SPECTRAL ANALYSIS OF AN OPERATOR ASSOCIATED

WITH LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS ^{∗}

SATORU MURAKAMI ^{†} and TOSHIKI NAITO, NGUYEN VAN MINH ^{‡}

Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan E-mail: murakami@xmath.ous.ac.jp

and

Department of Mathematics, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan E-mail: naito@e-one.uec.ac.jp ; minh@matha.e-one.uec.ac.jp

1. INTRODUCTION

In this paper, we treat the (autonomous) linear functional differential equation

ẋ(t) = L(x_{t}),  (1)

where L is a bounded linear operator mapping a uniform fading memory space B = B((−∞, 0]; C^{n}) into C^{n}, and study the admissibility of Eq. (1) for a translation invariant function space M which consists of functions whose spectrum is contained in a closed set Λ in R. In case of Λ = R or Λ = {2kπ/ω : k ∈ Z}, the problem for the admissibility is reduced to the one for the existence of bounded solutions, almost periodic solutions or ω-periodic solutions of the equation

ẋ(t) = L(x_{t}) + f(t)

with the forced function f(t) which is bounded, almost periodic or ω-periodic, and there are many results on the problem (e.g., [1], [3], [5], [7, 8] ). In this paper, we study the problem for a general set Λ. Roughly speaking, we solve the problem by determining the spectrum of an operator D_{M} − L_{M} which is associated with Eq. (1).

∗This paper is in final form and no version of it will be submitted for publication elsewhere.

†Partly supported in part by Grant-in-Aid for Scientific Research (C), No.11640191, Japanese Ministry of Education, Science, Sports and Culture.

‡On leave from the Department of Mathematics, University of Hanoi, 90 Nguyen Trai, Hanoi, Vietnam.

2. UNIFORM FADING MEMORY SPACES AND SOME PRELIMINARIES

In this section we explain uniform fading memory spaces which are employed throughout this paper, and give some preliminary results.

Let C^{n} be the n-dimensional complex Euclidean space with norm | · |. For any interval J ⊂ R := (−∞, ∞), we denote by C(J; C^{n}) the space of all continuous functions mapping J into C^{n}. Moreover, we denote by BC(J; C^{n}) the subspace of C(J; C^{n}) which consists of all bounded functions. Clearly BC(J; C^{n}) is a Banach space with the norm | · |_{BC(J;C^{n})} defined by |ϕ|_{BC(J;C^{n})} = sup{|ϕ(t)| : t ∈ J}. If J = R^{−} := (−∞, 0], then we simply write BC(J; C^{n}) and | · |_{BC(J;C^{n})} as BC and | · |_{BC}, respectively. For any function x : (−∞, a) ↦ C^{n} and t < a, we define a function x_{t} : R^{−} ↦ C^{n} by x_{t}(s) = x(t + s) for s ∈ R^{−}. Let B = B(R^{−}; C^{n}) be a complex linear space of functions mapping R^{−} into C^{n} with a complete seminorm | · |_{B}. The space B is assumed to have the following properties:

(A1) There exist a positive constant N and locally bounded functions K(·) and M(·) on R^{+} := [0, ∞) with the property that if x : (−∞, a) ↦ C^{n} is continuous on [σ, a) with x_{σ} ∈ B for some σ < a, then for all t ∈ [σ, a),

(i) x_{t} ∈ B,

(ii) x_{t} is continuous in t (w.r.t. | · |_{B}),

(iii) N|x(t)| ≤ |x_{t}|_{B} ≤ K(t − σ) sup_{σ≤s≤t} |x(s)| + M(t − σ)|x_{σ}|_{B}.

(A2) If {ϕ^{k}}, ϕ^{k} ∈ B, converges to ϕ uniformly on any compact set in R^{−} and if {ϕ^{k}} is a Cauchy sequence in B, then ϕ ∈ B and ϕ^{k} → ϕ in B.

The space B is called a uniform fading memory space, if it satisfies (A1) and (A2) with K(·) ≡ K (a constant) and M(β) → 0 as β → ∞ in (A1). A typical one for uniform fading memory spaces is given by the space

C_{γ} := C_{γ}(C^{n}) = {ϕ ∈ C(R^{−}; C^{n}) : lim_{θ→−∞} |ϕ(θ)|e^{γθ} = 0}

which is equipped with norm |ϕ|_{C_{γ}} = sup_{θ≤0} |ϕ(θ)|e^{γθ}, where γ is a positive constant.

It is known [2, Lemma 3.2] that if B is a uniform fading memory space, then BC ⊂ B and

|ϕ|_{B} ≤ K|ϕ|_{BC},  ϕ ∈ BC.  (2)

For other properties of uniform fading memory spaces, we refer the reader to the book [4].

We denote by BUC(R; C^{n}) the space of all bounded and uniformly continuous functions mapping R into C^{n}. BUC(R; C^{n}) is a Banach space with the supremum norm which will be denoted by || · ||. The spectrum of a given function f ∈ BUC(R; C^{n}) is defined as the set

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

where

u ∗ f(t) := ∫_{−∞}^{+∞} u(t − s)f(s)ds ; ũ(s) := ∫_{−∞}^{∞} e^{−ist}u(t)dt.

We collect some main properties of the spectrum of a function, which we will need in the sequel, for the reader’s convenience. For the proof we refer the reader to [6], [10-11].

Proposition 1 The following statements hold true:

(i) sp(e^{iλ·}) = {λ} for λ ∈ R.

(ii) sp(e^{iλ·}f) = sp(f) + λ for λ ∈ R.

(iii) sp(αf + βg) ⊂ sp(f) ∪ sp(g) for α, β ∈ C.

(iv) sp(f) is closed. Moreover, sp(f) is not empty if f ≢ 0.

(v) sp(f(· + τ)) = sp(f) for τ ∈ R.

(vi) If f, g^{k} ∈ BUC(R; C^{n}) with sp(g^{k}) ⊂ Λ for all n ∈ N, and if

lim_{k→∞} ||g^{k} − f|| = 0, then sp(f) ⊂ \overline{Λ}. (vii) sp(ψ ∗ f) ⊂ sp(f) ∩ supp ψ̃ for all ψ ∈ L^{1}(R).

In the following we always assume that B = B(R^{−}; C^{n}) is a uniform fading memory space. For any bounded linear functional L : B ↦ C^{n} we define an operator L by

(Lf)(t) = L(f_{t}), t ∈ R,

for f ∈ BUC(R; C^{n}). It follows from (2) that

|(Lf)(t) − (Lf)(s)| ≤ ||L|||f_{t} − f_{s}|_{B}
≤ K||L|||f_{t} − f_{s}|_{BC},

and hence Lf ∈ BUC(R; C^{n}). Consequently, L is a bounded linear operator on BUC(R; C^{n}).

For any closed set Λ ⊂ R, we set

Λ(C^{n}) = {f ∈ BUC(R; C^{n}) : sp(f) ⊂ Λ}.

From (iii)–(vi) of Proposition 1, we can see that Λ(C^{n}) is a translation-invariant closed subspace of BUC(R; C^{n}).

Proposition 2 Let Λ be a closed set in R. Then the space Λ(C^{n}) is invariant under the operator L.

Proof Let f ∈ BUC(R; C^{n}). It suffices to establish that sp(Lf) ⊂ sp(f). Let ξ ∉ sp(f). There is an ϵ > 0 with the property that u ∗ f = 0 for any u ∈ L^{1}(R) such that supp ũ ⊂ (ξ − ϵ, ξ + ϵ). Let v be any element in L^{1}(R) such that supp ṽ ⊂ (ξ − ϵ, ξ + ϵ). Since

∫_{−∞}^{∞} v(t − s)f_{s}(θ)ds = ∫_{−∞}^{∞} v(t − s)f(s + θ)ds
= (v ∗ f)(t + θ) = 0

for θ ≤ 0, (A2) yields that ∫_{−∞}^{∞} v(t − s)f_{s}ds = 0 in B. Hence

(v ∗ Lf)(t) = ∫_{−∞}^{∞} v(t − s)L(f_{s})ds
= L(∫_{−∞}^{∞} v(t − s)f_{s}ds)
= 0,

which shows that ξ ∉ sp(Lf).

3. SPECTRUM OF AN OPERATOR ASSOCIATED WITH FUNCTIONAL DIFFERENTIAL EQUATIONS

We consider the linear functional differential equation

ẋ(t) = L(x_{t}),  (1)

where L is a bounded linear operator mapping a uniform fading memory space B = B(R^{−}; C^{n}) into C^{n}. A translation-invariant space M ⊂ BUC(R; C^{n}) is said to be admissible with respect to Eq. (1), if for any f ∈ M, the equation

ẋ(t) = L(x_{t}) + f(t)

possesses a unique solution which belongs to M. Let Λ be a closed set in R. An aim in this section is to obtain a condition under which the subspace Λ(C^{n}) introduced in the previous section is admissible with respect to Eq. (1). To do this, we first introduce the operators D_{Λ} and L_{Λ} associated with Eq. (1):

D_{Λ} := (d/dt)|_{D(D_{Λ})}
L_{Λ} := L|_{Λ(C^{n})},

where

D(D_{Λ}) = {u ∈ Λ(C^{n}) : du/dt ∈ Λ(C^{n})}.

Clearly, the admissibility of Λ(C^{n}) with respect to Eq. (1) is equivalent to the invertibility of the operator D_{Λ}−L_{Λ} in Λ(C^{n}). In fact, we will determine the spectrum σ(D_{Λ}−L_{Λ}) of D_{Λ} −L_{Λ} in Theorem 1, and as a consequence of Theorem 1, we will obtain a condition for Λ(C^{n}) to be admissible with respect to Eq. (1).

Before stating Theorem 1, we prepare some notation. For any λ ∈ Λ, we define a function ω(λ) : R^{−} ↦ C := C^{1} by

[ω(λ)](θ) = e^{iλθ}, θ ∈ R^{−}.

Because B is a uniform fading memory space, it follows that ω(λ)a ∈ B for any (column) vector a ∈ C^{n}. In particular, we get ω(λ)e_{i} ∈ B for i = 1, ⋯, n, where e_{i} is the element in C^{n} whose i-th component is 1 and the other components are 0. We denote by I the n × n unit matrix, and define an n × n matrix by

(L(ω(λ)e_{1}), ⋯, L(ω(λ)e_{n})) =: L(ω(λ)I).

Theorem 1 Let Λ be a closed subset of R, and let D_{Λ} and L_{Λ} be the ones introduced above. Then the following relation holds:

σ(D_{Λ} − L_{Λ}) = {μ ∈ C : det[(iλ − μ)I − L(ω(λ)I)] = 0 for some λ ∈ Λ} (=: ˜ (iΛ)).

In order to establish the theorem, we need the following result for ordinary differential equations:

Lemma 1 Let Q be an n × n matrix such that σ(Q) ⊂ iR\iΛ. Then for any f ∈ Λ(C^{n}) there is a unique solution x_{f} in Λ(C^{n}) of the system of ordinary differential equations

ẋ(t) = Qx(t) + f(t).

Moreover, the map f ∈ Λ(C^{n}) ↦ x_{f} ∈ Λ(C^{n}) is continuous.

Proof. Without loss of generality, we may assume that Q is a matrix of Jordan canonical form

Q = ( \begin{matrix}iλ_{1} & δ_{1} & & 0 \\ & iλ_{2} & δ_{2} \\ & … & … \\ & & iλ_{n−1} & δ_{n−1} \\ 0 & & & iλ_{n} \end{matrix} ) ,

where {λ_{1}, ⋯, λ_{n}} ∩ Λ = ∅, and δ_{k} = 0 or 1 for k = 1, ⋯, n − 1. The equation for x_{n} is written as

ẋ_{n}(t) = iλ_{n}x_{n}(t) + f_{n}(t),

where f_{n} ∈ Λ(C). By setting z(t) = x_{n}(t)e^{−iλ_{n}t}, we get

ż(t) = e^{−iλ_{n}t}f_{n}(t) =: g(t).

It follows that 0 ∉ sp(g) because of sp(g) ⊂ Λ − {λ_{n}}. Then, by virtue of [6, Chapter 6, Theorem 3 and its proof] there exists an integrable function ϕ such that z = ϕ∗g satisfies ż(t) = g(t) (and hence x_{n}(t) = z(t)e^{iλ_{n}t} is a solution of the above equation). From (vii) of Proposition 1 it follows that sp(z) ⊂ sp(g), and hence sp(x_{n}) ⊂ sp(g) + {λ_{n}} ⊂ Λ. If y(t) is another solution in Λ(C) of the above equation, then x_{n}(t) − y(t) ≡ ae^{iλ_{n}t} for some a, and hence sp(x_{n} − y) ⊂ {λ_{n}}. Since λ_{n} ∉ Λ, we must have x_{n} − y ≡ 0. Thus the above equation possesses a unique solution x_{n} in Λ(C), which is represented as the convolution of f_{n} and an integrable function. For this x_{n}, let us consider the equation for x_{n−1}

ẋ_{n−1}(t) = iλ_{n−1}x_{n−1}(t) + δ_{n−1}x_{n}(t) + f_{n−1}(t).

Since the term δ_{n−1}x_{n}(t)+f_{n−1}(t) belongs to the space Λ(C), the above argument shows that the equation for x_{n−1} possesses a unique solution in Λ(C), too. In fact, the solution is represented as

ψ_{n−1} ∗ f_{n−1} + ψ_{n} ∗ f_{n}

for some integrable functions ψ_{n−1} and ψ_{n}. Continue the procedure to the equations for x_{n−2}, ⋯, x_{2} and x_{1}, subsequently. Then we conclude that the system possesses a unique solution in Λ(C^{n}), which is represented as the convolution Y ∗ f for some n × n matrix-valued integrable function Y.

Proof of Theorem 1. In case where | · |_{B} is a complete semi-norm of B, one can prove the theorem by considering the quotient space B/| · |_{B}. In order to avoid some cumbersome notation, we shall establish the theorem in case where | · |_{B} is a norm and consequently B is a Banach space.

Assume that μ ∈ C satisfies det[(iλ − μ)I − L(ω(λ)I] = 0 for some λ ∈ Λ. Then there is a nonzero a ∈ C^{n} such that iλa − L(ω(λ)a) = μa. Set ϕ(t) = e^{iλt}a, t ∈ R. Then ϕ ∈ D(D_{Λ}), and

[(D_{Λ} − L_{Λ})ϕ](t) = ϕ̇(t) − L(ϕ_{t})
= iλe^{iλt}a − L(e^{iλt}ω(λ)a)
= e^{iλt}(iλa − L(ω(λ)a))
= μϕ(t),

or (D_{Λ} − L_{Λ})ϕ = μϕ. Thus μ ∈ P_{σ}(D_{Λ} − L_{Λ}). Hence ˜ (iΛ) ⊂ σ(D_{Λ} − L_{Λ}).

Next we shall show that ˜ (iΛ) ⊃ σ(D_{Λ} − L_{Λ}). To do this, it is sufficient to prove the claim:

Assertion If det[(iλ − k)I − L(ω(λ)I)] ≠ 0 (∀λ ∈ Λ), then k ∈ ρ(D_{Λ} − L_{Λ}).

To establish the claim, we will show that for each f ∈ Λ(C^{n}), the equation

ẋ(t) = L(x_{t}) + kx(t) + f(t), t ∈ R  (3)

possesses a unique solution x_{f} ∈ Λ(C^{n}) and that the map f ∈ Λ(C^{n}) ↦ x_{f} ∈ Λ(C^{n}) is continuous. We first treat the homogeneous functional differential equation

ẋ(t) = L(x_{t}) + kx(t),  (4)

and consider the solution semigroup T (t) : B ↦ B, t ≥ 0, of Eq. (4) which is defined as

T (t)ϕ = x_{t}(ϕ), ϕ ∈ B,

where x(·, ϕ) denotes the solution of (4) through (0, ϕ) and x_{t} is an element in B defined as x_{t}(θ) = x(t+θ), θ ≤ 0. Let G be the infinitesimal generator of the solution semigroup T (t). We assert that

iR ∩ σ(G) = {iλ ∈ iR : det[(iλ − k)I − L(ω(λ)I)] = 0}.

Before proving the assertion, we first remark that the constant β introduced in [4, p. 127] satisfies β < 0 because B is a uniform fading memory space. In particular, if λ is a real number, then Re(iλ) = 0 > β, and hence ω(λ)b ∈ B for any b ∈ C^{n} by [4, p. 137, Th. 2.4].

Now, let iλ ∈ iR ∩ σ(G). Since iλ is a normal point of G by [4, p. 141, Th. 2.7], we must have that iλ ∈ P_{σ}(G). Then [4, p. 134, Th. 2.1] implies that there exists a nonzero b ∈ C^{n} such that iλb − L(ω(λ)b) − kb = 0, which shows that iλ belongs to the set of the right hand side in the assertion. Conversely, assume that iλ is an element of the set of the right hand side in the assertion. Then there is a nonzero a ∈ C^{n} such that iλa = ka + L(ω(λ)a). Set x(t) = e^{iλt}a, t ∈ R. Then x_{t} = e^{iλt}ω(λ)a and

ẋ(t) = iλe^{iλt}a = e^{iλt}(ka + L(ω(λ)a))
= kx(t) + L(x_{t}).

Thus x(t) is a solution of Eq. (4) satisfying x_{0} = ω(λ)a, and it follows that T (t)ω(λ)a = T (t)x_{0} = x_{t} = e^{iλt}ω(λ)a for t ≥ 0, which implies that ω(λ)a ∈ D(G) and G(ω(λ)a) = iλω(λ)a. Thus iλ ∈ σ(G) ∩ iR, and the assertion is proved.

Now consider the sets Σ_{C} := {λ ∈ σ(G) : Reλ = 0} and Σ_{U} := {λ ∈ σ(G) : Reλ > 0}. Then the set Σ = Σ_{C} ∪ Σ_{U} is a finite set [4, p. 144, Prop. 3.2]. Corresponding to the set Σ, we get the decomposition of the space B:

B = S ⊕ C ⊕ U,

where S, C, U are invariant under T (t), the restriction T (t)|_{U} can be extendable as a group, and there exist positive constants c_{1} and α such that

∥T (t)|_{S}∥ ≤ c_{1}e^{−αt} (t ≥ 0),
∥T (t)|_{U}∥ ≤ c_{1}e^{αt} (t ≤ 0)

([4, p. 145, Ths. 3.1, 3.3]). Let Φ be a basis vector in C, and let Ψ be the basis vector associated with Φ. From [4, p. 149, Cor. 3.8] we know that the C-component u(t) of the segment x_{t} for each solution x(·) of Eq. (3) is given by the relation u(t) = ⟨Ψ, Π_{C}x_{t}⟩ (where Π_{C} denotes the projection from B onto C which corresponds to the decomposition of the space B), and u(t) satisfies the ordinary differential equation

u̇(t) = Qu(t) − Ψ̂(0^{−})f(t),  (5)

where Q is a matrix such that σ(Q) = σ(G) ∩ iR and the relation T (t)Φ = Φe^{tQ} holds. Moreover, Ψ̂ is the one associated with the Riesz representation of Ψ. Indeed, Ψ̂ is a normalized vector-valued function which is of locally bounded variation on R^{−} satisfying ⟨Ψ, ϕ⟩ = ∫_{−∞}^{0} ϕ(θ)dΨ̂(θ) for any ϕ ∈ BC(R^{−}; C^{n}) with compact support. Observe that Σ_{C} ⊂ iR \ iΛ. Indeed, if μ ∈ Σ_{C}, then μ = iλ for some λ ∈ R, where det[(iλ − k)I − L(ω(λ)I)] = 0 by the preceding assertion. Hence we get λ ∉ Λ by the assumption of the claim, and μ ∈ iR \ iΛ, as required. This observation leads to σ(Q) ∩ iΛ = ⊘. Since sp(Ψ̂(0^{−})f) ⊂ Λ, lemma 1 implies that the ordinary differential equation (5) has a unique solution u satisfying sp(u) ⊂ Λ and ∥u∥ ≤ c_{2}∥Ψ̂(0^{−})f∥ ≤ c_{3}∥f∥ for some constants c_{2} and c_{3}. Consider a function ξ : R ↦ B defined by

ξ(t) = ∫_{∗−∞}^{t} T^{∗∗}(t − s)Π_{S}^{∗∗}Γf(s)ds + Φu(t) − ∫_{∗t}^{∞} T^{∗∗}(t − s)Π_{U}^{∗∗}Γf(s)ds,

where Γ is the one defined in [4, p. 118] and ∫_{∗} denotes the weak-star integration (cf. [4, p. 116]). If t ≥ 0, then

T (t)ξ(σ) + ∫_{∗σ}^{t+σ} T^{∗∗}(t + σ − s)Γf(s)ds

= T (t)[∫_{∗−∞}^{σ} T^{∗∗}(σ − s)Π_{S}^{∗∗}Γf(s)ds + Φu(σ) − ∫_{∗σ}^{∞} T^{∗∗}(σ − s)Π_{U}^{∗∗}Γf(s)ds]

+ ∫_{∗σ}^{t+σ} T^{∗∗}(t + σ − s)Γf(s)ds

= ∫_{∗−∞}^{σ} T^{∗∗}(t + σ − s)Π_{S}^{∗∗}Γf(s)ds + Φe^{tQ}u(σ) − ∫_{∗σ}^{∞} T^{∗∗}(t + σ − s)Π_{U}^{∗∗}Γf(s)ds
+ ∫_{∗σ}^{t+σ} T^{∗∗}(t + σ − s)(Π_{S}^{∗∗} + Π_{C}^{∗∗} + Π_{U}^{∗∗})Γf(s)ds
= ∫_{∗−∞}^{t+σ} T^{∗∗}(t + σ − s)Π_{S}^{∗∗}Γf(s)ds + Φ[e^{tQ}u(σ) + ∫_{σ}^{t+σ} e^{(t+σ−s)Q}(−Ψ̂(0^{−})f(s))ds]
− ∫_{∗t+σ}^{∞} T^{∗∗}(t + σ − s)Π_{U}^{∗∗}Γf(s)ds
= ∫_{∗−∞}^{t+σ} T^{∗∗}(t + σ − s)Π_{S}^{∗∗}Γf(s)ds + Φu(t + σ)
− ∫_{∗t+σ}^{∞} T^{∗∗}(t + σ − s)Π_{U}^{∗∗}Γf(s)ds
= ξ(t + σ),

where we used the relation T^{∗∗}(t)Π_{C}^{∗∗}Γ = T^{∗∗}(t)Φ⟨Ψ, Γ⟩ = Φe^{tQ}(−Ψ̂(0^{−})). Then [4, p. 121, Th. 2.9] yields that x(t) := [ξ(t)](0) is a solution of (3). Define a ψ ∈ B^{∗} × ⋯ × B^{∗} (n-copies) by ⟨ψ, ϕ⟩ = ϕ(0), ϕ ∈ B. Then

x(t) − Φ(0)u(t) = ⟨ψ, ξ(t) − Φu(t)⟩
= ⟨ψ, ∫_{∗−∞}^{t} T^{∗∗}(t − s)Π_{S}^{∗∗}Γf(s)ds − ∫_{∗t}^{∞} T^{∗∗}(t − s)Π_{U}^{∗∗}Γf(s)ds⟩
= ∫_{−∞}^{t}⟨ψ, T^{∗∗}(t − s)Π_{S}^{∗∗}Γ⟩f(s)ds − ∫_{t}^{∞}⟨ψ, T^{∗∗}(t − s)Π_{U}^{∗∗}Γ⟩f(s)ds
= ∫_{−∞}^{∞} Y (t − s)f(s)ds = Y ∗ f(t),

where Y (·) = ⟨ψ, T^{∗∗}(·)Π_{S}^{∗∗}Γ⟩χ_{[0,∞)} − ⟨ψ, T^{∗∗}(·)Π_{U}^{∗∗}Γ⟩χ_{(−∞,0]} and it is an n × n matrixvalued integrable function on R. Then σ(x − Φ(0)u) ⊂ σ(f) ⊂ Λ by (vii) of Proposition 1, and hence x − Φ(0)u ∈ Λ(C^{n}). Thus we get x ∈ Λ(C^{n}) because of sp(u) ⊂ Λ. Moreover, the map f ∈ Λ(C^{n}) ↦ x ∈ Λ(C^{n}) is continuous.

Finally, we will prove the uniqueness of solutions of (3) in Λ(C^{n}). Let x be any solution of (3) which belongs to Λ(C^{n}). By [4, p. 120, Th. 2.8] the B-valued function Π_{S}x_{t} satisfies the relation

Π_{S}x_{t} = T (t − σ)Π_{S}x_{σ} + ∫_{∗σ}^{t} T^{∗∗}(t − s)Π_{S}^{∗∗}Γf(s)ds

for all t ≥ σ > −∞. Note that sup_{σ∈R} |x_{σ}|_{B} < ∞. Therefore, letting σ → −∞ we get

Π_{S}x_{t} = ∫_{∗−∞}^{t} T^{∗∗}(t − s)Π_{S}^{∗∗}Γf(s)ds,

because

lim_{σ→−∞} ∫_{∗σ}^{t} T^{∗∗}(t − s)Π_{S}^{∗∗}Γf(s)ds = ∫_{∗−∞}^{t} T^{∗∗}(t − s)Π_{S}^{∗∗}Γf(s)ds

converges. Similarly, one gets

Π_{U}x_{σ} = − ∫_{∗σ}^{∞} T^{∗∗}(σ − s)Π_{U}^{∗∗}Γf(s)ds.

Also, since ⟨Ψ, x_{t}⟩ satisfies Eq. (5) and since sp(⟨Ψ, x_{t}⟩) ⊂ sp(x) ⊂ Λ, it follows that Π_{C}x_{t} = Φ⟨Ψ, x_{t}⟩ = Φu(t) for all t ∈ R by the uniqueness of the solution of (5) in Λ(C^{n}). Consequently, we have x_{t} ≡ ξ(t) or x(t) ≡ [ξ(t)](0), which shows the uniqueness of the solution of (3) in Λ(C^{n}).

Corollary 1 Suppose that det[iλI − L(ω(λ)I)] ≠ 0 for all λ ∈ Λ. Then Eq. (1) is admissible for M = Λ(C^{n}).

Proof. The corollary is a direct consequence of Theorem 1, since 0 ∉ σ(D_{M} − L_{M}).

Corollary 2 Let Λ be a closed set in R, and suppose that det[(iλ − k)I − L(ω(λ)I)] ≠ 0 for all λ ∈ Λ. Then there exists an n × n matrix-valued integrable function F such that

[(iλ − k)I − L(ω(λ)I)]^{−1} = F̃(λ) := ∫_{−∞}^{∞} F (t)e^{−iλt}dt  (∀λ ∈ Λ).  (6)

Furthermore, for any f ∈ Λ(C^{n}) Eq. (3) possesses a unique solution in Λ(C^{n}) which is explicitly given by F ∗ f.

Proof. As seen in the proof of Theorem 1, there exists an n×n matrix-valued integrable function Y such that (D_{M} − B_{M} − k)^{−1}f − Φ(0)u(t) = Y ∗ f for all f ∈ M := Λ(C^{n}). Furthermore, as pointed out in the proof of Lemma 1, there exists an integrable matrixvalued function F_{1} such that u = F_{1} ∗ f is a unique solution of (5) satisfying sp(u) ⊂ Λ for each f ∈ M. Set F = Y + Φ(0)F_{1}. Then F is an n × n matrix-valued integrable function on R, and F ∗ f is is a unique solution in M of Eq. (3) for each f ∈ M.

Now we shall prove the relation (6). Let λ ∈ Λ, and set x^{j}(t) = F (t) ∗ e^{iλt}e_{j} for j = 1, ⋯, n. We claim that

F̃(λ)e_{j} = \frac{1}{2T} ∫_{s−T}^{s+T} x^{j}(t)e^{−iλt}dt, j = 1, ⋯, n

for all s ∈ R. Indeed, we get

∫_{s−T}^{s+T} x^{j}(t)e^{−iλt}dt = ∫_{s−T}^{s+T} (∫_{−∞}^{∞} F (τ)e^{iλ(t−τ)}dτ)e^{−iλt}dt · e_{j}
= ∫_{s−T}^{s+T} ∫_{−∞}^{∞} F (τ)e^{−iλτ}dτdt · e_{j}
= 2T F̃(λ)e_{j}.

Since

\frac{1}{2T} ∫_{−T}^{T} x_{t}^{j}(θ)e^{−iλt}dt = \frac{1}{2T} ∫_{−T +θ}^{T +θ} x^{j}(τ)e^{−iλτ}dτ · e^{iλθ} = [ω(λ)](θ)F̃(λ)e_{j}

for θ ≤ 0, (A2) implies that

\frac{1}{2T} ∫_{−T}^{T} x_{t}^{j}e^{−iλt}dt = ω(λ)F̃(λ)e_{j}, j = 1, ⋯, n.

Then

\frac{1}{2T} (x^{j}(T)e^{−iλT} − x^{j}(−T)e^{iλT}) = \frac{1}{2T} ∫_{−T}^{T} \{−iλx^{j}(t) + ẋ^{j}(t)\}e^{−iλt}dt
= \frac{1}{2T} ∫_{−T}^{T} (−iλx^{j}(t) + L(x_{t}^{j}) + kx^{j}(t) + e^{iλt}e_{j})e^{−iλt}dt
= (k − iλ)F̃(λ)e_{j} + e_{j} + L(ω(λ)F̃(λ)e_{j})
= [(k − iλ)I + L(ω(λ)I)]F̃(λ)e_{j} + e_{j}.

Letting T → ∞ in the above, we get 0 = [(k−iλ)I+L(ω(λ)I)]F̃(λ)e_{j}+e_{j} for j = 1, ⋯, n, or F̃(λ) = [(iλ − k)I − L(ω(λ)I)]^{−1}, as required.

We denote by AP (C^{n}) or AP the set of all almost periodic (continuous) functions f : R ↦ C^{n}. The next result on the admissibility of Λ(C^{n}) ∩ AP (C^{n}) with respect to Eq. (1) is a direct consequence of Corollary 2, because F ∗ f ∈ AP whenever f ∈ AP and F is integrable.

Corollary 3 Suppose that det[iλI − L(ω(λ)I)] ≠ 0 for all λ ∈ Λ. Then Eq. (1) is admissible for Λ(C^{n}) ∩ AP (C^{n}).

The preceding corollary is a result in the non-critical case. In fact, if (1) is a scalar equation (that is, n = 1), our result is available even for the crical case.

Corollary 4 The following statements hold true for Eq. (1) with n = 1:

(i) Let f ∈ AP (C) with discrete spectrum, and assume the following condition:

lim_{T →∞} \frac{1}{T} ∫_{0}^{T} \overline{z(s)}f(s)ds = 0 for any almost periodic solution z(t)

of Eq. (1) satisfying sp(z) ⊂ sp(f).

Then the equation ẋ(t) = L(x_{t}) + f(t) has an almost periodic solution.

(ii) Let f ∈ BUC(R; C) be a periodic function of period τ > 0, and assume the following condition:

∫_{0}^{τ} \overline{z(s)}f(s)ds = 0 for any τ-periodic solution z(t) of Eq. (1).

Then the equation ẋ(t) = L(x_{t}) + f(t) has a τ-periodic solution.

Proof. (ii) is a direct consequence of (i). We shall prove (i). To do this, it suffices to show that iλ − L(ω(λ)) ≠ 0 for any λ ∈ sp(f). Suppose that iλ = L(ω(λ)) for some λ ∈ sp(f), and set z(t) = e^{iλt}, t ∈ R. As seen in the proof of Theorem 1, z(t) is a (periodic) solution of Eq. (1), and moreover sp(z) ⊂ sp(f). Therefore, by the condition in the statement (i) we get lim_{T →∞}(1/T) ∫_{0}^{T} f(s)e^{−iλs}ds = 0, which shows that λ is not an exponent of f(t). On the other hand, because sp(f) is discrete, any point in sp(f) must be an exponent of f(t). This is a contradiction.

4. APPLICATIONS

As an application, we consider the integro-differential equation

ẋ(t) = ∫_{0}^{∞}[dB(s)]x(t − s),  (7)

where B is an n × n matrix-valued function whose components are of bounded variation satisfying

∃γ > 0 : ∫_{0}^{∞} e^{γs}d|B(s)| < ∞.

In order to set up Eq. (7) as an FDE on a uniform fading memory space, we take the space C_{γ} introduced in Section 2, and define a functional L on C_{γ} by

L(ϕ) = ∫_{0}^{∞}[dB(s)]ϕ(−s), ϕ ∈ C_{γ}.

Then Eq. (7) is rewritten as Eq. (1) with B = C_{γ}, and our previous results are applicable to Eq. (7):

Theorem 2 Suppose that det[iλI − ∫_{0}^{∞}[dB(s)]e^{−iλs}] ≠ 0 for all λ ∈ Λ. Then Eq. (7) is admissible for the spaces Λ(C^{n}) and Λ(C^{n}) ∩ AP (C^{n}).

In fact, there exists an n × n matrix-valued integrable function F such that

[(iλ − k)I − ∫_{0}^{∞}[dB(s)]e^{−iλs}]^{−1} = F̃(λ) (∀λ ∈ Λ),

and for any f ∈ Λ(C^{n}), F ∗ f is a unique solution in Λ(C^{n}) of the equation

ẋ(t) = ∫_{0}^{∞}[dB(s)]x(t − s) + f(t).

Finally, we consider the following integro-differential equation

ẋ(t) = Ax(t) + ∫_{0}^{∞} x(t − s)db(s)  (8)

in a Banach space X, where A is the infinitesimal generator of an analytic strongly continuous semigroup of linear operators on X, and b : R^{+} ↦ C is a function of bounded variation satisfying

∃γ > 0 : ∫_{0}^{∞} e^{γs}d|b(s)| < ∞.

In a similar way for Eq. (7), one can define the operator L on the uniform fading memory space C_{γ}(X).

Now we denote by BUC(R; X), Λ(X), AP (X), D_{Λ(X)}, L_{Λ(X)}, ⋯ the ones corresponding to BUC(R; C^{n}), Λ(C^{n}), AP (C^{n}), D_{Λ(C^{n})}, L_{Λ(C^{n})}, ⋯, and set M(C) = Λ(C)∩AP (C) and M(X) = Λ(X) ∩ AP (X). Then M(X) is a translation invariant closed subspace of BUC(R; X), and one can consider the operator D_{M(X)} − L_{M(X)}, together with the operator D_{M(C)} − L_{M(C)}.

Lemma 2 Under the notation explained above, the following relation holds:

σ(D_{M(X)} − L_{M(X)}) = σ(D_{M(C)} − L_{M(C)})

Proof. The inclusion σ(D_{M(X)} − L_{M(X)}) ⊂ σ(D_{M(C)} − L_{M(C)}) is an immediate consequence of Corollary 2 (cf. [9, Lemma 3.6]). We shall establish the converse inclusion. Let k ∈ σ(D_{M(C)} − L_{M(C)}), and assume that k ∉ σ(D_{M(X)} − L_{M(X)}). It follows from Theorem 1 that k = iλ − ∫_{0}^{∞} e^{−iλs}db(s) for some λ ∈ Λ. Let a ∈ X be any nonzero element, and define a function f ∈ Λ(X) by f(t) = e^{iλt}a, t ∈ R. Then there is a unique solution x in M(X) of the equation

ẋ(t) = kx(t) + ∫_{0}^{∞} x(t − s)db(s) + f(t).  (9)

Since x ∈ AP (X), the limit

lim_{T →∞} \frac{1}{T} ∫_{−s}^{T −s} x(t)e^{−iλt}dt (=: x_{λ})

exists in X uniformly for s ∈ R. From (9) we get the relation

[x(T)e^{−iλT} − x(0)]/T = −(iλ/T) ∫_{0}^{T} x(t)e^{−iλt}dt + (k/T) ∫_{0}^{T} x(t)e^{−iλt}dt

+(1/T) ∫_{0}^{T} [∫_{0}^{∞} x(t − s)db(s)]e^{−iλt}dt + a, and hence letting T → ∞ we get [−iλ + k + ∫_{0}^{∞} e^{−iλs}db(s)]x_{λ} + a = 0, or a = 0. This is a contradiction. Hence we must have the inclusion σ(D_{M(X)} − L_{M(X)}) ⊃ σ(D_{M(C)} − L_{M(C)}).

For M(X) = Λ(X) ∩ AP (X), we denote by A_{M(X)} the operator f ∈ M(X) ↦ Af(·) with D(A) = {f ∈ M(X) : f(t) ∈ D(A), Af(·) ∈ M for ∀t ∈ R}. For two (unbounded) commuting operators D_{M(X)} − L_{M(X)} and A_{M(X)}, it is known (cf. [9, Theorem 2.2]) that

σ(\overline{D_{M(X)} − L_{M(X)} − A_{M(X)}}) ⊂ σ(D_{M(X)} − L_{M(X)}) − σ(A_{M(X)}),

here \overline{(⋯)} denotes the usual closure of the operator. Applying Lemma 2 and this relation, we get the following result on the admissiblity of M(X) with respect to Eq. (8).

Theorem 3 Assume that iλ − ∫_{0}^{∞} e^{−iλs}db(s) ∈ ρ(A) for all λ ∈ Λ. Then for any f ∈ Λ(X) ∩ AP (X) the equation ẋ(t) = Ax(t) + ∫_{0}^{∞} x(t − s)db(s) + f(t) has a unique (mild) solution in Λ(X) ∩ AP (X).

References

[1] J.K. Hale, ”Theory of Functional Differential Equations”, Springer-Verlag, New York-Heidelberg-Berlin, 1977.

[2] J.K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), 11–41.

[3] L. Hatvani and T. Krisztin, On the existence of periodic solutions for linear inhomogeneous and quasilinear functional differential equations, J. Diff. Eq. 97 (1992), 1–15.

[4] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math. 1473, Springer-Verlag, Berlin-New York 1991.

[5] C. Langenhop, Periodic and almost periodic solutions of Volterra integral differential equations with infinite memory, J. Diff. Eq. 58 (1985), 391–403.

[6] B.M. Levitan and V.V. Zhikov, ”Almost Periodic Functions and Differential Equations”, Moscow Univ. Publ. House 1978. English translation by Cambridge University Press 1982.

[7] J.J. Massera and J.J. Schäffer, ”Linear Differential Equations and Function Spaces”, Academic Press, New York, 1966.

[8] S. Murakami, Linear periodic functional differential equations with infinite delay, Funkcial. Ekvac. 29 (1986), N.3, 335–361.

[9] S. Murakami, T. Naito and Nguyen V. Minh, Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces, J. Diff. Eq. (in press).

[10] J. Prüss, ”Evolutionary Integral Equations and Applications”, Birkhäuser, Basel, 1993.

[11] Q.P. Vu, Almost periodic solutions of Volterra equations, Diff. Int. Eq. 7 (1994), 1083–1093.

[12] Q.P. Vu and E. Schüler, The operator equation AX − XB = C, stability and asymptotic behaviour of differential equations, J. Diff. Eq. 145 (1998), 394–419.