FAMILIES ON THE REAL LINE

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FAMILIES ON THE REAL LINE

ADINA LUMINIT¸A SASU

Received 20 October 2004; Accepted 26 September 2005

We give necessary and suﬃcient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair (L^p(R,X),L^q(R,X)). We show that the admissibility of the pair (L^p(R,X),L^q(R,X)) is equivalent to the uniform exponential dichotomy of an evolution family if and only ifp≥q. As applications we obtain charac- terizations for uniform exponential dichotomy of semigroups.

Copyright © 2006 Adina Luminit¸a Sasu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Exponential dichotomy is one of the most important asymptotic properties of evolution equations (see [1–5,7–10,12,15,19–25]). In the last few years new concepts of exponential dichotomy have been introduced and characterized, using discrete and continuous- time methods.

Integral equations have proved to be significant tools in the study of the asymptotic be- haviour ofC0-semigroups, evolution families, and linear skew-product flows, respectively (see [7–10,19–21,23,24]). For an evolution familyᐁ= {U(t,s)}t,s∈J,t≥s, one considered the integral equation

f(t)=U(t,s)f(s) + _t

sU(t,τ)v(τ)dτ, t≥s,t,s∈J, (E_ᐁ) whereJ∈ {R+,R}. In caseJ=R+, an important result has been proved by Van Minh et al. [24] and it is given by the following.

Theorem 1.1. Letᐁ= {U(t,s)}t≥s≥0be an evolution family such that for everyx∈Xthe mapping (t,s)→U(t,s)xis continuous. Then,ᐁis uniformly exponentially dichotomic if and only if for everyv_∈C0(R+,X) there is f _∈C0(R+,X) such that the pair (f,v) verifies (E_ᐁ) and the subspaceY1={x∈X: sup_t_≥₀U(t, 0)x<∞}is closed and complemented inX.

Hindawi Publishing Corporation Abstract and Applied Analysis

Volume 2006, Article ID 31641, Pages1–16 DOI10.1155/AAA/2006/31641

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Theorem 1.1has been generalized for the case of evolution families with nonuniform exponential growth in [8]. There we have proved that in the nonuniform case, the solvability inC0(R+,X) of (E_ᐁ) implies the nonuniform exponential dichotomy of the evolution familyᐁ= {U(t,s)}t≥s≥0. The discrete-time version ofTheorem 1.1has been obtained in [9] for the case of discrete and continuous evolution families. Characterizations for uniform exponential dichotomy of evolution families on the half-line withL^p-spaces were obtained in [19,23].

For the caseJ=R, a significant result has been obtained by Latushkin et al. [7], as shown in the following.

Theorem 1.2. Letᐁ= {U(t,s)}t≥sbe an evolution family such that for everyx∈X the mapping (t,s)→U(t,s)x is continuous, and letᏲ(R,X) be one of the spaces Cb(R,X), C0(R,X) orL^p(R,X), (p∈[1,∞)). Then,ᐁis uniformly exponentially dichotomic if and only if for everyv∈Ᏺ(R,X) there is a unique f ∈Ᏺ(R,X) such that the pair (f,v) verifies (E_ᐁ).

The main tool in [7] was the use of the evolution semigroup associated toᐁ.Theorem 1.2has been generalized in [10], where pointwise and global exponential dichotomy of a linear skew-product flowπ=(Φ,σ) is expressed in terms of the unique solvability in C0(R,X) of an associated integral equation:

f(t)=Φσ(θ,s),t−sf(s) + _t

sΦσ(θ,τ),t−τv(τ)dτ, t≥s. (Eπ) The purpose of the present paper is to give general characterizations for uniform exponential dichotomy of evolution families on the real line. The proofs are direct, the methods being based on input-output techniques, on the use of some specific operators associated to the integral equation (E_ᐁ), and on the properties of certain subspaces related to the evolution family. We will obtain that the admissibility of the pair (L^p(R,X),L^q(R,X)), withp,q∈[1,∞), is a suﬃcient condition for uniform exponential dichotomy of evolution families, and it becomes necessary forp≥q.

Finally, we apply our results in order to obtain necessary and suﬃcient conditions for uniform exponential dichotomy of aC0-semigroup in terms of the unique solvability of an integral equation associated to it.

2. Evolution families

Let X be a real or complex Banach space. The norm on X and onᏮ(X), the Banach algebra of all bounded linear operators onX, will be denoted by · .

Definition 2.1. A familyᐁ= {U(t,s)}t≥sof bounded linear operators onXis called an evolution family if the following properties hold:

(i)U(t,t)=I, for allt∈R;

(ii)U(t,s)U(s,t0)=U(t,t0), for allt≥s≥t0;

(iii) for everyx∈Xand everyt,t0, the mappings→U(s,t0)xis continuous on [t0,∞) and the mappings→U(t,s)xis continuous on (−∞,t];

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(iv) there existM≥1 andω >0 such that

Ut,t0≤Me^ω(t⁻^t⁰⁾, ∀t≥t0. (2.1)

Definition 2.2. An evolution familyᐁ= {U(t,s)}t≥sis said to be uniformly exponentially dichotomic if there are a family of projections{P(t)_}_t_∈R and two constantsK_≥1 and ν>0 such that

(i)U(t,t0)P(t0)=P(t)U(t,t0), for allt≥t0;

(ii)U(t,t0)x ≤Ke⁻^ν(t⁻^t⁰⁾x, for allx∈ImP(t0) and allt≥t0; (iii)U(t,t0)y ≥(1/K)e^ν(t⁻^t⁰⁾y, for ally∈KerP(t0) and allt≥t0;

(iv) the restrictionU(t,t0)_|: KerP(t0)→KerP(t) is an isomorphism, for allt≥t0. Lemma 2.3. If the evolution familyᐁ= {U(t,s)}t≥sis uniformly exponentially dichotomic relative to the family of projections{P(t)}t∈R, then sup_t_∈RP(t)<∞and for everyx∈X, the mappingt→P(t)xis continuous.

Proof. This is a simple exercise.

Letᐁ= {U(t,s)}t≥sbe an evolution family onXand letp∈[1,∞). For everyt0∈R, we consider the linear subspace

X1

t0

=

x∈X: _∞

t0

Ut,t0

x^pdt <∞

. (2.2)

We denote byᏲᐁ(t0) the set of all functionsϕ:R−→Xwith the property thatϕ(t)= U(t+t0,s+t0)ϕ(s), for alls≤t≤0.

Remark 2.4. Ifϕ∈Ᏺᐁ(t0), thenϕis continuous onR−.

For everyt0∈R, we denote byX2(t0) the linear space of allx∈Xwith the property that there is a functionϕx∈Ᏺᐁ(t0) such thatϕx(0)=xand _−∞⁰ ϕx(t)^pdt <∞. Lemma 2.5. Ifᐁ= {U(t,s)}t≥sis an evolution family, thenU(t,t0)Xk(t0)⊂Xk(t), for all t≥t0and allk∈ {1, 2}.

Proof. This is immediate.

Proposition 2.6. If the evolution familyᐁ= {U(t,s)}t≥sis uniformly exponentially di- chotomic relative to the family of projections{P(t)}t∈R, thenX1(t0)=ImP(t0) andX2(t0)= KerP(t0), for everyt0∈R.

Proof. Let M≥1, ω >0 be given by Definition 2.1 and let K≥1, ν>0 be given by Definition 2.2. Lett0∈R.

It is easy to see that ImP(t0)⊂X1(t0). Ifx∈X1(t0), letα_x:=(_t^∞₀ U(t,t0)x^pdt)^{1/ p}. Forτ≥t0+ 1, from

Uτ,t0

x≤Me^ωUt,t0

x, ∀t∈[τ−1,τ], (2.3)

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it follows that

Uτ,t0

x≤Me^ωαx, ∀τ≥t0+ 1. (2.4) This implies thatqx:=sup_t_≥_t₀U(t,t0)x<∞. Then from

x−Pt0

x≤Ke⁻^ν(t⁻^t⁰⁾Ut,t0

I−Pt0

x

≤Ke⁻^ν^(t⁻^t⁰⁾q_x+KPt0

x, ∀t≥t0,

(2.5) we obtain thatx∈ImP(t0).

Ifx∈KerP(t0), we defineψx:R−→X,ψx(t)=U(t0,t0+t)⁻_|¹x, where for everyt≤0, U(t0,t0+t)⁻_|¹denotes the inverse of the operatorU(t0,t0+t)_|: KerP(t0+t)→KerP(t0).

Then,ψx(0)=x,ψx∈Ᏺᐁ(t0), and

ψx(t)≤Ke^νtx, ∀t≤0, (2.6)

sox∈X2(t0).

Letx∈X2(t0). Then there isϕ_x∈Ᏺᐁ(t0) such that ϕ_x(0)=x, λ_x:=0

−∞

ϕ_x(t)^pdt 1/ p

<∞. (2.7)

Lett≤0. From

ϕx(t)=Ut+t0,s+t0

ϕx(s), ∀s∈[t−1,t], (2.8) it follows that

ϕ_x(t)≤Me^ωλ_x, ∀t≤0. (2.9)

Then from Pt0

x=Ut0,t0+tPt0+tϕ_x(t)≤Ke^νtPt0+tϕ_x(t)

≤KMe^ωλxsup

s∈R

P(s)e^νt, ∀t≤0, (2.10)

it follows thatP(t0)x=0, sox∈KerP(t0).

Remark 2.7. If an evolution family ᐁ= {U(t,s)_}_t_≥_s is uniformly exponentially dichotomic with respect to a family of projections, then according to the above result this family of projections is uniquely determined.

3. Exponential dichotomy and admissibility of the pair (L^p(R,X),L^q(R,X)) for evolution families

LetXbe a Banach space and letᏴ(R,X) be the space of all Bochner measurable func- tionsv:R→X, identifying the functions which are equal almost everywhere. For every

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p∈[1,∞), the linear space L^p(R,X)=

v∈Ᏼ(R,X) : _∞

−∞

v(τ)^pdτ <∞

(3.1) is a Banach space with respect to the norm

vp:= _∞

−∞

v(τ)^pdτ 1/ p

. (3.2)

Letᐁ= {U(t,s)}t≥sbe an evolution family onXand letp,q∈[1,∞). We consider the integral equation

f(t)=U(t,s)f(s) + _t

sU(t,τ)v(τ)dτ, ∀t≥s, (E_ᐁ) with f ∈L^p(R,X) andv∈L^q(R,X).

Definition 3.1. The pair (L^p(R,X),L^q(R,X)) is said to be admissible for the evolution familyᐁ= {U(t,s)}t≥sif for everyv∈L^q(R,X) there is a unique f ∈L^p(R,X) such that the pair (f,v) verifies (E_ᐁ).

If the pair (L^p(R,X),L^q(R,X)) is admissible for the evolution familyᐁ= {U(t,s)}t≥s, then it makes sense to define the operator

Γ:L^q(R,X)−→L^p(R,X), Γv=f . (3.3) It is easy to see thatΓis linear and it is closed. It follows thatΓis bounded, so there is γ >0 such thatΓvp≤γvq, for allv∈L^q(R,X).

Proposition 3.2. If the pair (L^p(R,X), L^q(R,X)) is admissible for the evolution family ᐁ= {U(t,s)}t≥s, then

(i)X1(t0)∩X2(t0)= {0}, for allt0∈R; (ii)X1(t0) +X2(t0)=X, for allt0∈R;

(iii) the restrictionU(t,t0)_|:X2(t0)→X2(t) is an isomorphism, for allt≥t0.

Proof. (i) Lett0∈Rand letx∈X1(t0)∩X2(t0). Then, there is a functionϕx∈Ᏺᐁ(t0) such thatϕx(0)=xand _−∞⁰ ϕx(t)^pdt <∞. We define

f :R−→X, f(t)=

⎧⎨

⎩ Ut,t0

x, t > t0, ϕ_xt−t0

, t≤t0. (3.4)

Then, it is easy to see that f(t)=U(t,s)f(s), for allt≥s. Sincex∈X1(t0), we obtain that f ∈L^p(R,X). It follows thatf =0, sox= f(t0)=0.

(ii) Letx∈Xand lett0∈R. We consider the function v:R−→X, v(τ)=χ_[t0,t0+1](τ)Uτ,t0

x, (3.5)

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whereχ[t0,t0+1]denotes the characteristic function of the interval [t0,t0+ 1]. From hypothesis, there is f ∈L^p(R,X) such that the pair (f,v) verifies (E_ᐁ). Then

f(t)=Ut,t0

ft0

+ _t

t0

U(t,τ)v(τ)dτ=Ut,t0

ft0

+x, ∀t≥t0+ 1. (3.6)

Since f ∈L^p(R,X), it follows that f(t0) +x∈X1(t0). Letϕ:R−→X,ϕ(t)= f(t+t0).

From the fact that the pair (f,v) verifies (E_ᐁ), it follows that

ϕ(t)=Ut0+t,t0+sϕ(s), ∀s≤t≤0 (3.7) which shows thatϕ∈Ᏺᐁ(t0). Since f ∈L^p(R,X), it follows that f(t0)∈X2(t0). Finally, we obtain thatx=(x+f(t0))−f(t0)∈X1(t0) +X2(t0).

(iii) Lett > t0. Lety∈KerU(t,t0)∩X2(t0), and letϕ_y∈Ᏺᐁ(t0) withϕ_y(0)=yand

0

−∞ϕy(s)^pds <∞. Considering the function h:R−→X, h(τ)=

⎧⎨

⎩ Uτ,t0

y, τ > t0, ϕ_yτ−t0

, τ≤t0, (3.8)

we have thath∈L^p(R,X). It is easy to observe that the pair (h, 0) verifies (E_ᐁ). This implies thath=0. In particular, it follows that y=h(t0)=0, so, the operatorU(t,t0)_|: X2(t0)→X2(t) is injective.

To prove the surjectivity, letx∈X2(t). We consider the functions v:R−→X, v(τ)= −χ[t,t+1](τ)U(τ,t)x, f : [t,∞)−→X, f(τ)=

⎧⎨

⎩

(t+ 1−τ)U(τ,t)x, τ∈[t,t+ 1],

0, τ > t+ 1.

(3.9)

We observe thatv∈L^q(R,X) and f(r)=U(r,s)f(s) +

_r

sU(r,τ)v(τ)dτ, ∀r≥s≥t. (3.10) From hypothesis there isg∈L^p(R,X) such that the pair (g,v) verifies (E_ᐁ). It follows that f(r)−g(r)=U(r,t)(f(t)−g(t)), for allr≥twhich implies thatx−g(t)= f(t)−g(t)∈ X1(t).

From (ii) there is y1∈X1(t0) and y2∈X2(t0) such thatg(t0)=y1+y2. Sinceg(t)= U(t,t0)g(t0), we obtain that g(t)=U(t,t0)y1+U(t,t0)y2, then x−U(t,t0)y2=(x− g(t)) +U(t,t0)y1. FromLemma 2.5and from (i), we deduce thatx−U(t,t0)y2=0, so x∈U(t,t0)X2(t0).

This shows that the operatorU(t,t0)_|:X2(t0)→X2(t) is surjective and completes the

proof.

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Lemma 3.3. Lett0< t1≤ ∞and letα: [t0,t1)→R+be a continuous function with the prop- erty that there areM≥1 andω,h∈(0,∞) such that

α(t)≤Me^ω(t⁻^s)α(s), ∀s,t∈ t0,t1

,s≤t, (3.11)

_t+2h

t+h α(τ)dτ≤1 e

_t+h

t α(τ)dτ, (3.12)

for everyt∈[t0,t1) witht+ 2h < t1. Then α(t)≤Ke⁻^ν(t⁻^t⁰⁾αt0

, ∀t∈ t0,t1

, (3.13)

whereK=(Me)²e^3ωhandν=1/h.

Proof. Lett∈[t0,t1),n∈N, andr∈[0,h) such thatt=t0+nh+r. Ifn≥2, then _t₀_+nh

t0+(n−1)hα(τ)dτ≤e⁻⁽ⁿ⁻¹⁾ _t₀_+h

t0

α(τ)dτ. (3.14)

Using the relation (3.11), we have that _t₀_+h

t0

α(τ)dτ≤Mhe^ωhαt0

, α(t)≤M he^2ωh

_t₀_+nh

t0+(n−1)hα(τ)dτ. (3.15) From relations (3.14)–(3.15), it follows that

α(t)≤M²e^3ωhe⁻⁽ⁿ⁻¹⁾αt0

. (3.16)

Denotingν=1/hand takingK=(Me)²e^3ωh, we obtain that α(t)≤Ke⁻^ν^(t⁻^t⁰⁾αt0

. (3.17)

Ifn∈ {0, 1}, thent−t0<2h. It follows that α(t)≤Me^2ωhαt0

≤Ke⁻^ν^(t⁻^t⁰⁾αt0

. (3.18)

Theorem 3.4. If the pair (L^p(R,X),L^q(R,X)) is admissible for the evolution familyᐁ= {U(t,s)}t≥s, then there existK≥1 andν>0 such that

Ut,t0

x≤Ke⁻^ν(t⁻^t⁰⁾x, ∀x∈X1 t0

,∀t≥t0. (3.19) Proof. From hypothesis there isγ≥1 such that

Γvp≤γvq, ∀v∈L^q(R,X). (3.20) We denoteh=(γe)^p.

Lett0∈R, letx∈X1(t0)\ {0}, and lett1=sup{t≥t0:U(t,t0)x=0}. We consider the functionϕ: [t0,t1)→X,ϕ(t)=U(t,t0)x.

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Ift1> t0+ 2h, for everyt≥t0witht+ 2h < t1, we consider the functions v:R−→X, v(τ)=χ[t,t+h](τ) ϕ(τ)

ϕ(τ), f :R−→X, f(τ)=

_τ

−∞

χ[t,t+h](s) ϕ(s) dsϕ(τ).

(3.21)

We have thatv∈L^q(R,X) and since x∈X1(t0), it follows that f ∈L^p(R,X). It is easy to see that the pair (f,v) verifies (E_ᐁ), soΓv= f. From (3.20) it follows thatfp≤ γvq=γh^1/q. In particular, this inequality shows that

t+2h t+h

f(τ)^pdτ 1/ p

≤γh^1/q. (3.22)

We denoteδ= _t^t+h(1/ϕ(s))ds. Then, from (3.22) we deduce that _t+2h

t+h

ϕ(τ)^pdτ 1/ p

≤γ

δh^1/q. (3.23)

Let

h=

⎧⎪

⎨

⎪⎩

1, forp=1,

h^{1/ p}, forp∈(1,∞), p= p

p−1. (3.24)

Then, we have

_t+2h

t+h

ϕ(τ)dτ≤h t+2h

t+h

ϕ(τ)^pdτ 1/ p

. (3.25)

Using (3.23), we deduce that _t+2h

t+h

ϕ(τ)dτ≤γ

δhh^1/q. (3.26)

Since

h²≤ t+h

t

ϕ(τ)1 dτ t+h

t

ϕ(τ)dτ

=δ t+h

t

ϕ(τ)dτ

(3.27) from (3.26) we obtain that

_t+2h

t+h

ϕ(τ)dτ≤γhh^1/q h²

_t+h

t

ϕ(τ)dτ≤γh h

_t+h

t

ϕ(τ)dτ. (3.28)

By the definition ofh, from (3.28) it follows that _t+2h

t+h

ϕ(τ)dτ≤1 e

_t+h

t

ϕ(τ)dτ. (3.29)

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LetM,ωbe given byDefinition 2.1. ApplyingLemma 3.3forα= ϕ, it follows that ϕ(t)≤Ke⁻^ν(t⁻^t⁰⁾ϕt0, ∀t∈

t0,t1

, (3.30)

whereK=(Me)²e^3ωhandν=1/h.

BecauseKandνdo not depend ont0orx, the proof is complete.

Corollary 3.5. If the pair (L^p(R,X),L^q(R,X)) is admissible for the evolution familyᐁ= {U(t,s)}t≥s, thenX1(t0) is a closed linear subspace, for allt0∈R.

Proof. Lett0∈Rbe fixed and let (xn)⊂X1(t0) be a sequence convergent tox∈X. It fol- lows that there isL >0 such thatxn ≤L, for alln∈N. IfK,νare given byTheorem 3.4, we deduce thatU(t,t0)x_n ≤KLe⁻^ν(t⁻^t⁰⁾, for all t≥t0 and all n∈N. Hence, we obtain thatU(t,t0)x ≤KLe⁻^ν^(t⁻^t⁰⁾, for all t≥t0, sox∈X1(t0). It follows thatX1(t0) is

closed.

Lemma 3.6. Letα: [t0,∞)→R+be a continuous function with the property that there are M≥1 andω,h∈(0,∞) such that

α(t)≤Me^ω(t⁻^s)α(s), ∀t≥s≥t0, (3.31) _t+2h

t+h α(τ)dτ≥e _t+h

t α(τ)dτ, ∀t≥t0. (3.32) Then

α(t)≥ 1

Ke^ν(t⁻^s)α(s), ∀t≥s≥t0+h, (3.33) whereK=M²e^3ωhandν=1/h.

Proof. Lett > s≥t0+h,n∈N, andr∈[0,h) such that t−s=nh+r. From (3.32) it follows that

_s+(n+2)h

s+(n+1)hα(τ)dτ≥eⁿ⁺² _s

s−hα(τ)dτ. (3.34)

Using the relation (3.31), we have that _s+(n+2)h

s+(n+1)hα(τ)dτ≤Mhe^2ωhα(t), hα(s)≤Me^ωh _s

s−hα(τ)dτ. (3.35) From (3.34)–(3.35), it follows that

α(t)≥ eⁿ⁺²

M²e^3ωhα(s)≥ 1

Ke^ν^(t⁻^s)α(s), (3.36)

whereν=1/handK=M²e^3ωh.

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Theorem 3.7. If the pair (L^p(R,X),L^q(R,X)) is admissible for the evolution familyᐁ= {U(t,s)}t≥s, then there existK≥1 andν>0 such that

Ut,t0

y≥ 1

Ke^ν(t⁻^t⁰⁾y, ∀y∈X2

t0),∀t≥t0. (3.37) Proof. From hypothesis there isγ≥1 such that

Γvp≤γvq, ∀v∈L^q(R,X). (3.38) We denoteh=(γe)^p.

Lett0∈Rand let y∈X2(t0)\ {0}. FromProposition 3.2(iii) there isz∈X2(t0−h)\ {0}such thatU(t0,t0−h)z=y. Denoting byϕ: [t0−h,∞)→X,ϕ(t)=U(t,t0−h)z, and usingProposition 3.2(iii), we have thatϕ(t)=0, for allt≥t0−h.

Lett≥t0−h. We consider the function

v:R−→X, v(τ)= −χ_[t+h,t+2h](τ) ϕ(τ)

ϕ(τ). (3.39)

Since

z1= t+2h

t+h

ϕ(s)ds

z∈X2

t0−h, (3.40)

there isλ∈Ᏺᐁ(t0−h) withλ(0)=z1and _−∞⁰ λ(s)^pds <∞. Let

f :R−→X, f(τ)=

⎧⎪

⎪⎨

⎪⎪

⎩ _∞

τ

χ[t+h,t+2h](s)

ϕ(s) dsϕ(τ), τ≥t0−h, λτ−t0+h, τ < t0−h.

(3.41)

We have thatv∈L^q(R,X), f ∈L^p(R,X), and the pair (f,v) verifies (E_ᐁ). SoΓv=f and from (3.38) it follows thatfp≤γvq=γh^1/q. In particular, from this inequality, we deduce that

_t+h

t

f(τ)^pdτ 1/ p

≤γh^1/q. (3.42)

We denoteδ= _t+h^t+2h(1/ϕ(s))ds. Then, from (3.42) we obtain that t+h

t

ϕ(τ)^pdτ 1/ p

≤γ

δh^1/q. (3.43)

Let

h=

⎧⎪

⎪⎨

⎪⎪

⎩

1, forp=1,

h^{1/ p}, forp∈(1,∞), p= p

p−1. (3.44)

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Using analogous arguments as in the proof ofTheorem 3.4, we immediately deduce that _t+h

t

ϕ(τ)dτ≤γhh^1/q h²

_t+2h

t+h

ϕ(τ)dτ≤1 e

_t+2h

t+h

ϕ(τ)dτ. (3.45)

LetM,ωbe given byDefinition 2.1. ApplyingLemma 3.6forα= ϕ, it follows that ϕ(t)≥ 1

Ke^ν^(t⁻^s)ϕ(s), ∀t≥s≥t0, (3.46) whereK=M²e^3ωhandν=1/h. This implies that

Ut,t0−hz_≥ 1

Ke^ν^(t⁻^t⁰⁾Ut0,t0−hz, ∀t_≥t0 (3.47) which means that

Ut,t0

y≥ 1

Ke^ν(t⁻^t⁰⁾y, ∀t≥t0. (3.48) SinceKandνdo not depend ont0ory, we obtain the conclusion.

Corollary 3.8. If the pair (L^p(R,X),L^q(R,X)) is admissible for the evolution familyᐁ= {U(t,s)}t≥s, thenX2(t0) is a closed linear subspace, for allt0∈R.

Proof. Lett0∈R. If y∈X2(t0) andϕ_y is a function given by the definition of the space X2(t0) withϕy(0)=y, then it is easy to see thatϕy(s)∈X2(t0+s), for alls≤0.

Let (xn)⊂X2(t0) be a sequence convergent tox∈X. For everyn∈Nthere is a func- tionϕn∈Ᏺᐁ(t0) such thatϕn(0)=xnand _−∞⁰ ϕn(τ)^pdτ <∞. Since

ϕ_n(0)=Ut0,t0+sϕ_n(s), ∀s≤0,∀n∈N, (3.49) forK,νgiven byTheorem 3.7, it follows that

x_n−x_m=Ut0,t0+sϕ_n(s)−ϕ_m(s)

≥ 1

Ke⁻^νsϕ_n(s)−ϕ_m(s), ∀s≤0,∀m,n∈N. (3.50) Using the fact that (xn) is fundamental, from (3.50) we obtain that for everys≤0 the sequence (ϕn(s)) is fundamental, so it is convergent. We denoteϕ(s) :=limn→∞ϕn(s), for alls≤0. Henceϕ(0)=xandϕ∈Ᏺᐁ(t0). From (3.50) we deduce that

ϕ(s)≤Ke^νsx_n−x+ϕ_n(s), ∀(s,n)∈R−×N. (3.51) This implies thatx∈X2(t0) and the proof is complete.

The first main result of this section is given by the following.

Theorem 3.9. If the pair (L^p(R,X),L^q(R,X)) is admissible for the evolution familyᐁ= {U(t,s)}t≥s, thenᐁis uniformly exponentially dichotomic.

(12)

Proof. FromProposition 3.2,Corollary 3.5, andCorollary 3.8, it follows that for everyt∈ R,X1(t)⊕X2(t)=X. LetP(t) be the projection corresponding toX1(t), that is, ImP(t)= X1(t) and KerP(t)=X2(t). UsingLemma 2.5, we have thatP(t)U(t,t0)=U(t,t0)P(t0), for allt≥t0. FromProposition 3.2, the restriction U(t,t0)_|: KerP(t0)→KerP(t) is an isomorphism, for allt≥t0. Finally, usingTheorem 3.4andTheorem 3.7, we obtain that

ᐁis uniformly exponentially dichotomic.

Theorem 3.9gives a suﬃcient condition for the uniform exponential dichotomy of an evolution family. In what follows, we will establish when the uniform exponential dichotomy of an evolution family implies the admissibility of the pair (L^p(R,X),L^q(R,X)).

Lemma 3.10. Letp,q∈[1,∞) withp≥q, letν>0, and letv∈L^q(R,R+). Then, the func- tions f1,f2:R→R+defined by

f1(t)= _t

−∞e⁻^ν(t⁻^s)v(s)ds, f2(t)= _∞

t e⁻^ν(s⁻^t)v(s)ds (3.52) belong toL^p(R,R+).

Proof. This follows using H¨older’s inequality.

Proposition 3.11. Letᐁ= {U(t,s)}t≥s be an evolution family and let p,q∈[1,∞). If X1(t0)∩X2(t0)= {0}, for allt0∈R, then for everyv∈L^q(R,X) there exists at most one

f ∈L^p(R,X) such that the pair (f,v) verifies (E_ᐁ).

Proof. Letv∈L^q(R,X). Suppose that there are f,f1∈L^p(R,X) such that the pairs (f,v) and (f1,v) verify (E_ᐁ). Then, we have

f1(t)−f(t)=U(t,s)f1(s)−f(s), ∀t≥s. (3.53) Lett0∈R. From f1(t)−f(t)=U(t,t0)(f1(t0)−f(t0)) and f,f1∈L^p(R,X), it follows that f1(t0)−f(t0)∈X1(t0).

Letψ:R−→X,ψ(s)= f1(t0+s)−f(t0+s). From (3.53) we obtain thatψ∈Ᏺᐁ(t0).

Because f1,f ∈L^p(R,X), it follows that ψ(0)= f1(t0)− f(t0)∈X2(t0). Using the hypothesis, we obtain that f1(t0)= f(t0). Since t0∈Rwas arbitrary, we deduce that f =

f1.

Theorem 3.12. Letᐁ= {U(t,s)}t≥sbe an evolution family and letp,q∈[1,∞) withp≥ q. Thenᐁis uniformly exponentially dichotomic if and only if the pair (L^p(R,X),L^q(R,X)) is admissible forᐁ.

Proof (Necessity). Let{P(t)}t∈Rbe the family of projections given byDefinition 2.2. For v∈L^q(R,X) we consider the function

f :R−→X, f(t)= _t

−∞U(t,s)P(s)v(s)ds− _∞

t U(s,t)⁻_|¹I−P(s)v(s)ds,

(3.54)