ON THE MILD SOLUTIONS OF HIGHER-ORDER DIFFERENTIAL EQUATIONS IN BANACH SPACES

ON THE MILD SOLUTIONS OF HIGHER-ORDER DIFFERENTIAL EQUATIONS IN BANACH SPACES

NGUYEN THANH LAN Received 28 January 2003

For the higher-order abstract differential equationu⁽ⁿ⁾(t)=Au(t) +f(t),t∈R, we give a new definition of mild solutions. We then characterize the regular ad- missibility of a translation-invariant subspaceᏹof BUC(R, E) with respect to the above-mentioned equation in terms of solvability of the operator equation AX−XᏰⁿ=C. As applications, periodicity and almost periodicity of mild so- lutions are also proved.

1. Introduction

The qualitative theory of mild solutions on the whole line of the differential equation of type

u(t)=Au(t) +f(t), t∈R, (1.1) whereAis a closed operator on a Banach spaceE, has been of increasing interest in the last decades. If Ais a bounded operator onE, mild solutions of (1.1), which are the same as the classical solutions, are defined by

u(t)=e^Atu(0) + _t

0e^A(t−s)f(s)ds, t∈R. (1.2) In [4], Dalec’ki˘ı and Kre˘ın made a systematic study on the asymptotic behavior of solutions of the form (1.2). For unbounded operatorA, where the situation changes dramatically, the first question is, which solutions of (1.1) are consid- ered asmild solutions? IfAis the generator of aC0-semigroupT(t),t≥0, it is logical to define mild solutions of (1.1) by

u(t)=T(t−s)u(s) + _t

sT(t−τ)f(τ)dτ, t≥s. (1.3)

Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:15 (2003) 865–880 2000 Mathematics Subject Classification: 34G10, 34K06, 47D06 URL:http://dx.doi.org/10.1155/S1085337503303057

With this definition in hand, many authors investigated the qualitative behavior of (1.3) in different ways (see [10,12,13,14,17] and references therein). The second-order differential equationu(t)=Au(t) +f(t), whereAis the genera- tor of a cosine family (C(t)), and for which mild solutions are defined by

u(t)=C(t−s)u(s) +S(t−s)u(s) + _t

sS(t−τ)f(τ)dτ, (1.4) has been also studied in [3,8,18].

Recently, Arendt and Batty [1], Schweiker [20], and Sch¨uler and Ph ´ong [19]

studied the first- and second-order differential equations, in which A is not the generator of aC0-semigroup or of a cosine family, respectively. Although their definitions of mild solutions are slightly different, they all showed that the existence and uniqueness of mild solutions, which belong to a subspaceᏹof BUC(R, E), are closely related to the solvability of the operator equation of the form

AX−XᏰ= −δ0, (1.5)

whereᏰis the differential operator inᏹandδ0is the Dirac operator defined by δ0(f) :=f(0).

Inspired by this rapid development, in this paper, we consider the higher- order differential equation

u⁽ⁿ⁾(t)=Au(t) +f(t), (1.6) whereAis a closed linear operator onEand f is a continuous function fromR toE. First, we give a general definition of mild solutions to (1.6). This definition is an extension of that introduced in [1], wheren=1,n=2, andAgenerally is neither the generator of aC0-semigroup nor of a cosine family, respectively.

Several properties of mild solutions are then shown inSection 2.

InSection 3, we consider the conditions for the solvability of operator equa- tionAX−XB=C, in particular, whenB=Ᏸⁿ, whereᏰis the differential oper- ator on a function space andC= −δ0.

Assume thatᏹis a closed, translation-invariant subspace of BUC(R, E). The subspaceᏹis said to beregularly admissible with respect to (1.6) if for every f ∈ᏹ, (1.6) has a unique mild solutionu∈ᏹ. InSection 4, we characterize the regular admissibility ofᏹin terms of solvability of the operator equation.

Namely, we show that the subspaceᏹis regularly admissible if and only if the operator equation of the form

AX−XᏰⁿ= −δ0 (1.7)

has a unique bounded solution. As applications, inSection 5we show that if the admissible subspaceᏹis the space of 1-periodic functions, then

sup

k∈Z

k^m(2πki)ⁿ−A⁻¹<∞ (1.8)

is a necessary condition, that each mild solution onᏹbelongs toC^(m)(R, E), where 0≤m≤n. Finally, we prove that, under some classical condition, ifσ(A)

∩(iR)ⁿis countable, then each bounded mild solution of the higher-order equa- tion is almost periodic provided f is almost periodic. This result, shown by a short proof, generalizes [1, Theorem 4.5].

2. Mild solutions of higher-order differential equations

First, we fix some notations. ByC⁽ⁿ⁾(R, E) we denote the space of continuous functions with continuous derivativesu, u, . . . , u⁽ⁿ⁾and by BUC(R, E) the space of bounded, uniformly continuous functions with values inE. The operatorI: C(R, E)→C(R, E) is defined byI f(t) :=_t

0 f(s)dsandIⁿf :=I(Iⁿ⁻¹f).

Definition 2.1. (a) We say thatu:R→Eis a classical solution of (1.6) ifu∈ D(A),u∈Cⁿ(R, E), and (1.6) is satisfied.

(b) A continuous functionu(t)∈C(R, E) is called a mild solution of (1.6) if I⁽ⁿ⁾u(t)∈D(A) for allt∈Rand there existnpointsv0, v1, . . . , vn−1inEsuch that

u(t)=

n−1 i=0

tⁱ

i!vi+AIⁿu(t) +Iⁿf(t) (2.1) for allt∈R.

Remark 2.2. Using the standard argument, we can prove the following state- ments:

(i) if a mild solution u ism-times differentiable, 0≤m < n, thenvi, i= 0,1, . . . , m, are the initial values, that is,u(0)=v0,u(0)=v1, . . ., andu^(m)(0)= v_m;

(ii) ifn=1 andAis the generator of aC0-semigroupT(t), then a continuous functionu:R→Eis a mild solution of (1.6) if and only if it has the form

u(t)=T(t−s)u(s) + _t

sT(t−r)f(r)dr; (2.2) (iii) similarly, ifn₌2 andAis a generator of a cosine family (C(t)) onE, any

continuously differentiable functionuonEof the form u(t)=C(t−s)u(s) +S(t−s)u(s) +

_t

sS(t−τ)f(τ)dτ, (2.3) where (S(t)) is the associated sine family, is a mild solution of (1.6);

(iv) if u is a bounded mild solution of (1.6) corresponding to a bounded inhomogeneity f andφ∈L¹(R, E), thenu∗φis a mild solution of (1.6) corresponding to f∗φ.

Directly from their definitions, we can collect some properties of mild solutions of (1.6).

Lemma2.3. Letube a mild solution of the higher-order differentiable equation (1.6). If

(i)uis inC⁽ⁿ⁾(R, E); or

(ii)u(t)∈D(A)for allt∈RandAu(·)∈C(R, E), thenuis a classical solution.

Proof. (i) Sinceuis a mild solution, we have AIⁿu(t)=u(t)−

n−1 0

tⁱ

i!vi−Iⁿf(t). (2.4) The right-hand side of (2.4) is n-time differentiable so is the left-hand side.

Hence, limh→0A1

h _t+h

t Iⁿ⁻¹u(s)ds=lim

h→0

1 h

A _t+h

0 Iⁿ⁻¹u(s)ds−A _t

0Iⁿ⁻¹u(s)ds

= d dt

AIⁿ(t)

(2.5)

exists. Since

limh→0

1 h

_t+h

t Iⁿ⁻¹u(s)ds=Iⁿ⁻¹u(t) (2.6) andAis closed, we obtain thatIⁿ⁻¹u(t)∈D(A) and

d dt

AIⁿu(t)=AIⁿ⁻¹u(t). (2.7)

By taking the derivative on both sides of (2.4), we obtain AIⁿ⁻¹(t)=u(t)−

n−2 0

tⁱ

i!vi+1−Iⁿ⁻¹f(t) (2.8) for allt∈R. Repeating this procedure (n−1) times, we obtain thatuisn-times differentiable andu⁽ⁿ⁾(t)=Au(t) +f(t), that is,uis a classical solution.

(ii) Ifu(t)∈D(A) for allt∈RandAu(·)∈C(R, E), thenAIⁿu(t)=IⁿAu(t).

Taking thenth derivative of the right-hand side of u(t)=

n−1 0

tⁱ

i!v_i+IⁿAu(t) +Iⁿf(t), (2.9)

we have thatuisn-times continuously differentiable andu⁽ⁿ⁾(t)=Au(t) + f(t),

that is,uis a classical solution.

In what follows we consider the spectrum of mild solutions of (1.6). For a bounded functionu∈L^∞(R, E), theCarleman transformuˆofuis defined by

ˆ u(λ)=









 _∞

0 e^−λtu(t)dt, Re(λ)>0,

− 0

−∞e^−λtu(t)dt, Re(λ)<0.

(2.10)

It is clear that ˆuis holomorphic onC\iR. A pointµ∈Ris called aregular point if ˆuhas a holomorphic extension in a neighborhood ofiµ. The spectrum ofuis defined as follows:

sp(u)= {µ∈R:µis not regular}. (2.11) The following lemma, whose proof can be found in [7,15], will be needed later.

Lemma2.4. Let f,gbe inBUC(R, E)andφ∈L¹(R, E). Then (i) sp(f)is closed andsp(f)= ∅if and only if f =0;

(ii) sp(f +g)⊂sp(f)∪sp(g);

(iii) sp(f ∗φ)⊂sp(f)∩suppᏲφ, whereᏲφis the Fourier transform ofφ.

The following lemma is the first result about the spectrum of mild solutions of (1.6).

Lemma2.5. Let f be a bounded continuous function and letube a bounded mild solution of (1.6). Then

sp(u)⊆

µ∈R: (iµ)ⁿ∈σ(A)∪sp(f). (2.12) Proof. It is easy to see thatIu(λ)=(1/λ) ˆu(λ), henceIⁿu(λ)=(1/λⁿ) ˆu(λ). Taking the Carleman transform on both sides of (2.1), we have

ˆ

u(λ)=Q(λ) + 1

λⁿAu(λ) +ˆ 1

λⁿfˆ(λ), (2.13)

where

Q(λ)= _∞

0 e^−λt



ⁿ⁻¹

i=0

tⁱ i!vi



dt=

n−1 i=0

u_i

λⁱ. (2.14)

From (2.13) we obtain

λⁿ−Au(λ)ˆ =λⁿQ(λ) + ˆf(λ) (2.15)

forλ /∈iR. Hence, forλⁿ∈ρ(A) we have ˆ

u(λ)=

λⁿ−A⁻¹λⁿQ(λ) + ˆf(λ). (2.16) Note thatλⁿQ(λ) is a holomorphic function in terms ofλ. It implies that ifµ∈R is a regular point of f and (iµ)ⁿ∈ρ(A), then ˆuhas holomorphic extension in a neighborhood ofiµ, that is,µis a regular point ofu. Hence, we have the inclusive

relation.

FromLemma 2.5, we directly have the following corollary.

Corollary2.6. Ifuis a bounded mild solution of (1.6) corresponding to f ≡0, thensp(u)⊆ {µ∈R: (iµ)ⁿ∈σ(A)}.

Corollary2.7. If(iR)ⁿ∩σ(A)= ∅, then (1.6) has at most one bounded mild solution.

3. The equationAX−XDⁿ=C

LetAandBbe closed, generally unbounded, linear operators on Banach spacesE andFwith dense domainsD(A) andD(B), respectively, and letCbe a bounded linear operator fromEtoF. A bounded operatorX:F→Eis called asolutionof the operator equation

AX−XB=C (3.1)

if for every f ∈D(B) we have X f ∈D(A) and AX f −XB f =C f. Equation (3.1) has been considered by many authors. It was first studied intensively for bounded operators by Dalec’ki˘ı and Kre˘ın [4], Rosenblum [16]. For unbounded case, (3.1) was studied in [2,11,12,13], whenAandBare generators ofC0- semigroups, and in [17,19] whenAandBare closed operators. We cite here some main results which will be used in the sequel.

Theorem3.1. (i)LetAandBbe generators ofC0-semigroups onEandF, one of which is analytic such thatσ(A)∩σ(B)= ∅. Then for every bounded operatorC, (3.1) has a unique bounded solution (see[11, Theorem 15]).

(ii)LetAbe a closed operator and letBbe a bounded operator such thatσ(A)∩ σ(B)= ∅. Then for every bounded operatorC, (3.1) has a unique bounded solution Xwhich has the following integral form:

X= 1 2πi

Γ(λ−A)⁻¹C(λ−B)⁻¹dλ, (3.2) whereΓis a closed Cauchy contour aroundσ(B)and is separated fromσ(A)(see [17, Theorem 3.1]).

(iii)If (3.1) has a unique bounded solution for every bounded operatorC, then σ(A)∩σ(B)= ∅(see[2, Theorem 2.1]).

We now consider the situation whenF=ᏹ, a translation-invariant subspace of BUC(R, E), and B=Ᏸⁿ_ᏹ, the restriction of Ᏸⁿ toᏹ, where Ᏸ:=d/dt on BUC(R, E). It is well known thatσ(Ᏸ)=iRandσ(Ᏸⁿ)=(σ(Ᏸ))ⁿ.

Let now ᏹk:= {f ∈ᏹ: sp(f)⊂[−ik, ik]}, k≥1. Then the following properties hold (see [5,19]):

(i)ᏹkare translation-invariant subspaces, (ii)ᏹk⊂ᏹk+1,

(iii)Ᏸᏹkis bounded.

We first need the following lemma which was proved in [19].

Lemma3.2. LetᏰᏹandᏰᏹkbe as above, then σᏰᏹ

= ∪^∞_k=1σᏰᏹk

. (3.3)

FromLemma 3.2we obtain the following lemma.

Lemma3.3. For any positive integern≥1, the following equality holds:

σᏰⁿ_ᏹ= ∪^∞_k=1σᏰⁿ_ᏹk. (3.4) Proof. We show that

σᏰⁿ_ᏹ⊆ ∪^∞_k=1σᏰⁿ_ᏹk. (3.5) Note thatσ(Ᏸⁿ)=(iR)ⁿ, hence σ(Ᏸⁿ_ᏹ)⊆(iR)ⁿ. Assume that (iλ)ⁿ∈σ(Ᏸⁿ_ᏹ), λ∈R. Then there is a sequence of vectors (fk)k⊂ᏹ such that fk∈D(Ᏸⁿ_ᏹ), f_k =1, and

klim→∞

(iλ)ⁿ−Ᏸⁿ_ᏹfk=0. (3.6)

Letλ1, λ2, . . . , λnbe thencomplex roots of the equationxⁿ=(iλ)ⁿ. Then we have (iλ)ⁿ−Ᏸⁿ_ᏹfk=

n j=1

λj−Ᏸᏹ

fk. (3.7)

We show that there is at least oneλjbelonging to the spectrum ofᏰᏹ. Assume contrarily that allλjbelong toρ(Ᏸᏹ), then

fk= n j=1

λj−Ᏸᏹ−1

(iλ)ⁿ−Ᏸⁿ_ᏹfk−→0 ask−→ ∞, (3.8) which is contradictory tofk =1. Hence, there is aλjwhich belongs toσ(Ᏸᏹ).

By Lemma 3.2, there is a number k such that iλj∈σ(Ᏸᏹk). Since Ᏸᏹk is bounded, (iλ)ⁿ=(iλ_j)ⁿ∈σ(Ᏸⁿ_ᏹk), and hence the inclusion (3.5) follows. Since the inverse of (3.5) is obvious, the lemma is proved.

From Lemmas3.2and3.3we have the following lemma.

Lemma3.4. For any positive integern≥1the following equality holds:

σᏰⁿ_ᏹ=

λⁿ:λ∈σᏰᏹ

. (3.9)

We now return to the operator equation

AX−XᏰⁿ_ᏹ=δ₀^ᏹ, (3.10)

whereδ^ᏹ₀ is the restriction of the Dirac operator toᏹ. Assume that σ(A)∩

λⁿ:λ∈σᏰᏹ

= ∅. (3.11)

Then, byLemma 3.4, it is equivalent to

σ(A)∩σᏰⁿ_ᏹ= ∅. (3.12)

Therefore, fork=1,2, . . ., we have

σ(A)∩σᏰⁿ_ᏹk= ∅. (3.13)

ByTheorem 3.1, the operator equation

AX−XᏰⁿ_ᏹk=δ^ᏹ₀^k (3.14)

has a unique bounded solutionXkwhich is of the form X_k= − 1

2πi

Γk

(λ−A)⁻¹δ₀^ᏹnλ−Ᏸⁿ_ᏹk⁻¹dλ, (3.15) whereΓkis a contour aroundσ(Ᏸⁿ_ᏹk) and is separated fromσ(A). Moreover, the uniqueness ofXkimplies

Xk|ᏹl=Xl forl < k. (3.16) We state a result about the existence and uniqueness of bounded solutions of (3.10), whose proof is similar to that of [19, Theorem 7] (forn=2) and is omit- ted.

Theorem 3.5. Assume that condition (3.11) holds. Then the operator equation (3.10) has a unique bounded solution if and only if

sup

n≥1

Xk<∞, (3.17)

whereX_kare defined by (3.15).

4. Admissible subspaces

Letᏹbe a closed translation-invariant subspace of BUC(R, E), which is regu- larly admissible with respect to (1.6). Define the linear operatorGonᏹsuch that for each f ∈ᏹ,G f is the unique mild solution of (1.6) inᏹ, we have the following lemma.

Lemma4.1. The operatorGis a linear, bounded operator onᏹ. Proof. We define operator ˜G:ᏹ→ᏹ⊗Eⁿby

G f˜ :=

u, v0, v1, . . . , vn−1

, (4.1)

whereuis the unique mild solution of (1.6) corresponding to f andv0, v1, . . . , v_n−1are contained in the mild solution

u(t)=

n−1 0

tⁱ

i!vi+AIⁿu(t) +Iⁿf(t). (4.2) We will show that ˜G is closed. Let (fk)k∈N⊆ᏹ with limkfk= f and ˜G fk= (u_k, v_0,k, . . . , v_n−1,k) with lim_k→∞G f˜ _k=(u, v0, . . . , v_n−1), that is, lim_k→∞u_k=uand limk→∞vj,k=vkforj=0,1, . . . , n−1. Then we have limk→∞Iⁿuk(t)=Iⁿu(t) and, by (4.2),

AIⁿuk(t)=uk(t)−

n−1 0

tⁱ

i!vi,k−Iⁿfk(t)

−→u(t)−

n−1 0

tⁱ

i!vk−Iⁿf(t) ask−→ ∞.

(4.3)

SinceAis closed we obtain thatIⁿu(t)∈D(A) and AIⁿu(t)=u(t)−

n−1 0

tⁱ

i!vi−Iⁿf(t). (4.4) That means that ˜G f =(u, v0, v1, . . . , vn−1). Hence, ˜Gis closed and thus bounded.

SinceG=G˜◦P, whereP:ᏹ⊗Eⁿ→ᏹis the projection on the first coordinate and thus is a bounded operator, we obtain thatGis bounded.

The operatorGis called thesolution operatorof (1.6) and is commuting with the translation and hence is commuting with the differential operator, as the following lemma shows.

Lemma4.2. LetAbe a closed operator onEwith nonempty resolvent set and letᏹ be an admissible subspace ofBUC(R, E). Then the following conditions hold:

(i)Sh·G=G·Sh, whereShis the translation operator onᏹ;

(ii)Ᏸᏹ·G=G·Ᏸᏹ.

Proof. (i) Letu=G f be the unique mild solution of the higher-order differen- tial equation (1.6). Ifuis a classical solution, then (G f)⁽ⁿ⁾(t+h)=A(G f)(t+ h) +f(t+h), and henceS_h·G f =G·S_hf. For the case thatuis not a classical solution, letλ∈ρ(A). Since

R(λ, A)u(t)=

n−1 0

tⁱ

i!R(λ, A)ui+AIⁿR(λ, A)u(t) +IⁿR(λ, A)f(t), (4.5) it is easy to see that ˜u(t)=R(λ, A)u(t) is the unique solution of (1.6) correspond- ing to ˜f =R(λ, A)f. But ˜u(t)∈D(A) for allt∈R. Hence, byLemma 2.3(ii), ˜u is a classical solution. From the above result for a classical solution and the fact thatShandR(λ, A) commute, we have

R(λ, A)ShG f =ShR(λ, A)G f =ShGR(λ, A)f

=GS_hR(λ, A)f =GR(λ, A)S_hf =R(λ, A)GS_hf , (4.6) from which it follows thatShG f =GShf for all f ∈ᏹ. Part (ii) is a direct con-

sequence of (i), and the lemma is proved.

Corollary4.3. Assume thatAis a closed operator with nonempty resolvent set.

Letᏹbe a regularly admissible subspace ofBUC(R, E)and letube the unique mild solution corresponding to f inᏹ. If f ∈Cⁿ(R, E)such that f, f, . . . , f⁽ⁿ⁾belong toᏹ, thenuis a classical solution.

In what follows, we assume thatᏹsatisfies the following additional assump- tion.

Assumption 4.4. For allC∈ᏸ(ᏹ, E) and f ∈ᏹ, the functionΦ(t)=CS(t)f belongs toᏹ.

The regular admissibility of a space is closely related to the solvability of oper- ator equation (3.1). This relation was shown in [13], whenn=1, and in [19,20], whenn=2. The following theorem is a generalization of those results.

Theorem4.5. LetAbe a closed operator onEwith nonempty resolvent set and let ᏹbe a translation-invariant subspace inBUC(R, E), which satisfiesAssumption 4.4. Then the following statements are equivalent:

(i)ᏹis a regularly admissible subspace;

(ii)the operator equation

AX−XᏰ⁽ⁿ⁾_ᏹ = −δ0 (4.7)

has a unique solution;

(iii)for every bounded operatorC:ᏹ→E, the operator equation

AX−XᏰ⁽ⁿ⁾_ᏹ =C (4.8)

has a unique solution.

Proof. (i)⇒(ii). Let G:ᏹ→ᏹbe the bounded operator defined by G f =u, whereuis the unique mild solution inᏹ. We define the operatorX:ᏹ→Eby

X f :=(G f)(0). (4.9)

ThenX is a bounded operator. Now let f ∈Ᏸⁿ_ᏹ. ByLemma 4.2,u=G f is a classical solution of (1.6), that is,

(G f)⁽ⁿ⁾(t)=A(G f)(t) +f(t). (4.10) Note that, byLemma 4.2, (G f)⁽ⁿ⁾=G f⁽ⁿ⁾. Takingt=0 from (4.10) and using this fact, we haveAX f −XᏰⁿf = −δ0f for f ∈Ᏸⁿ_ᏹ, that is,X is a bounded solution of (4.7).

To show the uniqueness, we assume thatX0 is a solution of (4.7). Then for every f _∈Ᏸⁿ_ᏹ, the functionu_∈ᏹ, defined byu(t)₌X0S(t)f, is a classical so- lution of (1.6). Indeed,

u⁽ⁿ⁾(t)=X0ᏰⁿS(t)f =

AX0+δ0

S(t)f =Au(t) +f(t) (4.11) for allt∈R. We will show thatu(t)=X0S(t)f is a mild solution of (1.6) for every f ∈ᏹ. To this end, let f ∈ᏹ and (fk)k∈N⊆D(Ᏸⁿ_ᏹ) with limkfk= f. ThenG f =lim_kG f_k=lim_kX0S(·)f_k=X0S(·)f. Hence,G f =X0S(·)f, that is, u=X0S(·)f is a mild solution of (1.6).

Assume now thatX1andX2are two solutions of (4.7). Then, for every f ∈ ᏹ,u=(X1−X2)S(·)f is a mild solution of the higher-order equationu⁽ⁿ⁾(t)= Au(t). By the uniqueness of the mild solution we haveu≡0, which impliesX1= X2.

(ii)⇒(iii). Let X be the unique solution of (4.7). Define the bounded op- eratorY:ᏹ→EbyY f :=Xf˜, where ˜f(t)= −CS(t)f. Let f ∈D(Ᏸⁿ_ᏹ), then (Ᏸⁿ_ᏹf)˜(t)= −CS(t)Ᏸⁿ_ᏹf =Ᏸⁿ_ᏹf˜(t). Hence, we have

AY f =AXf˜=XᏰⁿ_ᏹf˜+δ0f˜=XᏰⁿ_ᏹf˜+C f =YᏰⁿ_ᏹf +C f , (4.12) that is,Yis a bounded solution of (4.8).

The uniqueness of the solution of operator equationAX−XᏰⁿ_ᏹ=Cfollows directly from the uniqueness of the solution ofAX−XᏰⁿ_ᏹ= −δ0.

(iii)⇒(i). We have shown above that ifXis a bounded solution of (4.7), then u(t) :=XS(t)f is a mild solution of the higher-order equation (1.6). It remains to show that this solution is unique. In order to do it, assume thatuis a mild solution of the homogeneous equationu⁽ⁿ⁾(t)=Au(t),t∈R. ByCorollary 2.6,

(isp(u))ⁿ⊆σ(A). On the other hand, sinceu∈ᏹ,isp(u)⊆σ(Ᏸᏹ), which im- plies (isp(u))ⁿ⊆σ(Ᏸⁿ_ᏹ). ByTheorem 3.1(iii), it follows from (iii) thatσ(A)∩ σ(Ᏸⁿ_ᏹ)= ∅. Hence, sp(u)= ∅, sou≡0 and the theorem is proved.

5. Applications

In this section, we will apply the results ofSection 4to the space of periodic and of almost periodic functions. LetP(ω) be the space of periodic functions from RtoEwith the periodω. For the sake of simplicity, we assume the periodω=1.

We begin with the case in whichn=2 andAis the generator of a cosine family (C(t)). It is well known that

(1)Ais the generator of an analyticC0-semigroup given by e^Azx= 1

(πz) _∞

0 e^−t2/4zC(t)x dt, Re(z)>0; (5.1) (2)Ᏸ²is the generator of a cosine family given by

C(t)=1 2

᏿(t) +᏿(−t) (5.2) and hence is the generator of an (analytic)C0-semigroup inP(1).

By Theorems3.1(i) and 4.5,P(1) is regularly admissible if and only ifσ(A)∩ σ(Ᏸ²_P(1))= ∅. On the other hand,σ(Ᏸ²_P(1))= {(2kπi)²:k∈Z} = {−k²π²:k∈ Z}. Hence, we have the following theorem.

Theorem5.1. LetAbe the generator of a strongly continuous cosine family. Then P(1) is regularly admissible with respect to u(t)=Au(t) +f(t)if and only if {−4k²π²:k∈Z} ⊂ρ(A).

In general, however, the condition of the formσ(A)∩σ(Ᏸⁿ_ᏹ)= ∅does not imply the regular admissibility of subspaceᏹ. At least the operatorAmust sat- isfy some conditions, as the following theorem shows.

Theorem5.2. LetAbe a closed operator on a Banach spaceEwith nonempty re- solvent set and suppose thatP(1)is regularly admissible with respect to the equation u⁽ⁿ⁾(t)=Au(t) +f(t), t∈R. (5.3) Then

(1) (2πki)ⁿ∈ρ(A)andsup_k∈Z((2πki)ⁿ−A)⁻¹<∞,

(2)if each mild solution onP(1)belongs toC^(m)(R, E),0≤m≤n, then(2πki)ⁿ

∈ρ(A)andsup_k∈Zk^m((2πki)ⁿ−A)⁻¹<∞.

Proof. By assumption, P(1) is a regularly admissible function space, so, by Theorem 4.5, the equation AX−XᏰⁿ_P(1)=Chas a unique solution for every bounded operatorC. Hence, byTheorem 3.1(iii),σ(A)∩σ(ᏰⁿP(1))= ∅. On the

other hand, it is not hard to see thatσ(ᏰⁿP(1))= {(2kπi)ⁿ:k∈Z}. It follows that σ(A)∩ {(2kπi)ⁿ:k∈Z} = ∅or, in other words,{(2kπi)ⁿ:k∈Z} ⊂ρ(A).

To prove (1), letG:P(1)→P(1) be the solution operator and take f(t)= e^2kπitx0,x0∈E, as a 1-periodic function. It is not too hard to check thatG f(t)= e^2kπit·((2kπi)ⁿ−A)⁻¹x0is the (unique) mild solution of (5.3). Hence,

(2kπi)ⁿ−A⁻¹x0= G f ≤ G · f = G ·x0 (5.4) for allx0∈Eandk∈Z. Hence, sup_k∈Z((2kπi)ⁿ−A)⁻¹ ≤ G<∞.

To prove (2) observe that since each mild solution onP(1) belongs toC^(m)(R, E), the composite operatorᏰ^mP(1)Gis everywhere defined and closed. Hence, it is a bounded operator. Thus,

Ᏸ^m_P(1)G f=(2kπ)^m(2kπi)ⁿ−A⁻¹x0≤Ᏸ^m_P(1)G· f

=Ᏸ^m_P(1)G·x0 (5.5)

for allx0∈Eandk∈Z. Hence, sup_k∈Zk^m((2kπi)ⁿ−A)⁻¹ ≤C·Ᏸ^m_P(1)Gfor a certain constantC, and that completes the proof.

The converse ofTheorem 5.2generally does not hold (see [6] for a counterex- ample). However, we have the affirmative answer in certain special cases. IfEis a Hilbert space,n=1, andAis the generator of aC0-semigroup (T(t))t≥0, we have the following theorem whose proof of (b)⇒(a) can be found in [14].

Theorem5.3. LetAbe the generator of aC0-semigroup on a Hilbert spaceE. Then the following conditions are equivalent:

(a)for each1-periodic function f, the equation

u(t)=Au(t) +f(t) (5.6)

has a unique1-periodic mild solution;

(b){2πki:k∈Z} ⊂ρ(A)andsup_k∈Z(2πki−A)⁻¹<∞.

Also, ifn=2,m=1, andAis the generator of a cosine family (C(t)) on a Hilbert spaceE, we have a positive answer. Namely, we have the following theo- rem whose proof of the converse part (b)⇒(a) can be found in [8].

Theorem5.4. IfAis the generator of a cosine family on a Hilbert spaceE, then the following statements are equivalent:

(a)for each1-periodic function f, the equation

u(t)=Au(t) +f(t) (5.7)

has a unique1-periodic mild solution which belongs toC¹(R, E);

(b){−4π²k²:k∈Z} ⊂ρ(A)andsup_k∈Zk(4π²k²+A)⁻¹<∞.

We now apply the results ofSection 4to AP(R, E), the space of almost peri- odic functions fromRtoE. As a preparation, we recall some basic concepts and results about almost periodic functions. (For more details, readers are referred to [1,9].) A pointλ∈Ris called a point of almost periodicity of the functionuif there is a neighborhoodᐁofλsuch that for everyφ∈L¹(R) with suppᏲφ⊂ᐁ, whereᏲφis the Fourier transform ofφ, the functionφ_∗uis almost periodic.

The complement inRof the set of points of almost periodicity ofuis called the almost periodic spectrum of f and is denoted by sp_AP(u).

We say thatu∈BUC(R, E) istotally ergodicif

Tlim→∞

1 2T

_T

−Te^−iνsu(s)ds (5.8)

exists for allν∈R. The following theorem can be found in [9] (parts (a) and (b)) and [17] (part (c)).

Theorem5.5. Letu∈BUC(R, E)such thatsp_AP(u)is countable. Assume that (a)E⊇c0; or

(b)the range ofu(t)is weakly relatively compact; or (c)uis totally ergodic.

Thenuis almost periodic.

We now return to our higher-order equation. LetΓbe a compact set inR and let ᏹ=X(Γ) be the subspace of BUC(R, E) consisting of all functions f with sp(f)⊂Γ. It is easy to see thatᏹsatisfiesAssumption 4.4. Moreover,Ᏸᏹis bounded,σ(Ᏸᏹ)=iΓ, and thusσ(Ᏸⁿ_ᏹ)=(iΓ)ⁿ. Assume now thatσ(A)∩(iΓ)ⁿ=

∅; then, byTheorem 3.1(ii), the equationAX−XᏰⁿ_ᏹ= −δ0has a unique so- lution. ByTheorem 4.5,ᏹis regularly admissible and for any almost periodic function f, the mild solutionu(t)=XS(t)f is also almost periodic. Using these facts, we have the following theorem.

Theorem5.6. For the equation

u⁽ⁿ⁾(t)=Au(t) +f(t), t∈R, (5.9) assume that f is almost periodic andσ(A)∩(iR)ⁿis countable. Letu∈BUC(R, E)be a mild solution of (5.9). Thenu is almost periodic if one of the following conditions is satisfied:

(a)E⊇c0; or

(b)the range ofu(t)is weakly relatively compact; or (c)uis totally ergodic.

Proof. In view ofTheorem 5.5, we only have to show that sp_AP(u) is countable.

Sinceσ(A)∩(iR)ⁿis countable, it suffices to prove that (isp_AP(u))ⁿ⊂σ(A).

Letλbe any point inRsuch that (iλ)ⁿ∈ρ(A); we will show thatλ∈sp_AP(u).

Sinceρ(A) is an open set, there exists>0 such that (iΓ)ⁿ⊂ρ(A), whereΓ=[λ− , λ+]. SinceΓis compact andσ(A)∩(iΓ)ⁿ= ∅,X(Γ) is regularly admissible with respect to (5.9).

Letφbe a function inL¹(R, E) with suppᏲφ⊂Γand define ˜u:=u∗φand f˜:=f ∗φ. Then ˜uand ˜f are inX(Γ) (Lemma 2.4(iii)) and ˜f is an almost pe- riodic function. Moreover, ˜uis the unique mild solution of (5.9) corresponding to ˜f inX(Γ) (Remark 2.2). By the reasoning preceding this theorem, ˜uis also almost periodic. So,λis a point of almost periodicity ofu, that is,λ∈sp_AP(u),

and the theorem is proved.

Acknowledgments

This paper was written while the author was visiting the Department of Mathe- matics, Ohio University. The author thanks Dr. Vu Quoc Ph ´ong for many valu- able discussions and suggestions.

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Nguyen Thanh Lan: Department of Mathematics, Western Kentucky University, Bowl- ing Green, KY 42101, USA

E-mail address:[email protected]