisfy condition P, then Aþ B is A-closable, and

�ððAþ BÞAÞ � �ðAÞ þ �ðBÞ: ð23Þ

In particular, if D(A) is dense in X, then ðAþ BÞA¼ Aþ B, where Aþ B denotes the

usual closure of Aþ B.

Proof For the proof we refer the reader to [2, Theorems 7.2, 7.3]. g

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