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arXiv:1108.3392v1 [math.FA] 17 Aug 2011

EXISTENCE OF BOUNDED UNIFORMLY CONTINUOUS

MILD SOLUTIONS ON R OF EVOLUTION

EQUATIONS AND SOME APPLICATIONS.

Bolis Basit and Hans G¨ unzler

Abstract.

u(0) = x has on R a mild solution u ∈ Cub(R,X) (that is bounded and uniformly

continuous) with u(0) = xφ, where A is the generator of a holomorphic C0-semigroup

(T(t))t≥0on X with supt≥0||T(t)|| < ∞, φ ∈ L∞(R,X) and isp(φ) ∩ σ(A) = ∅.

As a consequence it is shown that if F is the space of almost periodic AP, almost

automorphic AA, bounded Levitan almost periodic LAPb, certain classes of recurrent

functions RECband φ ∈ L∞(R,X) such that Mhφ := (1/h)?h

each h > 0, then u ∈ F ∩Cub. These results seem new and generalize and strengthen

several recent Theorems.

We prove that there is xφ∈ X for which (*)du(t)

dt

= Au(t) + φ(t),

0φ(· + s)ds ∈ F for

§1. Introduction, Definitions and Notation

In this paper we study solutions of the inhomogeneous abstract Cauchy problem

(1.1)

du(t)

dt

= Au(t) + φ(t), u(t) ∈ X, t ∈ J ∈ {R+,R},

(1.2)u(0) = x,

where A : D(A) → X is the generator of a C0-semigroup (T(t))t≥0on the complex

Banach space X and φ ∈ L1

loc(J,X), x ∈ X.

By [15, Corollary 2.5, p. 5], it follows that D(A) is dense in X and A is a closed

linear operator.

By a classical solution of (1.1) we mean a function u : J → D(A) such that

u ∈ C1(J,X) and (1.1) is satisfied.

By a mild solution of (1.1), (1.2) we mean a ω ∈ C(J,X) with?t

0ω(s)ds ∈ D(A)

for t ∈ J and

1991 Mathematics Subject Classification. Primary 47D06, 43A60 Secondary 43A99, 47A10.

Key words and phrases. Evolution equations, non-resonance, global solutions, bounded uni-

formly continuous solutions, almost periodic, almost automorphic.

Typeset by AMS-TEX

1

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2 BASIT AND G¨UNZLER

(1.3)ω(t) = x + A?t

0ω(s)ds +?t

0φ(s)ds, t ∈ J.

For J = R+this is the usual definition [1, p. 120].

A classical solution is always a mild solution (see [1, (3.1), p. 110], for J = R+,

φ = 0). Conversely, a mild solution with ω ∈ C1(J,X) and φ ∈ C(J,X) is a

classical solution (as in [1, Proposition 3.1.15]).

With translation and the case J = R+[1, Proposition 3.1.16] one can show, for

J,A,T, φ as in (1.1) and after (1.2)

Lemma 1.1. ω : J → X is a mild solution of (1.1) if and only if

(1.4)ω(t) = T(t − t0)ω(t0) +?t

t0T(t − s)φ(s)ds, t ≥ t0, t0∈ J.

When J = R mild solutions of (1.1) may not exist. In this paper we establish the

existence of a bounded uniformly continuous mild solution on R to (1.1), (1.2) for

φ ∈ L∞(R,X), x = x(φ) ∈ X and isp(φ) ∩ σ(A) = ∅ when A is the generator of a

holomorphic C0-semigroup (T(t)) satisfying supt∈R+||T(t)|| < ∞. (Theorem 3.5).

In Theorem 3.6, we show that all mild solutions of (1.1), (1.2) are bounded and

uniformly continuous when J = R+and φ and A as in Theorem 3.5. Theorem 3.6(i)

should be compared with the ”Non-resonance Theorem 5.6.5 of [1]” : the result of

[1] is more general since the half-line spectrum is used (⊂ Beurling spectrum).

However, our main result (Theorem 3.5) can not be deduced from Theorem 5.6.5 of

[1]. Moreover, in the important cases of almost periodic, almost automorphic and

recurrent functions, the half-line spectrum coincides with the Beurling spectrum

(see [8, Example 3.8] and [10, Corollary 5.2]).

By mild admissibility of F with respect to equation (1.1) we mean that for every

φ ∈ F, where F ⊂ L∞(R,X) is a subspace with (4.2)- (4.5) below equations (1.1),

(1.2) have a unique mild solution on R which belongs to F (see [14, Definition 3.2,

p. 248]). In the case F ⊂ Cub(R,X), several methods have been used to prove

admissibility for (1.1), (1.2) (see [16], [4], [14], [12] and references therein). The

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ADMISSIBILITY3

method of sums of commuting operators was introduced in [14], and then extended

to the case F = AA(R,X) in [12], to prove admissibility for (1.1). As consequences

of Theorem 3.5 we generalize and strengthen some recent results on admissibility.

Namely, we prove that if φ ∈ MF ∩ L∞(R,X) (see (4.1)) and isp(φ) ∩ σ(A) = ∅,

then there is a mild solution uφ ∈ F ∩ Cub(R,X) of (1.1), (1.2). For example

if φ ∈ MAP ∩ L∞(R,X), then uφ ∈ AP or if φ ∈ MAA ∩ L∞(R,X), then

uφ∈ AA ∩ Cub.

Further notation and definitions. R+= [0,∞), L(X) = {B : X → X,B linear

bounded } with norm ||B||, S(R) contains Schwartz’s complex valued rapidly de-

creasing C∞-functions, Cub(R,X) = {f : R → X : f bounded uniformly continuous

}, AP = AP(R,X) almost periodic functions, AA = AA(R,X) almost automorphic

functions [19], BAA = BAA(R,X) Bochner almost automorphic functions [21, p.

66], [12]; for f ∈ L1

Fourier transform?

loc(J,X) (Pf)(t) := Bochner integral

?t

0f(s)ds,?f(λ) = L1-

Rf(t)e−iλ tdt, fa(t) = translate f(a + t) where defined, a real,

sp = Beurling spectrum (2.3), Proposition 2.3.

§2. Preliminaries

In this section we collect some lemmas and propositions needed in the sequel.

Proposition 2.1. Let F ∈ L1(R,L(X)) and φ ∈ L∞(R,X) respectively F ∈

L1(R,X) and φ ∈ L∞(R,C) respectively F ∈ L1(R,C) and φ ∈ L∞(R,X).

(i) F ∗ φ(t) :=?∞

−∞F(s)φ(t − s)ds

exists as a Bochner integral for t ∈ R and F ∗ φ ∈ Cub(R,X).

(ii) If moreover Pφ ∈ L∞(R,X) respectively Pφ ∈ L∞(R,C), then

(2.1)F ∗ (Pφ) ∈ C1(R,X), (F ∗ (Pφ))′= F ∗ φ,

(2.2)F ∗ (Pφ) = P(F ∗ φ) + (F ∗ (Pφ))(0).

Proof. (i) The integrand of (i) is measurable (approximate F and φ by continuous

functions a.e), it is dominated by ||φ||∞|F| ∈ L1(R,R), |F|(t) := ||F(t)||, so this

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4 BASIT AND G¨UNZLER

integrand ∈ L1(R,L(X)) for any fixed t ∈ R. To F there is a sequence (Hn) of

L(X)-valued step-functions with ||F−Hn||L1 → 0. It follows ||F∗φ−Hn∗φ||∞→ 0,

so with the Hn∗ φ also F ∗ φ is bounded and uniformly continuous. Similarly in

the other cases.

(ii) By part (i) we conclude that F ∗Pφ,F ∗φ ∈ Cub(R,X); [1, Proposition 1.2.2]

and the Lebesgue convergence theorem give existence and (F ∗ Pφ)′= F ∗ φ,∈

Cub(R,X), so (F ∗ Pφ) ∈ C1(R,X). It follows F ∗ Pφ = P(F ∗ φ) + F ∗ Pφ(0). ¶

In the following sp denotes the Beurling spectrum, sp(φ) = sp{0|R}(φ) as defined

for example in [8, (3.2), (3.3)] case S = φ ∈ L∞(R,X), V = L1(R,C), A = {0|R}:

(2.3) sp{0|R}(φ) := {ω ∈ R : f ∈ L1(R,C),φ ∗ f = 0 imply?f(ω) = 0}.

spBis defined in [1, p. 321], spCis the Carleman spectrum [1, p.293/317].

Proposition 2.2. If φ ∈ L∞(R,X), then

(2.4) sp(φ) = sp{0|R}(φ) = spB(φ) = spC(φ).

See also [8, (3.3), (3.14)].

Proof. sp(φ) ⊂ spBφ: Assuming λ ∈ sp(φ) and h ∈ L1(R,C) with?h(λ) ?= 0, we

conclude φ ∗ h ?= 0. This implies λ ∈ spB(φ).

spB(φ) ⊂ sp(φ): Assume λ ∈ spB(φ) but λ ?∈ sp(φ). Then there is h ∈ L1(R,C)

with?h(λ) ?= 0 and φ ∗ h = 0. With Wiener’s inversion theorem [11, Proposition

1.1.5 (b), p. 22], there is hλ ∈ L1such that k = h ∗ hλ satisfies?k = 1 on some

neighbourhood V = (λ−ε,λ+ε) of λ. Now let g ∈ L1(R,C) be such that supp? g ⊂

V . Then k ∗ g = g and 0 = φ ∗ h = φ ∗ k = φ ∗ (k ∗ g) = φ ∗ g. This implies

λ ?∈ spB(φ) by [1, p. 321]. This is a contradiction.

spB(φ) = spC(φ): See [1, Proposition 4.8.4, p. 321]. ¶

Lemma 2.3. Let φ ∈ L∞(R,X).

(i) If Pφ ∈ L∞(R,X), then sp(φ) ⊂ sp(Pφ) ⊂ sp(φ) ∪ {0}.

(ii) If F ∈ L1(R,C) or F ∈ L1(R,L(X)), then sp(F ∗ φ) ⊂ sp(φ)∩ supp?F.

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(iii) If f ∈ L1(R,C), then sp(φ − φ ∗ f) ⊂ sp(φ) \ U, where U is the interior of

the set {λ ∈ R :?f(λ) = 1}.

See also [3, Theorem 4.1.4], [14, Theorem 2.1], [17, Proposition 0.4].

Proof. (i) If f ∈ L1(R,C) with (Pφ) ∗ f = 0, then (2.4) yields φ ∗ f = 0 and

so the first inclusion follows from (2.3((see also [17, Proposition 04 (iv), p. 20]).

The second inclusion follows by [9, Proposition 1.1. (f)] valid also for X-valued

functions.

(ii) F ∈ L1(R,L(X)): to λ ?∈ sp(φ) exists f ∈ L1(R,C) with φ∗f = 0,?f(λ) = 1,

then (F ∗φ)∗f = F ∗(φ∗f) = 0, so λ ?∈ sp(F ∗φ), yielding sp(F ∗φ) ⊂ sp(φ). (The

associativity of convolution used here can be shown as in [1, Proposition 1.3.1, p.

22]). If λ ?∈ supp?F, there exists f ∈ S(R) with (supp?f)∩supp?F = ∅ and?f(λ) = 1,

then?

F ∗ f =?F?f = 0, and so F ∗ f = 0; this gives 0 = (F ∗ f) ∗ φ = F ∗ (f ∗ φ) =

F∗(φ∗f) = (F∗φ)∗f, and so λ ?∈ sp(F∗φ), yielding sp(F∗φ) ⊂ supp?F. (Existence

of F ∗ f,F ∗ f ∈ L1(R,L(X)), and?

F ∗ f =?F?f follows with Fubini-Tonelli similar

as for Proposition 2.1(i).)

The proof for the case F ∈ L1(R,C) is similar.

(iii) If λ ∈ sp(φ) ∩ U, then there is fλ ∈ S(R) with supp?

fλ ⊂ U such that

?

fλ(λ) ?= 0 and f ∗ fλ= fλ. It follows λ ?∈ sp(φ − φ ∗ f). ¶

In the following if U,V ⊂ R, then d(U,V ) := infu∈U,v∈V|u − v|. If U,V are

compact and U ∩ V = ∅, then d(U,V ) > 0.

Lemma 2.4. Let ψ ∈ Cb(R,X). Then there exists a sequence of X-valued trigono-

metric polynomials πnsuch that

(i)supn∈N||πn||∞< ∞.

(ii) πn(t) → ψ(t) as n → ∞ locally uniformly on R.

(iii) If the Beurling spectrum sp(ψ)∩[α,β] = ∅, then there exists δ > 0 such that

the Fourier exponents of πncan be selected from R \ (α − δ,β + δ).

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(iv) If K1,K2are compact subsets of (α,β), where K1= sp(ψ) is the Beurling

spectrum of ψ and K1∩K2= ∅ and η = d(K1,K2), then there exist W, ϕ so that the

Fourier exponents of πncan be selected from W, where W is an open set containing

K1, and ϕ ∈ D(R) with ϕ = 1 on W and d(K2,W) ≥ d(K2, suppϕ) ≥ η/2.

Proof. (i), (ii): Similar to the proof of Lemma 3.1 (3.1) of [4] with [1, Theorem

4.2.19].

(iii) Choose (α1,β1) ⊂ R such that [α,β] ⊂ (α1,β1) and sp(ψ) ⊂ R \ (α1,β1).

Let f ∈ S(R) be such that supp?f ⊂ (α1,β1) and?f = 1 on [α − δ,β + δ] for some

δ > 0. Let (Tn) satisfy (i), (ii). Then πn = Tn− Tn∗ f satisfy (i), (ii) since by

Lemma 2.3 (ii) sp(ψ ∗ f) ⊂ (R \ (α1,β1)) ∩ (α1,β1) = ∅ implies ψ ∗ f = 0 by [1,

Theorem 4.8.2 (a)] and Proposition 2.2, and the Fourier exponents of πnbelong to

R \ (α − δ,β + δ).

(iv) Let Γ = (It)t∈K1, Ω = ∪t∈K1It be a system of open intervals with the

properties

It= (t − δ,t + δ), It⊂ (α,β) and 0 < δ < η/2 for each t ∈ K1.

With a partition of unity [18, Theorem 6.20, p. 147] there is a sequence (ψi) ⊂ D(Ω)

with ψisupported in some Iti∈ Γ, ψi≥ 0 and Σ∞

i=1ψi(t) = 1, t ∈ Ω. Moreover, for

K1there is a positive integer m and an open set W ⊃ K1such that

(2.5)ϕ := Σm

i=1ψi(t) = 1 for each t ∈ W.

Since supp ψi⊂ Iti, d(K2,W) ≥ d(K2, suppϕ) ≥ η/2. Now consider the system

ΓW:= (Jt)t∈K1, where Jt= It∩ W for each t ∈ K1. As above there is an open set

U ⊃ K1and ϕW ∈ D(R) such that ϕW = 1 on U and supp ϕW ⊂ W. It follows

ϕW =?

fW with fW ∈ S(R). Let (Tn) be a sequence of X-valued trigonometric

polynomials satisfying (i), (ii). Take πn = Tn∗ fW, n ∈ N. Then sp(πn) ⊂ W,

Lemma 2.3 (iii) gives sp(ψ − ψ ∗ fW) = ∅ and so ψ = ψ ∗ fW as in (iii). ¶

Proposition 2.5. If f,?f ∈ L1(R,X) ∩ Cb(R,X), then ([1, Theorem 1.8.1 d), p.

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45])

f(t) = (1/2π)?

R?f(λ)eiλtdλ, t ∈ R.

§3. Holomorphic C0-semigroups

In this section we study (1.1) when A is the generator of a holomorphic C0-

semigroup (T(t))in the sense [1, Definition 3.7.1]. By [1, Corollary 3.7.18], it follows

that there is a > 0 such that

(3.1) {is, s ∈ R : |s| > a} ⊂ ρ(A) and

(3.2) sup|s|>a||sR(is,A)|| < ∞

Lemma 3.1. Let b > 0, m ∈ N. There are fk∈ Cm([0,b]), k = 0,1,··· ,n such

that

f(k)

k(b) = 1, f(j)

k(b) = 0, k ?= j, k,j = 0,1,··· ,m,

f(j)

k(0) = 0, k,j = 0,1,2,··· ,m, (f(0)= f).

Proof. The case m = 2 : Let f(t) = (π/2)sin(π(t/b)3/2), t ∈ [0,b]. Then

f0(t) = sinf(t), f1(t) = (t − b)f0(t), f2(t) = (1/2)(t − b)2f0(t)

satisfy all the requirements of the statement.

The general case follows by the Lagrange Interpolation Theorem ([2, p. 395]). ¶

Lemma 3.2. Let A : D(A) → X be the generator of a holomorphic C0-semigroup

(T(t)).

(i) There exists a > 0 such that R(iλ,A) = (iλI−A)−1is infinitely differentiable

on R \ (−b,b) for each b > a. Moreover, R(iλ,A)|(R \ (−b,b)) can be extended to

a function Hm ∈ Cm(R,L(X)) for any m ∈ N with H(j)

m (0) = 0, 0 ≤ j ≤ m,

Hm(t)X ⊂ D(A) for t ∈ R.

(ii) H(k)

m ∈ L1(R,L(X)), 1 ≤ k ≤ m.

Proof. (i) By (3.1) σ(A) ∩ iR ⊂ i[−a,a] for some a > 0. It follows R(iλ,A) =

(iλI − A)−1is infinitely differentiable on R \ (−b,b) for each b > a. Fix one such

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8 BASIT AND G¨UNZLER

b and let fk(t) satisfy the conditions of Lemma 2.1 and gk(−t) = fk(t), t ∈ [0,b],

k = 0,1,··· ,m. Set

F(t) = f0(t)R(ib,A) + f1(t)R′(ib,A) + ··· + fm(t)R(m)(ib,A), t ∈ [0,b],

F(t) = g0(t)R(−ib,A) + g1(t)R′(−ib,A) + ··· + gn(t)R(m)(−ib,A), t ∈ [−b,0].

Then F ∈ Cm([−b,b],L(X)). Set

(3.3)Hm(λ) = R(iλ,A) on R \ (−b,b) and Hm= F on [−b,b].

Then Hm∈ Cm(R,L(X)). Since R(λ,A)X ⊂ D(A) by definition ([1, p. 462]) and

D(A) is linear, Hm(R)X ⊂ D(A) with Lemma 3.1.

(ii) By(3.2) there is M = M(m) > 0 with

(3.4)

||(R(iλ,A))k|| ≤ M/|λ|kon R \ (−b,b), 1 ≤ k ≤ m.

Since ( [1, p. 463, Corollary B.3])

(3.5)H(k)

m (λ) = k!(−i)k(R(iλ,A))k+1if b ≤ |λ|, 1 ≤ k ≤ m,

with (3.4),(3.5) and part (i) one has H(k)

m ∈ L1(R,L(X)), 1 ≤ k ≤ m. ¶

Proposition 3.3. Let m ≥ 2 and Hmbe as in (3.3).

(i) The function Gmdefined by

Gm(t) := (1/2π)?

(3.6)

RHm(λ)eiλtdλ,

exists as an improper Riemann integral for each t ?= 0.

(ii)tkGm∈ L1(R,L(X)), k = 1,2,··· ,m − 2.

Proof. With Lemma 3.2 (ii), one has H′

m,H′′

m∈ L1(R,L(X)). Hmbeing continuous

on R, for t ?= 0 and T > 0 one can use partial integration and gets

?T

0Hm(λ)eiλtdλ = (1/it)Hm(T)eitT− (1/(it))?T

0H′

m(λ)eiλtdλ,

similarly for the integral from −t to 0. This shows that Gm(t) given by (3.6) exists

as an improper Riemann integral and is given by

(i/(2tπ))(?∞

−∞H′

m(λ)eiλtdλ, t ?= 0.

Induction gives with Lemma 3.2 (ii), for 1 ≤ k ≤ m,

Gm(t) = (1/2iπ)(i/t)k?

(3.7)

RH(k)

m (λ)eiλtdλ, t ?= 0.

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Hence Gmis continuous and ||Gm(t)|| ≤ 1/(2π|t|k)?

R||H(k)

m (λ)||dλ in each point

t ?= 0.

(ii) This follows by (3.7) using Lemma 3.2 (ii). ¶

Lemma 3.4. (i) I(t) =?∞

b

R(iλ,A)cosλtdλ exists as improper Riemann integral

for each t > 0. Moreover, ||I(t)|| = O(|lnt|) for all 0 < t < π/2b.

?∞

b

R(iλ,A)sinλtdλ exists as improper Riemann integral for each (ii) J(t) =

t > 0. Moreover, J(t) is bounded for all t > 0.

(iii) For Gm of Proposition 3.3 and m ≥ 2, one has ||Gm(t)|| = O(|ln|t||) in

some neighbourhood of 0, Gm ∈ L1(R,L(X)) (Bochner-Lebesgue integrable func-

tions).

(iv)?

Gm= Hmif m ≥ 3 for the Hmof Lemma 3.2.

(v) Gm∗ x = 0 for each x ∈ X, m ≥ 2.

Proof. (i) For t ≥ π/2b, I(t) exists similarly as in Proposition 3.3. For 0 < t < π/2b,

we have I(t) = (1/t)?∞

a0(t) =?π/2

bt

R(iλ/t,A)cosλdλ,

ak(t) =?π(1/2+2k)

btR(iλ/t,A)cosλdλ = (1/t)[a0(t) +?∞

k=1ak(t)], where

π(1/2+2(k−1))R(iλ/t,A)cosλdλ =?π

0(Bk(λ,t)−Ak(λ,t))sinλdλ, with

Bk(λ,t) = R(i(λ + π(3/2 + 2(k − 1)))/t,A),

Ak(λ,t) = R(i(λ + π(1/2 + 2(k − 1)))/t,A).

Since R(µ,A) −R(λ,A) = (λ−µ)R(µ,A)R(λ,A) for any λ,µ ∈ ρ(A) [1, B.4, p.

464], we have Bk(λ,t) − Ak(λ,t) =

−(iπ/t)R(i(λ + π(3/2 + 2(k − 1)))/t,A)R(i(λ + π(1/2 + 2(k − 1)))/t,A).

Using (3.4) we conclude that ||ak(t)|| = O(t

series (1/t)?∞

(1/t)||a0|| ≤ M?π/2

bt

(ii) we have J(t) = (1/t)?∞

where b0(t) =?π

k2) for all t > 0, k ∈ N. It follows that

k=1akis convergent and bounded on (0,∞). Again by (3.4),

dλ ≤ M?π/2

bt

btR(iλ/t,A)sinλdλ = (1/t)[b0(t) +?∞

btR(iλ/t,A)sinλdλ, bk(t) =?π(1+2k)

cosλ

λ

1

λdλ = M ln(π/2) − ln(bt) = O(|lnt|).

k=1bk(t)],

π(1+2(k−1))(R(iλ/t,A)sinλdλ.

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10 BASIT AND G¨UNZLER

Sincesinλ

Similarly as in part (i), we conclude that the series (1/t)?∞

λ

is bounded on R\{0}, we conclude that b0(t)/t is bounded for all t > 0.

k=1bk(t) is convergent

for all t > 0 and bounded on (0,∞).

(iii) Follows as in parts (i), (ii) and Proposition 3.3 noting that

2πGm(t) =?b

[I(b,t) +?∞

where I(b,t) =?b

−bHm(λ)eiλtdλ +?−b

−∞R(iλ,A)eiλtdλ +?∞

b

R(iλ,A)eiλtdλ) =

b(R(−iλ,A) + R(iλ,A))cosλt + (R(iλ,A) − R(−iλ,A))sinλt)dλ],

−bHm(λ)eiλtdλ.

(iv) Since H′

−i(tGm)(t) = (1/2π)?

m,tGm∈ L1(R,L(X)) ∩ C(R,L(X)) by Proposition 3.3(ii) and

ReiλtH′

m(λ)dλ,

one gets existence of?

H′

Gm

′and by Proposition 2.5 for λ ∈ R

m(λ) = −i?

Re−iλttGn(t)dt =?

Gm

′(λ).

This implies Hm=?

(v) Gm∗ x(t) =?

Gm.

RGm(t)xdt =?

Gmx(0) =?

Gm(0)x = Hm(0)x = 0, with Lemma

3.4, Proposition 2.1 and Lemma 3.2. ¶

In the following we use:

For any G ∈ L1(R,L(X)), λ ∈ R with?G(λ) = R(iλ,A) and x ∈ X one has

Gx ∈ L1(R,X) and?

(Gx) =?

(G)x [1, Proposition 1.1.6], so for eλ(t) := eiλtx, the

following LB-integrals exist and

(G ∗ eλ)(t) = eiλt?

RG(s)xe−iλsds = eiλt?

(Gx)(λ) = eiλt?G(λ)x = eiλtR(iλ,A)x,

t ∈ R, with R(iλ.A)x ∈ D(A).

So for x ∈ X and the λ and the G above, the equation

(3.8)u′(t) = Au(t) + xeiλt, t ∈ R, u(0) = R(iλ,A)x

has a classical solution u(t) = eiλtR(iλ,A)x = (G ∗ eλ)(t), t ∈ R.

Let A, a,b be as in Lemma 3.2, δ = b−a and c = b+3δ. Choose f ∈ S(R) such

that

(3.9)

?f(λ) = 1 on [−(b + δ),b + δ] and supp?f ⊂ (−(b + 2δ),b + 2δ).

Let φ ∈ L∞(R,X). Set

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(3.10)φ1= φ ∗ f and φ2= φ − φ ∗ f.

Then (3.9) and Lemma 2.3 (ii) give

(3.11)sp(φ1) ⊂ supp?f ⊂ (−(b + 2δ),b + 2δ) and sp(φ2) ∩ [−b,b] = ∅.

Theorem 3.5. Let (T(t)) be a holomorphic C0-semigroup on X with generator A

such that supt≥0||T(t)|| < ∞ and let φ ∈ L∞(R,X) with Beurling spectrum sp(φ)

satisfying isp(φ) ∩ σ(A) = ∅. Then there is xφ∈ X such that the equations (1.1),

(1.2) with x = xφhave a bounded uniformly continuous mild solution on R.

Proof. Let φ1,φ2be given by (3.10).

Case φ1: Let K1= sp(φ1), iK2= σ(A)∩iR and η = d(K1,K2). Then K1,K2⊂

(−c,c), where c is defined above (3.9). By Lemma 2.4 (iv) there exist W, ϕ and

a bounded sequence of trigonometric polynomials (πn) such that πn(t) → φ1(t) as

n → ∞ uniformly on each bounded interval of R and the Fourier exponents of (πn)

belong to W, where W is an open set W ⊃ K1with d(K2,W) ≥ d(K2, suppϕ) ≥

η/2, ϕ ∈ D(R) and ϕ = 1 on W. Set Q(λ) = ϕ(λ)R(iλ,A) on R \ K2 and

0 on K2. Since supp ϕ ∩ K2 = ∅, Q is infinitely differentiable on R and has

compact support, with?Q ∈ Cb(R,L(X)) ∩ L1(R,L(X)) by partial integration,

so with Proposition 2.5 the Q is the Fourier transform of a bounded continuous

function F ∈ L1(R,L(X)). We claim that uφ1(t) = (F ∗ φ1)(t) is a mild solution

on R of the equation u′(t) = Au(t) + φ1(t), u(0) = (F ∗ φ1)(0). Using (3.8)and

sp(πn) ⊂ W, one can show that un(t) := (F ∗ πn)(t) is a classical solution on R

of the equation u′(t) = Au(t) + πn(t), u(0) := (F ∗ πn)(0) and therefore is a mild

solution. So by Lemma 1.1, un(t) = T(t−t0)(F ∗πn(t0))+?t

t0T(t−s)πn(s)ds for

each t ≥ t0∈ R, un(0) = (F ∗πn)(0), n ∈ N. Passing to the limit when n → ∞, we

get with Lemma 2.4 (i), (ii) and the dominated convergence theorem

(F ∗ φ1)(t) = T(t − t0)(F ∗ φ1)(t0) +?t

t0T(t − s)φ1(s)ds for each t ≥ t0∈ R.

Now Lemma 1.1 shows that uφ1(t) is a mild solution on R of the equation (1.1)

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12 BASIT AND G¨UNZLER

with φ = φ1and x = (F ∗ φ1)(0).

Case φ2: By (3.11) case φ2, one has sp(φ2) ∩ [−b,b] = ∅. It follows Pφ2 ∈

Cub(R,X) by [9, Corollary 4.4 valid also for X-valued functions]. By Lemma 2.3,

sp(φ2) ⊂ sp(Pφ2) ⊂ sp(φ2)∪{0}. With h ∈ S(R) such that?h(λ) = 1 on [−a/2,a/2]

and supp?h ⊂ (−a,a), one has sp((Pφ2)∗h) ⊂ {0}, (Pφ2)∗h = x0∈ X [3, Theorem

4.2.2], so by Lemma 2.3 (ii) the spectrum of ψ := Pφ2−(Pφ2)∗h satisfies sp(ψ) ⊂

sp(φ2). By Lemma 2.4 (iii) there exists a sequence of trigonometric polynomials

(πn) such that πn(t) → ψ(t) as n → ∞ uniformly on each bounded interval of R

and the Fourier exponents of (πn) belong to R \ [−b,b]. Using (3.8), one can show

that un(t) := (G∗πn)(t) is a classical and then a mild solution on R of the equation

u′(t) = Au(t)+πn(t), u(0) := G∗πn(0), where G = G3defined by (3.7) case m = 3.

So by Lemma 1.1 (1.4), un(t) = T(t−t0)(G∗πn(t0))+?t

t0T(t−s)πn(s)ds for each

t ≥ t0∈ R, un(0) = G ∗ πn(0), n ∈ N. Passing to the limit when n → ∞, we get

with Lemma 2.4 (i), (ii), G ∈ L1(R,L(X)) by Lemma 3.4(iii)

(G ∗ ψ)(t) = T(t − t0)(G ∗ ψ)(t0) +?t

t0T(t − s)ψ(s)ds for each t ≥ t0∈ R.

By Proposition 2.1 (i) we conclude that G ∗ ψ,G ∗ φ2 ∈ Cub(R,X). By (2.4),

(G ∗ ψ) ∈ C1(R,X).By Lemma 1.1 and [1, Proposition 3.1.15 valid also for

J = R]), G ∗ ψ is a classical solution on R of the equation w′(t) = Aw(t) + ψ(t),

w(0) = G∗ψ(0). So, G∗ψ(t) ∈ D(A) for each t ∈ R. Since G∗ψ = G∗Pφ2−G∗x0

and G ∗ x0= 0 by Lemma 3.4 (v), it follows G ∗ Pφ2(t) ∈ D(A) for each t ∈ R,

with Proposition 2.1 then

(G∗ψ)′(t) = G∗φ2(t) = A(G∗(Pφ2−x0)(t))+Pφ2(t)−x0= A(P(G∗φ2)(t))+

A(G∗(Pφ2)(0))+(Pφ2)(t)−x0= −x0+A(G∗(Pφ2))(0))+AP(G∗φ2)(t)+Pφ2(t),

with P(G ∗ φ2)(t) ∈ D(A) for each t ∈ R. It follows that G ∗ φ2(0) = −x0+

A(G ∗Pφ2)(0)). Hence v = G∗φ2is a mild solution of v′(t) = Av(t) + φ2(t) on R,

v(0) = G ∗ φ2(0) by (1.3).

Finally, uφ:= F ∗ φ1+ G ∗ φ2is a mild solution of (1.1) on R with x = uφ(0) =

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F ∗ φ1(0) + G ∗ φ2(0) = xφbounded and uniformly continuous by the above. ¶

Theorem 3.6. Let A, φ be as in Theorem 3.5 and let φ1,φ2be as in (3.10). There

exist F,G ∈ L1(R,L(X)) so that

(i) For each x ∈ X, u(t) = T(t)x+F ∗φ1−F ∗φ1(0)+G∗φ2−G∗φ2(0) is the

unique mild solution on R+of (1.1). Moreover, u ∈ Cub(R+,X).

(ii) In addition, if J = R, (T(t)) is a bounded C0-group (supt∈R||T(t)|| < ∞),

then for each x ∈ X equations (1.1),(1.2) have a unique mild solution on R given

by u(t) = T(t)x+F ∗φ1−F ∗φ1(0)+G∗φ2−G∗φ2(0). Moreover, u ∈ Cub(R,X).

Proof. (i) By [1, Proposition 3.1.16] the solutions of v′= Av on R+are given by

v(t) = T(t)x, (i) follows by Theorem 3.5.

(ii) With Theorem 3.5 only u′= Au, u(0) = x , has to be considered.

Uniqueness: (1.4) gives u(0) = T(0 − (−n))u(−n) = T(n)u(−n),u(−n) =

T(−n)u(0), u(t) = T(t − (−n))T(−n)u(0) = T(t)u(0), t > −n, n ∈ N.

Existence: u(t) := T(t)x, t ∈ R, gives u(t0) = T(t0)x or x = T(−t0)u(t0), so

u(t) = T(t)T(−t0)u(t0) = T(t − t0)u(t0), with Lemma 1.1 u is a mild solution on

R. ¶

§4. Existence of generalized almost periodic

solutions of equation (1.1) in the non-resonance case

For A ⊂ L1

loc(R,X) we define mean classes MA by ([5, p. 120, Section 3 ])

(4.1)

MA := {f ∈ L1

Mhf(t) = (1/h)?h

loc(R,X) : Mhf ∈ A, h > 0}, where

0f(t + s)ds.

Usually A ⊂ MA ⊂ M2A ⊂ ··· with the ⊂ in general strict (see [6, Proposition

2.2, Example 2.3]).

We denote by F any class of functions having the following properties:

(4.2) F linear ⊂ L1

loc(J,X) ⊂ XR.

(4.3) (φn) ⊂ F ∩ Cuband φn→ ψ uniformly on R implies ψ ∈ F.

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14 BASIT AND G¨UNZLER

(4.4) φ ∈ F, a ∈ R implies φa∈ F.

(4.5) B ◦ φ ∈ F for each B ∈ L(X), φ ∈ F ∩ Cub.

Lemma 4.1. If F satisfies (4.2)-(4.5) and φ ∈ L∞(R,X)∩MF, F ∈ L1(R,L(X))

respectively L1(R,C) then F ∗ φ ∈ F ∩ Cub(R,X).

Proof. By Proposition 2.1 (i), F ∗ φ exists and ∈ Cub. To F there is a sequence

of L(X)-valued step-functions Hn =?mn

j=1Bjχ[αj,βj)with ||F − Hn||L1 → 0, so

||F ∗ φ − Hn∗ φ||∞ ≤ ||φ||∞||F − Hn||L1 → 0 as n → ∞.With (4.2), (4.3)

it is enough to show (BχI) ∗ φ ∈ F, I = [α,β) for each B ∈ L(X). With [1,

Proposition 1.6] one has (BχI)∗φ = B(χI∗φ), with (4.5) we have to show ψα,β:=

?β

0

φ ∈ MF gives?h

−?−h

0

φ(· + s)ds ⊂ F, which gives ψα,β ∈ F. The proof of F ∈ L1(R,C) is the

αφ(· − s)ds =

?−α

0φ(·+s)ds ∈ F if h > 0, (4.4) for a = −h then?h

−βφ(· + s)ds =

?−α

φ(· + s)ds −?0

−βφ(· + s)ds ∈ F. Now

0φ(·−h+s)ds =

same. ¶

Examples of F satisfying (4.2)-(4.5) are the spaces of almost periodic func-

tions AP = AP(R,X), almost automorphic functions AA(R,X), Bochner almost

automorphic functions BAA(R,X), bounded Levitan almost periodic functions

LAPb(R,X) [7, p. 430], linear subspaces with (4.2) of bounded recurrent functions

of RECb(R,X) = RC of [7, p. 427], Eberlein almost periodic functions EAP(R,X)

and so on.

Theorem 4.2. Let A, φ be as in Theorem 3.5 and φ ∈ MF with F satisfying

(4.1), (4.2)-(4.5). Then there is xφ∈ X such that the equation (1.1), (1.2) with

x = xφhas a bounded uniformly continuous mild solution on R which belongs to F.

Proof. By Theorem 3.5,uφ:= F ∗ φ1+ G ∗ φ2is a mild solution of (1.1) on R with

x = uφ(0) = F ∗φ1(0)+G∗φ2(0) = xφ, bounded and uniformly continuous. Since

φ ∈ MF ∩ L∞, φ1 = φ ∗ f ∈ F ∩ Cub by Lemma 4.1; since F ∩ Cub ⊂ MF by

[6, Proposition 2.2, p.1011], φ1∈ MF. Again with Lemma 4.1 one gets F ∗ φ1∈

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ADMISSIBILITY15

F ∩ Cub. Similarly, φ2= φ − φ1∈ MF ∩ L∞and so G ∗ φ2∈ F ∩ Cubby Lemma

4.1. ¶

We should remark for example that if φ is bounded Stepanoff Sp-almost periodic,

the uφ∈ AP [5, (3.8), p. 134]. Also, if φ is a Veech almost automorphic function

[19], then uφis a uniformly continuous Bochner almost automorphic function [21,

p. 66], [7, (3.3)].

Example 4.3. X = Yn, Y complex Banach space, A = complex n × n matrix, u,

φ X-valued in (1.1), φ ∈ L∞(R,X). Then, if sp(φ) contains no purely imaginary

eigen-value of A and φ ∈ MF, F with (4.1)-(4.5), then (1.1) has a mild solution

on R which belongs to F ∩ Cub. This extends a well known result of Favard [13, p.

98-99].

Another example would be a result on the almost periodicity of all solutions

of the inhomogeneous wave equation in the non-resonance case [20, p. 179, 181

Th´ eor` eme III.2.1], [1, Proposition 7.1.1], here one has a C0-group, all solutions of

the homogeneous equation are almost periodic.

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E-mail ”guenzler@math.uni-kiel.de”.

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