ALMOST PERIODICITY OF INHOMOGENEOUS PARABOLIC EVOLUTION EQUATIONS

LAHCEN MANIAR AND ROLAND SCHNAUBELT

Abstract. We show the (asymptotic) almost periodicity of the bounded solution to the parabolic evolution equation u′(t) = A(t)u(t) + f(t) on ℝ (on ℝ_{+}) assuming that the linear operators A(t) satisfy the ‘Acquistapace–Terreni’ conditions, that the evolution family generated by A(·) has an exponential dichotomy, and that R(ω, A(·)) and f are (asymptotically) almost periodic.

1. Introduction

In the present work we investigate the almost periodicity of the solutions to the parabolic inhomogeneous evolution equations

u′(t) = A(t)u(t) + f(t), t ∈ ℝ,  (1.1)
u′(t) = A(t)u(t) + f(t), t > 0,  u(0) = x,  (1.2)

in a Banach space X. It is assumed that the linear operators A(t) satisfy the ‘Acquistapace–Terreni’ conditions, that the evolution family U solving the homogeneous problem has an exponential dichotomy, and that the functions t ↦ R(ω, A(t)), for an ω ≥ 0, and f are almost periodic (compare Section 2). In Theorem 4.5 we show that then the unique bounded mild solution u : ℝ → X of (1.1) is almost periodic. This fact is a consequence of the almost periodicity of Green’s function corresponding to U, established also in Theorem 4.5. In Theorem 5.4 we prove the analogous results for (1.2) on ℝ_{+} in the context of asymptotic almost periodicity imposing a necessary compatibility condition on x and f.

The almost periodicity of inhomogeneous problems has been studied by many authors in the autonomous case, where A(t) = A, and in the periodic case, where A(t) = A(t+p), see [3], [4], [6], [11], [12], [16], [21], and the references therein. Equations with almost periodic A(·) are treated in, e.g., [8] and [10] for X = ℂ^{n} and in [13] for a certain class of parabolic problems, see also [5], [15], [17]. For general evolution families U (but subject to an extra condition not assumed here), it is shown in [14] that U has an exponential dichotomy with an almost periodic Green’s function if and only if there is a unique almost periodic mild solution u of (1.1) for each almost periodic f, see also [8, Prop.8.3]. Our main theorems extend [8, Prop.8.4], [10, Thm.7.7], and [13, p.240], and complement [14, Thm.5.3] in the case of parabolic evolution equations. The initial value problem (1.2) was not studied in [8], [10], [13], and [14] in the context of asymptotic almost periodicity.

1991 Mathematics Subject Classification. 34G10, 47D06.

Key words and phrases. Parabolic evolution equation, (asymptotic) almost periodicity, exponential dichotomy, Green’s function, robustness.

Our strategy is similar to Henry’s approach in [13, §7.6] who derived the almost periodicity of Green’s function Γ corresponding to U (and thus of the dichotomy projections P (·)) from a formula for Γ(t + τ, s + τ) − Γ(t, s) (compare (4.2)). In the context of [13], this equation allows to verify Bohr’s definition of almost periodicity by straightforward estimates in operator norm if τ is a pseudo period of A(·). However, in the present more general situation one obtains such a formula only on a subspace of X so that one cannot proceed in this way. To overcome this difficulty, we employ Yosida approximations. This requires some preparations given in Section 3 and a somewhat delicate limiting process presented in Section 4. We point out that we can not estimate the relevant quantities for the approximating problems independently of n, cf. Lemma 4.1. The almost periodicity of the mild solution u of (1.1) then follows from the standard formula (2.11) which expresses u in terms of Γ, see Theorem 4.5. We also give a straightforward application to a second order parabolic equation. The initial value problem (1.2) is treated in Section 5 by essentially the same methods.

In the next section we collect several concepts and preliminary results. We refer to [9] for unexplained notation. By c = c(α, β, ⋯) we denote a generic constant only depending on the constants in the hypotheses involved and the quantities α, β, · · ·.

2. Prerequisites

Let X be a Banach space. A set U = {U(t, s) : t ≥ s, t, s ∈ ℝ} of bounded linear operators on X is called an evolution family if (E1) U(t, s) = U(t, r)U(r, s) and U(s, s) = I for t ≥ r ≥ s and (E2) (t, s) ↦ U(t, s) is strongly continuous for t > s. We say that an evolution family U has an exponential dichotomy (or is hyperbolic) if

there are projections P (t), t ∈ ℝ, being uniformly bounded and strongly continuous in t and constants δ > 0 and N ≥ 1 such that

(a) U(t, s)P (s) = P (t)U(t, s),

(b) the restriction U_{Q}(t, s) : Q(s)X → Q(t)X of U(t, s) is invertible (and we set U_{Q}(s, t) := U_{Q}(t, s)^{−1}),

(c) ∥U(t, s)P (s)∥ ≤ Ne^{−δ(t−s)} and ∥U_{Q}(s, t)Q(t)∥ ≤ Ne^{−δ(t−s)} for t ≥ s and t, s ∈ ℝ. Here and below we let Q = I − P for a projection P. Exponential dichotomy is a classical concept in the study of the long–term behaviour of evolution equations, see e.g. [7], [8], [9], [13], [15]. If U is hyperbolic, then the operator family

Γ(t, s) := \{ \begin{matrix}U(t, s)P (s), & t ≥ s, t, s ∈ ℝ, \\ −U_{Q}(t, s)Q(s), & t < s, t, s ∈ ℝ, \end{matrix}

is called Green’s function corresponding to U and P (·). If P (t) = I for t ∈ ℝ, then U is exponentially stable. The evolution family is called exponentially bounded if there are constants M > 0 and γ ∈ ℝ such that ∥U(t, s)∥ ≤ Me^{γ(t−s)} for t ≥ s. For computations involving Green’s function it is useful to observe that

(t, s) ↦ U_{Q}(t, s)Q(s) is strongly continuous for t, s ∈ ℝ, t ≠ s,
U_{Q}(t, s)Q(s) = U_{Q}(t, r)U_{Q}(r, s)Q(s) for t, r, s ∈ ℝ,

cf. [9, Lemma VI.9.17].

Let U be an exponentially bounded evolution family on X. It is a well known and important fact that the exponential dichotomy of U persists under small perturbations, see e.g. [7], [8], [9, §VI.9], [13], [20]. Our approach also relies on this property. More precisely, we will use Proposition 2.1 below which is a refinement of [19, Prop.2.3]. This result is based on a characterization of exponential dichotomy by means of the evolution semigroup

(T_{U}(t)f) (s) := U(s, s − t)f(s − t), t ≥ 0, s ∈ ℝ, f ∈ C_{0}(ℝ, X),

on C_{0}(ℝ, X) (the space of continuous functions vanishing at infinity). Note that T_{U} is an exponentially bounded semigroup being strongly continuous on (0, ∞), but not necessarily at t = 0. We then have

U has exponential dichotomy ⟺ I − T_{U}(1) is invertible,  (2.1)

see the equivalence (a)⇔(b) in [9, Thm.VI.9.18] or [7, §3.2.3] and the references therein. There it is also shown that then T_{U} is hyperbolic and that the dichotomy projections of U are given by

P (·) = \frac{1}{2πi} ∫_{|λ|=1} R(λ, T_{U}(1)) dλ.  (2.2)

To be precise, in these works it is assumed that (t, s) ↦ U(t, s) is strongly continuous for t ≥ s (and not only for t > s as in our paper), but for the proof of the above mentioned facts this does not matter, see [20]. Proposition 2.1. Let U and V be evolution families with ∥U(t, s)∥, ∥V (t, s)∥ ≤ Me^{γ(t−s)} for t ≥ s. Assume that U has an exponential dichotomy with projections P_{U}(s) and constants N, δ > 0 and that

q := sup_{s∈ℝ}∥U(s + 1, s) − V (s + 1, s)∥ ≤ \frac{(1 − e^{−δ})^{2}}{8N^{2}} .  (2.3)

Then V has an exponential dichotomy with projections P_{V} (s) and constants 0 < δ′ < − log(2qN + e^{−δ}) and N′, where N′ only depends on M, γ, N, δ, δ′. Moreover,

∥P_{U}(t) − P_{V} (t)∥ ≤ q \frac{16N^{2}}{3(1 − e^{−δ})^{2}} .  (2.4)

Proof. The result is a consequence of [19, Prop.2.3] and its proof except for the uniformity of N′. Equations (2.6) and (2.7) of [19] combined with (2.3) yield

∥R(1, T_{V} (1))∥ ≤ \frac{8N}{3(1 − e^{−δ})} .

The uniformity of N′ thus follows from the next lemma (a variant of [18, Lem.4]). □

Lemma 2.2. Let U be an evolution family on a Banach space X such that ∥U(t, s)∥ ≤ Me^{γ(t−s)}, t ≥ s, and ∥R(1, T_{U}(1))∥ ≤ C for the evolution semigroup T_{U}(·) on C_{0}(ℝ, X). Then U has an exponential dichotomy with constants N, δ > 0 depending only on M, γ, C.

Proof. We first recall that if λ ∈ ρ(T_{U}(1)), then λe^{iξ} ∈ ρ(T_{U}(1)) and ∥R(λe^{iξ}, T_{U}(1)∥ = ∥R(λ, T_{U}(1))∥, where ξ ∈ ℝ and λ ∈ ℂ, see [7, (3.8)] or [9, p.483]. This fact implies

∥R(λ, T_{U}(1))∥ ≤ C̃ := \frac{C}{1 − (e^{δ} − 1)C}  (2.5)

for |λ| = e^{α} and 0 ≤ |α| ≤ δ < log(1 + \frac{1}{C}). By (2.1) and a simple rescaling argument, we obtain the exponential dichotomy of U for every exponent 0 < δ < log(1 + \frac{1}{C}). Fix such a δ. If the result were false, then there would exist evolution families U_{n} on Banach spaces X_{n} satisfying the asumptions, real numbers t_{n} and s_{n}, and elements x_{n} ∈ X_{n} such that ∥x_{n}∥ = 1 and

e^{δ|t_{n}−s_{n}|} ∥Γ_{n}(t_{n}, s_{n})x_{n}∥ ⟶ ∞ as n → ∞,  (2.6)

where Γ_{n} is Green’s function of U_{n}. By assumption and (2.2), the projections P_{n}(t) are uniformly bounded for t ∈ ℝ and n ∈ ℕ. Hence the operators Γ_{n}(t, s), s ≤ t ≤ s + 1, n ∈ ℕ, are also uniformly bounded. Thus we have either t_{n} > s_{n} + 1 or t_{n} < s_{n} in (2.6).

In the first case, we write t_{n} = s_{n} + k_{n} + τ_{n} for k_{n} ∈ ℕ and τ_{n} ∈ (0, 1]. Otherwise, t_{n} = s_{n}−k_{n}+τ_{n} for k_{n} ∈ ℕ and τ_{n} ∈ (0, 1]. Take continuous functions φ_{n} with 0 ≤ φ_{n} ≤ 1, supp φ_{n} ⊂ (s_{n} + \frac{τ_{n}}{2} , s_{n} + \frac{3τ_{n}}{2} ), and φ_{n}(t_{n} ∓ k_{n}) = 1 (here t_{n} − k_{n} is used in the first case, and t_{n} + k_{n} in the second). Set λ = e^{∓δ} and f_{n}(s) = e^{±δ(s−s_{n})} φ_{n}(s) U_{n}(s, s_{n})x_{n}. (We let U(t, s) := 0 for t < s.) Using [9, (VI.9.4)] in the second inequality, we compute

[R(λ, T_{U_{n}}(1))f_{n}](t_{n}) = ∑_{k=0}^{∞} λ^{−(k+1)}[T_{U_{n}}(k)P_{n}(·)f_{n}](t_{n}) − ∑_{k=1}^{∞} λ^{k−1}[T_{U_{n},Q}(k)^{−1}Q_{n}(·)f_{n}](t_{n})

= ∑_{k=0}^{∞} λ^{−(k+1)}U_{n}(t_{n}, t_{n} − k)P_{n}(t_{n} − k)f_{n}(t_{n} − k)

− ∑_{k=1}^{∞} λ^{k−1}U_{n,Q}(t_{n}, t_{n} + k)Q_{n}(t_{n} + k)f(t_{n} + k)

= ∑_{k=0}^{∞} e^{±δ(k+1)}e^{±δ(t_{n}−k−s_{n})}φ_{n}(t_{n} − k)U_{n}(t_{n}, s_{n})P_{n}(s_{n})x_{n}

− ∑_{k=1}^{∞} e^{∓δ(k−1)}e^{±δ(t_{n}+k−s_{n})}φ_{n}(t_{n} + k)U_{n,Q}(t_{n}, s_{n})Q_{n}(s_{n})x_{n}.

Here exactly one term does not vanish, namely k = k_{n} in the first sum if t_{n} > s_{n} and k = k_{n} in the second sum if t_{n} < s_{n}. Therefore

R(λ, T_{U_{n}}(1))f_{n}(t_{n}) = e^{±δ}e^{δ|t_{n}−s_{n}|} Γ_{n}(t_{n}, s_{n})x_{n} ,

and the assumptions and (2.5) imply

∥e^{−δ}e^{δ|t_{n}−s_{n}|}Γ_{n}(t_{n}, s_{n})x_{n}∥ ≤ ∥R(λ, T_{U_{n}}(1))f_{n}∥_{∞} ≤ C̃Me^{2(γ+δ)}.

This estimate contradicts (2.6). □

In the present work we study operators A(t) on X subject to the following hypotheses. (H1) There is an ω ≥ 0 such that the operators A(t), t ∈ ℝ, satisfy Σ_{ϕ} ∪ {0} ⊆

ρ(A(t) − ω), ∥R(λ, A(t) − ω)∥ ≤ \frac{K}{1+|λ|} , and

∥(A(t) − ω)R(λ, A(t) − ω) [R(ω, A(t)) − R(ω, A(s))]∥ ≤ L |t − s|^{μ} |λ|^{−ν}

for t, s ∈ ℝ, λ ∈ Σ_{ϕ} := \{λ ∈ ℂ \ \{0\} : | arg λ| ≤ ϕ\}, and constants ϕ ∈ (\frac{π}{2}, π), L, K ≥ 0, and μ, ν ∈ (0, 1] with μ + ν > 1.

This assumption was introduced by P. Acquistapace and B. Terreni in [2] (for ω = 0). It implies that there exists a unique evolution family U on X such that (t, s) ↦ U(t, s) ∈ L(X) is continuous for t > s, U(·, s) ∈ C^{1}((s, ∞), L(X)), ∂_{t}U(t, s) = A(t)U(t, s), and

∥A(t)^{k}U(t, s)∥ ≤ C (t − s)^{−k},  (2.7)
∥A(t)U(t, s)R(w, A(s))∥ ≤ C,  (2.8)
∥U(t, s)(ω − A(s))^{α}x∥ ≤ C (μ − α)^{−1} (t − s)^{−α} ∥x∥  (2.9)

for 0 < t − s ≤ 1, k = 0, 1, 0 ≤ α < μ, x ∈ D((ω − A(s))^{α}), and a constant C depending only on the constants in (H1). Moreover, ∂_{s}^{+}U(t, s)x = −U(t, s)A(s)x for t > s and x ∈ D(A(s)) with A(s)x ∈ \overline{D(A(s))}. This follows by an obvious rescaling from [1, Thm.2.3] and [23, Thm.2.1], see also [2], [22]. We say that A(·) generates U. Note that U is exponentially bounded by (2.7) with k = 0. We further suppose that

(H2) the evolution family U generated by A(·) has an exponential dichotomy with constants N, δ > 0, dichotomy projections P (t), t ∈ ℝ, and Green’s function Γ.

We point out that it is quite difficult to find conditions on A(·) implying (H2). Such results are usually based on (2.1) and variants of it, see [8], [10], [13], [15], [18], [19], [20].

Assume that (H1) and (H2) hold and that f : ℝ → X is bounded and continuous. A classical solution of (1.1) is a function u ∈ C^{1}(ℝ, X) satisfying u(t) ∈ D(A(t)) for t ∈ ℝ and (1.1). It is known that then

u(t) = U(t, s)u(s) + ∫_{s}^{t} U(t, τ)f(τ) dτ  for all t ≥ s,  (2.10)

see [1, Prop.3.2, 5.1]. On the other hand, there is a unique bounded continuous function satisfying (2.10), namely

u(t) = ∫_{ℝ} Γ(t, τ)f(τ) dτ,  t ∈ ℝ,  (2.11)

see e.g. (the proof of) [7, Thm.4.28]. We call the function u given by (2.11) the mild solution of (1.1). Observe that the mild solution is a classical one if, for instance, f is Hölder continuous due to [2, Thm.6.3] and (2.10).

It remains to introduce the concept of almost periodicity, see e.g. [4], [10], [15], [16]. The next definition due to H.Bohr is the most convenient one for our purposes.

Definition 2.3. Let Y be a Banach space. A continuous function g : ℝ → Y is called almost periodic if for every ϵ > 0 there exist a set P (ϵ) ⊆ ℝ and a number ℓ(ϵ) > 0 such that each interval (a, a + ℓ(ϵ)), a ∈ ℝ, contains an almost period τ = τ_{ϵ} ∈ P (ϵ) and the estimate ∥g(t + τ) − g(t)∥ ≤ ϵ holds for all t ∈ ℝ and τ ∈ P (ϵ). The space of almost periodic functions is denoted by AP (ℝ, Y).

We recall that AP (ℝ, Y) is a closed subspace of the space of bounded and uniformly continuous functions, see [15, Chap.1]. Our last assumption reads as follows.

(H3) R(ω, A(·)) ∈ AP (ℝ, L(X)) with pseudo periods τ = τ_{ϵ} belonging to sets P (ϵ, A).

It is not difficult to verify that then R(ω′, A(·)) ∈ AP (ℝ, L(X)) if ω′ ≥ ω.

3. Yosida approximations

Assume that (H1) holds and define the Yosida opproximations A_{n}(t) = nA(t)R(n, A(t)) of A(t) for n > ω and t ∈ ℝ. These operators generate an evolution family U_{n} on X. We want to show that A_{n}(·) satisfies the same assumptions as A(·). The elementary (but tedious) proof of the next result is omitted, cf. [2, Lem.4.2] or [22, Prop.2.1].

Lemma 3.1. Assume that (H1) holds. Fix ω′ > ω and ϕ′ ∈ (\frac{π}{2}, ϕ). Then there are constants n_{0} > ω, L′ ≥ L, and K′ ≥ K (only depending on the constants in (H1), ω′, and ϕ′) such that the operators A_{n}(t), t ∈ ℝ, satisfy (H1) with constants K′, ϕ′, ω′, L′, μ, ν for all n ≥ n_{0}.

Since (H1), (H2), and (H3) still hold for A(·) with constants K′, L′, ω′, ϕ′, we can assume that A(·) and A_{n}(·) satisfy (H1) with the same constants, denoted by K, L, ω, ϕ.

Lemma 3.2. If (H1) and (H3) hold, then there is a number n_{1} ≥ n_{0} such that R(ω, A_{n}(·)) ∈ AP (ℝ, L(X)) for n ≥ n_{1}, with pseudo periods belonging to P (ϵ/κ, A), where κ := 2 + 4K.

Proof. Let τ > 0 and t ∈ ℝ. We first observe that

R(λ, A_{n}(t)) = \frac{1}{λ+n}(n − A(t))R(\frac{λn}{λ+n}, A(t)) = \frac{n^{2}}{(λ+n)^{2}} R(\frac{λn}{λ+n}, A(t)) + \frac{1}{λ+n}  (3.1)

if n ≥ n_{0} and | arg(λ − ω)| ≤ ϕ. Equation (3.1) yields

R(ω, A_{n}(t + τ)) − R(ω, A_{n}(t)) = \frac{n^{2}}{(ω + n)^{2}} (R(\frac{ωn}{ω + n}, A(t + τ)) − R(\frac{ωn}{ω + n}, A(t)))

= \frac{n^{2}}{(ω + n)^{2}} R(ω, A(t + τ))[1 − \frac{ω^{2}}{ω + n}R(ω, A(t + τ))]^{−1}

− \frac{n^{2}}{(ω + n)^{2}} R(ω, A(t))[1 − \frac{ω^{2}}{ω + n}R(ω, A(t))]^{−1}, (3.2)

where we have used that

∥\frac{ω^{2}}{ω + n}R(ω, A(s))∥ ≤ \frac{ω^{2}}{ω + n} \frac{K}{1 + ω} ≤ \frac{ωK}{n} ≤ \frac{1}{2}

for n ≥ n_{1} := max{n_{0}, 2ωK} and s ∈ ℝ. In particular,

∥[1 − \frac{ω^{2}}{ω + n}R(ω, A(s))]^{−1}∥ ≤ 2.  (3.3)

Hence, (3.2) implies

∥R(ω,A_{n}(t + τ)) − R(ω, A_{n}(t))∥

≤ 2 ∥R(ω, A(t + τ)) − R(ω, A(t))∥

+ \frac{K}{1 + ω} ∥[1 − \frac{ω^{2}}{ω + n}R(ω, A(t + τ))]^{−1} − [1 − \frac{ω^{2}}{(ω + n)^{2}} R(ω, A(t))]^{−1}∥.

Employing (3.3) again, we obtain

∥[1 − \frac{ω^{2}}{ω + n}R(ω, A(t + τ))]^{−1} − [1 − \frac{ω^{2}}{ω + n}R(ω, A(t))]^{−1}∥

≤ 4 ∥[1 − \frac{ω^{2}}{ω + n}R(ω, A(t + τ))] − [1 − \frac{ω^{2}}{ω + n}R(ω, A(t))]∥

≤ 4ω ∥R(ω, A(t + τ)) − R(ω, A(t))∥.

Putting everything together, we arrive at

∥R(ω, A_{n}(t + τ)) − R(ω, A_{n}(t))∥ ≤ (2 + 4K)∥R(ω, A(t + τ)) − R(ω, A(t))∥  (3.4)

for n ≥ n_{1} and t ∈ ℝ. The assertion thus follows from (H3). □

In order to see that A_{n}(·) satisfies also (H2), we need the following result which is of interest in itself. For ω = 0 it is shown in [5, Prop.4.4] (note that our result does not simply follow by rescaling). We give here a different, more elementary proof leading to a different rate of convergence. Proposition 3.3. Let (H1) hold and fix 0 < t_{0} < t_{1}. Then ∥U(t, s) − U_{n}(t, s)∥ ≤ c(t_{1}, θ) n^{−θ} for 0 < t_{0} ≤ t − s ≤ t_{1}, n ≥ n_{2}(t_{0}) := max{n_{0}, t_{0}^{−2/μ}}, and any 0 < θ < min{μ/2, 1−μ/2, μ(μ+ν−1)/2}. Moreover, ∥(U(t, s)−U_{n}(t, s))R(ω, A(s))∥ ≤ c(t_{1}, α) n^{−α} for 0 ≤ t − s ≤ t_{1}, n ≥ n_{0}, and α ∈ (0, μ).

Proof. Let 0 < h < t_{0} and 0 < t_{0} ≤ t − s ≤ t_{1}. Then we have

U(t, s) − U_{n}(t, s) = (U(t, s + h) − U_{n}(t, s + h))U(s + h, s) − U_{n}(t, s + h)·

· [U(s + h, s) − e^{hA(s)} + e^{hA(s)} − e^{hA_{n}(s)} + e^{hA_{n}(s)} − U_{n}(s + h, s)]

=: S_{1} − S_{2}.  (3.5)

Due to Lemma 4.3 in [2] and equation (2.6) and Lemma 2.2 in [1], we obtain

∥S_{2}∥ ≤ c(t_{1}) (h^{μ+ν−1} + (hn)^{−1}).  (3.6)

The other term can be transformed into

S_{1} = ∫_{s+h}^{t} U_{n}(t, σ)(A(σ) − A_{n}(σ))U(σ, s) dσ
= ∫_{s+h}^{t} U_{n}(t, σ)(ω − A_{n}(σ))^{α}(ω − A_{n}(σ))^{1−α}((A(σ) − ω)^{−1} − (A_{n}(σ) − ω)^{−1})·
· (A(σ) − ω)U(σ, s) dσ.

where α ∈ (0, μ). The estimates (2.7), (2.9), and

\begin{aligned}∥(A_{n}(σ) − ω)^{−1} − (A(σ) − ω)^{−1}∥ & = ∥\frac{1}{ω+n}A(σ)R(\frac{ωn}{ω+n}, A(σ))A(σ)R(ω, A(σ))∥ ≤ \frac{c}{n}, \\ ∥(ω − A_{n}(σ))^{1−α}∥ & ≤ c ∥ω − A_{n}(σ)∥^{1−α} ≤ cn^{1−α} \end{aligned} (3.7)

(use the moment inequality [9, Thm.II.5.38] in the second line) lead to

∥S_{1}∥ ≤ c(t_{1}, α) h^{−1}n^{−α}.  (3.8)

Combining (3.5), (3.6), and (3.8), we deduce

∥U(t, s) − U_{n}(t, s)∥ ≤ c(t_{1}, α) ((nh)^{−1} + h^{μ+ν−1} + n^{−α}h^{−1}).

The first assertion now follows if we set h := n^{−μ/2}. The second one can be shown using the formula

(U(t, s) − U_{n}(t, s))R(ω, A(s)) = ∫_{s}^{t} U_{n}(t, σ)(ω − A_{n}(σ))^{α}(ω − A_{n}(σ))^{1−α}·
· ((A(σ) − ω)^{−1} − (A_{n}(σ) − ω)^{−1})(A(σ) − ω)U(σ, s)R(ω, A(s)) dσ,

and the estimates (2.9), (3.7), and (2.8). □

Propositions 2.1 and 3.3 immediately imply that A_{n}(·) fulfills also (H2). Corollary 3.4. Let (H1) and (H2) hold. Then there is a number n_{3} ≥ n_{2}(1) such that U_{n} has an exponential dichotomy for n ≥ n_{3} with constants δ′ ∈ (0, δ) and N′ = N′(δ′) independent of n. Moreover, the dichotomy projections P_{n}(t) of U_{n} satisfy ∥P_{n}(t)− P (t)∥ ≤ c(θ) n^{−θ} for t ∈ ℝ, where θ ∈ (0, 1) and n_{2}(1) are given by Proposition 3.3.

4. Main results for equations on ℝ

We first establish the almost periodicity of Green’s function Γ_{n} for the evolution family U_{n} generated by the Yosida approximation A_{n}(·). Lemma 4.1. Assume that (H1), (H2), and (H3) hold. Let n ≥ max{n_{1}, n_{3}}, η > 0, and τ ∈ P (η/κ, A), where n_{1}, n_{3}, and κ were given in Section 3. We then have

∥Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s)∥ ≤ c η n^{2} e^{− \frac{δ}{2} |t−s|} , t, s ∈ ℝ.

Proof. The operators Γ_{n}(t, s) exist by Corollary 3.4. It is easy to see that

g_{n}(σ) := \frac{d}{dσ} (Γ_{n}(t, σ)Γ_{n}(σ + τ, s + τ))
= Γ_{n}(t, σ)(A_{n}(σ + τ) − A_{n}(σ))Γ_{n}(σ + τ, s + τ)
= Γ_{n}(t, σ)(A_{n}(σ) − ω)((A_{n}(σ) − ω)^{−1} − (A_{n}(σ + τ) − ω)^{−1})·
· (A_{n}(σ + τ) − ω)Γ_{n}(σ + τ, s + τ),

for σ ≠ t, s, τ ≥ 0, and n ≥ n_{3}. Hence,

∫_{ℝ} g_{n}(σ) dσ = \begin{cases}∫_{−∞}^{s} g_{n}(σ)dσ + ∫_{s}^{t} g_{n}(σ)dσ + ∫_{t}^{∞} g_{n}(σ)dσ, & t ≥ s, \\ ∫_{−∞}^{t} g_{n}(σ)dσ + ∫_{t}^{s} g_{n}(σ)dσ + ∫_{s}^{∞} g_{n}(σ)dσ, & t < s, \end{cases}

= \begin{cases}−Γ_{n}(t, s)Q_{n}(s + τ) + P_{n}(t)Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s)P_{n}(s + τ) + Q_{n}(t)Γ_{n}(t + τ, s + τ), \\ P_{n}(t)Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s)Q_{n}(s + τ) + Q_{n}(t)Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s)P_{n}(s + τ), \end{cases}

= Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s).

We have shown that

Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s) = ∫_{ℝ} Γ_{n}(t, σ)(A_{n}(σ) − ω) [R(ω, A_{n}(σ + τ)) − R(ω, A_{n}(σ))]·
· (A_{n}(σ + τ) − ω)Γ_{n}(σ + τ, s + τ) dσ  (4.1)

for s, t ∈ ℝ and n ≥ n_{3}. Lemma 3.2 and Corollary 3.4 now yield

∥Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s)∥ ≤ cηn^{2} ∫_{ℝ} e^{− \frac{3δ}{4} |t−σ|} e^{− \frac{3δ}{4} |σ−s|} dσ

if also n ≥ n_{1} and τ ∈ P (η/κ, A), which gives the asserted estimate. □

Lemma 4.2. Assume that (H1) and (H2) hold. Fix 0 < t_{0} < t_{1} and let θ > 0, n_{2}(t_{0}), and n_{3} be given by Proposition 3.3 and Corollary 3.4. Then ∥Γ(t, s) − Γ_{n}(t, s)∥ ≤ c(t_{1}, θ) n^{−θ} holds for t_{0} ≤ |t − s| ≤ t_{1} and n ≥ max{n_{3}, n_{2}(t_{0})}.

Proof. If t_{0} ≤ t − s ≤ t_{1}, we write

Γ_{n}(t, s) − Γ(t, s) = (U_{n}(t, s) − U(t, s))P_{n}(s) + U(t, s)(P_{n}(s) − P (s)).

So the assertion is a consequence of Proposition 3.3 and Corollary 3.4. For −t_{1} ≤ t − s ≤ −t_{0}, we have

Γ(t, s) − Γ_{n}(t, s) = U_{n,Q}(t, s)Q_{n}(s) − U_{Q}(t, s)Q(s)
= U_{Q}(t, s)Q(s)(Q_{n}(s) − Q(s)) + (Q_{n}(t) − Q(t))U_{n,Q}(t, s)Q_{n}(s)
− U_{Q}(t, s)Q(s)(U_{n}(s, t) − U(s, t))U_{n,Q}(t, s)Q_{n}(s).

Again the asserted estimate follows from Proposition 3.3 and Corollary 3.4. □

Employing (4.1), we extend a formula given on [13, p.240] to the present setting. Corollary 4.3. Let (H1) and (H2) hold, t, s, τ ∈ ℝ, and x ∈ D((ω − A(s))^{β}) for some β > 0 (or x contained in a suitable interpolation space). We then have

Γ(t + τ, s + τ)x − Γ(t, s)x = ∫_{ℝ} Γ(t, σ)(A(σ) − ω) [R(ω, A(σ + τ)) − R(ω, A(σ))]
· (A(σ + τ) − ω)Γ(σ + τ, s + τ)x dσ.  (4.2)

Proof. We want to obtain (4.2) by taking the limit as n → ∞ in (4.1) evaluated at x ∈ D((ω − A(s))^{β}). The left hand side converges as required due to Lemma 4.2 and Corollary 3.4. For r ≠ ρ, n ≥ n_{3}, and α ∈ (1 − ν, μ), we write

Γ_{n}(r, ρ)(ω − A_{n}(ρ))^{α} = \begin{cases}P_{n}(r)U_{n}(r, ρ)(ω − A_{n}(ρ))^{α},  ρ < r ≤ ρ + 1, \\ U_{n}(r, ρ + 1)P_{n}(ρ + 1)U_{n}(ρ + 1, ρ)(ω − A_{n}(ρ))^{α}, & ρ + 1 ≤ r, \\ −U_{n,Q}(r, ρ + 1)Q_{n}(ρ + 1)U_{n}(ρ + 1, ρ)(ω − A_{n}(ρ))^{α}, & r < ρ, \end{cases}

A_{n}(r)Γ_{n}(r, ρ) = \begin{cases}A_{n}(r)U_{n}(r, ρ)P_{n}(ρ), & ρ < r ≤ ρ + 1, \\ A_{n}(r)U_{n}(r, r − 1)U_{n}(r − 1, ρ)P_{n}(ρ), & ρ + 1 ≤ r, \\ −A_{n}(r)U_{n}(r, r − 1)U_{n,Q}(r − 1, ρ)Q_{n}(ρ)x, & r < ρ. \end{cases}

By (the proof of) [22, Prop.3.1] and Lemma 4.2 these terms converge strongly to the analogous terms without n as n → ∞. Using also [22, Prop.2.1], one deduces the pointwise convergence of the integrand in (4.1). Moreover, we observe that

A_{n}(r)U_{n}(r, ρ)P_{n}(ρ)x = A_{n}(r)U_{n}(r, ρ)(ω − A_{n}(ρ))^{−β}((ω − A_{n}(ρ))^{β}x
− (ω − A_{n}(ρ))^{β}U_{n}(ρ, ρ − 1)U_{n,Q}(ρ − 1, ρ)Q_{n}(ρ)x).

Due to (2.7) and [23, Thm.2.1], the norm of the right hand side is smaller than

c (r − ρ)^{β−1}(∥(ω − A_{n}(ρ))^{β}x∥ + ∥x∥).

The moment inequality, see e.g. [9, Thm.II.5.38], further yields

∥(ω − A_{n}(ρ))^{β}x∥ = ∥[(ω − A_{n}(ρ))(ω − A(ρ))^{−1}]^{β} (ω − A(ρ))^{β}x∥ ≤ c ∥(ω − A(ρ))^{β}x∥.

Combining these estimates with (2.9), [22, Prop.2.1], and (2.7), we obtain an integrable, n–independent bound of the integrand in (4.1) evaluated at x ∈ D((ω − A(s))^{β}). The assertion thus follows from the theorem of dominated convergence. □

Though the above formula is quite interesting, the proofs of our main results only use the two preceding lemmas. Proposition 4.4. Assume that (H1), (H2), and (H3) hold. Let ϵ > 0 and h > 0. Then

∥Γ(t + τ, s + τ) − Γ(t, s)∥ ≤ ϵ e^{− \frac{δ}{2} |t−s|}

holds for |t − s| ≥ h and τ ∈ P (η/κ, A), where κ = 2 + 4K and η = η(ϵ, h) → 0 as ϵ → 0 and h is fixed.

Proof. Let ϵ > 0 and h > 0 be fixed. Then there is a t_{ϵ} > h such that

∥Γ(t + τ, s + τ) − Γ(t, s)∥ ≤ ϵ e^{− \frac{δ}{2} |t−s|}

for |t − s| ≥ t_{ϵ}. Let h ≤ |t − s| ≤ t_{ϵ} and τ ∈ P (η/κ, A). Lemmas 4.1 and 4.2 show that

∥Γ(t + τ, s + τ) − Γ(t, s)∥ ≤ (c(t_{ϵ})e^{\frac{δ}{2} t_{ϵ}} n^{−θ} + cηn^{2}) e^{− \frac{δ}{2} |t−s|}

for n ≥ max{n_{1}, n_{2}(h), n_{3}}. We now choose first a large n and then a small η > 0 (depending on ϵ and h) in order to obtain the assertion. □

Theorem 4.5. Assume that (H1), (H2), and (H3) hold. Then r ↦ Γ(t + r, s + r) belongs to AP (ℝ, L(X)) for t, s ∈ ℝ, where we may take the same pseudo periods for t, s with |t − s| ≥ h > 0. If f ∈ AP (ℝ, X), then the unique bounded mild solution u = ∫_{ℝ} Γ(·, s)f(s) ds of (1.1) is almost periodic.

Proof. In Lemma 4.1 we have seen that P_{n}(·) ∈ AP (ℝ, L(X)). Corollary 3.4 then shows that P (·) ∈ AP (ℝ, L(X)). Thus the first assertion follows from Proposition 4.4. Further, for τ, h > 0 and t ∈ ℝ, we write

u(t + τ) − u(t) = ∫_{ℝ} Γ(t + τ, s + τ)f(s + τ) ds − ∫_{ℝ} Γ(t, s)f(s) ds

= ∫_{ℝ} Γ(t + τ, s + τ)(f(s + τ) − f(s)) ds + ∫_{|t−s|≥h}(Γ(t + τ, s + τ) − Γ(t, s))f(s) ds

+ ∫_{|t−s|≤h}(Γ(t + τ, s + τ) − Γ(t, s))f(s) ds.

For \overline{ϵ} > 0 let η = η(\overline{ϵ}, h) be given by Proposition 4.4. Let P (ϵ, A, f) be the set of pseudo periods for the almost periodic function t ↦ (f(t), R(ω, A(t))), cf. [15, p.6]. Taking τ ∈ P (η/κ, A, f), we deduce from Proposition 4.4 and (H2) that

∥u(t + τ) − u(t)∥ ≤ \frac{2N}{δκ} η(\overline{ϵ}, h) + (\frac{4}{δ} \overline{ϵ} + 4Nh)∥f∥_{∞}.

for t ∈ ℝ. Given an ϵ > 0, we can take first a small h > 0 and then a small \overline{ϵ} > 0 such that ∥u(t + τ) − u(t)∥ ≤ ϵ for t ∈ ℝ and τ ∈ P (η/κ, A, f) =: P (ϵ). □

Remark 4.6. For g ∈ AP (ℝ, Y) and λ ∈ ℝ the means

lim_{t→∞} \frac{1}{2t} ∫_{−t}^{t} e^{−iλs}g(s) ds

exist. They are different from zero for at most countable many λ, which are called the frequencies of g, see e.g. [4, §4.5], [15, §2.3]. The module of g is the smallest additive subgroup of ℝ containing all frequencies of g. By [15, p.44] (see also [10, Thm.4.5]) and the proof of Theorem 4.5 the module of the solution u to (1.1) is contained in the joint module of f and R(ω, A(·)) (which is the smallest additive subgroup of ℝ containing the frequencies of the function t ↦ (f(t), R(ω, A(t)))). Similarly, the modules of Γ(t+·, s+·), t ≠ s, are contained in that of R(ω, A(·)). Example 4.7. Consider the parabolic problem

\begin{matrix}D_{t} u(t, x) = ∑_{k,l=1}^{n} D_{k} a_{kl}(t, x) D_{l}u(t, x) + a_{0}(t, x)u(t, x) + f(t, x), t ∈ ℝ, x ∈ Ω, \\ ∑_{k,l=1}^{n} n_{k}(x) a_{kl}(t, x) D_{l}u(t, x) = 0, t ∈ ℝ, x ∈ ∂Ω, \end{matrix} (4.3)

on the time interval ℝ. Here Ω ⊆ ℝ^{n} is a bounded domain with C^{2}–boundary ∂Ω being locally on one side of Ω, D_{t} = d/dt, D_{k} = d/dx_{k}, and n(x) is the outer unit normal vector. We assume that the coefficients satisfy

a_{kl} ∈ C_{b}^{μ}(ℝ, C(\overline{Ω})) ∩ C_{b}(ℝ, C^{1}(\overline{Ω})) ∩ AP (ℝ, L^{n}(Ω)), k, l = 1, ⋯, n,
a_{0} ∈ C_{b}^{μ}(ℝ, L^{n}(Ω)) ∩ C_{b}(ℝ, C(\overline{Ω})) ∩ AP (ℝ, L^{n/2}(Ω))

for some \frac{1}{2} < μ ≤ 1, where n/2 is replaced by 1 if n = 1. Moreover, (a_{kl}) is supposed to be symmetric and real and to satisfy ∑_{k,l} a_{kl}(t, x) y_{k} y_{l} ≥ η |y|^{2} for a constant η > 0, x ∈ \overline{Ω}, t ∈ ℝ, y ∈ ℝ^{n}. Let

A(t, x, D) = ∑_{k,l=1}^{n} D_{k} a_{kl}(t, x) D_{l} + a_{0}(t, x).

We then define on X = L^{p}(Ω), 1 < p < ∞, the operator A(t)φ = A(t, ·, D)φ with domain

D(A(t)) = {φ ∈ W^{2,p}(Ω) : ∑_{k,l=1}^{n} n_{k}(·) a_{kl}(t, ·) D_{l}φ = 0 on ∂Ω},

where the boundary condition is understood in the sense of traces if necessary. It is shown in [22, §4] that A(t), t ∈ ℝ, fulfill (H1) for μ and each ν ∈ (0, \frac{1}{2}). Thus there exists an evolution family U on X solving the Cauchy problem corresponding to (4.3) for f = 0. In the same way, one shows that (H3) holds (with the same pseudo periods as the coefficients). Therefore (4.3) has a unique mild solution u ∈ AP (ℝ, X) provided that f ∈ AP (ℝ, X) and U has an exponential dichotomy. The solution is classical if, e.g., f is also Hölder continuous in time.

5. Main results for equations on ℝ_{+}

The arguments and results of the previous sections can be extended to the problem (1.2) on ℝ_{+}. Here we will concentrate on the necessary modifications for the sake of brevity. We first introduce the space

AP (ℝ_{+}, Y) := {g : ℝ_{+} → Y : ∃ g̃ ∈ AP (ℝ, Y) s.t. g̃|ℝ_{+} = g}

of almost periodic functions on ℝ_{+}. We remark that the function g̃ in the above definition is uniquely determined, cf. [4, Prop.4.7.1]. The following concept is more important for our investigations. Definition 5.1. A continuous function g : ℝ_{+} → Y is called asymptotically almost periodic if for every ϵ > 0 there exists a set P (ϵ) ⊆ ℝ_{+} and numbers s(ϵ), ℓ(ϵ) > 0 such that each interval (a, a + ℓ(ϵ)), a ≥ 0, contains an almost period τ = τ_{ϵ} ∈ P (ϵ) and the estimate ∥g(t + τ) − g(t)∥ ≤ ϵ holds for all t ≥ s(ϵ) and τ ∈ P (ϵ). The space of asymptotically almost periodic functions is denoted by AAP (ℝ_{+}, Y). Due to e.g. [4, Thm.4.7.5], these spaces are related by the equality

AAP (ℝ_{+}, Y) = AP (ℝ_{+}, Y) ⊕ C_{0}(ℝ_{+}, Y).  (5.1)

Evolution families and exponential dichotomy on time intervals [a, ∞) are defined by restricting the definitions on ℝ to parameters t, s ≥ a. So we can make the following assumptions. (H1’) The operators A(t), t ≥ −1, satisfy (H1) for t, s ≥ −1. (H2’) The evolution family U generated by A(·) has an exponential dichotomy on [−1, ∞)

with projections P (t), t ≥ −1, constants N, δ > 0, and Green’s function Γ. (H3’) R(ω, A(·)) ∈ AAP (ℝ_{+}, L(X)) with constants s(ϵ, A) and pseudo periods τ = τ_{ϵ}

belonging to sets P (ϵ, A). (We have to involve the interval [−1, ∞) for technical reasons, see (5.6). Each interval [b, ∞) with b < 0 would do the job.) Assume that (H1’) and (H2’) hold and let f : ℝ_{+} → X be bounded and continuous. Then the mild solution of (1.2) is given by

u(t) := U(t, 0)x + ∫_{0}^{t} U(t, s)f(s) ds, t ≥ 0.

This function is a classical solution (i.e., u ∈ C^{1}((0, ∞), X) ∩ C(ℝ_{+}, X), u(t) ∈ D(A(t)), and (1.2) holds for t > 0) if x ∈ \overline{D(A(0))} and, e.g., f is Hölder continuous, see [2, Thm.6.3]. By [1, Prop.3.2,5.1], each classical solution is a mild one if x ∈ \overline{D(A(0))}. Writing f(s) = P (s)f(s) + Q(s)f(s), one sees that a mild solution u satisfies

u(t) = U(t, 0)(x + ∫_{0}^{∞} U_{Q}(0, s)Q(s)f(s) ds) + ∫_{0}^{∞} Γ(t, s)f(s) ds, t ≥ 0.  (5.2)

Thus u is bounded if and only if the term in brackets belongs to P (0)X if and only if

Q(0)x = − ∫_{0}^{∞} U_{Q}(0, s)Q(s)f(s) ds.  (5.3)

In this case the mild solution of (1.2) is given by

u(t) = U(t, 0)P (0)x + ∫_{0}^{∞} Γ(t, s)f(s) ds,  t ≥ 0.  (5.4)

Again we consider the Yosida approximations A_{n}(t), t ≥ 0, and the evolution family U_{n} generated by A_{n}(·). Lemma 3.1 and Proposition 3.3 clearly hold on ℝ_{+} if we replace (H1) by (H1’). Since also the estimate (3.4) remains valid for t, τ ≥ 0, Lemma 3.2 is still true if we replace (H1) and (H3) by (H1’) and (H3’) and AP (ℝ, L(X)) by AAP (ℝ_{+}, L(X)). Moreover, we can take s(ϵ, A_{n}) = s(ϵ, A).

However, it is not immediate that Corollary 3.4 can be extended to the half line case because the proof of the perturbation result Proposition 2.1 only works on ℝ. But one can overcome this obstacle using a suitable extension of U. Lemma 5.2. Assume that (H1’) and (H2’) hold. Then there is a number n′_{3} ≥ n_{0} such that, for n ≥ n′_{3}, the evolution family U_{n} generated by A_{n}(·) has an exponential dichotomy on ℝ_{+} with dichotomy projections P_{n}(t) and constants δ′ ∈ (0, δ) and N′ = N′(δ′) independent of n. Moreover, ∥P_{n}(t) − P (t)∥ ≤ c n^{−θ} for t ≥ 0, where θ ∈ (0, 1) is a fixed number given by Proposition 3.3.

Proof. Let d ≥ δ, n ≥ n_{0}, and set R = δQ(0) − dP (0). We define

U^{d}(t, s) := \begin{cases}U(t, s), & t ≥ s ≥ 0, \\ U(t, 0)e^{−sR}, & t ≥ 0 ≥ s, \\ e^{(t−s)R}, & 0 ≥ t ≥ s, \end{cases} U_{n}^{d}(t, s) := \begin{cases}U_{n}(t, s), & t ≥ s ≥ 0, \\ U_{n}(t, 0)e^{−sR}, & t ≥ 0 ≥ s, \\ e^{(t−s)R}, & 0 ≥ t ≥ s. \end{cases}

Clearly, U^{d} and U_{n}^{d} satisfy (E1). Observing that e^{rR} = e^{−rd}P (0) + e^{rδ}Q(0), it is easy to see that U^{d} and U_{n}^{d} are exponentially bounded independent of d and n and that U^{d} has an exponential dichotomy on ℝ with constants N, δ and projections P^{d}(t) = P (t) for t ≥ 0 and P^{d}(t) = P (0) for t ≤ 0. Moreover, (t, s) ↦ U_{n}^{d}(t, s) is norm continuous for t ≥ s, s ↦ U^{d}(s + t, s) is norm continuous from the left for each t > 0, and t ↦ U^{d}(t, s) is norm continuous from the left for t > s ((E2) only holds for U^{d} if D(A(0)) is dense). As shown in [20], Proposition 2.1 remains valid under these conditions. To apply this result, we want to find n′_{3} ≥ n_{0} and d_{n} ≥ δ such that

∥U^{d_{n}}(s + 1, s) − U_{n}^{d_{n}}(s + 1, s)∥ ≤ c n^{−θ}  (5.5)

for s ∈ ℝ, n ≥ n′_{3}, and some θ > 0. Due to Proposition 3.3, we have

∥U^{d}(s + 1, s) − U_{n}^{d}(s + 1, s)∥ ≤ \begin{cases}c n^{−θ}, & s ≥ −\frac{1}{2}, \\ 0, & −1 ≥ s, \end{cases}

for a fixed θ ∈ (0, 1) and n ≥ n_{2}(1/2). If s ∈ (−1, −1/2), then Proposition 3.3 and (2.7) yield

∥U^{d}(s + 1, s) − U_{n}^{d}(s + 1, s)∥  (5.6)

≤ ∥(U(s + 1, 0) − U_{n}(s + 1, 0))(R(ω, A(0))(ω − A(0))U(0, −1)U_{Q}(−1, 0)Q(0)e^{−sδ}∥

+ ∥(U(s + 1, 0) − U_{n}(s + 1, 0))P (0)e^{sd}∥
≤ c (n^{−θ} + e^{−d/2}) ≤ 2c n^{−θ}

if we choose a sufficiently large d =: d_{n}. Thus (5.5) holds for n ≥ n_{2}(1/2) =: n′_{3} and these d_{n}. The assertions then follow from Proposition 2.1 by restricting U_{n}^{d} to ℝ_{+}. □

We can now proceed almost as in the previous section; we only have to take care of certain additional exponentially decaying terms. Proposition 5.3. Assume that (H1’), (H2’), and (H3’) hold. Let ϵ > 0, h > 0, and |t − s| ≥ h. Then there are numbers η = η(ϵ, h) and s̃(ϵ) ≥ s(η, A) such that

∥Γ(t + τ, s + τ) − Γ(t, s)∥ ≤ ϵ e^{− \frac{δ}{2} |t−s|}

for τ ∈ P (η/κ, A) and t, s ≥ s̃(ϵ), where κ = 2 + 4K and η = η(ϵ, h) → 0 as ϵ → 0 and h is fixed. Moreover, P (·) ∈ AAP (ℝ_{+}, L(X)).

Proof. Let ϵ > 0 and h > 0 be fixed. Let t, s ≥ 0. Then there is a t_{ϵ} > h such that

∥Γ(t + τ, s + τ) − Γ(t, s)∥ ≤ ϵ e^{− \frac{δ}{2} |t−s|}

for |t − s| ≥ t_{ϵ}. For h ≤ |t − s| ≤ t_{ϵ} we deduce as in Lemma 4.2 from Proposition 3.3 and Lemma 5.2 that

∥Γ(t, s) − Γ_{n}(t, s)∥ ≤ c(t_{ϵ}, θ) n^{−θ}e^{− \frac{δ}{2} |t−s|}  (5.7)

if n ≥ max{n′_{3}, n_{2}(h)}. Using the same function g_{n}, the arguments given in the proof of Lemma 4.1 lead to

Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s) = Γ_{n}(t, a)Γ_{n}(a + τ, s + τ) + ∫_{a}^{∞} g_{n}(σ) dσ

for t, s ≥ a ≥ 0 and τ ≥ 0. Taking η > 0, τ ∈ P (η/κ, A), and t, s ≥ b ≥ a := s(η, A), this equality yields as in Lemma 4.1

∥Γ_{n}(t + τ, s + τ) − Γ_{n}(t, s)∥ ≤ cηn^{2} ∫_{a}^{∞} e^{− \frac{3δ}{4} |t−σ|} e^{− \frac{3δ}{4} |σ−s|} dσ + ce^{− \frac{3δ}{4} (t−a)} e^{− \frac{3δ}{4} (s−a)}
≤ ce^{− \frac{δ}{2} |t−s|}(ηn^{2} + e^{− \frac{3δ}{2} (b−a)}),  (5.8)

where we use Lemma 5.2. We first choose a sufficiently large n = n(ϵ, h), then a small η = η(ϵ, h), and finally a large b =: s̃(ϵ) ≥ s(η, A) in order to obtain the asserted estimate for h ≤ |t − s| ≤ t_{ϵ} from (5.7) and (5.8). The last claim is shown as in Theorem 4.5. □

Theorem 5.4. Assume that (H1’), (H2’), and (H3’) hold and that x ∈ X and f ∈ AAP (ℝ_{+}, L(X)) satisfy (5.3). Then the mild solution u of (1.2) is asymptotically almost periodic.

Proof. Using (5.4), we write

u(t + τ) − u(t) = (U(t + τ, t) − I)U(t, 0)P (0)x + U(t + τ, τ)P (τ) ∫_{0}^{τ} U(τ, s)P (s)f(s) ds
+ ∫_{0}^{∞} Γ(t + τ, s + τ)(f(s + τ) − f(s)) ds
+ ∫_{0}^{∞}(Γ(t + τ, s + τ) − Γ(t, s))f(s) ds =: S_{1} + S_{2} + S_{3} + S_{4}

for t, τ ≥ 0. Clearly, ∥S_{1}∥ + ∥S_{2}∥ ≤ ce^{−δt}. Let ϵ > 0 be given and set a := s(ϵ, f). For t ≥ a and τ ∈ P (ϵ, f), the asymptotic almost periodicity of f yields

S_{3} = U(t + τ, a + τ)P (a + τ) ∫_{0}^{a} U(a + τ, s)P (s)(f(s + τ) − f(s)) ds
+ ∫_{a}^{∞} Γ(t + τ, s + τ)(f(s + τ) − f(s)) ds,
∥S_{3}∥ ≤ \frac{2N^{2}}{δ} ∥f∥_{∞} e^{−δ(t−a)} + \frac{2N}{δ} ϵ.

For \overline{ϵ} > 0 and h > 0, let η = η(\overline{ϵ}, h) and b := max\{s̃(\overline{ϵ}), s(ϵ, f)\} be given by Proposition 5.3. Choosing t ≥ b and τ ∈ P (η/κ, A), we deduce from Proposition 5.3 that

S_{4} = U(t + τ, b + τ)P (b + τ) ∫_{0}^{b} U(b + τ, s + τ)P (s)f(s) ds
− U(t, b)P (b) ∫_{0}^{b} U(b, s)P (s)f(s) ds
+ ∫_{b}^{t−h}(Γ(t + τ, s + τ) − Γ(t, s))f(s) ds + ∫_{t−h}^{t+h} ⋯ ds + ∫_{t+h}^{∞} ⋯ ds,
∥S_{4}∥ ≤ (\frac{2N^{2}}{δ} e^{−δ(t−b)} + \frac{4\overline{ϵ}}{δ} + 4Nh)∥f∥_{∞} .

We now take first small h > 0 and \overline{ϵ} > 0 and then a large ŝ(ϵ) ≥ b such that ∥u(t + τ) − u(t)∥ ≤ cϵ for t ≥ ŝ(ϵ) and τ ∈ P (η/κ, A, f), the joint set of pseudo periods of f and R(ω, A(·)). □

We conclude with some remarks concerning solutions of (1.2) in AP (ℝ_{+}, X). Assume that (H1), (H2), and (H3) hold and f ∈ AP (ℝ_{+}, X). Let f̃ ∈ AP (ℝ, X) be the extension of f. By Theorem 4.5 the function

ũ(t) = ∫_{ℝ} Γ(t, s)f̃(s) ds,  t ∈ ℝ,  (5.9)

belongs to AP (ℝ, X). Its restriction u ∈ AP (ℝ_{+}, X) satisfies

u(t) = U(t, 0) ∫_{−∞}^{0} U(0, s)P (s)f̃(s) ds + ∫_{0}^{∞} Γ(t, s)f(s) ds, t ≥ 0.

In view of (5.2) this function is a mild solution of (1.2) with the initial value

x = ũ(0) = ∫_{ℝ} Γ(0, s)f̃(s) ds.  (5.10)

(Note that (5.10) implies (5.3).)

Conversely, if u ∈ AP (ℝ_{+}, X) is a mild solution of (1.2), then it is given by (5.4). The formula (5.4) gives

u(t) = U(t, 0)(P (0)x − ∫_{−∞}^{0} U(0, s)P (s)f̃(s) ds) + ∫_{ℝ} Γ(t, s)f̃(s) ds, t ≥ 0.

Since the second summand is almost periodic, the decomposition (5.1) shows that

P (0)x = ∫_{−∞}^{0} U(0, s)P (s)f̃(s) ds.

On the other hand, (5.3) must hold so that x has to satisfy (5.10).

Theorem 5.5. Assume that (H1), (H2), and (H3) hold and that f ∈ AP (ℝ_{+}, X). Then (1.2) has a mild solution u ∈ AP (ℝ_{+}, X) if and only if the initial value x is given by (5.10). In this case u is the restriction of ũ defined in (5.9).

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L. Maniar, Université Cadi Ayyad, Faculté des Sciences Semlalia, B.P. 2390, 40000 Marrakech, Morocco

E-mail address: maniar@ucam.ac.ma

R. Schnaubelt, FB Mathematik und Informatik, Martin–Luther–Universität, 06099 Halle, Germany.

E-mail address: schnaubelt@mathematik.uni-halle.de