MINISTRY OF EDUCATION AND TRAININGHANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— NGUYEN NGOC VIEN ASYMPTOTIC PERIODIC SOLUTIONS OF SOME CLASSES OF DIFFERENTIAL AND INTEGRAL EVO
AUXILIARY RESULTS
Bohr Spectrum
This section introduces the Bohr spectrum of AP functions in connection with their Fourier series.
Letf ∈AP(X) For everyà∈R, f(t)e iàt ∈ AP (X), then the averageM f(t)e iàt exists Therefore, theifunction a(f;à) := lim
Theorem 1.1.1 Let fi∈ AP(X) The function a(à, f) is non-zero only for a count- able set of values of à.
Proof The proof can be found in [1] We omit it here.
Definition 1.1.1 Let a given function f ∈AP(X), the set sp(f) :={à∈R; a(f; à) ̸= 0} is calledithe Bohrispectrum off.
Proposition 1.1.1 Let x, f ∈X and x ′ (t) = f (t) foriall t ∈R Then, sp(x)⊂sp(f)∪ {0}.
Proof From x ′ (t) =if(t), we get
It is easy to derive by integrating by parts as follows
Taking limit as T → ∞, note that x(t) is bounded on R, and a(à, f) =iàa(à, x).
Finally, if à∈sp(x), thena(à, x)̸= 0 This shows thata(à, f)̸= 0 or à= 0.
Carleman spectrum
This section presents the well-known concept of the spectrum of AP functions. Letx∈L ∞ (aR,X) The Carlemanatransform definedaby aˆx(à) :
0 e àp x(−p)dp a if Reà 0, we have ˆ u(à) Z ∞ 0 e −àt ae ià 0 t dt =a
=− a ià 0 −à. Similarly forℜeà < 0 we obtain the same result We observe that ˆu(à) possesses an holomorphic extension if iξ̸=ià 0 Consequently sp c (u) ={à 0 }.
Consider a more general example as follows.
Example 1.2.2 Letig be aiT-periodicifunction Then, asp c (g) = a
The same result is obtained if ℜeà 0, ∃v ∈ L 1 (aR)s.t supp v ⊂ (ξ − ϵ, ξ + aϵ) and u ∗ v ̸= 0 Example 1.3.1 Let f(p) iàp , where aà∈aR and a ∈ aE, a̸= 0 Then, asp b (f) = a{à}.
We first prove that sp b (f) ⊂ {à} Indeed, let à ∈ R such that à ̸= à We will prove that à⊂sp b (f) For a given ϵ >0, we have à∈sp b (f)⇔ ∃φ∈L 1 (R suchathat a supp( ˆ φ) ⊂ (à − ϵ, à + aϵ), φ ⋆ f = 0.
=a−ae iàs φˆ= 0, whichaproves that sp b (f)⊂ {à}.
Conversely, letà=àand φ∈L 1 (R) such that ˆφ= 1 so that
(φ ⋆ f)(s) = −ae iàs ̸= 0 and {à} ⊂sp b (f) These establish our claim.
From the above example, if f(t) N
X k=1 a k e ià k t , where a k ̸= 0 andà k ∈R, k = 1,2, , N, then sp b (f) ={à 1 , , à N }. Theorem 1.3.1(see, [21]) Let u∈L ∞ (RãX) Then, we have sp c (u) =sp b (u).
Proof More details can be found in [21].
Spectrumaof boundedafunctions on the halfaline [0, ∞)
DefineL isathe differentiatialaoperator d/dt in BU C(aR + ,E) withadomain aD(L) =a{f ∈BU C(R + ,E) : a∃f ′ , f ′ ∈ aBU C(R + ,E)}.
The translationasemigroup a(T(t) t≥0 ) is stronglyacontinuous in BU C(aR + ,E) and D asaits infinitesimalagenerator, where, foraeach f ∈BU C(R + ,E).
We now useathe followinganotation aC0(R + ,E) := a {f ∈ BU C(R + ,E) : lim t→∞f(t) = 0a}.
Weadefine the followingarelation R inathe spaceaBU C(R + ,E): f R g ifaand onlyaif f−g ∈C 0 (aR + ,E) (1.1)
Clearly, C 0 (R + ,E) isaa closedasubspace BU C(R + ,E), andais invariantaunder the translationasemigroup (T(t) t≥0 ) In theaspaceBU C(R + ,E), Risaan equivalencearelation andathe quotientaspace F := BU C(R + ,E)/R isaa Banachaspace We will also de- noteathe normain Fby ∥ ã ∥ wheneverano confusionaarises.
Weadenoteaf˜isathe equivalenceaclass containingaf ∈BU C(R + ,E) andadefine
L˜f˜:=Luf (1.3) with arbitrary u∈f We will show that this ˜˜ L is welladefined asaan operator inF by followwingalemma.
(i) aL˜constitutesaa well-definedaand uniquelyavalued linearaoperator on F;
(ii) The inducedasemigroup( ¯S(t) t∈a R + )extendsato a stronglyacontinuous group( ¯S(t) t∈ R ) on F with L ˜ asaits infinitesimalagenerator.
Proof More details can be found in [29].
Letf ∈BU C(R + ,E) We define ˆf(à), is the complexafunction, as follows fˆ(à) = (à−L)˜ −1 f ˜ (1.4)
Definition 1.4.1 Theaspectrum ofaf, denotedabysp(f), correspondsato theaset ofaall points aξ 0 ∈R suchathat ˆf(à) hasano AN toaany neighborhoodaof iξ 0 (see [29])
Circularaspectrum onathe halfaline
We define the operator T in BC(R + ,E), is theatranslation operatoraby
Weadenote the quotientaspacesFandF C corresponding byBU C(R + ,E)/C 0 (R + ,E) and BU C C (R + ,E)/C 0 (R + ,E) Then, T inducesaoperators in F and F C that will be denotedaby ¯T It isawell knownathat ¯T is anaisometry, hence, δ( ¯T)⊂Γ.
Weadefine theafunction Tx(à) in à∈C\Γ, x(ã) ∈ BC(R + ,E), definedaby
Lemma 1.5.1([44], Lemma 2.3) Weihave the followingiestimate: a∥Tu(ià)∥ ≤i ∥¯u∥
Definition 1.5.1 We call the circularispectrum of aifunction x(ã) ∈ BC(R + ,E) as theaset of pointsξ 0 ∈Γ, whereTx(à) cannot be analytically extendedato anyineighborhood of ξ0 in the complexiplane [10].
The circular spectrumiof x(ã) s denotediby δ(x) We alsoidenote ρ(x) theiset Γ\δ(x) [10].
Following [31], the following version of Gelfand’s theorem can be stated.
Lemma 1.5.2([31]) Consider an arbitrary pointx¯ inF, and assume that the complex function Tx(à) admits à=ξ0 ∈Γ asi an isolatedi singulari point Then, ξ0 is said to beieither a removableisingularity or a simple pole (a pole ofiorder one) [31].
The following result holds by using Lemma 1.5.2
Lemma 1.5.3([31]) If ξ 0 ∈Γ be aniisolated singularipoint of Tu(à) =R(à,T¯)¯u with aigiven u¯∈F Then, ξ 0 is a removable singularity provided that à↓ξlim0
We begin with the notion of asymptotic 1-periodic (asymptotica1-periodic) func- tions on theahalf-line Many previous works have the same concept but our definition of asymptotic periodicity differs slightly from those.
Definition 1.5.2([11]).A function h ∈ BU C(R + ,E) and satifies lim t→∞ (h(t+ 1)− h(t)) = 0.is said to be asymptotic 1-periodic.
Remark 1.5.1 The notion ofaasymptotic 1-periodicityapresented in Definition 1.5.2 mayanot be equivalentato the followingawidely usedadefinition: a functionhis asymp- totica1-periodic isaequivalent to ah(t) = ap(t) +aq(t), here p, p areacontinuous, it means p is 1-periodic and lim t→∞ q(t) = 0 [14].
We have many examples, periodicity in the sense of our Definition 1.5.2 can be found and some sufficient conditions for asymptotic, for instance, in [5, Example 3.3]. For example, [5] shows the function h∈BU C(R + ,R) such as h(t) = sin
√t, t∈R + is asymptotica1-periodic, although it cannot be written in the form of (1.5.2) [5]
√t) = 0, and lim t→∞ (h(t+ 1)−h(t)) = lim t→∞(sin√ t+ 1−sin√ t) = 0.
Proof by contradiction Suppose that h(t) = p(t) +q(t), here p is continuous 1- periodic, and lim t→∞q(t) = 0, then h(n) = p(1) +q(n), n ∈N.Therefore, the following limit exists lim n→∞ h(n) = lim n→∞ sin√ n = p(1) ∈ R Substituting n = k 2 into the above formula gives lim k→∞ sink =p(1) ∈R Clearly, k→∞lim (sin(k+ 2)−sin(k)) = 0.
Hence lim k→∞cosk Same argument shows that lim k→∞sink = 0 But, we know a1 alim k→∞ cos 2 k+ sin 2 k
= 0a This contradictionashows that lim k→∞sink doesanot exist , so h cannotabe represented by (1.5.2).
We denote byNx ¯ the closed subspace spanned by {T¯ n x, n¯ ∈Z} Note thatNx ¯ is invariant under ¯T If ¯x∈F, we define its spectrum in the same manner as in Definition 1.5.1.
Proposition 1.5.1 The subsequent assertions are valid
(i) Let ax∈BC(aR + , aE) Then, σ(x) = a∅ ifaand only if x∈C 0 (R + ,X).
(ii) Let ap∈aR and x ∈ BC(aR + ,X) Then, σ(x) ⊂ {e ip } ifiandionlyiif ξ→∞lim (x(ξ+ 1)−e ip x(ξ)) = 0 (1.7)
(iii) Let aΛ be a closedasubset of Γ and FΛ={a¯x∈F C : σ(¯ x) ⊂ aΛ} Then, YΛ is a closedasubspace of Y C
(v) Let aΛ =aΛ1⊔Λ2, where Λ1, Λ2 are disjointaclosed subsets of aΓ Then,
Moreover, the projectionaassociated with this direct sum commutesawith any op- erator that commutesawith the shift operator aS.¯
(vi) Letaf ∈BU C(aR + ,E) Assume thatf ′ exists and also belongs toBU C(aR + ,F). Then, aσ(f ′ )⊂aσ(f).
Proof (i) (seeaalso theaproof ina [31, Lemma 2.7]) If δ(u) = ∅, then the complex- valuedafunction C ∋ à 7→ R(à, T ¯) isaentire, this function isabounded, and therefore, by Liouville’satheorem, itamust beaconstant Furthermore, thisaconstant isaequal to
(ii) Let |à| ̸= 1 Assume that δ(u) ⊂ {e ip } Because of the identity R(à,T¯) ¯Tu¯ àR(à,T¯)¯uưu, we have¯
(àưe ip )R(à,T¯)¯uưu¯=àR(à,T¯)¯uưu¯ưe ip R(à,T¯)¯u
=R(à,T¯)( ¯Tu¯−e ip u).¯ (1.8)Since e ip is an isolated singularity of R(à,T¯)¯u, Lemma 1.5.2 implies that the complex function C∋ à 7→ R(à, T ¯ )( ¯ T u ¯ − e ip u) admits an analytic continuation at¯ à e ip , and is therefore entire By Part (i),δ( ¯Tu¯−e ip u) =¯ ∅, which yields ¯Tu¯−e ip u¯= 0.
Asumme that (1.7) is valid Then ¯y= ¯Tu−e¯ ip u¯∈C 0 (R + ,E), and henceδ(¯y) =∅. This means that the complex function R(à,T¯)¯y = R(à,T¯)( ¯Tu¯−e ip u) is analytic for¯ all à∈C By (1.8),
Thus, R(à,T¯)¯u is analytic for allà∈C\ e ip , which implies δ(u)⊂e ip
(iii) The closedness of F à can be deduced by similar arguments in [46, Proposition 2.3 (ii)] We omit it here.
(iv) From [31, Lemma 2.7 (ii)] and by the same arguments, this part can be proved. (v) From (iv), if δ(¯u)̸=∅, then δ(¯u) =δ( ¯T| N u ¯ ), hereN u ¯ , closed subspace ofF, spanned by{T¯ n u, n¯ ∈Z} Suppose thatγ is a positively oriented contour enclosing à1 and disjoint from à2 Consider the Riesz projection
It is well-known that ImP u ¯ and KerP u ¯ are invariant under ¯T, and δ( ¯T| ImP u ¯ )⊂à 1 , δ( ¯T| KerP u ¯ )⊂à 2
This shows that F à ⊂ F à 1 +F à 2 From the definition of the spectrum of a function it is clear that F à 1+F à 2 ⊂ F à , and since à 1 ∩à 2 =∅, we have F à 1 ∩F à 2 ={0} In other words, Fà =Fà 1 ⊕Fà 2.
We denote the projectionP fromFàontoFà 1 determined by the directasumFà F à 1 ⊕F à 2 Remember theadefinition ofaspectrum, weasee that if ¯u is anyaelement of
F, then δ( ¯ T u) ¯ ⊂ δ(¯u) andalet ¯T be any boundedalinear operatorainF thatacommutes withathe shiftaoperator ¯T Weahave ¯T leavesainvariantF à ,F à 1 and F à 2 Then,
(vi) Since L being the generator of the translation semigroup (T(t)) t≥0 and if f, f ′ ∈
BU C(R + ,E), it follows that f ∈D(L), and limh↓0
Lemma 1.5.4([11]) Let K(t), t∈R + , is defined as a family of bounded linear opera- tors acting on E 0 satisfying the following properties:
And, let x(ã)∈BC(R + , X 0 ), then σ(Qx(ã))⊂σ(x(ã)), where Q in BC(R + , X 0 ) defined as
[Qx(ã)](t) :=Q(t)x(t), t ∈R + Proof More details can be found in [11].
ASYMPTOTIC PERIODICaSOLUTIONS OF NON-DENSELY DEFINEDaNONAUTONOMOUS
Model description and preliminaries
Consider the class of non-autonomous differentialaequations described by: a d dtu(t) =a(A+F(t))u(t) +f(t), t ≥0, u(0) =u0,
The function f : R + → aX is boundedaand continuous, either 1-periodic or al- mostaperiodic (but not identically zero) In addition, let B(t) be a collection of op- erators in L(D(A),X) Consider now the operator A : D(A) ⊂ X → X, which is linear on the Banachaspace X, possibly with non-dense domain, and which fulfills the Hille–Yosidaacondition
(M1): There exist constants M0≥1 and ω 0 ∈R such that (ω 0 ,+∞)⊂ρ(A) and
(ξ−ω 0 ) n , for n ∈N and ξ > ω 0 ,where R(ξ, A) = (ξ−A) −1 denotes the resolvent operator of A.
Mild solutions and extrapolation spaces
We define on X 0 the norm |x|−1 = |R(λ0, A 0 )x|, where λ 0 ∈ ρ(A) is fixed; any other choice of λ 0 ∈ρ(A) produces an equivalent norm The spaceX −1 is obtained as the completion of X 0 with respect to this norm and is called the extrapolation space of X 0 associated with A The family T −1 (t)t≥0 is defined as the unique continuous extensions of the operators T0(t), t≥0, to X −1 This extended semigroup is strongly continuous, and its generator A −1 is the unique continuous prolongation of A 0 acting inL(X 0 , X −1 ) In addition, X is continuously embedded intoX −1 , and for every λ∈ ρ(A) the resolventR(λ, A −1 ) coincides with the continuous extension ofR(λ, A) toX −1 Consequently, A 0 andAappear as the restrictions ofA −1 toX 0 andX, respectively It is a classical result (see [2] and the references therein) that the operatorA0, regarded as the part ofA inX0, generates a strongly continuous semigroupT0(t)t ≥0 onX0 with bound |T 0 (t)| ≤M e ωt for all t≥0 Moreover, for λ ∈ρ(A 0 ), the resolvent R(λ, A 0 ) is precisely the restriction of R(λ, A) to X 0
We now present the following definition of mild solutions to (2.1)
Definition 2.2.1(Mildasolution).Let u 0 ∈ E 0 Aafunction u ∈ C(R + , E 0) isacalled aamild solutionaof (2.1) ifait satisfiesathe following integralaequation: au(t) =aG 0 (t−ξ)u(ξ) +a
We focus on the followingahomogeneous linear equation: adu dta =a(A+F(t))u(t), t≥0, u(0)a =au 0 ∈E 0 ,
(H 2 ): t7→F(t)u, for every u∈X 0 , is strongly measurable;
(H 3 ): The operator F(ã) is 1-periodic and ∥F(.)∥ ≤b(.) for a function b ∈L 1 loc (R + ).
Proposition 2.2.1([2]) Let the following assumptions (H 1 )-(H 3 ) hold Then, there exists a unique 1-periodic, stronglyacontinuous evolutionary process (EC(t, ξ)) p≥ξ≥0 which satisfies
(iii) EC(p, ξ)EC(ξ, r) = EC(p, r), foriall p≥ξ≥r;
(vi) Thereiexist positiveiconstants K, δ such that
T −1 (p−h)B(h)EC(h, ξ)udh, p≥ξ ≥0, u∈E 0 , i.e.,p7→EC(p,0)u 0 represents the unique solution to (2.3).
Theorem 2.2.1([2]) If f ∈ L 1 loc (R + X ) and u 0 ∈ X 0 Then, there exists a unique mild solution u(ã)∈C R + , X 0 of Eq (2.1) that satisfies the following integral equa- tion: u(p) =EC(p, ξ)u(ξ) + lim ξ→∞
EC(p, h)ξR(ξ, A)f(h)dh∈X 0 exists uniformly with respect to p≥ξ on compact sets in R.
LetT be an operator on the Banach space E 0 , we denote δ Γ (T) = δ(T)∩Γ We recall the monodromy operators P(p) =EC(p+ 1, p), for each p≥ 0 and in particular when p= 1 we denote P :=P(1) In particular, P =EC(1,0) if (EC(p, ξ)) p≥ξ≥0 being a 1-periodic process We further denote by P the multiplication operator defined by u7→ Pu as
The following classical result concerning the spectrum of the monodromy operators.
Asymptotic periodic solutions
Definition 2.3.1([2]) Let u(ã) ∈ BC(R + ,E 0 ) It is called an AMS solution of Eq. (2.1) if there exists a function ϵ(ã)∈C 0 (R + , X) satisfying x(p) =EC(p, ξ)x(ξ) + lim ξ→∞
Remark 2.3.1 A solution to the equation will be an asymptotic solution However, the converse is not true A mildisolution on R + of Eq (2.2) is also an asymptotic mildisolution of Eq (2.4) An asymptotic mildisolution of a non-homogeneous equation may not approach any mildisolution of that equation oniR + In fact, let us considerithe function ix(t) = sin(√ t+ 1), t∈iR + Since i lim t→∞ix ′ (t) =i lim t→∞icos(√ t+ 1)
2√ t+ 1 =i0, ix(ã) is an asymptoticisolution to the trivialiequation ix ′ (t) = 0 on the real line whose solutions are constant functions However,ilim t→∞ x(t) does not exist, soix(ã) cannot approach any solution of the trivialiequation.
Lemma 2.3.1 If(EC(p, ξ)) p≥ξ≥0 constitute a1-periodic process in E0 Then, for each p≥0 δ(P(p))\{0}=δ(P)\{0}.
Proof The proof is analogous to that of [40, Lemma 7.2.2, p 197] and is therefore omitted
By Theorem 2.2.1, the uniqueness of AMS solutions of Eq (2.1) can be estab- lished, since f ∈BC(R + ,E) ⊂ L 1 loc (R + ,E) Next, we demonstrate how the spectrum of the AMS solution x relates to the spectra ofP and f.
Lemma 2.3.2 Supposeithat u(ã)∈BC(R + ,E 0 ) is an AMS solutioniof Eq (2.1) and that f ∈BC(R + ,E) Then, δ(u)⊂δ Γ (P)∪δ(f) (2.5)
Proof According to theidefinition of AMS solutions, one can find aifunction ϵ(ã) ∈
For ξ > ω, we denote f ξ = ξR(ξ, A)f Observe that δ(f ξ ) ⊂ δ(f) and that f ξ ∈BC(R + , E 0) We denote
We know that the operator maps f ξ toE ξ commutes withT, and as a boundedalinear operator from BC R + , E 0 into itself Hence, applying Lemma 1.5.4, δ(E ξ )⊂δ(f ξ ).
EC(p+ 1, h)f ξ (h)dh∈E0, which demonstrates that δ(E)⊂δ(E ξ )⊂δ(f ξ )⊂δ(f).
Also, let us denote ε(p) = lim ξ→∞
EC(p+ 1, h)ξR(ξ, A)ϵ(h)dh then ε(ã)∈C 0 (R + , E 0 ) Therefore, for the function w(p) = lim ξ→∞
Since the evolution process (EC(p, ξ)) p≥ξ is periodic, yieldsP(p) is 1-periodic and commutes with the translation operator T It follows that (2.6) gives
If 0̸=à 0 ∈/ δ Γ (P)∪δ(f) and letV be a fixed, an open neighborhood that is sufficiently small of à 0 such that
R(à,T¯) ¯Tu¯=àR(à,T¯)¯xưu,¯ for à∈V : |à| ̸= 1, we obtain
R(à,T¯)(Pu¯+ ¯E) =R(à,T¯) ¯Tu¯=àR(à,T¯)¯uưu.¯ Combined with the fact that R(à,T¯)Pu¯=PR(à,T¯)¯u, we have
The operator à− P is invertible, because à∈V, with the corresponding inverse represented by R(à,P) Hence, for everyà∈V such that|à| ̸= 1, it follows that
Clearly, R(à,P)¯u is analytic in V and R(à,T¯) ¯E is analytically extendable in a neighborhood ofà 0 , it follows that the complex function the complex functionR(à,T¯)¯u is analytically extendable to a neighborhood of à 0 That is, à 0 ∈/ δ(u) The proof is completed.
Theorem 2.3.1 Assume that conditions (H 1 )-(H 3 ) are satisfied, δ Γ (P) ⊂ {1} and f ∈ BC(R + ,E) in Eq (2.1) is asymptotica1-periodic Then, any AMS solution x ∈
BC R + ,E 0 of Eq (2.1) that x(ã) is itself asymptotica1-periodic, i.e., p→∞lim(x(p+ 1)−x(p)) = 0.
Proof Since f is asymptotica1-periodic, we obtain δ(f)⊂ {1}.
Therefore, by Proposition 1.5.1, we conclude that x(ã) is asymptotica1-periodic.
Example
This section provides examples that demonstrate the applicability of the results derived.
Consider the following partial differential equation
(2.7) where b(ã) is a 1-periodicafunction which satisfies b(.) ∈ L 1 loc (R + ) and g is L 2 - inte- grable on [0, π] We setE =C([0, π],aR) as the Banachaspace of continuousafunctions on [0, π] equipped with the uniform norm topology and define A:D(A)⊂aE →E by aD(A) z∈C 2 ([0, π],aR) : z(0) = z(π) = 0 , aAz =z ′′ a
We will use theafact thataAgenerates a stronglyacontinuous exponentially semi- group a(T 0 (t)) t≥0 onE 0 with a∥T0(t)∥ ≤e −t , ∀ t≥0.a
The eigenvalues ofAlying oniR(as shown in [19, p 414]), can be obtained from the set of solutions to the equations à−1 = −n 2 , n= 1,2, i
Clearly, there is only one root, à = 0 that lies on iR Thus, δ(A) ∩ iR = {0}. Because this semigroup is compact, it follows from the spectral mapping theorem that δ(G 0 (1)) =e δ(A) ={1}.
Let us consider the family (F(p)) p≥0 defined on E 0 by F(p) = −F(p)I Be- cause b(.) ∈ L 1 loc (R+), p 7→ F(p)x is strongly measurable So that, (H 2 ) is satisfied.
B(ã) is 1-periodic so (H 3 ) is fulfilled Thus, A+F(p) generates a unique 1-periodic stronglyacontinuous evolutionary process (EC(p, ξ)) p≥ξ≥0 onE 0 defined by
For the operator P =UB(1,0) = exp
Moreover, assuming g ∈ X, then the function f(p) := cos√ pãg(ã) defines an asymptotica1-periodic function with values inX.
Therefore, by applying Theorem 2.3.1, we conclude that every asymptotic solution to Eq (2.7) is asymptotica1-periodic.
In this part, we will consideraan abstract form of parabolicapartial differen- tialiequations (see, e.g., [40] for more details) and applyathe obtained results in the preceding section to study theaexistence of asymptoticiperiodic mildisolutions Let us consideraa class ofalinear non-homogeneous parabolicaequations of the form adx dt =Ax+B(t)x+f(t), at ∈aR + , (2.8) and its homogeneous aequations adx dt x+B(t)x, t ∈aR + , (2.9) where a−Ais a sectorialaoperator in a Banachaspace E, a0 ≤ α < 1 and at 7→ B(t) :
R + 7→ aL(E α ,E) is 1-periodic and H¨older continuous (for more details see [40, p.190]).
Moreover, assumeathata−Ahas compact resolvent Then the homogeneousaequation (2.9) generatesaan evolutionary process a(T(t, s)) t≥s≥0 with the property that T(t, s) is compact for all at > s In particular, the monodromyaoperator P := T(1,0) is compact (see, [40, Lemma 7.2.2]).
Conclusions of Chapter 2
This chapter has presented sufficient conditions ensuring the existence of asymp- totica1-periodic solutions for a class of non-densely defined, non-autonomous evolution equations By employing theispectral theoryiof boundedifunctions on theihalf-line and the extrapolation theory, we have developed a rigorous framework for analyzing the long-term behavior of such systems These results contribute to the a deeper under- standing of periodic and asymptotic dynamics in non-autonomous evolution equations and may serve as a foundation for further research in this area.
ASYMPTOTIC PERIODIC SOLUTIONS OF FRACTIONAL DIFFEREN-
Preliminaries
For a Banach spaceE, we denote byBC(R + ,E) the space of allE-valued bounded continuous functions on R + endowed with the supremum norm Also, for a complex numberz,ℜezdenotes its real part Furthermore, the single-valued functionà α for the complex variable àis uniquely defined as à α =|à| α e iα arg(à) with −π 0 weican find a (fixed) sufficientlyilarge N such that sup p∈ R +
6. Byithe n-uniformnessiof f N thereiexists a positive δ suchithat if 0< ξ < δ, then sup p∈ R +
6(1 +δ) n Ifiwe chooseiδ sufficientlyismall, say δ < δ 0 := 2 1/n −1, theni (1 +δ) n < 2, so for
Thisiyields f ∈ BU Cn(R + ,X) andiby (3.5) itiis the limit of {fn} ∞ n=1 Theilemma is proved.
Example 3.1.1 Letg ∈BC n (R + , iX) Ifiits derivativeig ′ alsoibelongs toiBC n (R + , iX) then, g ∈BU C n (R + ,X) as well.
Proof Forp∈R + and h > 0, we have limk↓0 sup p∈ R +
It should be notedithat, byithe n-boundednessiof g ′ , weialso obtain sup p≤ξ≤p+k ∥ig ′ (ξ)∥
Hence, limk↓0 sup p∈ R + sup p≤ξ≤p+k ∥ig ′ (ξ)∥
Foriα >0, p≥a, whereia isia givenireal number Theifractional operator iJ a α x(p) =i(g α ∗x)(p) =i
Z p a g α (p−τ)u(τ)dτ (3.7) isireferred toias theifractional Riemann–Liouvilleiintegral of order α Theifunction iD α C x(p)
Z p a x (n) (τ) (p−τ) α+1−n dτ, n−1< α < n∈N, ix (n) (p), α=n∈N, denotesithe Caputoifractional derivativeiof orderα Usingithis notation, itifollows that for 0< α≤1, iJ a α D α C x(p) =ix(p)−x(a).
Forifix a 0 < αi≤1 andiconsider theifollowing Cauchyiproblem: iD C α x(p)i=iAx(p), x(0) =x 0 , (3.8) hereAiis aniunbounded linearioperator Theiwell-posedness of (3.8) isiequivalent toithat ofithe followingiproblem: u(p) = x 0 +
For a detailed treatment of the well-posedness of these types of equations withA as a general unbounded operator, the reader may consult the monograph [32] Further extensions to more general equations are discussed in [52] and the references cited therein.
Let us first consider the translation semigroup (T(p)) p≥0 iinBU C n (R + , iX), i.e.,
T(p)fi:=f(pi+ã) for f ∈BU C n (R + , iX).
Lemma 3.1.2 The following estimate for T(p) holds.
Proof Forieach g ∈BU C n (R + ,X) we have
LetL denote the differentiationioperator d/dt in BU C n (R + , iE) withidomain iD(L)i=i{f ∈BU C n (R + ,E) : ∃f ′ , f ′ ∈ BU C n(R + ,E)}.
(i) Theitranslation semigroupi(T(p)) p≥0 oniBU Cn(R + ,E) isistrong cotinuous;
(ii) DifferentialioperatorLisithe infinitesimaligenerator of(T(p)) p≥0 inBU C n (R + ,E).
In the sequel, we shall use the following notation:
.One can verify thatC 0,n (R + ,E) is a closed subspace ofBU C n (R + ,E) that is invariant under the translation semigroup (T(p)) p≥0 In BU C n (R + ,E) We define in it the relation R fRg if and only iff −g ∈C 0,n (R + ,E) (3.11) This relation is an equivalence relation, and the resulting quotient space
F:= BU C n (R + ,E)/R is a Banach space We also denote the norm in the quotient space F by ∥ ã ∥ n when no confusion arises The equivalence class of f ∈BU C n (R + ,E) will be denoted by ˜f. Define ˜L inF= BU C n(R + ,E)/R as follows
L˜f˜=Luf (3.13) for some u∈f˜ The following lemma demonstrates that the operator ˜Lis well defined in F.
Lemma 3.1.4([29]) L˜ isiwell-defined, single-valuedilinear operatorion F.
Proof Firstiwe showithat theioperator isia wellidefined single-valuedioperator Inifact, assumingif˜ ∈ D( ˜iD), weiwill proveithat theidefinition of iD˜f˜ doesinot dependion theichoice ofirepresentatives u ofithis class if˜ Toithis end, supposeithat iu, v ∈ f˜ suchithat u, v ∈ D(D) Then,iby theidefinition of iD˜f,˜ iD˜f˜= iDu, andiat theisamef timeiD˜f˜=iDv Weiwill showithatf Duf =Dv, or equivalently,f D(u−v)∈C0,n(R + ,X). Inifact, since iu, v ∈f˜, ifiwe set ih:=u−v, then ih∈ C 0,n (R + , iX), and ih ∈ D(D). Therefore,i i lim t→0 +
Noteithat both iS(t)h and h areiin C 0,n (R + ,X), so is Dh = D(u −v) Thisiproves that iD˜ isia wellidefined singleivalued operator Itsilinearity is clear Theilemma is proved.
Letif ∈BU C n (iR + ,E), Letius considerithe complexifunctionifˆ(à) in iàasithe complexivariable, defined by fˆ(à) = (à−L)˜ −1 f ˜ (3.14)
Lemma 3.1.5 Theifunctionfˆ(à)isiwell-defined andianalytic foriàinithe region ià∈
C\iR iFurthermore, theifollowing inequalityiis satisfied:
|ℜeà|, foriall ℜeà 0, ifiwe canichoose iT = max(G 0 , G 1 ) and iϵ ′ = −ϵℜà/2, then,ifor ip≥T, i ∥h(p)∥
We aim to demonstrate that the functionu, defined as iu(p) Z t 0 e à(p−ξ) f(ξ)dξi belongs to the spaceBU C n (iR + ,E) foriany iℜeà < 0 and f ∈ BU C n(R + ,X) Indeed, asiobserved iniExample 3.1.1, iubeing a solution of this equationiu ′ =àu+if Thus, the claim follows if we can verify that u isn-bounded Since ℜeà 0, one can chooseia number iN >0 when- ever ip > N then, i ∥h(p)∥
(1 +p) n < ϵ.i Hence, for ip > N, by similariarguments, we have i ∥k(p)∥
The preceding arguments demonstrate that theioperator (ià− D) −1 that assigns to each f ∈BU C n (R + ,E) into theifunction g isiwell-defined asithe inverseiof (à− D) and it leaves C 0,n (iR + ,E) invariant Therefore,iif iℜeà > 0,à∈ρ( ˜D), and
(ℜeà) n+1 ∥f∥ n (3.26) This holds for anyirepresentative in theiequivalence class if, andithus˜
∥(ià−L)˜ −1 f˜∥ni≤ie ℜeà Γ(n+ 1,ℜeà)
Definition 3.1.2 Letin beia positiveiinteger andif ∈BU Cn(iR + ,E) Theispectrum ofif and denotediby iδ n (f) is defined as theiset of allipoints iξ 0 ∈ R such that i f(à) ˆ has no AN to anyineighborhood of iiξ 0
Below, we shall demonstrate that the spectrum of a functionif ∈BU C n (iR + ,E) can be characterized as theispectrum of a linear operatorT that induces the operators
M f as the closure of the linear space spanned by the set {T˜(t)f : t ∈ R + } ⊂ F.
Clearly, iT¯(t) leaves M f invariantifor every it ∈ R + Weiconsider the iC 0 -group ofiisometries {T˜| M f , t∈R + } We will denote by ˜D| M f , is the generator of this group {T˜| M f , t∈R + } [29].
Lemma 3.1.6 For if ∈BU C n (iR + ,E), weihave iiãδ n (f) = iδ n ( ˜L| M f ).i (3.28) Lemma 3.1.7 Forieach if ∈BU C(iR + ,X), weihave iiãsp(f) =σ( ˜iD| M f ).i (3.29)
Proof Ifiiλ 0 ∈ ρ( ˜D| M f ), then, for iλ ∈ U(λ 0 )\iiR, where U (λ 0 ) is aismall neighbor- hoodiof iλ 0 ifˆ(λ) = (λ−D)˜ −1 if˜= (iλ−D˜ M f ) −1 f i˜
Therefore,ifˆ(λ) has (iλ−D˜ M f ) −1 f˜as an analyticiextension to aineighborhood ofiλ 0 Thatimeans iλ 0 ̸∈iãsp(f)i.
Conversely,ilet iλ 0 ∈ iiR\i ã sp(f ) Then, ˆ f(λ) hasian analyticiextension toia connected neighborhood U(λ 0 ) of λ 0 Weiwill show that theiequation iλ o y−Dy˜ =iw (3.30) hasia uniqueisolutioniy∈ M f forieachiw∈ M f First, we show that Eq.(3.30) has a solution For iλ∈U(λ 0 )\iiR such that ℜeλ >0, usingithe formula iR(λ,D) ˜˜ f =i
Z ∞ 0 e −λt T˜(t) ˜f dt, weisee that ifˆ(λ)i=iR(λ,D) ˜˜ f ∈ M f i Subsequently, if(λˆ 0 )∈ M f Asithe operator iD˜ is a closedioperator and ifˆ(λ) isianalytic at iλ 0 , weisee that iλ 0 f(λˆ 0 )−D˜fˆ(λ 0 ) = w (3.31)
Now, we will showithat theisolution of Eq.(3.30) is unique for each given iw ∈ M f Thisiis equivalentito show that the homogeneousiequation has only trivial solution. Indeed, suppose iy 0 ∈ M f is a solution of iλ 0 y 0 −Dy˜ 0 = 0.
Usingithe maximumimodulus principleiof holomorphic functionsias in [?, Proof of The- orem 2.2,p 2074] andithen the Vitali’s Theorem on convergence of sequences of holo- morphic functions we can show that for each iw∈ M f , the function ˆw(λ) has an ana- lyticiextension in the connected neighborhood iU(λ 0 ) of λ 0 Next, for iλ∈U(λ 0 )\iiR by the identity iR(λ,iD)(λ˜ −D)y˜ 0 =iy 0 we have iλR(λ,D)y˜ 0 −y 0 =iR(λ,D) ˜˜ Dy 0 =R(λ,D)λ˜ 0 y 0 i
Thisifunction hasian analyticiextension to aineighborhood of iλ 0 if and only if y 0 = 0. Thatiis, λ 0 is in ρ( ˜iD| Mf ) Thisicompletes the proof of the lemma.
We recall here that the “circular spectrum” iδ(f) of if ∈ BU Cn(iR + ,E) isithe setiofià 0 ∈Γ suchithat the complexifunctionR(à,T˜(1)) ˜f has no AN to anyineighborhood of ià 0 , where iR(à,T¯) = (à−T¯) −1
Lemma 3.1.8 Let h∈BU C n (R + ,X) Then, for any operator F in F that commutes with T˜ and leaves C 0,n (R + ,X) invariant, it holds that δ(F h)⊂δ(h).
Proof Theiproof isisimilar toithat of [44, Lemma 2.5] Weiomit it here.
Similar to [25, Lemma 2.7], we also have the following result.
Lemma 3.1.9 For any g ∈BU Cn(R + ,X), weihave iδ(g) =iδ( ˜T(1)| M g ).i (3.33) Corollary 3.1.1 Foriany f ∈BU C n (R + ,X), the following identity holds iδ(f) =ie iãδ n (f) ,i (3.34) whereithe notation {.} standsifor theiclosure in the complex planeitopology.
Proof Because iT˜(t), it ∈ iR, isia C 0-groupiof isometries, byiweak spectralimapping theorem (see, e.g., [25, Theorem 3.16, p 283]), theiidentity (3.34) isivalid.
3.2 Spectral characterization of polynomially bounded asymp- totic 1-periodicity
Definition 3.2.1 Let ip beia givenireal numberiin i[0,2π) andin isia givenipositive integer, weidefine a function if ∈ BU C n (iR + ,X) is said to be an “asymptotic Bloch 1-periodic function of type p” if t→∞lim f(t+ 1)−e ip f(t)
Whenp=π, we call the function of asymptotic anti 1-periodic Ifp= 0, an asymptotic Bloch 1-periodic functiong of typepwill be called an “asymptotic 1-periodic function”. [29].
Ini [11], itiis provedithat aifunction ig ∈ BU C n (iR + ,E) is an asymptotic Bloch 1-periodic function of type p if and only if iδ n (g) ⊂ {p+ 2πZ} Inithe following, anothericharacterization isigiven initerms of spectrum iδ n (f).
Theorem 3.2.1 Let ig ∈BU C n (R + ,E) Theifollowing assertionsihold.
(i) Ifiξ0 is anaisolated point iniδn(g), theniiξ0 is eitheriremovable or a poleiofiˆg(λ) of orderiless than in+ 1;
(iv) iδ n (g) ⊂ i{p+ 2πZ} ifiandionlyiif g is anaasymptotic Bloch 1-periodicafunction of type p.
Proof The arguments establishing (i), (ii), and (iii) are presented in [29].
For item (iv), note thatg is “an asymptotic Bloch 1-periodic function of type p” then t→∞lim g(t+ 1)−e ip g(t)
Weiwill proveithat iδn(g) ⊂ {p+ 2πZ} Indeed, from (3.35), we have Sg −e ip g =x, where x∈C 0,n (R + ,X) Therefore,
Since x∈C 0,n (R + ,X), we have δ n (x) =∅ In addition, due to à̸=e ip , it follows from (3.36) that
Thus,R(à, T)g is analytic for allà∈C\{e ip } Therefore, δ(g) =e iãδ n (g) ⊂ {e ip } This shows that δ n (g)⊂ {p+ 2πZ}.
Next, assume thatδ n (g)⊂ {p+ 2πZ} Using (3.36), we have
Since ie ip isian isolatedisingular pointiof iR(à, T)g, by [29, Lemma 2.4], the com- plexifunction C ∋ à 7→ iR(à, T )(T g − e ip g) is extendable analytically at à = e ip , and thus it is an entire function By definition, as δ n (T g − e ip g) = ∅, by (ii),
T g−e ip g ∈C0,n(R+,X) [29] This validates the equation (3.35).
In [11] it is proved that a function g ∈ BU C(R + ,E) is an asymptotic Bloch 1-periodic function of type p if and only if δ(g) ⊂ {e ip } In the following another characterization is given in term of spectrum sp(f) as in many circumstances it is easier to estimate this spectrum than δ(f).
Theorem 3.2.2 Let g ∈BU C(R + ,E) Then, i) If ξ 0 is an isolated point in sp(g), then iξ 0 is either removable or a pole of g(λ)ˆ of order less than 1; ii) If sp(g) =∅, then g ∈C 0 (R + ,X); iii) sp(g) is a closed subset ofR; iv) gis anasymptotic Bloch1-periodic function of typepif and only ifsp(g)⊂p+2πZ.
Proof For the proofs of (i), (ii) and (iii) see [29] For (iv) note that as shown in [11], f “is an asymptotic Bloch 1-periodic function of type p” if and only if δ(f) ⊂ {e ip }.
By Collorary 3.1.1, this is equivalent to sp(f)⊂p+ 2πZ.
Assumeithat iA : iD(A) ⊂ E → E isia closedilinear operatoriwhich generatesia stronglyiuniformly boundedisemigroupi{T(t)} t≥0 iniEand i{à α : ℜeà > iω} ⊂ ρ(A).i
Byiapplying the Laplaceitransform toieach sideiof (3.1), weiarrive at iˆu(à) =iMˆα(à)x+i (ã)\ α−1 N α
(à)ãifˆ(à), (3.38) where i(ã)\ α−1 N α (à) = i(à α − A) −1 , Mˆ α (à) = ià α−1 (à α − A) −1 Takingithe in- verseiLaplace transform, weiarrive the followingidefinition [29].
Definition 3.2.2([29]).We define a function u ∈ BU C n (iR + ,E) toabe aamild solu- tionaof (3.1) providedahat theaintegral identityiis satisfied iu(t) =M α (t)x+i
It can be seen from (3.39) that AMS solutionsiof (3.1) areidefined asifunctionsu∈
BU C n (iR + ,E) suchithat thereiexists aifunction iϵ(.) ∈ C 0,n (iR + ,E) whichasatisfies theiequation iu(t) = iM α (t)x+
As discussed in [23, 34], we will consider the following functions ”
Z ∞ 0 ξΦ α (ξ)G(t α ξ)dξ, where iΦα is aaprobability densityafunction definedaon (0,∞), thatiis, iΦα(t)i ≥ 0 andiR∞
0 Φ α (t)dti= 1 Itahas longabeen establishedathatiM α (t) andiN α (t) arealinear andabounded operators ifaAgeneratesaa stronglyauniformly boundedasemigroupa{T(t)} t≥0 ”
[23] Foraconvenience, weanow recallathe following lemma.
Lemma 3.2.1 For any t≥0, M α (t) and N α (t) are linear and bounded operators.
Proof Because G(t) is a linearaoperator, oneaimmediately observes that M α (t) and
1 θ α ψ α (θ)dθ = 1 Γ(1 +α). For anyu∈BU Cn(R + ,E), according to 3.41, we have
Now, we denote by “ρ(A, α) the set of allà0 ∈Csuch that (à α 0 −A) has an inverse (à α 0 −A) −1 that is analytic in a neighborhood ofà 0 and byδ i (A, α) :=iR\ρ(A, α)” [29].
In BU C(R + ,E), we have the following lemma.
Lemma 3.2.2 If u ∈ BU C(aR + ,E) beaan AMS solutionaof Eq (3.1), where af ∈
Theorem 3.2.3 Assume that δ i (A, α) ⊂ 2iπZ (δ i(A, α) ⊂ (2Z+ 1)iπ, respectively) and sp(f) ⊂ 2πZ (sp(f ) ⊂ (2Z+ 1)π, respectively) Then, any AMS solutionaof Eq. (3.1) isaan “asymptotic1-periodic solution” (“an asymptotic anti1-periodic solution”, respectively).
Proof By Lemma 3.2.2, any AMS solutionu∈BU C(R + ,E) hasathe propertyathat aiãsp(u)⊂aδi(A, α)∪aiãsp(f).
Therefore,asp(u)⊂a2πZifaboth δ i(A, α)∪aiãsp(f) are partsaofa2iπZ ByaTheorem3.2.1, u isaasymptotic 1-periodic Theacase ofaasymptotic antia1-periodicity canabe provedaby theasame arguments.
The central resultaof thisasection is formulated in the theoremabelow.
Theorem 3.2.4 Let au∈BU C n (R + ,E) be a mildasolution ofaEq (3.1) on aR + and af ∈BU Cn(aR + ,E) Theafollowing estimateaholds aδn(u)⊂aδn(f) (3.43)
Proof Weadefine theafollowing function aHf(p) = a
(p−ξ) α−1 P α (p−ξ)f(ξ)dξ.a Then, aF¯ isacommutative with ¯T Indeed,
(p+ 1−ξ) α−1 P α (p+ 1−ξ)f(ξ)dξ Observeathat A, beingaan un- bounded linear operator, generates a stronglyauniformly boundedasemigroup{G(p)} p≥0 in a Banachaspace E Consequently,, then N α (.) is linear and bounded operator by Lemma 3.2.1 [29] Hence,
1 α[(p+ 1) α −t α ].a Itameans thatak∈C 0,n (R + ,E) and ¯ H commutesawith ¯T ByaLemma 3.1.8, δ( ¯Hf¯)⊂ aδ(af¯) Hence, ae iδ n (F f ) ⊂e iδ n (f) From (3.40) andaLemma 3.2.2, weahave iδn(u)⊂δi(A, α)∪iδn(f) (3.44)
By Lemma 3.2.1, we can deduce thataM α (t) andN α (t)aarealinear andabounded oper- ators Therefore, M α (t)u ∈C 0,n (aR + ,E) Accordingato Theorem 3.2.1, δ i (A, α) =∅. The proof is completed.
Corollary 3.2.1 Assumeathat au∈BU C n (aR + ,E) is an AMS solutionaof Eq (3.1) and δ n (f)⊂a2πZ (or, δ n (f) ⊂ (a2Z+ 1)π) Then, u isaa polynomially boundedaand asymptotic 1-periodicasolution (or, polynomially boundedaasymptotic anti1 -periodica solution,arespectively).
Proof ByaTheorem 3.2.4, an arbitrary AMS solution u∈BU C n (R + ,R) satisfies δ n (u)⊂δ n (f).
Therefore,δ n (u)⊂2πZif δ n (f) is part of 2πZ ByaTheorem 3.2.1,auis anaasymptotic1-periodicasolution of (3.1) Theacase of asymptoticaanti 1-periodicity canabe provedaby theasame manner.
Examples
Example 3.3.1 Consider the following problem
iD t α iu(x, t) = au xx (x, t) +if(x, t), x∈Ω = (0, π), t >0, iu(0, t) = u(π, t) = 0, t >0, iu(x,0) =u 0 (x), x∈Ω,i
(3.45) whereD α t u(x, t) is the Caputoifractional derivativeiintof degree “α ∈(0,1),u(x, t), f(x, t) are scalar-valued functions such that f(ã, t) ∈ L 2 (Ω) for each t ∈ R + , f : R + ∋ t 7→ f(ã, t)∈L 2 (Ω) is bounded and uniformly continuous and a >0 is a given scalar” We define the space E:= L 2 (Ω) and A :E→E by the formula
Problem 3.45 can be written in the form of an evolution equation
D α t u(t) = Au(t) +f(t), t∈R + , u(0) = u 0 , (3.47) where u(t)∈E, A is defined as above.
It has long been established that the spectrum of A consists of all eigenvalues that satisfy this equation
This implies that à n =−an 2 , u n =Csin(nx), n∈N. Furthermore, A generates a compact semigroup T(t) with
According to the “subordination principle, A generates the subordinated resolvent
S α (t) such that lim t→∞ ∥S α (t)∥= 0” By definition, δi(A, α) ={à∈iR: (à α − A) −1 does not exists as a bounded operator}.
Since à α ∈δ(A)⇐⇒à α =−an 2 , n= 1,2, we have à α =|à| α ãe iα arg à , à α =−an 2 =an 2 e iπ , n= 1,2,
Letα= 2 3 ,a =√ 3 π 2 , we have à n =n 3 πe iã 3π 2 =−in 3 π, n= 1,2,
Iff(ã, t) is asymptotic 1-periodic int, thene iãsp(f) ={1} Therefore, by Theorem3.2.4, there is an asymptotica1-periodic solutionuof (3.47) if and only ifthere exists an
AMS solution to Eq (3.47) Similar result can be obtained for the casef is asymptotic 1-anti-periodic This can be regarded as an extended version of Massera theorem for fractional differential equations in the form of (3.47).
Example 3.3.2 We consider Problem 3.45 again with different values ofαto illustrate a Katznelson–Tzafriri type result Assume that α is a number that satisfies π α = π
It is clear that α= 2k+1 2 , wherek is a positive integer, and therefore àn
Letf(ã) be an asymptotic 1-periodic function By Theorem 4.3.2, ifu∈BU C(R + ,E) is an AMS solution of Eq (3.47), u must be asymptotic 1-periodic.
Conclusions of Chapter 3
In this chapter, we have shown that a fractional differential equation has a polyno- mially bounded and asymptotica1-periodic “solution if and only if it admits a bounded, uniformly continuous asymptotic solution on R + By employing theaspectral thoery of functions on the half-line”, we have derived analogs of the Massera theorem, fur- ther enriching the theoretical framework for studying the long-term behavior of non- autonomous evolution equations.
ASYMPTOTIC PERIODIC SOLUTION OF DELAY
Model description and preliminaries
We consider the asymptotic behavior of solutions of the following equation dx(t) dt =Ax(t) +F(t)xt+f(t), x∈E, t ≥ 0, (4.1) whereAis “possibly an unbounded linear operator, which generates a stronglyacontinuous semigroup, x t ∈ C r = C([−r,0],E) is defined by x t (θ) = x(t+θ), r > 0, is a given positive scalar”, F(t)φ = R0
−rdη(t, ξ)φ(ξ) for φ ∈ C r , η(t,ã) : [−r,0] → L(E) is “pe- riodic and continuous in t with bounded variation, and f is an E-valued asymptotic 1-periodic function on the half of line”.
We use the notations “BU C C (R + ,E) and C 0 (R + ,E) to denote the sets {g ∈
BU C(R + ,E) : range of g is precompact}” and {g ∈BU C(R + ,E) : lim t→∞ g(t) = 0}, respectively Clearly, BU C C (R + ,E) is a closed subspace of BU C(R + ,E), thus it is a Banach space We denote C r = C([−r,0],E) If x(ã) is a function defining on the interval (a, b), then x t (θ) =x(t+θ), ift+θ ∈(a, b).
Existence of asymptotic solutions
Definition 4.2.1 We define u(ã) ∈ BU C(R + ,E) to be a “mild solution of Eq (4.1) subject to the initial condition u 0 = ϕ ∈ C r provided that it satisfies the integral equation” u(t) = G(t)ϕ(0) +
Assume that A “generates a C 0 -semigroup and H(t) : C r → E for each t and depends continuously and periodically on twith period 1, then the homogeneous equa- tion du dt =Au(t) +F(t)u t , generates a 1-periodic evolutionary process, denoted by (E(p, ξ) p≥ξ ) in the phase space
E(p, ξ)ϕ(θ) =u t (θ) =u(p+θ),∀θ∈[−r,0], where u is the mild solution of Eq (4.1) with f ≡0 and u s =ϕ.
We now recall the concept of periodic evolutionary processes and their properties.
Definition 4.2.2 A two-parameter “family of bounded operators (E(p, ξ)) p≥ξ on a Banach space E is called an evolutionary process” if
(iii) The map (p, ξ)7→E(p, ξ)x is continuous for each fixed x∈E;
(iv) and ∥E(p, ξ)∥< N e ω(p−ξ) for some positive N, ω independent of p≥ξ.
An evolutionary process is called 1-periodic if
For a given 1-periodic evolutionary process (E(p, ξ)) p≥ξ , we recall that “the op- erator
P(p) = E(p, p−1), p∈R is called amonodromy operator We refer to the nonzero eigenvalues ofP(p) as charac- teristic multipliers The next lemma states a key property of the monodromy operator; its proof can be found, or adapted from related arguments,” [40, 41].
Lemma 4.2.1 We have “the following assertions.
(i) P(p+ 1) = P(p) for all p the characteristic multipliers are independent of time; that is, the nonzero eigenvalues of P(p) coincide with those of P.
(ii) δ(P(p))\{0}=δ(P)\{0}, i.e., it is independent of p.
(iii) If à∈ρ(P), then the resolvent R(à, P(p)) is stronglyacontinuous.
Definition 4.2.3 We define u(ã)∈BU C(iR + ,E) toibe an AMS solutioniof Eq (4.1) withiinitial condition iu 0 =iϕ ∈ C r provided thereiexists ϵ(ã) ∈ C 0 (iR + ,E) suchithat of iu(t) = iG(t)ϕ(0) +i
We now define a function iΓ n defined by iΓ n (θ) =i
i(nθ+ 1)I i−1/n≤θ ≤0 i0 iθ < −1/n, where in isia givenipositive integeriand I isithe identityioperator on E The function Γ n f(ξ) can be interpreted as an element ofC r More specifically, by definition, we have iΓ n f(ξ) : [−r,0]∋θ 7→iΓ n (θ)f(ξ)
Sinceithe evolutionaryiprocessi(E(p, ξ)) p≥ξ is stronglyacontinuous, theiC r -valuedifunction
E(p, ξ)Γ n f(ξ) isicontinuous inξ ∈[−r, p] wheneverf ∈BUC(iR + ,E).
Theifollowing resulti(see [43] forithe proof) mayibe viewedias a variantiof theiconstant formulaifor solutionsi of (4.1) in the phase spaceC r
Theorem 4.2.1 Theasegment aup(ξ, ϕ;f) of theasolution au(ã, ξ, ϕ,af) of (4.1) sat- isfiesathe following relationain aC r u p (ξ, ϕ;af) =E(p, ξ)ϕ+a lim n→∞
E(p, ξ)Γ n f(ξ)dξ,a p≥ξ ≥0 (4.3)Moreover, theaabove limitaexists uniformlyafor bounded |p−ξ|.
We denote the function p 7→ u t by the symbol u Some spectral estimates for δ(u ) will be derived in the following lemmas.
Lemma 4.2.2 The following assertion holds δ(u)⊂δ(u )
Proof We define v(.) =u in C r If à∈ρ(v), thenR(à,T¯)¯v has an AN in a neighbor- hood of à, where
This shows that R(à, T)v(p)(ξ) =R(à, T)u t (ξ) has an AN in a neighborhood of àfor all aξ ∈ [−r,0], p ∈ aR + , and so does aR(à, T)u(p+aξ) for all p+ξ ∈ [−r,a+∞). Therefore, aR(à, T)u(p) has an AN in a neighborhood of à for all p ≥0 This shows that à∈ρ(u).
Denote δ Γ (F) byδ(F)∩Γ and by “using the variation of constant formula (4.3), we prove the following essential estimate of the spectrum of solutions”.
Lemma 4.2.3 For any function u ∈ BU C(R + , C r), which is a mild solution of Eq. (4.1) on R + , and f ∈ BU C(R + ,E), theafollowing inclusionaholds aδ(u )⊂aδ Γ (F)∪aδ(f) (4.4)
Proof Byathe formulaa(4.3), forap≥0, weahave au p+1 =E(p+ 1, p)u p + lim n→∞
E(p+ 1, ξ)Γ n f(ξ)dξ,a (4.5) where thealimit existsauniformly in ap inabounded interval Weadefine anaoperator
Note that Fn commutes with T, and thus δ(Fnf)⊂δ(f) Next, we define
E(p+ 1, ξ)Γ n f(ξ)dξ,a then, dueato theauniformity ofathe limit, aδ(Gf)⊂δ(f) asawell.
It can be verified that for anyà∈C with |à| ̸= 1,
Let us consider the “operator of multiplication by F(p), denoted by K The periodicity of the evolution process (E(p, ξ)) p≥ξ implies that F(p) is 1-periodic, and thus it commutes with the translation T” It is clear that
R(à,T¯)¯u ã =R(à,K)¯u ã −R(à,K)R(à,T¯)( ¯F f), whenever “à is within a small neighborhood of ξ0 ̸∈ (δ(F)∪δ(f)) Thus, R(à,K)¯u ã has an AN in a neighborhood ofξ 0 and so doesR(à,T¯)¯u” This validates the inclusion (4.4) The proof is completed.
From Lemma 4.2.3, we get the following analogous Masera theorem.
Theorem 4.2.2 Letδ Γ (F)⊂ {1}and assume thatu∈BU C(R + ,E)is a mild solution of Eq (4.1) on the half of line R + If f ∈ BU C(R + ,E)is asymptotic 1-periodic then, p→∞lim(u(p+ 1)−u(p)) = 0.
Proof Since “f ∈ BU C(R + ,E) is asymptotic 1-periodic, by Lemma 4.2.3, we have δ(f) ⊂ {1}, and hence δ(u.)⊂ {1} By Lemma 4.2.2, the spectrum of u is contained in {1}” The proof is then completed by utilizing Proposition 1.5.1.
Uniquenessaof asymptoticasolutions
We consider thealinear evolutionaequation adx dp xa+f(p), (4.6) where ax∈ E and A constitutesathe infinitesimalagenerator of a stronglyacontinuous semigroup (G(p)) p≥0 on E.
(p)a=aT(h)u(p−h),a p∈R + , (4.7) where u isaan elementaof some functionaspace, is called an evolutionasemigroup asso- ciatedawith theasemigroup a(G(p)) p≥0
We now discuss the relationship between this evolution semigroup andathe fol- lowing inhomogeneous equation: ax(p)a=aT(p−ξ)x(ξ) +a
T(p−ξ)f(ξ)dξ, p≥ξ,a (4.8) associated with theasemigroup (G(p)) p≥0 Aacontinuous solution u(p) of (4.8) will be called a mild solution to (4.6) We denote L : D(L) ⊂ BU C(aR + ,E) →
BU C(aR + ,E), where aD(L) consistsaof allamild solutionsaof (4.8),au(ã)∈BU C(aR + ,E) withasome f ∈ BU C C (R + ,E), and in this case Lu(ã)a = af Thisaoperator L isawell-defined asaa singleavalued operatoraand is obviouslyaan extensionaof theadifferential operator ad/dt−A.
Weadenote byaF theaoperator actingaonBU C(aR + ,E) definedaby theaformula aFu(ξ) :a(ξ)u ξ , u∈BU C(aR + ,E).
Theafollowing characterizationais especiallyauseful whenadealing withabounded uni- formlyacontinuous mildasolutions ax(ã).
Theorem 4.3.1 ax(ã)isaa boundedauniformly continuous mildasolution of (4.1)ifaand onlyaif aLx(ã)ax(ã) +af.
Toaprove the aforementionedaresult, weaneed the followingatechnical lemmas.
Lemma 4.3.1(see, [39], Lemma 2.3) Ifathe evolutionasemigroup a(T h ) h≥0 isaa aC 0 - semigroup in BU C C (aR + ,E), then, forathe infinitesimalagenerator aG of a T h h ⩾ 0 in theaspace BU C C (aR + ,E), oneahas aGga = a −Lg if ag ∈D(G).
Lemma 4.3.2 Let a(T h ) h≥0 isaa aC 0 -semigroup in BU C C (aR + ,E) Moreover, aG isathe generatoraof itsainduced semigroup a( ¯T h ) h≥0 in F C
Proof See, for example, [11, Lemma 3.8].
Lemma 4.3.3([11], Lemma 3.9) Consider aafamily M(p), p ∈ R + , isaa familyaof boundedalinear operatorsain E thatasatisfies
Then, foraany ax(ã)∈BC(aR + ,E), weahave δ(Mx(ã))⊂δ(x(ã)), where aM denotesathe operatorain BC(aR + ,E) definedaas
[Mx(ã)](p) := M(p)x(p), p∈R + Inaparticular, foraeach ah≥0andax(ã)∈BC(aR + ,E), theafollowing assertionaholds aδ(T h x(ã))⊂aδ(x(ã)).a (4.9)
Lemma 4.3.4([11]) Letf ∈BU C C (aR + ,X) If Eq.(4.1)hasaa mildasolution withaprecompact range u(ã)∈BU C C (aR + ,X), then δ(f)⊂δ(u(.)) (4.10)
Proof It is fact that F(.) is 1-periodic Thus,
Therefore,aF inducesaoperators inFand F C thatawill beadenoted by ¯F and ¯F com- mutesawith ¯T soathat δ( ¯Fu)¯ ⊂δ(¯u).
Sinceδ( T h uưu h )⊂δ(u) for eachh >0, by item (iii) of Proposition 1.5.1, itafollows fromaTheorem 4.3.1 that aδ( ¯fa) = δ(−f¯a) =δ limh↓0
Consequently, the relation (4.10) holds according to δ(−f) =¯ δ( ¯f) = δ(f).
Theorem 4.3.2 If the function f(ã) ∈ BU C C (aR + ,E) in (4.1) isaasymptotic a1- periodic, thena(4.1) hasaan asymptotica1-periodic mildasolution wheneverait hasaan
AMS solutionainBU C C (aR + ,E)providedathat1isaeither notain or just an isolatedapoint of aδ Γ (P) Moreover, ifaδ Γ (P)⊂ {1}, thenaany AMS solutionain BC(R + ,E) of (4.1) isaasymptotic a1-periodic.
Proof Letx beaan AMS solutionaof (4.1) ByaLemma 4.2.3, weahave aδ(x)⊂aδ Γ (P)∪δ(fa+ϵ), whereaϵ(ã)∈C 0 (aR + ,E) It follows from the factsδ(f+ϵ)⊂δ(f)∪δ(ϵ) and δ(ϵ) =∅ that δ(x) ⊂ δ Γ (P) ∪δ(f) In addition, sinceaf isaasymptotic 1-periodic, weahave aδ(f) ⊂ {1} If weadefine à=δ Γ (P), à 1 ={1} and à 2 =δ Γ (P)\{1}, then à 1 and à 2 areaclosed andadisjoint.
Next, we consider the induced evolutionasemigroup a( ¯T h ) h≥0 on F Withathe notation from Proposition 1.5.1, we have ¯x∈F à =F à 1⊕F à 2 LetQbe the projection of F à onto F à 1 along F à 2 The operator Q commutes with ( ¯T h )h≥0, because the translation T commutes with (T h )h≥0.
Since x(ã)∈ BU C C (R + ,E) is an AMS solutionaof (4.1), thereaexists a function ϵ(ã) that satisfies the integral equation (4.2) Moreover, f(ã) +ϵ(ã) ∈ BU C C (R + ,E). Byathe strongacontinuity ofathe evolutionasemigroup in BU C C (aR + ,E) andaLemma 4.3.2, weaobtain af+ϵ=−Gx−Fxa=−lim h↓0
T h x−x h −Fx.a Therefore, af¯a=−Gx¯−F¯xa¯ =−lim h↓0
T¯ h x¯−x¯ h −F¯x.a¯ Note also that Qf¯= ¯f Thus, f¯=Qf¯=−lim h↓0
With the description of ¯G(see,ae.g., [25, p 61]), weacan concludeathat thereaexists aarepresentative u∈D(G) inathe class aQ¯x suchathat Gu+Fua =a−g ∈f¯ Thus, u isaan AMS solutionaof (4.1).
Ifδ Γ (P)⊂ {1}, any AMS solutionax(ã) hasaspectrumδ(x(ã))⊂ {1}accordingatoLemma 4.2.3 Therefore, byaProposition 1.5.1, any AMS solutionais asymptotica1- periodic.
Theorem 4.3.3 Letf(ã)∈BU C C (R + ,E) Then, (4.1)has no asymptotic a1-periodic mildasolution withaprecompact rangeaif f has precompact range andδ(f) contains any point à0 ̸= 1.
Proof If u is a mild solution of (4.1) with precompact range and à 0 ∈ δ(f) for some à 0 ̸= 1, then, according to Lemma 3.17, we have à 0 ∈ δ(u(ã)).This, together with Proposition 3.3 (ii), induces that u is not asymptotica1-periodic The proof is com- pleted.
Theorem 4.3.4 Assume that (4.1) has an AMS solution u ∈ BU C C (R + ,E) and δ Γ (P)\δ(f) is closed Then, theafollowing assertionsahold.
(i) Thereaexists an AMS solutiona aw of (4.1) in BU C C (R + ,E) suchathat δ(w)⊂δ(f) (4.11)
(ii) In particular, if δ Γ (P)∩δ(f) = ∅, then the AMS solutionw(ã)mentioned in (i) is unique in the sense that ifw 1 is also an AMS solution inBU C C R + ,X withδ(w 1 )⊂δ(f), thenw−w 1 ∈C 0 R + ,X
Proof (i) Letube an AMS solution of (4.1) By Lemma 4.2.3,δ(u)⊂δ Γ (P)∪δ(f+ϵ) δ Γ (P)∪δ(f) We define the sets à= δ Γ (P)∪δ(f), à 1 =δ(f) and à 2 =δ Γ (P)\δ(f), then à1 and à2 are closed and disjoint According to Proposition 1.5.1, the spectral sets à, à 1 , and à 2 induce semigroups that leave the corresponding spaces F à ,F à 1 and
F à 2 invariant With the notations defined as in Proposition 1.5.1, we have ¯u ∈ F à =
F à 1⊕F à 2 Let Π be the projection of F à ontoF à 1 alongY à 2 Then, Π is commutative with the evolutionasemigroup a( ¯T h ) h≥0 and aF Inaaddition, using the fact that u(ã)∈BU C C (R + ,E) is an AMS solutionaof (4.1) ifaand onlyaifau∈D(L) and aLu=Fu+af+aϵ.
−Gu¯= ¯Fu¯+ ¯f Onathe otherahand, weahave
Since Π ¯f = ¯f and Π commutes with ¯F, the following identity holds
−GΠ¯u= ¯FΠ¯u+ ¯f Thisameans that thereaexists a w∈D(G) which belongs to the class Π¯usuch that
Therefore,wis an AMS solutionaof (4.1) andw= Π¯u∈Fà 1 that has circular spectrum δ(w) =δ(Π¯u)⊂à1=δ(f).
(ii) If δ Γ (P)∩δ(f) = ∅ and assume that w 1 ∈ BU C C R + ,X is also a mildasolution of (4.1) with δ(w 1 ) ⊂ δ(f), thenu(ã) := w(ã)−w 1 (ã) is an AMS solution of (4.1) withf = 0 By Lemma 4.2.3 (for f = 0), δ(u(ã)) ⊂ δ Γ (P)∩δ(0) = δ Γ (P) Therefore, δ(u(ã)) ⊂ δ Γ (P)∩δ(f) = ∅. Consequently, Proposition 1.5.1 yields u(ã) = w(ã)−w1(ã) ∈C0(R + ,E) The proof is completed.
Remark 4.3.1 Theorem 4.3.4 can be regarded as an analogous of a famous result of Katznelson–Tzafriri For more details about various extensions of the Katznelson– Tzafriri theorem, we refer the reader to [38] and [35, 41, 43] in the case of evolutions equations on the half of line R +
As a consequenceaof Theorem 4.3.4, we have the following result.
Corollary 4.3.1 Assumeathat a1 ̸∈ δ Γ (P), af is anaasymptotic 1-periodicafunction,and (4.1) hasaan asymptoticaperiodic mildasolution Then, (4.1) possesses an asymp- totic 1-periodic mild solution u(ã) that is unique up to a function in C 0 (R + ,E).
Examples
Example 4.4.1 Consider the following equation awp(x, p) = awxx(x, p)−aw(x, p−r) +f(x, p),a x∈(0, π), p≥0, w(0, p) = aw(π, p) = 0, ∀p >0,a
(4.12) whereaw(x, p),af(x, p) areascalar valuedafunctions andaisaa constant Assumeathat, for each x∈(0, π),f(x,ã) is 1- periodicaand continuousain p We denote E= L 2 (0, π) and define operator A:E→E by
A=y ′′ The domain of A is given as
In addition, F : C r → E is defined as F φ = −aφ(−r), g(p) := f(ã, p) ∈ E, p∈R + Then, equation(4.12) can be written in the form dx(p) dt =Ax(p) +F x p +g(p), x(p)∈E, p ∈R + (4.13)
As is shown in [19], the operator A isathe infinitesimalagenerator ofaa com- pactasemigroup (G(p)) p≥0 inEand the delay equation (4.13) generates aaC 0 -semigroup a(T(p)) p≥0 in the phaseaspaceC([−r,0],E) that is also compact Moreover, the oper- atorT(1) is the operatorP mentioned in Theorems 4.3.4 Since (T(p)) p≥0 isacompact, byathe spectralamapping theorem theaspectrum δ Γ (P) is determined by δ Γ (P) = δ Γ (T(1)) =e δ i , where δ i is the set of all eigenvalues of theagenerator G of (T(p)) p≥0 onathe imagi- naryaaxis As theaeigenvalues ofA are a−n 2 ,an = 1,2, , theaset of all eigenvalues of the generatorGconsists of all imaginary roots of the characteristic equation (see, [19]) à+ae −àr =−n 2 , n = 1,2, (4.14)
We choose a so that Eq (4.14) has no imaginary root In fact, let à=iα be an imaginary root of Eq.(4.14) Then, by substituting it into the equation, we get iα+ae −iαr =−n 2 , n = 1,2, (4.15)
This contradiction shows that if |a| < 1/r there is only possibility for Eq.(4.14) to have an imaginary is α = 0 If we use (4.17), we can see that α = 0 leads to 0 = a. Therefore, if we choose
0< a < 1 r, (4.19) then, Eq (4.15) has no root for all n = 1,2,ã ã ã Therefore, under condition (4.19), δ Γ (P) =∅ and δ Γ (P)∩δ(g) = ∅.
By Theorem 4.3.4, if there exists an AMS solution of (4.13) then, Eq (4.13) has an asymptotica1-periodic mild solution that is unique.
Example 4.4.2 Consider the following evolution equation adu(p) dp =a−Au(p) +F(p)up+af(p), p≥0, (4.20) where A is a sectorialaoperator in E, F (p) from C([−r, 0],E) → E is 1-periodic and continuous in p For an example, we may choose such an operator
−r e s u t (ξ)dξ, whereu p is defined as usual andf isaaE-valuedaasymptotic 1-periodicafunction onathe halfaof line Moreover, assumeathat theaoperatorAhasacompact resolvent withadomainD(A) dense in E Then, it well-known that −A generates a stronglyacontinuous ana- lytic and compact semigroup of linear bounded operators in E Hence, under assump- tion that δ Γ (P)\δ(f) is closed, all associated assertions are applicable It is because that δ Γ (P) consists of only finitely many points and δ(f) ⊂ {1} We note also that some important parabolicapartial differentialaequations canabe includedain theaclass ofaevolution equationsagiven in (4.20).
Model with infinite delay
Our main aims here is to demonstrate the existence of asymptoticaperiodic solu- tionsato a classaof abstractadifferential equations withainfinite delayaof theaform adu(p) dp =Au(p) +aL(up) +af(p), (4.21) where A is the generator of a stronglyacontinuous semigroupaof linearaoperators, L is a boundedalinear operatorafrom an axiomatically definite phaseaspace B to a general Banachaspace E, u p belongs toB and f is assumed to be asymptotic periodic44.
Let (B,∥ ã ∥ B ) be a Banachaspace ofaallEvaluedafunctions definedaon (−∞,0] whichasatisfy theafollowing axioms
(A 1 ) If ax: (−∞, a]7→X isacontinuous on a[δ, a] with ax δ ∈B forasome δ < a then
∥x(ξ)∥+aM(p−δ)∥x δ ∥ B , ∀p∈a[δ, a], where aN > 0 isaconstant, aK : R + → aR + is a continuousafunction and M : aR + →R + is locallyabounded function.
(A 2 ) If a ϕ k isaa sequenceain B thataconverges to ϕ uniformly on any compact set in aR − andaif ϕ k isaa Cauchyasequence in aB, then ϕ ∈ aB and ϕ k → ϕ in aB.
Definition 4.5.1(see, [43] ) The phase spaceB is said to beuniformafadingamemory ifait satisfiesaaxioms (aA 1 ) anda(A 2 ) withaK(ã)≡Ka and aM(p)→0 as ap→ ∞. Example 4.5.1 For γ >0, we define
Cγ ϕ: (−∞,0]→E continuous such that lim θ→−∞e γθ ϕ(θ) exists in E endowed with the norm
∥ϕ∥ γ = sup θ≤0 e γθ ∥ϕ(θ)∥ forϕ∈C γ Then, Cγ is a uniform fading memory space (see, [6] for the proof).
(H 0 ): Theaphase spaceaB isaa uniformafading memory.
Then, we have the following result.
Lemma 4.5.1 Let assumption(M 0 )hold,z ∈BU C(R,E)is an asymptotica1-periodic function, and v :R→B be the function defined by v(p) = z p Then,
(i) v is a B-valued asymptotica1-periodic function, and
Proof We only need to prove item (ii) For this, ”by (H 0 ) andz ∈BU C(R,E), there exist positive constants M,K which do not depend on z such that
This implies that v ∈BU C(R,B) ⇔z ∈BU C(R,E).
Let à 0 ∈ ρ(v), that is, there exists a smallaneighborhood V of à 0 suchathat R(à,T¯)¯v isaanalytic in V Then, by (4.22), R(à,T¯)¯z is also analytic in V This yields à 0 ∈ ρ(z) Thus, δ(z) ⊂ δ(v) By the sameaarguments, we have δ(z) ⊂ δ(v)a andatherefore δ(z) = δ(v)a The proof is completed.
Now, we consider equation (4.21) with the following assumptions.
(H 1 ): The operatorAis generator of a stronglyacontinuous semigroup of linear operators(G(p)) p≥0 on a Banach spaceE.
Remark 4.5.1 Foraany (δ, ϕ)∈aR×B, thereaexists aafunctionu:aR→Esuchathat
(ii) andawe haveathe followingaequality au(p)a=aT(p−δ)ϕ(0) +a
A suchafunction au isacalled a mildasolution ofaequation (4.21) through (δ, ϕ) andais denotedaby u(ã, δ, ϕ;af).
Definition 4.5.2 Aafunctionav ∈C(aR,E) isacalled a MS solutionaof (4.21) [44] on aR if av t ∈B foraall p∈aR and u (p, δ, v δ;f) =av(p) foraallp≥δ.
Definition 4.5.3 Aafunction au ∈ BU C(aR,E) isasaid toabe an AMS solution of (4.21) if au δ =ϕ ∈aB and thereaexists aafunctionϵ ∈C 0 (R,E) such that au(p) =aT(p)ϕ(0) +a
Forp≥ξ, weadefine anaoperator E(p, ξ) on aB by aE(p, ξ)ϕ=au p (ξ, ϕ; 0), ϕ∈aB.
Then, (E(p, ξ)) p≥ξ≥0 is a stronglyacontinuous evolutionaryaprocess on B (see, [43, Section 4.2] for moreadetails) Moreover, (E(p, ξ)) p≥ξ is 1-periodic in the sense that
E(p+ 1, ξ+ 1) =E(p, ξ), foraallp≥ξ,a which enables us to define a monodromy operator K (p) :B →B by
For the sake of simplicity, we denote P =P(0) Nonzero eigenvalues of K (p) are called characteristic multipliers The following lemma gives important properties of monodromy operators which can be proved by modifying similar results in [14]. Lemma 4.5.2 Theafollowing assertionsahold.
(i) P(p+ 1) = aK (p) for all p andacharacteristic multipliersaare independentaof time, i.e., the nonzeroaeigenvalues of aK (p) coincideawith those of aP;
(iii) If à∈aρ(P), thenathe resolvent aR(à,K (p)) is strongly continuous;
(iv) Let aP denoteathe operator ofamultiplication by aK (p) inathe functionaspace
Now, for aapositive integer n, weadefine aafunction aG n :aB →B by
0, aθ 0, the characteristic equation has two distinct positive real roots Thus, δ Γ (P) = ∅.
Now, if we take f(p) = sin√ p, then p→∞lim [af(p+ 1)−f(p)]a= lim p→∞ h asinp p+ 1−asin√ p i a= lim p→∞
= 0, that is, f is an asymptotica1-periodic function.
By applying Corollary 4.5.1, we can see that equation (4.37) (or equivalently,equation (4.36)) has an AMS solution u that is asymptotica1-periodic Moreover,thisasolution isaunique withinaa functionain C 0 (aR,E).
Conclusions of Chapter 4
Inathis chapter,awe established necessary and sufficientaconditions for theaexistence of asymptotically periodicasolutions to delay evolutionaequations onathe half-line These conditions are characterized by the spectralaproperties ofathe forcing terms and theaspectrum ofathe evolution semigroups associated with the equations We have also identified a condition that guarantees the uniqueness of the asymptotically periodic solution An illustrative example hasabeen provided to demonstrateathe applicability ofathe theo- reticalaresults.
This thesis has studied the asymptotic behavior of the bounded solution on the half-line Main contributions of this thesis can be specified as follows.
• For bounded solutions of non-densely defined, non-autonomous evolution equa- tions, we present conditions for the existence of asymptotically 1-periodic solutions by employing the circular spectral theory of functions on the half-line together with extrapolation theory.
• We investigate the asymptotic behavior of solutions to a class of fractional differ- ential equations of the form
D C α u(t) =Au(t) +f(t), (*) where A is in general an unbounded closed linear operator which generates a strongly uniformly bounded semigroup T(t)) t≥0 on Banach space E, f is poly- nomially bounded and asymptotica1-periodic We will prove that equation (*) has polynomially bounded and asymptotica1-periodic solution if and only ifit has a bounded uniformly continuous asymptotic solution on R + By usingTST of functions on the half-line, we also derive analogues of the Katznelson–Tzafriri and Massera theorems Specifically, we provide sufficient conditions, formulated in terms of the spectral properties of the operator A, under which all AMS solu- tions of equation (*) are asymptotica1-periodic, or there exists at least one AMS solution that is asymptotica1-periodic.
• We establish necessary and sufficient conditions for the existence of asymptotically periodic solutions to delay evolution equations on the half-line These conditions are determined by the spectrum of the forcing terms and the spectrum of the evolution semigroups associated with the equations Our goal is to extend Massera theorem by introducing a new concept of solutions, namely, asymptotic solutions for evolution equations in Banach spaces, which have potential applications to partial differential equations as well as abstract functional differential equations.
Future works: Potential further extensions
The results obtained in this thesis can be extended and further improved in future works Some specific topics can be
• Establish spectral criteria for the asymptotic constancy of solutions to the im- plicit difference equationCx(n+ 1) = T x(n) +y(n) in a Banach space X, where the bounded sequence {y(n)} n is asymptotically constant, T is a boundedalinear operator acting in X, C is an injective operator in L(X).
• Derive conditions for the periodic linear evolution equations of the form x ′′ (t) A(t)x(t) +f(t) to have an asymptotic periodic solutions on R + , where A(t) is a family of (unbounded) linear operators in a Banach space X, strongly and periodically depending ont, f is an asymptotic periodic function.
• Extending the results of Chapter 4
[P1] Le Anh Minh and Nguyen Ngoc Vien, Circular spectrum and asymptotic periodic solutions to a class of non-densely defined evolution equations, Commun Korean Math Soc., vol 38, no 4 (2023), pp 1153–1162.
[P2] Vu Trong Luong, Nguyen Duc Huy, Nguyen Van Minh, and Nguyen Ngoc Vien, On asymptotic periodic solutions of fractional differential equations and applications, Proc Amer Math Soc., vol 151, no 12 (2023), pp 5299–5312.
[P3] Nguyen Ngoc Vien, Vu Trong Luong, and Le Van Hien, On asymptotic periodic solutions of delay evolution equations on the half line,Funkcialaj-Ekvacioj, vo 67
[P4] Nguyen Duc Huy, Vu Trong Luong, Anh Minh Le, and Nguyen Ngoc Vien, Asymp- totic periodic solutions of differential equations with infinite delay (submitted),2024.
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