On theM ildSolutions of H igher O rder Differenti al E quations inBanachSpaces§ N guyen T hanh Lan ∗

D epartmentofMathem atics Western K entucky U niversity.

Abstract For the higher order differenti al equation u(n )(t) = A u (t) + f (t), t∈ IR (*) on a B anach space E , w e give a new definition of m ild solutions of (*). W e then characterize the regular adm issiblity o f a tran slation invariant subspace M ofB U C (IR,E)with resp ect to (*) intermsofsolvability of th e op erator eq u ation A X − X D n = C.As applications, periodicity a n d a lm ost p eriodicity o f m ildsolutions of (*) are also p roved .

1Introduction

Thequalitative th eory of m ildsolutions on the w holeline ofthe differenti al equation of the type

u (t) = A u (t) + f (t), t ∈ I R, (1.1) whereA isaclosed operator on a B anach space E , has been of increasing interestinthelast decades. IfA is a bounded operator on E ,m ildsolutions of (1.1),w hich are the sam e as the classical solutions,are defined by

 t u(t)= eAtu(0)+ eA(t−s)f(s)ds, t∈ I R. (1.2) 0 In th eirbook[3],D aleckiiand K rein m ad e a sy stem atic stu d y on th e asy m p - toticbehavior of solutions of the form (1.2). For unbounded operator A , §2000 A M S Subject C lassification: P rim ary 34 G 10,34 K 06,Secondary 47 D 06. ∗ Thispaperwaswritten , w h ile the author w as visiting the D epartm ent ofM athem atics, OhioUniversity. T he author thanks D r. V u Q uoc P hong for m any valuablediscu ssions and suggestions

1 wherethesitu ation changes dram atically, th e fi rst q u estion is, w h ich solu- tions of (1.1) are considered as m ildsolutions. If A is th e gen erator of a

C 0-sem igroup T (t),t ≥ 0,itislogical to define m ildsolutions of (1.1) by

 t u (t) = T (t − s)u (s) + T (t− τ)f(τ)dτ, t≥ s. (1. 3) s With th isdefinition in hand,m any authorsinvestigated the qualitative be- havior of (1.3) indifferent w ays (see [ 8],[11],[14],[19],[20],and references therein).T he second orderdifferenti alequation,u  (t) = A u (t) + f (t), w h ere Ais th e gen erator of a cosine fam ily (C (t)), an d for w hich m ildsolutions are defined by

 t u (t) = C (t − s)u (s) + S (t − s)u  (s) + S(t− τ)f(τ)dτ, s has b een also stu d ied in[9],[10]and [15]. R ecently, A rendt and B atty [2],Schweiker [16],andVu QuocPhongand Schuler [21]studied the first and second order differenti alequation,inwhich

Ais not the generator of a C 0-sem igroup or ofa cosine fam ily (resp ectively). Although theirdefinitions of m ildsolutions are slightlydifferent,they al l sh ow ed th at th e ex istence and uniqueness of m ildsolutions,w hich b elong to a subspace M of B U C (IR,E),arecloselyrelated to the solvability o f th e operator equation of the form

AX − X D = −δ 0, whereD isthedifferenti aloperatorinM andδ 0 istheDirac op erator d efi n ed by δ 0(f ) :=f(0). In sp ired b y th israpid developem ent, inthis paper, w e consider the higher order differenti alequation

u(n )(t) = A u (t) + f (t), (1.4) whereA isaclosed linear operator on E and f isacontinuous function from IRtoE.First, w e give a general definition of m ildsolutions to E quation (1.4). T hisdefinition isanextension of that in tro d u ced in[2],wheren = 1

2 and n = 2, and A generallyis not the generator of a C 0-sem igroup (and of acosine fam ily, resp ectively). Several prop eries of m ildsolution s are th en sh ow n in Section 2. In S ection 3,w e considertheconditions for the solvability ofoperator equa- tion A X − X B = C ,in p articular,w hen B = D n,w hereD is thedifferenti al operator on a function space,and C = − δ 0. A ssu m e th at M isaclosed, translation-invariant subspace of B U C (IR,E). Missaidtoberegularlyadmissiblewith resp ect to E q u ation (1.4), iffor every f ∈ M E quati on (1.4) has a unique m ildsolution u ∈ M . In Secti on 4 w e characterize the regular adm issibility o f M intermsofsolvability o f th e op erator eq u ation. N am ely, w e sh ow th at th e su b sp ace M isregularly adm issibleifand onlyif the operator equation of the form

n AX − X D =−δ0 (1.5) has a unique bounded solution. A s applications,in Section 5 w e show that iftheadmissible subspace M isthespaceof1-periodicfunctions, then su p k m ((2π k i )n −A)−1 < ∞ i s a necessary condition, that each m ildso- k∈Z Z lution on M belongs to C (m )(IR,E),where0 m n.Finally, w e prove that, under som e classical condition,ifσ(A)∩ (i IR)n iscountable, then each bounded m ildsolution of the higher order equation isalmostperiodic, pro- vided f isalmostperiodic. T hisresult, sh ow n b y a sh ort p ro of, generalises [2,T heorem 4.5].

2MildSolutions ofH igher O rder D ifferenti al Equations

First letusfix som e notations.ByC(n )(IR ,E )w e denotethe spaceofcontin- uous functions w ith con tinuous derivatives u ,u   ,...u(n ),and by B U C (IR,E) the space ofbounded,uniform lycontinuous functions w ith valuesinE.The  t operator I : C (IR,E) → C(IR,E) is defined by If(t) := 0 f(s)ds and Inf:=I(In− 1f).

3 Definition 2.1a)Wesaythat u :IR→ Eisaclassical solution of(1.4),if u∈D(A),u∈C n(IR,E)and(1.4) issatisfi ed. b) A continuous function u(t)∈ C (I R,E)iscalled a m ildsolution of(1.4), (n ) ifI u(t)∈ D (A) f or allt∈ I Randthere exist n pointsv0,v1,...,vn− 1 in Esuchthat n− 1 i t n n u(t)= vi +AIu(t)+ I f(t) (2.1) i= 0 i! for allt∈ I R.

Remark.Usin g th e stan d ard argu m en t, w e can p rove th e follow ing.

(i)Ifamildsolution u ism timesdifferenti able, 0 m < n , then vi,  i = 0,1,...,m , are th e initial values,i.e. u(0) = v0,u (0) = v 1,...,and (m ) u (0) = v m .

(ii)Ifn= 1andAis thegenerator ofa C 0 sem igrou p T (t), th en a con tin- uous function u :IR→ Eisam ildsolution of(1.4) ifand onlyifithas th e form  t u (t) = T (t − s)u (s) + T(t− r)f(r)dr. s (iii)Similarly, if n = 2 and A a generator of a cosine fam ily(C(t))onE, any continuouslydifferenti ablefunction u on E of the form

 t u (t) = C (t − s)u (s) + S (t − s)u  (s) + S(t− τ)f(τ)dτ, s where(S(t))is th e asso ciated sine fam ily, isamildsolution of (1.4).

(iv) Ifu is a bounded m ildsoution of (1.4) corresp on d ing to a bounded inhom ogenity f a n d φ ∈ L 1(IR,E)thenu∗φi sam ildsolution solution of (1.4) corresp on d ing to f ∗ φ.

Directlyfrom th eirdefinition,w e can collect som e p rop erties ofm ildsolutions of E quation (1.4).

Lem m a 2.2Letubeamildsolution ofthe higher order different iableequa- tion (1.4). If

(i)uisinC(n )(IR,E);or

4 (ii)u(t)∈D(A)for allt∈ I RandAu(·)∈ C(I R,E), then u isaclassical solution. Proof.(i)Since u isamildsolution w e have

n− 1 i n t n AI u(t)= u(t)− vi −If(t). (2.2) 0 i! Theright hand side of(2.2) isn-timedifferenti able, so isthelefthandside. Hence,   1  t+ h 1  t+ h  t lim A In− 1u(s)ds = lim A In− 1u(s)ds− A In− 1u(s)ds h→ 0 h t h→ 0 h 0 0 d = (A I n(t)) dt

1  t+ h n− 1 n− 1 exists. S ince lim h→ 0 h t I u(s)ds = I u(t)and A isclosed,w e obtain n− 1 d n n− 1 th at I u(t)∈ D (A)and dt(A I u(t))= A I u(t).B y taking thederivative on both sidesof(2.2),w e obtain

n− 2 i n− 1  t n− 1 AI (t) = u (t) − vi+ 1 −I f(t) 0 i! for allt∈ I R.Repeating this p ro ced u re (n − 1) timesweobtainuisn times differenti ableandu(n )(t) = A u (t) + f (t), i.e. u isaclassical solution. (ii)Ifu(t)∈ D(A)f or allt∈ I RandAu(·)∈ C(IR,E),thenAInu(t)= InAu(t).Taking the nth derivative of the right hand side of

n− 1 i t n n u(t)= vi +IAu(t)+ I f(t), 0 i! wehaveu isn timescontinuouslydifferenti ableandu(n )(t) = A u (t) + f (t), i.e.,u isaclassical solution. ♣

In th e follow ing w e consid er th e sp ectru m of m ildsolutions of (1.4). For a bounded function u ∈ L ∞ (IR,E),the C arlem an tra n formuofui ˆ sdefinedby    ∞ −λt  0 e u(t)dt,R e(λ)> 0, u(λ)=ˆ  (2.3)  0 −λt − −∞ e u(t)dt,R e(λ)< 0.

5 It isclear thatui ˆ sholom orphiconC \iIR. A point µ ∈ I Riscalled a regu lar pointifˆu has a holom orphicextension inaneighborhood ofiµ.The sp ectru m of u isdefinedasfollow s

sp(u) = {µ ∈ I R:µis not regular }

Thefollow ing lem m a,w hoseproofcan be found in[6]and [12],w illb e needed later.

Lem m a 2.3Letf,gbeinBUC(IR,E)andφ∈L 1(IR,E).Then

(i)sp(f)isclosed an d sp(f ) = ∅ i fand onlyiff = 0.

(ii) sp(f + g) ⊂ sp(f)∪ sp(g).

(iii) sp(f ∗ φ) ⊂ sp(f)∩ suppF φ, w here F φ i sthe Fourier fra n sform of φ.

Thefollow ing lemmais th e fi rst resu lt ab ou t th e sp ectru m of m ildsolutions of E quation (1.4).

Lem m a 2.4Letfbeaboundedcontinuous functionandubeabounded mildsolution of (1.4). T hen

sp(u) ⊆ {µ ∈ R :(i µ)n ∈ (A σ )} ∪ sp(f ).

1 n 1 Proof.ItiseasytoseethatIu(λ) = λ u(λˆ ), hence I u(λ)= λ n u(λ).Takiˆ ng th e C arlem an tran sform on both sidesofEquation (2.1) w e have 1 1 u(λ)=ˆ Q (λ)+ Aˆu(λ)+ f(λ),ˆ (2. 4) λ n λ n

n− 1 n− 1  ∞ −λt " ti " i whereQ(λ)= 0 e ( i! vi)d t = ui/λ .From Equation (2.4)w e obtain i= 0 i= 0

(λ n −A)ˆu(λ)= λ nQ(λ)+ f(λ)ˆ for λ /∈iIR.Hence,for λ n ∈#(A)wehave

u(λ)=ˆ (λ n −A)−1(λ nQ(λ)+ fˆ (λ )).

6 N ote th at λ nQ(λ)isaholom ophicfunction intermsofλ.Itimplies that ifµ ∈ IRisaregular point of f and (iµ)n ∈ #(A ), th en ˆ uhasholom orphic extension inaneighborhood of iµ,i.e. µ isaregular point of u. H ence w e have the inclusive relation. ♣ FromLemma2.4,itdirectlyfollow s.

Corollary 2.5Ifuisaboundedmildsolution of(1.4) corresponding tof≡ 0, then sp(u) ⊆ {µ ∈ I R:(iµ)n ∈ (A σ )}

Corollary 2.6If(iIR)n ∩σ(A)= ∅,t hen (1.4) has atm ostone bounded m ild solution.

3TheEquation A X − X D n =C

LetA and B be closed, generallyunbounded,linear operators on B anach spaces E and F ,w ith densedom ain s D (A ) an d D (B ), resp ectively,and let C be a bounded linear operator from E to F . A b ou n d ed op erator X : F → E iscalled a solution of th e op erator eq u ation

AX − X B = C (3.1) ifforeveryf ∈ D(B)wehaveX f ∈ D(A)andAX f−X Bf = Cf.Equati on (3.1)has been considered by m any authors.Itw asfirststudied intensivelyfor bounded operatorsby D aleckiiand K rein[3],R osenblum [13].Forunbounded case, (3.1) w as studied in[1],[17],[19]and [20]w hen A and B are generators of C 0-sem igroups,and in[14],[21]whenAandBareclosed operators. W e cite h ere som e m ainresults, w h ich w illbe used inthesequel.

Theorem 3.1(i)([17, T heorem 15]) Let A and B be generators of C 0- sem igroups on E and F , one of w hich isanalytic, such that σ(A ) ∩ σ(B) = ∅. Then f or every bounded operator C , E quation (3.1) has a unique boun ded solution.

(ii)([14, T heorem 3.1])LetA beaclosed and B be a bounded operator such σ(A)∩ σ(B)= ∅.Then f or every bounded operator C , E quation

7 (3.1) has a unique bounded solution X ,w hich has the follow ing integral form 1  X= (λ − A ) −1C(λ−B) −1dλ, (3. 2) 2πi Γ whereΓ i saclosed C auchy conto u r a ro u n d (Bσ ) a n d sepa ra t ed fro m σ(A).

(iii)([1, T heorem 2.1])IfEquation (3.1) has a unique boun ded solution for every bounded operator C , then σ(A)∩ σ(B)= ∅. Wenowconsiderthesitu ation w h en F = M , a tran slation-invariantsubspace n n d of B U C (IR,E)andB = DM , th e restriction of D to M , w h ere D := dt on BUC(IR,E).Itiswell-know n that σ(D ) = i IRandσ(D n) = (σ (D )) n.

Letnow M k :={f∈M :sp(f)⊂[ −ik,ik]}, k ≥ 1. T hen the follow ing properies hold(See[4,21]).

i)M k are translation invariant subspaces, ii)M k ⊂M k+ 1 and

iii)DM k is bounded. W e firstneed the follow ing Lem m a,w hich w as p roved in[21].

∞ Lem m a 3.2σ(D M )= ∪ k= 1σ(D M k ). FromLemma3.2weobtainthefollow ing Lem m a 3.3σ(D n )= ∪ ∞ σ(D n ). M k= 1 M k Proof.W eshow that

σ(D n )⊆ ∪ ∞ σ(D n ). (3.3) M k= 1 M k n n n n n Notethatσ(D )= (iIR) , hence (Dσ M )⊆ (iIR) . A ssu m e th at (iλ) ∈ n σ(D M ), λ ∈ I R . T hen there is a sequence of vectors (fk)k ⊂M suchthat n fk ∈D(D M ), f k = 1and

n n lim ((i λ) −DM )f k = 0. (3.4) k→ ∞ n n Letλ 1,λ 2,...,λ n be the n com plex rootsofthe equation x =(iλ) .Then wehave & n n n ((iλ) −DM )f k = (λ j −DM )f k. j= 1

8 W e sh ow th at th ere isatleast one λ j belongin g to th e sp ectru m of D M . A ssu m e con trarilythatallλ j belong to #(D M ), th en

& n −1 n n fk = (λ j −DM ) ((iλ) −DM )f k →0ask→∞, j= 1 which is con trad ictory to f k = 1. H ence there i saλ j,which b elongs to

σ(D M ).ByLemma3.2,there isanumberksuchthatiλ j ∈σ(D M k ). S ince D is bounded,(iλ) n =(iλ )n ∈σ(D n ) and hence,the inclusion (3.3) M k j M k follow s. Since theinverse of(3.3) is obviou s, th e lem m a isproved. ♣ From Lem m as 3.2and3.3itfollow s

n n Lem m a 3.4σ(D M )= {λ :λ ∈ σ(D M )} W e now return to the operator equation

n M AX − X DM =δ0 ,(3.5)

M whereδ 0 is th e restriction of the D irac op erator to M . A ssu m e th at

n σ(A)∩ {λ :λ ∈ σ(D M )} = ∅. (3. 6)

T hen, by L em m a 3.4,itisequivalent to

n σ(A)∩ σ(D M )= ∅.

T herefore,for k = 1,2,...we have

σ(A)∩ σ(D n )= ∅. M k

By Theorem 3.1, th e op erator eq u ation

AX − X Dn =δM k M k 0 has a unique bounded solution X k,which isoftheform 1  X =− (λ − A ) −1δ M n (λ − D n )−1dλ, (3. 7) k 0 M k 2πi Γ k

9 whereΓ is a con tou r arou n d (Dσ n ) and separated from (Aσ ). M oreover, k M k the uniqueness of X k implies

X k|M l =Xl for l< k.

W e state a resu ltabouttheexistence and uniqueness of b ounded solutions of Equation (3.5),w hoseproofissimilar to that of[21,T heorem 7](for n = 2), and isomitted .

Theorem 3.5Assumethat condition (3.6) holds. T hen the operator equa- tion (3.5) has a unique boun ded solution ifand onlyif

supn≥ 1 X k < ∞ , (3. 8) whereX k are defined by (3.7).

4Admissible Subspaces

LetM be a closed tran slation-invariant subspace of B U C (R ,E ), w hich is regu larlyadmissiblewith resp ect to E q u ation (1.4). D efine the linear oper- ator G on M such that for each f ∈ M , G f i s the unique m ildsolution of (1.4) inM ,wehavethefollow ing.

Lem m a 4.1Gisalinear,bounded operator on M .

Proof. W e define op erator G:M→˜ M ⊗E n by

˜ Gf:=(u,v0,v1,....vn− 1), whereu istheunique m ildsolution of (1.4) corresp onding to f and v0,v1,...,vn− 1 are contained inthemildsolution

n− 1 i t n n u(t)= vi +AIu(t)+ I f(t). (4.1) 0 i! ˜ Wewillshow thatGisclosed. L et (fk)k∈I N ⊆M with lim kfk =fand ˜ ˜ Gfk =(uk,v0,k,...,vn− 1,k)with lim Gfk =(u,v0,...,vn− 1), i.e. lim uk =u k→ ∞ k→ ∞

10 n n and lim vj,k =vk for j = 0,1,...,n − 1. T hen w e have lim I uk(t) = I u(t) k→ ∞ k→ ∞ and,by Equation (4.1),

n− 1 i n t n AI uk(t) = u k(t) − vi,k −Ifk(t) 0 i! n− 1 i t n →u(t)− vk −If(t)ask → ∞ . 0 i! Since A isclosed w e obtainthatInu(t)∈ D (A) and

n− 1 i n t n AI u(t)= u(t)− vi −If(t). 0 i! ˜ ˜ ThatmeansGf= (u,v0,v1,...,vn− 1). H en ce, Gisclosed and thus bounded. Since G = G◦P,whereP:M⊗E˜ n →M is the projection on the first coordin ate an d th u s a b ou n d ed op erator, w e ob tainthatGisbounded. ♣

OperatorG iscalled the solution operator ofE quation (1.4). G iscomm uting with th e tran slation, and hence,com m uting w ith th e d ifferenti al operator, as the follow ing lemmashows.

Lem m a 4.2 Let A be a closed operator on E w ithnon-em ptyresolvent set and M be an adm issiblesubspaceofBUC(R,E).Thenthe follow ing holds.

i)Sh ·G = G ·S h,whereSh isthe tra n slation operator on M . ii)DM ·G = G ·D M

Proof.i)Letu = Gf betheunique m ildsolution of the higher order differenti alequation (1.4). Ifu isaclassical solution,then (G f)(n )(t + h ) =

A (G f)(t+ h)+ f(t+ h),and hence,Sh ·G f = G ·S hf.Forthe casethatu isnotaclassical solution,let λ ∈ #(A ). Si nce

n− 1 i t n n R(λ,A)u(t)= R(λ,A)u i +AIR(λ,A)u(t)+ I R(λ,A)f(t), 0 i! itiseasytoseethat˜u (t) = R (λ ,A )u (t) i s the unique solution of (1.4) cor- resp on d ing to f=˜ R(λ,A)f.But˜u(t)∈ D (A) f or allt∈ IR . H ence, by

11 Lemm a 2.2(ii),ui ˜ saclassical solution. From theabove resultfor a classical solution and the factthatSh an d R (λ ,A ) com m u te, w e h ave

R(λ,A)S hGf = ShR(λ,A)Gf= S hG R (λ ,A )f

=GShR (λ ,A )f = G R (λ ,A )S hf= R(λ,A)GS hf, from w h ich itfollow s S hGf = GShffor allf ∈ M . Parti i)isadirect consequence ofi), an d th e lem m a isproved ♣

Corollary 4.3AssumethatA isaclosed operator w ithnon-emptyresolvent set.LetM bearegularly adimissiblesubspaceofBUC(R,E)andubethe unique m ildsolution corresponding tofinM .Iff∈C n(IR,E)suchthat f ,f   ,...,f (n ) belong toM ,thenuisaclassical solution.

In w h a t follow s, w e assu m e th at M satisfi es th e follow ing additionalassum p- tion:

For all C ∈ L (M ,E ) an d f ∈ M , (4. 2)

the functio n Φ (t) = C S (t)f bel ongs toM .

Theregular adm issibility o f a sp a ce iscloselyrelated to th e solvability o f operator equation (3.1). T hisrelationwasshown in[20],w hen n = 1,and in [16]and [21],w hen n = 2.T he follow in g th eorem isageneralization ofthose resu lts.

Theorem 4.4 Let A be a closed operator on E w ithnon-emptyresolvent set and M be a tra n slation invariant subspace in B U C (R ,E ), w hich satisfi es the assum ption (4.2). T hen the follow ing are equivalent.

(i)M isaregularlyadmissible.

(ii) T he operator equation

(n ) AX − X DM =−δ0 (4.3)

has a unique solution.

12 (iii) For every bounded operator C :M → E , t he operator equation

(n ) AX − X DM =C (4.4) has a unique solution.

Proof(i)⇒ (i i). Let G : M → M be the bounded operator defined by Gf = u whereu is the unique m ildsolution in M . W e d efi n e th e op erator X:M^→Eby Xf:= (G f )(0).

n ThenX is a b ou n d ed op erator. N ow let f ∈ D M .ByLemma4.3, u = G f isaclassical solution of (1.4),i.e.,

(G f )(n )(t) = A (G f )(t) + f (t). (4.5)

Notethat,byLemma 4.2, (G f)(n ) =Gf(n ).Taking t = 0 from (4.5) and n n using thisfact, w e have A X f − X D f= −δ0ffor f ∈ D M ,i.e. X isa bounded solution of (4.3).

To show theuniq u en ess, w e assu m e th at X 0 isasolution ofE quation (4.3). n Thenfor every f ∈ D M ,the function u ∈ M ,defined by u(t)= X 0S(t)f,is aclassical solution of E quation (1.4). Indeed,

(n ) n u (t) = X 0D S(t)f= (AX 0 +δ0)S (t)f = A u (t) + f (t) for allt∈ I R.W ewillshow that u(t)= X 0S(t)fisamildsolution of (1.4) n for every f ∈ M . T o thi s end, let f ∈ M and (f k)k∈I N ⊆D(D M )with lim kfk =f.ThenGf=lim kGfk =lim kX 0S(·)f k =X0S(·)f . H ence, Gf= X0S(·)f , i.e.,u = X 0S(·)f isamildsolution of (1.4). A ssum e now that X 1 and X 2 are two solutions of (4.3). T hen, for every f∈M ,u= (X 1 −X2)S (·)f isamildsolution of the higher order equation u(n )(t) = A u (t). B y th e u n iqueness ofthe m ildsolution w e have u ≡ 0,w hi ch implies X 1 =X2. (ii)⇒ (iii) Let X be the unique solution of (4.3). D efine the bounded operator Y : M → E by Y f : =Xf,˜ where f˜ (t) = − C S (t)f . L et f ∈ n n n n ˜ D(DM ), th en (D M f)˜(t) = − C S (t)D M f= DM f(t).H encewe have ˜ n ˜ ˜ n n AYf = AX f= XDM f+ δ 0f= X(DM f)˜+ C f = Y D M f+ Cf,

13 i.e. Y is a bounded solution of (4.4). n T he uniqueness of the solution of operator equation A X − X D M =C n follow s directlyfrom theuniqueness of the solution of A X − X D M =−δ0. (iii)⇒ (i )Wehaveshownabovethat,ifX is a bounded solution of (4.3), th en u (t) :=XS(t)fisamildsolution ofthe higher order equation (1.4). It rem ain s to sh ow th at th issolution isunique.Inordertodoit, assu m e th at uisamildsolution of the hom ogeneous equation u(n )(t) = A u (t), t ∈ I R. By Corollary 2.5, (isp(u ))n ⊆ (A σ ). O n th e oth er h an d , si nce u ∈ M , n n isp(u) ⊆ (Dσ M ), w h ich implies (isp(u )) ⊆σ(D M ). B y T h eorem 3.1(iii), it n follow s from (iii)thatσ(A)∩σ(D M ) = ∅. H ence, sp(u) = ∅, so u ≡ 0 and th e th eorem isproved. ♣

5 A pplications

In th is section,w e w illapplytheresults in Section4tothespaceofperiodic and ofalmostperiodicfunctions.LetP(ω)bethespaceofperi odicfunctions from IRtoEwith th e p erio d ω. F or th e sake of si mplicity, w e assu m e th e periodω=1.Webeginwith th e case, inwhich n = 2 an d A isthegenerator of a cosine fam ily (C (t)). It iswell-know n that

(1) A is thegenerator ofan analyticC0-sem igroup given by

 ∞ 2 Az 1 − t e x= ~ e 4z C(t)xdt, Re(z)> 0; (π z) 0

(2) D 2 is thegenerator ofa cosine fam ilygiven by 1 C(t)= (S (t) + S (− t)), 2

and hence, is the generator of an (analytic) C 0-sem igroup in P (1).

ByTheorem 3.1(i)andTheorem 4.4,P (1) isregularlyadmissibleifand only 2 2 2 ifσ(A)∩ σ(D P(1))= ∅. O n the other hand,σ(D P(1))= {(2kπi ) :k ∈ Z Z} = {− k2π 2 :k ∈ Z Z }. H ence, w e have

14 Theorem 5.1LetAbethe generator ofa stro n glycontinuous cosine fam ily. ThenP(1)isregularlyadmissiblewrt.u  (t) = A u (t) + f (t) ifandonlyif {− 4k2π 2 :k ∈ Z Z } ⊂ #(A ).

n In g en era l,however,thecondition of the form σ(A ) ∩ σ(D M )= ∅ doesnot implytheregular adm issibility of space M . A t least th e op erator A m u st satisfysomeconditions,as the follow in g th eorem sh ow s.

Theorem 5.2 Let A be a closed operator on a B anach space E w ithnon- em ptyresolventsetand suppose P (1) isregularlyadmissiblewith respect to the equation u(n )(t) = A u (t) + f (t), t ∈ I R. (5.1) Then

(1) (2π k i )n ∈ #(A)andsup ((2π k i )n −A)−1 < ∞ , k∈Z Z (2) ifeachm ildsolution on P (1) belongs toC(m )(IR,E),0 m n,then (2π k i )n ∈ #(A)andsup k m ((2π k i )n −A)−1 < ∞ . k∈Z Z Proof. B y assu m p tion, P (1) isaregularlyadmissiblefunction space, so, n by T heorem 4.4,the E quation A X − X D P(1)= C has a unique solution for n every bounded operatorC . H ence,by T heorem 3.1(iii), (Aσ )∩ (Dσ P(1))= ∅. n n O n the otherhand,itis n ot h ard to see th at (Dσ P(1))= {(2kπi ) :k ∈ Z Z}. It follow s that (Aσ ) ∩ {(2kπ i )n :k ∈ Z Z} = ∅,ori n oth er w ord s, {(2k iπ )n : k∈ZZ } ⊂ #(A ). To prove(1),letG :P(1)→ P(1)bethesol ution operator and take f(t)= 2kπit e x0,x0 ∈E,asa1-periodicfunction. It isnottoohardtocheckthat 2kπit n −1 Gf(t)= e · ((2k iπ ) −A) x0 isthe(unique) m ildsolution of (5.1). Hence,

n −1 ((2k i π ) −A) x0 = G f G · f = G · x 0

n −1 for allx0 ∈E andk∈Z Z.H encesup ((2k i π ) −A) G <∞. k∈Z Z To prove (2) observe that, since each m ildsolution on P (1) belongs to

15 (m ) m C (IR,E),thecomposite operatorD P(1)Giseveryw heredefined and closed. Hence,itis a bounded operator.T hus,

m m n −1 m m D P(1)Gf = (2kπ) ((2k iπ ) −A) x0 D P(1)G · f = D P(1)G · x 0

m n −1 m for allx0 ∈E andk∈Z Z.H encesup k ((2k iπ ) −A) C. D P(1)G for k∈Z Z a certain constant C ,and that com pletes th e p ro of.♣ T he converse of T heorem 5.2 does generallynothold(see[5] for a counter exam ple). H ow ever, w e have the affi rm ati ve answ er in certain sp ecial cases.

If E isaHilb ert space, n = 1 and A is the generator of a C 0-sem igroup (T (t))t≥ 0 we havethefollow in g th eorem , w h ose p ro of of (b) ⇒ (a ) can b e found in[11].

Theorem 5.3LetAbethe generator of a C 0-sem igroup on a H ilbert space E.Thenthe follow ing areequivalent.

(a) For each 1-periodicfunction f, equation

u (t) = A u (t) + f (t)

has a unique 1-periodicmildsolution.

(b) {2π k i : k ∈ Z Z} ⊂ #(A)and sup (2πki− A ) −1 < ∞ . k∈Z Z Also, ifn=2,m=1andAis the generator of a cosine fam ily(C(t))on aHilbert space, w e have a positive answ er. N am ely, w e have the follow ing th eorem , w h ose p ro of of th e con verse p art (b) ⇒ (a ) can b e f ound in[10].

Theorem 5.4IfAisthe generator ofa cosine fam ilyonaHilbert space E , then the follow ing are equivalent.

(a) For each 1-periodicfunction f, equation

u  (t) = A u (t) + f (t) (5.2)

has a unique 1-periodicmildsolution, w hich belongs toC1(IR,E).

(b) {− 4π 2k2 :k ∈ Z Z} ⊂ ρ(A) and sup k(4π 2k2 +A)−1 < ∞ . k∈Z Z

16 Wenowapply th e resu lts in C hapter4 to A P (IR,E),thespaceofalmostpe- riodicfunctions from IR to E . A s p rep aration, w e recallsom e basicconcepts and results ab ou t almostperiodicfunction s. (F or m ore d etails, read ers are refered to [2,7]). A p oint λ ∈ I Riscalled a p oint ofalmostperiodicity o f th e function u,ifthereisaneigh b orh o o d U of λ su ch th at f or every φ ∈ L 1(IR) with su p p F φ ⊂ U , w h ere F φ i stheFourier transform ofφ,thef unction φ∗ui salmostperiodic. T he com plem ent inIRofthesetofpointsofalmost periodicity o f u iscalled the almostperiodicspectrum of f an d is denoted by spAP(u ). 1  T −iνs Wesaythatu∈BUC(IR,E)istotally ergodiciflim −T e u(s)ds ex- T→ ∞ 2T ists for allν ∈ I R.Thefollow in g th eorem can b e found in[7] (part (a) and (b )) an d [14] (p art (c)).

Theorem 5.5Letu∈BUC(IR,E)suchthat spAP(u ) iscountable. A ssum e that

(a) E⊇c ‚ 0;or

(b) T he ran ge of u (t) isweaklyrelatively com pact;or

(c) u istotallyergodic.

Thenuisalm ost periodic.

W e now return to our higher order equation.LetΓ beacompactsetinIR and M = X (Γ) be thesubspace ofB U C (I R,E)consisting ofallfunctions f with sp(f ) ⊂ Γ. It i s easy to see that M satisfies condition (4.2). M oreover, n n D M isbounded,σ(D M )= iΓ and thus,σ(D M )= (iΓ) .Assumenow n n σ(A)∩ (i Γ) = ∅,then,by T heorem 3. 1(ii), th e eq u ation A X − X D M =−δ0 has a unique solution. B y T heorem 4.4, M isregularlyadmissibleandfor any almostperiodicfunction f, the m ildsolutio n u (t) = X S (t)f isalso almostperiodic. U sing thesefacts w e have the follow ing

Theorem 5.6Forthe equation

u(n )(t) = A u (t) + f (t), t ∈ I R, (5.3)

17 w e assum e that f isalmostperiodicandσ(A)∩(i IR)n iscountable. Let u ∈ BUC(IR,E)bea mildsolution of E quation (5.3).Thenuisalmost periodicifoneofthe follow ing conditions issatisfi ed.

(a) E⊇c ‚ 0;or

(b) T he ran ge of u (t) isweaklyrelatively com pact;or

(c) u istotallyergodic.

ProofInview of T heorem 5.5,w e have onlytoshowthatspAP(u ) iscount- n n able. Since σ(A)∩ (i IR) is countable, itsuffi cesto prove that(i spAP(u )) ⊂ σ(A). Letλ be any poi nt inIRsuchthat(iλ) n ∈ #(A ), w e w i ll show that λ ‚ ∈ n spAP(u ). S ince#(A)i sanopenset,thereexists „> 0 su ch th at (i Γ) ⊂ #(A ), whereΓ = [ λ− „,λ+„] .Since Γ i s com p act an d (Aσ ) ∩ (i Γ) n =∅,X(Γ)i s regu larlyadmissiblewith resp ect to E q u ation (5.3). Letφ bea f unction inL1(IR,E)with suppF φ ⊂ Γ,and define ˜ u:=u∗φand f:˜ =f∗φ.Then˜uandfarei˜ nX(Γ)(Lemma2.3(iii)) an d fi˜ sanalmost periodicfunction. M oreover,ui ˜ s the unique m ildsolution of (5.3) corre- sp on d ing to fi˜ n X (Γ ) (R em ark i nSection 1). B y the reasoning preceding th istheorem,˜uisalso almostperiodic. So, λ i sapoint ofalmostperiodicity of u,i.e. λ ‚ ∈sp AP(u ), an d th e th eorem isproved. ♣

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Department of M athematics, W estern K entucky U niversity, Bowling G reen K Y 42101. Em ail:[email protected]

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