On a Spectral Criterion for Almost Periodicity of Solutions of Periodic Evolution Equations

Toshiki Naito^{∗}, Nguyen Van Minh^{†}and Jong Son Shin^{‡}

Abstract

This paper is concerned with equations of the form: u′ = A(t)u + f(t) , where A(t) is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if u is a bounded uniformly continuous mild solution and P is the monodromy operator, then their spectra satisfy e^{isp_{AP} (u)} ⊂ σ(P) ∩ S^{1} , where S^{1} is the unit circle. This result is then applied to find almost periodic solutions to the above-mentioned equations. In particular, parabolic and functional differential equations are considered. Existence conditions for almost periodic and quasi-periodic solutions are discussed.

AMS (1991) Mathematics Subject Classification: 34C27, 34G10. Key words and Phrases: almost periodic solution, quasi-periodic solution, periodic equation, monodromy operator, spectrum of a function, spectral criterion.

Abbreviated title: A Spectral Criterion for Almost Periodicity

∗Department of Mathematics, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan, Email: naito@matha.e-one.uec.ac.jp

†Department of Mathematics, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182, Japan, Email: minh@im.uec.ac.jp

‡Department of Mathematics, Korea University,Kodaira, Tokyo 187, Japan.

1 Introduction and Preliminaries

Let us consider the following linear evolution equations

\frac{dx}{dt} = A(t)x + f(t), ∀t ∈ R  (1)

and

\frac{dx}{dt} = A(t)x, ∀t ∈ R,  (2)

where x ∈ X , X is a complex Banach space, A(t) is a (unbounded) linear operator acting on X for every fixed t ∈ R such that A(t) = A(t + 1) for all t ∈ R , f : R → X is an almost periodic function. In the paper we always assume that Eq.(1) is well-posed in the sense of [P], i.e. one can associate with it an evolution operator (U(t, s))_{t≥s} which satisfies the conditions below.

The asymptotic behavior of solutions to well-posed evolution equations has been of particular interest for the last two decades. We refer the reader to the books [Hal1], [He], [LZ], [Na], [Ne] and the surveys [Bat], [V3] and the references therein for more complete information on the subject. In this direction, one of the most interesting topics to be discussed in recent works is to find spectral conditions for almost periodicity of solutions (see e.g. [LZ], [HMN], [Pr] and recent papers [AB], [AS] , [Bas], [RV], [V1], [V2], [V3] , [NM] ).

A central result of this theory is as follows: let Eq.(1) be autonomous, i.e A(t) = A, ∀t . Then the spectrum sp_{AP} (u) of a bounded uniformly continuous mild solution to Eq.(1) satisfies the inclusion sp_{AP} (u) ⊂ σ(A)∩iR (see e.g. [LZ], [Bas], [AB]). It is interesting that starting from this inclusion one can not only find spectral conditions for almost periodicity for a bounded uniformly continuous solution of Eq.(1) but also establish a stability theory of solutions to Eq.(1) (see [Bas]).

As is known, an ordinary periodic differential equation with finite dimensional phase space can be transformed into an equation with constant coefficients (see e.g. [DK, chap. V]). Unfortunately, this is not the case for infinite dimensional equations. In fact, it is known [Hal2, Chap. 8], [He, Chap.7] that even for very simple infinite dimensional equations the above procedure fails, i.e. there are no Floquet transformations which reduce the periodic equations to equations with bounded coefficient operators although, for some particular cases one can show the existence of transformations which reduce the periodic equations under consideration to equations with (unbounded) coefficient operators (see e.g. [CLM], and also [Ku] for more information on Floquet theory for partial differential equations). Hence, in the infinite dimensional case, it is not relevant to treat periodic equations as autonomous ones especially for the almost periodicity of solutions (see e.g. [V1] for additional information).

The main purpose of this paper is to extend the above spectral inclusion to some classes of periodic evolution equations such as equations which monodromy operators either depend analytically on time or are compact. These classes of equations can be found among those having analytic solutions (see e.g. [KT1], [KT2], [M1], [M2] for basic results) and parabolic 1-periodic equations dx/dt + Ax = B(t)x + f(t) such that A has compact resolvent (see e.g. [He, p.196]). The spectral inclusions will be then used to find spectral conditions for the almost periodicity of a bounded uniformly continuous solution to the underlying equation. Our proofs of main results are mainly based on analyzing difference equations corresponding to Eq.(1). Our main results are Theorems 3, 4, 5 in Section 2. In Section 3 we find three classes of equations satisfying the conditions of Theorems 3, 4, 5 respectively. Our results in these classes are natural generalizations of well - known ones from ordinary differential equations.

Below we recall some notions and results of the spectral theory of uniformly continuous and bounded functions on the real line. Throughout the paper we shall denote by Z, R, C the set of integers, real and complex numbers, respectively. For every z ∈ C, ℜz will stand for the real part of z . L(X) will denote the space of bounded linear operators from X to itself . σ(A), ρ(A) will denote the spectrum and resolvent set of an operator A . BUC(R, X) will stand for the space of all uniformly continuous and bounded functions on the real line with sup-norm. C_{0}(R, X) will denote the subspace of BUC(R, X) consisting of functions v such that lim_{t→∞} v(t) = 0 . The subspace of BUC(R, X) consisting of all almost periodic functions in the sense of Bohr will be denoted by AP (X) which can be defined to be the smallest closed subspace of BUC(R, X) containing the functions of the form {e^{iη·}x, η ∈ R, x ∈ X} (see e.g. [LZ]). We denote by F the Fourier transform, i.e.

(Ff)(s) := ∫_{−∞}^{+∞} e^{−ist}f(t)dt  (3)

(s ∈ R, f ∈ L^{1}(R)) . Then the spectrum of u ∈ BUC(R, X) is defined to be the following set

sp(u) := {ξ ∈ R : ∀ϵ > 0∃f ∈ L^{1}(R), suppFf ⊂ (ξ −ϵ, ξ +ϵ), f ∗u ≠ 0} (4)

where

f ∗ u(s) := ∫_{−∞}^{+∞} f(s − t)u(t)dt.

It coincides with the set σ(u) consisting of ξ ∈ R such that the Fourrier-Carleman transform of u

û(λ) = \{ \begin{matrix}∫_{0}^{∞} e^{−λt}u(λ)dt, & (ℜλ > 0); \\ − ∫_{0}^{∞} e^{λt}u(−t)dt, & (ℜλ < 0) \end{matrix}  (5)

has a holomorphic extension to a neighborhood of iξ (see e.g. [Pr, Proposition 0.5, p.22]). We collect some main properties of the spectrum of a function, which we will need, in the following theorem for the reader’s convenience.

Theorem 1 Let f, g_{n} ∈ BUC(R, X) such that lim_{n→∞} ∥g_{n} − f∥ = 0 . Then

i) σ(f) is closed ,

ii) σ(f(· + h)) = σ(f) ,

iii) If α ∈ C\{0} σ(αf) = σ(f) ,

iv) If σ(g_{n}) ⊂ Λ for all n ∈ N then σ(f) ⊂ \overline{Λ} .

Proof For the proof we refer the reader to [Pr, Proposition 0.4, p. 20 and Theorem 0.8, p. 21] .

Similarly, one defines sp_{AP} (u) as follows.

Definition 1 Let u be a bounded uniformly continuous function. Then

sp_{AP} (u) := {λ ∈ R : ∀ϵ > 0 ∃ϕ ∈ L(R^{1})such that
suppFϕ ⊂ (λ − ϵ, λ + ϵ), ϕ ∗ f is not almost periodic}. (6)

By definition, similarly to sp(u) , sp_{AP} (u) has the following properties:

Proposition 1 Let u, v be bounded uniformly continuous functions on the real line. Then

i) sp_{AP} (u + v) ⊂ sp_{AP} (u) ∪ sp_{AP} (u)

ii) sp_{AP} (u(· + h)) = sp_{AP} (u).

2 Main Results

We begin this section by considering the 1-periodic strongly continuous process associated with the given 1-periodic evolution equation (1). We recall this concept in the following definition.

Definition 2 A family of bounded linear operators (U(t, s))_{t≥s}, (t, s ∈ R) from a Banach space X to itself is called 1-periodic strongly continuous evolutionary process if the following conditions are satisfied

i) U(t, t) = I for all t ∈ R,

ii) U(t, s)U(s, r) = U(t, r) for all t ≥ s ≥ r,

iii) The map (t, s) ↦ U(t, s)x is continuous for every fixed x ∈ X,

iv) U(t + 1, s + 1) = U(t, s) for all t ≥ s.

v) ∥U(t, s)∥ < Ne^{ω(t−s)} for some positive N, ω independent of t ≥ s .

Below if it does not cause any danger of confusion, for the sake of simplicity, we shall use the terminology periodic evolutionary process to designate 1periodic strongly continuous evolutionary process.

This notion comes naturally from the well-posed evolution equations (see e.g. [P]). And from now on we always assume that Eq.(1) is well-posed and there exisits a unique evolutionary process (U(t, s))_{t≥s}, (t, s ∈ R) associated with it having the properties listed in the above definition. Recall that a X valued function u defined on the real line is said to be a mild solution to Eq.(1) if it satisfies the following integral equation

u(t) = U(t, s)u(s) + ∫_{s}^{t} U(t, ξ)f(ξ)dξ, ∀t ≥ s, t, s ∈ R,  (7)

In connection with periodic evolutionary processes the following will be used as a basic property.

Lemma 1 Let (U(t, s)_{t≥s}) be a periodic evolutionary process and f be an almost periodic function. Then the function

t ↦ ∫_{t−h}^{t} U(t, ξ)f(ξ)dξ,

is almost periodic for every fixed h ≥ 0 .

Proof For the proof see [NM].

To deal with periodic evolution equations, the notion of monodromy operator will play the key role. And we shall try to state our conditions in terms of spectral properties of monodromy operators. Recall that for the evolutionary process (U(t, s))_{t≥s}, (t, s ∈ R) associated with the 1-periodic equation (1) the monodromy operators are defined to be P (t) := U(t, t − 1), t ∈ R . Note that the map t ↦ P (t) is also 1-periodic, and usually one takes P (0) as the monodromy operator because the above family of monodromy operators has ”similar properties” which are stated in the following

Lemma 2 ([He, Lemma 7.2.2 , p. 197]) P (t + 1) = P (t) for all t; characteristic multipliers are independent of time, i.e. the nonzero eigenvalues of P (t) coincide with those of P (s). Moreover, σ(P (t))\{0} is independent of t .

As usual, in this paper we will characterize conditions for almost periodicity of solutions in terms of spectral properties of the monodromy operators. Here, we will allow the spectrum of the monodromy operators to intersect the unit circle because otherwise the problem is trivial as is shown in the following

Theorem 2 Let (U(t, s)_{t≥s}) be an 1-periodic evolutionary process. Then the following assertions are equivalent:

i) σ(P (0)) ∩ S^{1} = ⊘ ,

ii) For every bounded function f Eq.(7) has a unique bounded mild solution,

iii) For every almost periodic function f Eq.(7) has a unique almost periodic mild solution.

Proof For the proof see [NM].

2.1 The case of analytic monodromy operators

In this subsection we often assume that Eq.(1) satisfy the following hyperthesis Definition 3 Eq.(1) is said to satisfy Condition H if the monodromy operators P (t) of the strongly continuous 1-periodic evolutionary process (U(t, s)_{t≥s}) associated with its homogeneous equation (2), as a function of t, is analytic in t .

Note that Condition H is satisfied automatically, if (U(t, s))_{t≥s} is a C_{0}- semigroup, and in general, it is not required that (U(t, s))_{t≥s} be norm continuous.

Since the map t ↦ P (t) is 1-periodic, Condition H will guarantee the following:

Lemma 3 Let Eq.(1) satisfy condition H. Then the Fourier series

∑_{k=−∞}^{+∞} P_{k}

associated with P (t) is absolutely convergent and

P (t) = ∑_{k=−∞}^{+∞} P_{k}e^{2πkit}  (8)

is convergent uniformly in t .

Proof For the proof see e.g. [Ka, p.32].

Another consequence of condition H which we will use below is concerned with a spectral estimate. To make it more clear we introduce the function space H(S, X) consisting of all X valued bounded holomorphic functions w on S := {z ∈ C : |ℜz| < 1} .

The following results are obvious from the theory of holomorphic functions and will be needed in the sequel.

Lemma 4 With the above notations the following assertions hold true:

i) H(S, X) is a Banach space with sup-norm, ii) Let μ ↦ V (μ) ∈ L(H(S, X)) be any holomorphic function defined on a

neighborhood U(z) of z ∈ S . Then for every given g ∈ H(S, X) , the function

λ ↦ [V (λ)g](λ)  (9)

is holomorphic.

Examples The function

h(λ) = ∫_{−1}^{0} e^{−λt}u(t)dt  (10)

for any continuous function u(·) is an element of H(S, X) . More generally,

g(λ) := ∑_{k=−∞}^{+∞} P_{k} ∫_{−1}^{0} e^{(λ−2πki)ξ}u(ξ)dξ,  (11)

where P_{k} is defined by (8), is also an element of H(S, X) .

In the paper we shall need the following spectral property of multiplication operators.

Lemma 5 Let R ∋ t ↦ Q(t) ∈ L(X) be norm continuous and periodic such that

σ(Q(t))\{0} = σ(Q(s))\{0} ∀t, s ∈ R.

Then

σ(Q̂)\{0} = σ(Q(t))\{0}, ∀t,

where Q̂ ∈ L(BUC(R, X)), [Q̂v](t) := Q(t)v(t), ∀t ∈ R, v ∈ BUC(R, X) .

Proof Consider the equation (λ−Q̂)u = v for given v ∈ BUC(R, X), λ ≠ 0 . It is equivalent to the equation (λ−Q(t))u(t) = v(t), t ∈ R . If λ ∈ ρ(Q̂)\{0} , and for a given y ∈ X , then take v(t) = y ∈ X, ∀t . Thus there is a unique u ∈ BUC(R, X) which satisfies the first equation, i.e.

λu(t) − Q(t)u(t) = y, ∀t.

Moreover,

∥u(t)∥ ≤ sup_{ξ} ∥u(ξ)∥ ≤ ∥R(λ, Q̂)v∥ ≤ ∥R(λ, Q̂)∥ sup_{ξ} ∥v(ξ)∥ = ∥R(λ, Q̂)∥∥y∥.

Thus the second equation has a unique solution for every fixed t ∈ R . This shows that λ ∈ ρ(Q(t))\{0}. Conversely, let λ ∈ ρ(Q(t))\{0} . Then, by assumption, for a given v ∈ BUC(R, X) and t ∈ R the second equation has a unique solution u(t) = R(λ, Q(t))v(t). It may be seen that the operator R(λ, Q(t)) is continuous and periodic in t. Hence, u ∈ BUC(R, X). Moreover,

sup_{t} ∥u(t)∥ ≤ sup_{t} ∥R(λ, Q(t))v(t)∥ ≤ sup_{t} ∥R(λ, Q(t))∥ sup_{t} ∥v(t)∥ ≤ r∥v∥,

where, by the above remark, sup_{t} ∥R(λ, Q(t))∥ < ∞ . This shows that λ ∈ ρ(Q̂)\{0} . □

In light of this lemma below we often identify the multiplication operator Q̂ in case Q(t) = constant with the operator Q itself if there is no danger of confusion.

For a fixed λ_{0} ∈ S we consider the space of bounded sequences Q(λ_{0}) := {g : {λ_{0} + 2πki, k ∈ Z} ↦ X, sup_{z} ∥g(z)∥ < ∞} . We denote by T_{Q}^{2π} the translation on Q(λ_{0}) defined as

[T_{Q}^{2π}g](z) = g(z + 2πi), ∀ z ∈ {λ_{0} + 2πki, k ∈ Z}.

We shall denote by T^{t} the translation group on H(S, X) , i.e.

T^{t}w(λ) := w(it + λ), ∀t ∈ R , λ ∈ S.

We shall denote by Q_{k} the operator acting on BUC(R, X) defined as Q_{k}v(s) := e^{2πiks}P_{k}v(s), s ∈ R .

Lemma 6 Let Eq.(1) satisfy condition H. Then the following estimates hold true:

σ( ∑_{k=−∞}^{+∞} P_{k}T^{2kπ}) ⊂ σ(P (0)) ⋃{0} ,  (12)

σ( ∑_{k=−∞}^{+∞} Q_{k}T^{2kπ}) ⊂ σ(P (0)) ⋃{0} ,  (13)

σ( ∑_{k=−∞}^{+∞} P_{k}T_{Q}^{2kπ}) ⊂ σ(P (0)) ⋃{0} .  (14)

Proof Consider the function

f(λ) := ∑_{k=−∞}^{+∞} P_{k}λ^{k},

where λ ∈ S^{1} . Since P (t) is analytic with respect to t ∈ R , f(λ) is holomorphic in a neighbourhood of S^{1} in the complex plane. Thus applying [S,Theorem 1] to function f(λ) and operator T^{2π} we obtain

σ(f(T^{2π})) ⊂ ⋃_{λ∈σ(T^{2π})} σ(f(λ)).

Since (T^{t})_{t∈R} is the translation group,

σ(T^{2π}) ⊂ S^{1} := {z ∈ C : ∥z∥ = 1}.

Thus, in view of Lemma 2,

σ(f(T^{2π})) = σ( ∑_{k=−∞}^{+∞} P_{k}T^{2πk})

⊂ ⋃_{λ∈S^{1}} σ( ∑_{k=−∞}^{+∞} P_{k}λ^{2πk})

⊂ ⋃_{t∈R} σ( ∑_{k=−∞}^{+∞} P_{k}e^{2πikt}) =
= ⋃_{t∈R} σ(P (t)) ⊂ σ(P (0)) ⋃{0}.

Hence (12) holds true. (13), (14) are proved in a similar manner. □

Now we consider first the homogeneous equation (2). Thus, if u is a mild solution to the homogeneous equation, then by (7)

u(t) = P (t)u(t − 1), ∀t .  (15)

We are going to prove the following Theorem 3 Let Eq.(1) satisfy condition H and u(·) is a bounded uniformly continuous mild solution of the homogeneous equation. Then the inclusion

e^{i sp(u)} ⊂ σ(P (0)) ∩ S^{1}  (16)

holds true.

Proof Let us first compute the Carleman transform of u(·) . For ℜλ > 0 ,

û(λ) = ∫_{0}^{∞} e^{−λt}u(t)dt = ∫_{0}^{∞} e^{−λt}P (t)u(t − 1)dt
= ∫_{0}^{∞} e^{−λt} ∑_{k=−∞}^{+∞} e^{2πikt}P_{k}u(t − 1)dt

= ∑_{k=−∞}^{+∞} P_{k} ∫_{0}^{∞} e^{−λt}e^{2πikt}u(t − 1)dt

= ∑_{k=−∞}^{+∞} P_{k}e^{−(λ−2πki)} ∫_{−1}^{∞} e^{−(λ−2πki)t}u(t)dt

= e^{−λ} ∑_{k=−∞}^{+∞} P_{k} ∫_{−1}^{∞} e^{−(λ−2πki)t}u(t)dt

= e^{−λ} ∑_{k=−∞}^{+∞} P_{k} ∫_{0}^{∞} e^{−(λ−2πki)t}u(t)dt

+e^{−λ} ∑_{k=−∞}^{+∞} P_{k} ∫_{−1}^{0} e^{−(λ−2πki)t}u(t)dt .

On the other hand, for ℜλ < 0,

û(λ) = − ∫_{0}^{∞} e^{λt} ∑_{k=−∞}^{+∞} e^{−2πikt}P_{k}u(−t − 1)dt

= − ∫_{0}^{∞} ∑_{k=−∞}^{+∞} P_{k}e^{(λ−2πik)t}u(−t − 1)dt

= − ∑_{k=−∞}^{+∞} P_{k} ∫_{1}^{∞} e^{(λ−2πik)(ξ−1)}u(−ξ)dξ

= −e^{−λ} ∑_{k=−∞}^{+∞} P_{k} ∫_{0}^{∞} e^{(λ−2πik)ξ}u(−ξ)dξ +

+e^{−λ} ∑_{k=−∞}^{+∞} P_{k} ∫_{0}^{1} e^{(λ−2πik)ξ}u(−ξ)dξ.

Finally, for ℜλ ≠ 0

û(λ) = e^{−λ} ∑_{k=−∞}^{+∞} P_{k}û(λ − 2πki) +

+e^{−λ} ∑_{k=−∞}^{+∞} P_{k} ∫_{−1}^{0} e^{−(λ−2πki)ξ}u(ξ)dξ.

Equivalently,

e^{λ}û(λ) = ∑_{k=−∞}^{+∞} P_{k}û(λ − 2πki) +

+ ∑_{k=−∞}^{+∞} P_{k} ∫_{−1}^{0} e^{−(λ−2πki)ξ}u(ξ)dξ.  (17)

Now , if ζ ∈ R such that e^{iζ} ∈ ρ(P (0)) ∩ S^{1} , then the function

w(λ) := [R(e^{λ}, f(T^{2π}))g](λ),

where

g(λ) := ∑_{k=−∞}^{+∞} P_{k} ∫_{−1}^{0} e^{−(λ−2πki)ξ}u(ξ)dξ,

in view of Examples and Lemma 4, is holomorphic on an open neighborhood U(iζ) ⊂ S such that e^{U(iζ)} ⊂ ρ(P (0)) . We show now that w(·) is a holomorphic extension of û at iζ , i.e.

w(λ) = û(λ), ∀λ ∈ U(iζ), ℜλ ≠ 0.

To this purpose, we fix a λ_{0} ∈ U(iζ) such that ℜ(λ_{0}) ≠ 0 . Since

e^{U(iζ)} ⊂ ρ(f(T_{Q}^{2π}))

and for every λ ∈ Q(λ_{0})

e^{λ_{0}} û(λ) = ∑_{k=−∞}^{+∞} P_{k}û(λ − 2πki) + g(λ)  (18)

we see that

û(·)|_{Q(λ_{0})} = R(λ_{0}, f(T_{Q}^{2π}))g|_{Q(λ_{0})}

and Eq.(17) is uniquely solvable in Q(λ_{0}). On the other hand, if we put

v(λ) = [R(λ_{0}, f(T^{2π}))g](λ), λ ∈ S,

then

e^{λ_{0}}v(λ) = ∑_{k=−∞}^{+∞} P_{k}w(λ − 2πki) + g(λ)

for all λ ∈ S and in particular for λ ∈ Q(λ_{0}) .

Since e^{λ_{0}} ∈ e^{U(iζ)}, from the unique solvability of (17), we have

û(λ) = v(λ) , ∀λ ∈ Q(λ_{0}) ,

in particular,

û(λ_{0}) = v(λ_{0}) = [R(λ_{0}, f(T^{2π}))g](λ_{0}) = w(λ_{0}).

Hence, û(λ_{0}) has a holomorphic extension at iζ where e^{iζ} ∈ ρ(P (0)) , i.e. ζ ∉ sp(u) . □

Now we consider the inhomogeneous equation

\frac{du}{dt} = A(t)u(t) + f(t)

with the same assumptions as for the homogeneous equation except that f is almost periodic with respect to t . We will use the so-called factorization technique developed in [AB]. We recall that the main idea of the method is as follows. Let us denote by (T_{t})_{t∈R} the translation group on BUC(R, X) , i.e.

(T_{t}u)(s) := u(t + s), ∀ξ ∈ R . Since (T_{t})_{t∈R} leaves AP (R, X) invariant there is an induced C_{0}-group on the quotient space Y := BUC(R, X)/AP (R, X) given by

\overline{T}_{t} = p(u) = p(T_{t}u)  (19)

where p : BUC(R, X) ↦ Y is the quotient mapping. Its generator is denoted by B̃ . For g ∈ Y we denote by M(g) := \overline{span}\{\overline{S}(t)g : t ∈ R\} the smallest closed subspace of Y which is invariant under (\overline{T}_{t})_{t∈R} . Using the notation B̃_{g} for the part of B̃ in M(g) we have

isp_{AP} (u) = σ(B̃_{p(u)}).  (20)

For more details see [AB].

For μ ∈ R, P ∈ L(X) let us denote by M_{μ} the multiplication operator in BUC(R, X) defined as

M_{μ}v(t) := e^{iμt}P v(t), ∀t ∈ R.

Then, since M_{μ} leaves the space AP (X) invariant it is clear that if w ∈ p(v), M_{μ}w ∈ p(M_{μ}v) . This remark allows us, by abuse of notations for the sake of simplicity, to write

p(e^{2πik·}P_{k}T^{2πk}R(λ, B)u) = e^{2πik·}P_{k}T^{2πk} p(R(λ, B)u).

Moreover, since the multiplication operator P̂ (recall that P̂v(t) := P (t)v(t), ∀t, v ) leaves AP (X) invariant it induces an operator ˜ P̂ on the quotient space Y := BUC(R, X)/AP (R, X). From the periodicity of the resolvent R(λ, P (s)) with respect to s , it may be seen that R(λ, P̂) leaves AP (X) invariant. By this one can show easily that σ( ˜P̂) = σ(P̂) . Thus

ρ( ˜P̂)\{0} = ρ(P (0))\{0} ⊂ ρ( ∑_{k=−∞}^{∞} Q_{k}T^{2πk}) = ρ( ∑_{k=−∞}^{∞} Q̃_{k}T^{2πk}),

where Q_{k} = e^{2πik·}P_{k} .

Theorem 4 Let Eq.(1) satisfy condition H, u be a bounded uniformly continuous mild solution to the inhomogeneous equation (1) with f almost periodic. Then

e^{i sp_{AP} (u)} ⊂ σ(P (0)) ∩ S^{1}.  (21)

Proof Put u_{s} := T_{s}u, s ∈ R . Then

u_{s}(t) = u(t + s) = U(t + s, t + s − 1)u(t + s − 1) + ∫_{t+s−1}^{t+s} U(t + s, ξ)f(ξ)dξ
= P (t + s)u_{s}(t − 1) + ψ_{s}(t).

Let us compute the Carleman transform of u_{s}(·) . For ℜλ > 0 ,

û_{s}(λ) = ∫_{0}^{∞} e^{−λt}u_{s}(t)dt = ∫_{0}^{∞} e^{−λt}P (t + s)u_{s}(t − 1)dt + ∫_{0}^{∞} e^{−λt}ψ_{s}(t)dt
= ∫_{0}^{∞} e^{−λt} ∑_{k=−∞}^{+∞} e^{2πik(t+s)}P_{k}u_{s}(t − 1)dt + ψ̂_{s}(λ)

= ∑_{k=−∞}^{+∞} e^{2πiks}P_{k} ∫_{0}^{∞} e^{−λt}e^{2πikt}u_{s}(t − 1)dt + ψ̂_{s}(λ)

= ∑_{k=−∞}^{+∞} e^{2πiks}P_{k} ∫_{0}^{∞} e^{−(λ−2πik)t}u_{s}(t − 1)dt + ψ̂_{s}(λ)

= ∑_{k=−∞}^{+∞} e^{2πiks}P_{k} ∫_{−1}^{∞} e^{−(λ−2πik)(t+1)}u_{s}(t)dt + ψ̂_{s}(λ)

= ∑_{k=−∞}^{+∞} e^{2πiks}e^{−(λ−2πki)}P_{k} ∫_{−1}^{∞} e^{−(λ−2πki)t}u_{s}(t)dt + ψ̂_{s}(λ)

= e^{−λ} ∑_{k=−∞}^{+∞} e^{2πiks}P_{k} ∫_{−1}^{∞} e^{−(λ−2πki)t}u_{s}(t)dt + ψ̂_{s}(λ)

= e^{−λ} ∑_{k=−∞}^{+∞} e^{2πiks}P_{k} ∫_{0}^{∞} e^{−(λ−2πki)t}u_{s}(t)dt

+e^{−λ} ∑_{k=−∞}^{+∞} e^{2πiks}P_{k} ∫_{−1}^{0} e^{−(λ−2πki)t}u_{s}(t)dt + ψ̂_{s}(λ) .

Hence, we have

û_{s}(λ) = e^{−λ} ∑_{k=−∞}^{∞} e^{2πiks}P_{k}û_{s}(λ − 2πki) + g_{s}(λ) + ψ̂_{s}(λ),

where

ψ_{s}(t) = ψ(t + s), ψ(t) = ∫_{t−1}^{t} U(t, ξ)f(ξ)dξ
g_{s}(λ) = e^{−λ} ∑_{k=−∞}^{∞} e^{2πiks}P_{k} ∫_{−1}^{0} e^{(λ−2πik)ξ}u_{s}(ξ)dξ.

This equation is also valid for ℜλ < 0.

As û_{s}(λ) = (R(λ, B)u)(s) , where B is the generator of the group (T_{t})_{t∈R} , from the above equality

(R(λ, B)u)(s) = e^{−λ} ∑_{k=−∞}^{∞} e^{2πiks}P_{k}(T^{2πk}R(λ, B)u)(s)+g_{s}(λ)+(R(λ, B)ψ)(s).

By Lemma 1, ψ(t) is almost periodic. Hence, since

R(λ, B)ψ = ∫_{0}^{∞} e^{−λt}T_{t}ψdt ∈ AP (X), ∀ ℜλ > 0 ,

and similarly for ℜλ < 0 , we have

p(R(λ, B)ψ) = 0.

Thus

p(R(λ, B)u) = e^{−λ} ∑_{k=−∞}^{∞} p(e^{2πik·}P_{k}T^{2πk}R(λ, B)u) + h(λ),

where

h(λ) = pg_{·}(λ) =
= e^{−λ} ∑_{k=−∞}^{∞} P_{k} ∫_{−1}^{0} e^{(λ−2πik)ξ}p(e^{2πik·}u_{·}(ξ))dξ

which is, obviously, a holomorphic function.

In general, from here we can follow the proof of Theorem 2 with some remarks concerning the procedure “passage to the quotient space”. We have

R(λ, B̃)p(u) = e^{−λ} ∑_{k=−∞}^{∞} T^{2πk}Q̃_{k}R(λ, B̃)p(u) + h(λ).

for all λ with ℜλ ≠ 0 .

Now let λ = iζ such that e^{iζ} ∈ ρ(P (0)) . We are going to show that ζ ∉ sp_{AP} (u) . To this purpose, it is equivalent to show that R(λ, B̃)p(u) has an holomorphic extension at iζ (see [AB, p.371]). This will be done in the same way as in the proof of Proposition 2. In fact,

w(λ) := R(e^{λ}, j(T^{2π}))h(λ),

where

j(λ) := ∑_{k=−∞}^{∞} Q̃_{k}λ^{k}.

is an analytic extension at λ = iζ of R(λ, B̃)p(u) . This means that λ ∉ sp_{AP} (u). □

2.2 The case of compact monodromy operators

In case where P (t) is compact, we can drop condition H, i.e. analyticity of the monodromy operator P (t) with respect to t , to get the spectral criteria (16) and (21). Indeed, it can be done as follows. Let us denote by

σ_{1} = {λ ∈ σ(P (0)) : |λ| ≥ 1}.

Thus, σ_{1} is finite. As is known that there exists a partial Floquet representation (see e.g. [He, p.198-200]), by a periodic transform we can reduce the equation

\frac{du}{dt} = A(t)u(t) + f(t)

to the system of equations of the form

\{ \begin{matrix}dy/dt = Cy + F (t)E_{1}(t)f(t) \\ dx_{2}/dt = A_{2}(t)x_{2} + E_{2}(t)f(t) \end{matrix}  (22)

where F (t), E_{1}(t), E_{2}(t) are 1-periodic. Moreover, letting P_{2}(t) denote the monodromy operators associated with the second component equation of (22), we have

σ(P_{2}(0)) = {λ ∈ σ(P (0)) : |λ| < 1}.

This, by Theorem 2, as E_{2}(t)f(t) is almost periodic implies that there exists a unique mild almost periodic solution x_{2} . Thus we are in a position to prove the following.

Theorem 5 Let P (0) be compact. Then for every bounded uniformly continuous mild solution to the homogeneous equation (inhomogeneous equation, respectively) the inclusion

e^{i sp(u)} ⊂ σ(P (0)) ∩ S^{1}
(e^{i sp_{AP} (u)} ⊂ σ(P (0)) ∩ S^{1} , respectively)

holds true.

Proof We prove only the assertion for the inhomogeneouos equations. The assertion for the homogeneous equations can be carried out in a similar manner. First we recall the property that

sp_{AP} (f + g) = sp_{AP} (f) ∪ sp_{AP} (g)

for f, g bounded uniformly continuous . Now suppose that (y(t), x_{2}(t)) is the bounded uniformly continuous mild solution of the decoupled equation (22) which by a 1-periodic transformation corresponds to u(t) . Obviously,

sp_{AP} (y(·) + x_{2}(·)) ⊂ sp_{AP} (y(·)).

On the other hand from the first component equation of (22) (see e.g. [Bas], [AB])

isp_{AP} (y(·)) ⊂ iR ∩ σ(C).

Thus

e^{i sp_{AP} (y(·))} ⊂ σ(P (0)) ∩ S^{1} .  (23)

As u(·) = Q(·)(y(·) + x_{2}(·)) for some 1-periodic transformation Q(t) the following estimate holds

sp_{AP} (u) ⊂ \overline{\{λ + 2πk, λ ∈ sp_{AP} (y), k ∈ Z\}}

(see e.g. [V1, Lemma 4.3]). Hence, by (24)

e^{i sp_{AP} (u)} ⊂ e^{i \overline{\{λ+2πk,λ∈sp_{AP} (y),k∈Z\}}}
⊂ \overline{e^{i sp_{AP} (y)}} ⊂ σ(P (0)) ∩ S^{1}.

□

2.3 Applications to Almost Periodicity of Bounded Solutions

Below we shall derive from the spectral inclusions proved in Theorems 3, 4, 5 some consequences by a standard argument. To this purpose we recall now the concept of total ergodicity. A function u ∈ BUC(R, X) is said to be totally ergodic if

M_{η}u := lim_{τ→∞} \frac{1}{2τ} ∫_{−τ}^{τ} e^{iηs}u(s + ·)ds

exists in BUC(R, X) for all η ∈ R . Now we are in a position to state our main results on the spectral criteria for almost periodicity of solutions to Eq.(1).

Corollary 1 Let f be almost periodic. Moreover, let Eq.(1)either satisfy condition H and σ(P (0))∩S^{1} be countable or its monodromy operators P (t) be compact for some t_{0}. Then every bounded uniformly continuous mild solution to Eq.(1) is almost periodic provided one of the following conditions holds

i) X does not contain any subspace isomorphic to c_{0},

ii) Range of u(·) is relatively compact,

iii) u is totally ergodic.

Proof The corollary is an immediate consequence of Theorems 3, 4 and [LZ,Theorem 4, p.92] and [RV, Section 3].

3 Examples

In this section we will give examples in which the conditions of Theorems 3, 4, 5 are satisfied. Moreover, we will discuss the relations of our results with other well known ones. In particular, our assertions on the existence of quasi - periodic solutions for parabolic and functional differential equations seem to be new. Note that the conditions of Theorems 3, 4 are automatically fulfilled in the trivial case of nonautonomousness, i.e. the homogeneous equation of Eq.(1) is autonomous and generates a C_{0}-semigroup. Nontrivial examples of nonautonomous evolution equations will be mainly found in the class of evolution equations having holomorphic solutions. For information on this topics we refer the reader to the following basic results [KT1], [KT2], [M1] [M2].

Let Δ be a complex convex open neighborhood of the closed real interval [0, 1] and {A(t), t ∈ Δ} be a family of operators in X such that

(a) For each t ∈ Δ , A(t) is densely defined closed linear operator in X;

(b) For any compact set K ⊂ Δ and any ϵ > 0 there exist constants λ_{0} and M such that

(b-1) the sector

Σ_{ϵ} := Σ(λ_{0}; −\frac{π}{2} − θ + ϵ, \frac{π}{2} + θ − ϵ)

is contained in the resolvent set ρ(A(t)) of A(t) for t in some neighborhood of K ,

(b-2) ∥(z − A(t))^{−1}∥ ≤ M/|z| for z ∈ Σ_{ϵ}, t ∈ K ,

(b-3) for each fixed z ∈ Σ_{ϵ}, (z − A(t))^{−1} is holomorphic for t in some neighborhood of K .

We will use the following notations in the sequel:

Ω_{θ} := {(s, t); s, t ∈ Δ, |arg(t − s)| < θ}, for 0 < θ < π/2.  (24)

Proposition 2 Let {A(t), t ∈ R} be a 1-periodic family of operators in X, i.e. A(t + 1) = A(t), ∀t such that on the interval [0, 1] it has an extension to the neighborhood Δ and conditions (a), (b) are satisfied. Then there exists an 1-periodic evolutionary process (U(t, s))_{t≥s} associated with the equation

\frac{dx}{dt} = A(t)x, ∀t ∈ R,

which has the monodromy operators P (t) := U(t, t − 1) analytic in t ∈ R .

Proof The proposition is an immediate consequence of [KT1], [KT2]. In fact, since U(t, s) is analytic in (s, t), s ≠ t the monodromy operator P (t) := U(t, t − 1) is analytic in t ∈ R .□

Let us consider the equation

\frac{dx}{dt} + Ax = B(t)x + f(t)  (25)

where A(·) is sectorial in X, 0 ≤ α < 1 and t ↦ B(t) : R ↦ L(X^{α}, X) is Hölder continuous. Moreover, suppose that A has compact resolvent. Then as shown in [He, Chap. 7], the homogeneous equation corresponding to Eq.(25) has an evolution operator T (t, s) which is compact as an operator on X^{β} for each β < 1, t > s . Observe that in this case if dimX = ∞ there is no Floquet representation, i.e. there does not exist a bounded operator B such that T (1, 0) = e^{B} . In fact, using the spectral mapping theorem one sees easily that for this representation to exist it is necessary that 0 ∈ ρ(T (1, 0)) . As is known, in this case there exists partial Floquet representation (see e.g. [He, Chap.7]). Hence we have the following:

Corollary 2 Let Eq.(25) satisfy the above mentioned conditions. Then every bounded uniformly continuous solution is almost periodic provided one of the following conditions holds

i) X does not contain any subspace isomorphic to c_{0},

ii) Range of u(·) is relatively compact,

iii) u is totally ergodic.

Proof This corollary is an immediate consequence of Corollary 1.□

Remarks It is shown in [DM, Theorem 6.11, p.81] that the inhomogeneous periodic equation (25) with f 1-periodic has a 1-periodic solution if and only if it has a bounded solution. This is a generalization of the well-known result from ordinary differential equations. However, the assertion does not carry over to the mentioned bounded solution itself. In view of Corollary 2, we claim that the bounded solution itself is almost periodic. Corollary 2 is in fact a generalization of a well - known result for ordinary differential equations [F, Corollary 6.5, p.101].

To illustrate another usefulness of Theorem 5 we consider now the existence of quasi-periodic solutions to the homogeneous equation of Eq.(25). It turns out that we do not need any further geometrical condition on the Banach space X to get the quasi-periodicity of a bounded solution.

Corollary 3 Let X be any complex Banach space and u(·) be a bounded solution to the homogeneous equation corresponding to Eq.(25). Then there are real numbers λ_{1} < ⋯ < λ_{n} in [0, 2π) and 1-periodic continuous functions u_{1}, u_{2}, ⋯ u_{n} such that

x(t) = ∑_{k=1}^{n} e^{iλ_{k}t}u_{k}(t), t ∈ R.  (26)

Proof In this case, the boundedness of solution x(·) implies the uniform continuity (see [He, Theorem 7.1.3 iii), p.190]). From Theorem 5 it follows that e^{i sp(x(·))} ∩ S^{1} consists of only finitely many points. Hence,

sp(x(·)) ⊂ ∪_{k=1}^{n}{λ_{k} + 2πp, p ∈ Z}.

This yields that the spectrum sp(x(·)) is discrete. And then by [AS, Theorem 3.5], the almost periodicity of x follows. By the Approximation Theorem (see [LZ, Chap.2]) we see that there is a sequence of functions of the form (26)

u^{(m)}(t) = ∑_{k=1}^{n} e^{iλ_{k}t}u_{k}^{(m)}(t), t ∈ R  (27)

which converges uniformly in t to x . The proof is complete if the following lemma holds true.

Lemma 7 With above notations if u^{(m)}(t) converges to 0 uniformly in t , then u_{k}^{(m)}(t) converges uniformly to 0 in t as well.

Proof of Lemma 7 We will prove the lemma by induction. The lemma holds true for n = 1 . Now suppose that the lemma holds true for n = p − 1 . We shall show that the lemma holds true for n = p as well. In fact, without loss of generality we can assume that λ_{1} = 0 , otherwise we can multiply both sides of (27) by e^{−iλ_{1}t} . Now set

w^{(m)}(t) := u^{(m)}(t) − u^{(m)}(t − 1)
= [u^{(m)}(t) + ∑_{k=2}^{p} e^{iλ_{k}t}u_{k}^{(m)}(t)] − [u^{(m)}(t − 1) + ∑_{k=2}^{p} e^{iλ_{k}(t−1)}u_{k}^{(m)}(t − 1)] .

Since u_{k}^{(m)}(t) is 1-periodic, we have

w^{(m)}(t) = ∑_{k=2}^{p} e^{iλ_{k}t}(1 − e^{−iλ_{k}})u_{k}^{(m)}(t)]

= ∑_{k=2}^{p} e^{iλ_{k}t}w_{k}^{(m)}(t),  (28)

where w_{k}^{(m)}(t) := (1 − e^{−iλ_{k}})u_{k}^{(m)}(t)] is 1-periodic. Obviously, since u^{(m)}(t) converges to 0 uniformly in t , w^{(m)}(t) converges to 0 uniformly in t . By induction assumption, w_{k}^{(m)}(t) converges to 0 uniformly in t . As λ_{k} ∈ [0, 2π) , this yields that u_{k}^{(m)}(t) converges to 0 uniformly in t .□

We now state a similar assertion to Corollary 3 for functional differential equations which proof is the same as that of Corollary 3. To this end, let us consider linear homogeneous functional differential equations of the form

\frac{dx}{dt} = L(t)x_{t}, ∀t ∈ R, x ∈ R^{n},  (29)

where L(t) is a bounded linear operator from C[−r, 0] to R^{n} for every fixed t and depends continuously and 1-periodically on the time t , x_{t}(θ) := x(t + θ), ∀θ ∈ [−r, 0], t ∈ R and r > 0 is given. For more details on this subject see e.g. [Hal2], [HL].

Corollary 4 Let x(·) be a bounded solution to Eq.(29). Then x(·) is quasiperiodic. More precisely, there are real numbers λ_{1}, ⋯, λ_{k} and periodic functions u_{1}(·), ⋯, u_{k}(·) with period τ > r such that

x(t) = ∑_{j=1}^{k} e^{iλ_{j}t}u_{j}(t), ∀t ∈ R.

Proof Consider the solution operator T (t, s), t ≥ s; t, s ∈ R (see [Hal2] for more details). As is known, for sufficiently large t−s (for instance t−s > r ) the operator T (t, s) is compact. Thus we are in a position to apply Theorem 5 in a similar manner as in Corollary 3 to assert that there are λ_{1}, ⋯, λ_{k} and C[−r, 0]-valued periodic functions v_{1}(·), ⋯, v_{k}(·) with period τ > r such that

x_{t} = ∑_{j=1}^{k} e^{iλ_{j}t}v_{j}(t), ∀t ∈ R.

Hence,

x(t) = x_{t}(0) = ∑_{j=1}^{k} e^{iλ_{j}t}v_{j}(t)(0),

= ∑_{j=1}^{k} e^{iλ_{j}t}u_{j}(t), ∀t ∈ R,

where u_{j}(t) := v_{k}(t)(0), j = 1, ⋯, k .□

Acknowledgement. The paper was written while N.V.M was the fellow of the Japan Society for the Promotion of Science. The support of the Society is gratefully acknowledged. The authors wish to thank the referee for his remarks and suggestions to improve the presentation of the paper.

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