SEPARATIONS OF TYPE II WITH APPLICATIONS TO RANDOM DELAY DIFFERENTIAL EQUATIONS.
JANUSZ MIERCZY ´NSKI, SYLVIA NOVO, AND RAFAEL OBAYA
Abstract. This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to nonautonomous random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.
1. Introduction
This paper continues the study of the existence of principal Lyapunov exponents, principal Floquet subspaces and generalized exponential separations for positive random linear skew-product semiflows in ordered Banach spaces. In particular, the concept of generalized exponential separation of type II is introduced as a natural modification of the classical concept, to later show the applicability of this new theory in the context of nonautonomous functional differential equations with finite delay.
Lyapunov exponents play an important role in the study of deterministic and random skew-product semiflows. Oseledets [28] obtained important results on Lyapunov exponents and measurable invariant families of subspaces for finite-dimensional linear dynamical systems, currently referred to as Oseledets multiplica-tive ergodic teorem. An important number of alternamultiplica-tive proofs of this theorem as well as extensions of the theory to relevant infinite dimensional dynamical sys-tems have been provided (see Arnold [3], Doan [7], Gonz´alez-Tokman and Quas [9], Johnson et al. [14], Krengel [15], Lian and Lu [16], Ma˜n´e [17], Millionˇsˇcikov [23], Raghunathan [31], Ruelle [33,34], Thieullen [37], and the references therein).
The largest finite Lyapunov exponent (or top Lyapunov exponent) and its as-sociated invariant subspace play a relevant role in this theory. Classically, the top finite Lyapunov exponent of a positive deterministic or random dynamical system
2010Mathematics Subject Classification. Primary: 37H15, 37L55, 34K06, Secondary: 37A30, 37A40, 37C65, 60H25.
Key words and phrases. Random dynamical systems, focusing property, generalized principal Floquet bundle, generalized exponential separation, random delay differential equations.
The first author is supported by the NCN grant Maestro 2013/08/A/ST1/00275 and the last two authors are partly supported by MEC (Spain) under project MTM2015-66330-P and EU Marie-Sk lodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS).
in an ordered Banach space is called the principal Lyapunov exponent if the as-sociated invariant family of subspaces, where this Lyapunov exponent is reached, is one-dimensional and spanned by a positive vector. In this case the invariant subspace is called theprincipal Floquet subspace.
The exponential separation theory was initiated for positive discrete-time de-terministic dynamical systems by Ruelle [32], and developed later by Pol´aˇcik and Tereˇsˇc´ak [29, 30]. Given a strongly ordered Banach space X and θ: Ω → Ω a homeomorphism of a compact set Ω, if Ω×X → Ω×X, (ω, u) 7→ (θ(ω), Tωu)
defines a vector bundle map withTωcompact and strongly positive for eachω∈Ω,
then it admits a continuous decomposition X =E(ω)⊕F(ω) where E(ω) is the principal Floquet subspace,F(ω) does not contain any strictly positive vector and the bundle map exhibits an exponential separation on the sum. This statement was generalized by Shen and Yi [36] to continuous-time deterministic (topological) skew-product semiflowsR+×Ω×X 7→Ω×X,(t, ω, u)→(θtω, Uω(t)u), whereUω(t)
is strongly positive for eachω∈Ω andt >0. In a typical instance of application, Ω is the translation hull of the coefficients of some linear differential equation, that is, the closure, in an appropriate topology, of the set of all time-translates of the coefficients, and Uω(t) is the solution operator of the equation. Applications and
extensions of this theory in the context of nonautonomous ordinary and parabolic partial differential equations can be found in H´uska and Pol´aˇcik [12], H´uska [11], H´uska et al. [13], and Novo et al. [24, 25], among other references. The theory of principal Floquet spaces and exponential separations was developed further by Mierczy´nski and Shen [18, 19]. Among others, they consider random families of parabolic linear partial differential equations whose coefficients are evaluated along the trajectories of a measurable dynamical system on a probability space: a random family is of such a type that it is embedded into a continuous deterministic (nonau-tonomous) family of linear equations which in its turn generates a (topological) skew-product flow exhibiting an exponential separation.
Novoet al.[24] introduced a modification of the notion of exponential separation. A linear skew-product semiflowR+×Ω×X →Ω×X,(t, ω, u)→(θtω, Uω(t)u) is
considered and the strong monotonicity condition is substituted by the following dichotomy behaviour: there exist times 0< t1 < T such thatUω(t) is a compact
operator fort≥T−t1 andω∈Ω, and for each vectoru≥0 eitherUω(t1)u= 0 or
Uω(t1)u0. Under these assumptions, the existence of a continuous decomposi-tionX =E(ω)⊕F(ω) is proved, whereE(ω) is the principal Floquet subspace and the semiflow exhibits an exponential separation on the sum, but now F(ω)∩X+ is not void and contains those positive vectors u satisfying Uω(t1)u = 0. This dynamical behaviour is called exponential separation (or continuous separation) of type II, implicity referring to the classical concept as exponential separation of type I. Novo et al. [25], Calzada et al. [5] and Obaya and Sanz [26, 27] show the importance of the exponential separation of type II in the study of linear and nonlinear nonautonomous functional differential equations with finite delay.
In the case where the coefficients of the linear differential equation are driven by trajectories of a measurable flowθon a probability space (Ω,F,P), the natural setting is that of positive measurable linear skew-product semiflows: Uω(t) is a
positive linear operator depending measurably on ω ∈ Ω. The definition of the
thatE(ω) andF(ω) now depend measurably onω. Also, thegeneralized principal Lyapunov exponent is the largest Lyapunov exponent forP-a.e. ω∈Ω.
Arnoldet al. in [4] were the first to prove the existence of generalized exponential separation for discrete-time positive random dynamical systems generated by ran-dom families of positive matrices. Later, Mierczy´nski and Shen [20] provided the as-sumptions required for general random positive linear skew-product semiflows (with both discrete and continuous time) in order to admit generalized principal Floquet subspaces and generalized exponential separation of type I (in contrast to [18,19], no embedding into topological semiflows was used in the proofs). The applica-tion of this theory to a variety of random dynamical systems arising from Leslie matrix models, cooperative linear ordinary differential equations and linear para-bolic partial differential equations can be found in Mierczy´nski and Shen [21, 22]. In particular, the existence of generalized principal Floquet subspaces for random skew-product semiflows generated by cooperative families of delay differential equa-tions is also obtained in [22].
In this paper, the positivity conditions satisfied by the linear random dynamical systems are weakened, in order to assure the existence of generalized principal Flo-quet subspaces and the existence of generalized exponential separations of type II. The structure and main results of the paper is as follows. In Section 2 the notions and assumptions used throughout the paper are introduced. In particular,
X will be an ordered separable Banach space with dual X∗ separable and positive coneX+normal and reproducing. Conditions of integrability and positivity for the random linear skew-product semiflow are imposed, and a new focusing assumption is considered. More precisely, there is a positive time T > 0 such that if u ∈
X+ and ω ∈ Ω then U
ω(T)u = 0 or Uω(T)u is strictly positive and satisfies a
classical focusing inequality in the terms stated by Mierczy´nski and Shen [20]. For simplicity we fix the scale T = 1 throughout the paper. From these assumptions, the integrability, positivity and an alternative focusing property for the measurable dual skew-product semiflow are obtained.
Under the new focusing condition, in Section3the existence of a family of gener-alized principal Floquet subspaces is shown. Section4is devoted to the introduction of the new concept of generalized exponential separation of type II and the proof of the existence under the previously considered assumptions. Finally, Section 5 illustrates the application of the theory to random dynamical systems generated by scalar linear random delay differential equations with finite delay.
2. Preliminaries
A probability space is a triple (Ω,F,P), where Ω is a set, F is a σ-algebra of subsets of Ω, and P is a probability measure defined for all F ∈ F. We always assume that the measurePis complete.
Ameasurable dynamical system on the probability space (Ω,F,P) is a (B(R)⊗
F,F)-measurable mappingθ: R×Ω→Ω such that • θ(0, ω) =ω for anyω∈Ω,
• θ(t1+t2, w) =θ(t2, θ(t1, ω)) for anyt1,t2∈Rand anyω∈Ω.
We writeθ(t, ω) asθtω. Also, we usually denote measurable dynamical systems by
Ametric dynamical systemis a measurable dynamical system ((Ω,F,P),(θt)t∈R) such that for eacht∈Rthe mappingθt: Ω→Ω isP-preserving (i.e.,P(θ−1t (F)) =
P(F) for anyF ∈Fandt∈R). It is said to beergodic if for any invariantF ∈F, eitherP(F) = 1 orP(F) = 0.
2.1. Measurable linear skew-product semidynamical systems. We consider a separable Banach spaceX such that its dualX∗ is separable.
We writeR+for [0,∞). By ameasurable linear skew-product semidynamical
sys-tem or semiflow, Φ = ((Uω(t))ω∈Ω,t∈R+,(θt)t∈R) onX covering a metric dynamical system (θt)t∈Rwe understand a (B(R+)⊗F⊗B(X),B(X))-measurable mapping
[R+×Ω×X3(t, ω, u)7→Uω(t)u∈X]
satisfying
Uω(0) = IdX for each ω∈Ω, (2.1)
Uθsω(t)◦Uω(s) =Uω(t+s) for each ω∈Ω andt, s∈R
+, (2.2) [X 3u7→Uω(t)u∈X]∈ L(X) for each ω∈Ω andt∈R+.
Sometimes we write simply Φ = ((Uω(t)),(θt)). Eq. (2.2) is called the cocycle property.
Forω∈Ω, by anentire orbit ofUωwe understand a mapping vω:R→X such thatvω(s+t) =Uθsω(t)vω(s) for eachs∈Randt≥0.
Next we introduce the dual of Φ. For ω ∈ Ω, t ∈ R+ and u∗ ∈X∗ we define
Uω∗(t)u∗ by
hu, Uω∗(t)u∗i=hUθ−tω(t)u, u
∗i for eachu∈X (2.3) (in other words,Uω∗(t) is the mapping dual to Uθ−tω(t)).
As explained in [20], sinceX∗ is separable, the mapping [R+×Ω×X∗3(t, ω, u∗)7→Uω∗(t)u∗∈X∗]
is (B(R+)⊗F⊗B(X∗),B(X∗))-measurable. The measurable linear skew-product semidynamical system Φ∗ = ((Uω∗(t))ω∈Ω,t∈R+,(θ−t)t∈R) onX∗ covering (θ−t)t∈R will be called thedual of Φ. The cocycle property for the dual takes the form
Uθ−∗ tω(s)◦Uω∗(t) =Uω∗(t+s) for each ω∈Ω andt, s∈R+ (2.4) Let Ω0 ∈ F. A family {E(ω)}ω∈Ω0 of l-dimensional vector subspaces of X is
measurable if there are (F,B(X))-measurable functions v1, . . . , vl: Ω0 → X such
that (v1(ω), . . . , vl(ω)) forms a basis ofE(ω) for eachω∈Ω0.
Let {E(ω)}ω∈Ω0 be a family of l-dimensional vector subspaces of X, and let
{F(ω)}ω∈Ω0 be a family ofl-codimensional closed vector subspaces ofX, such that
E(ω)⊕F(ω) = X for all ω ∈Ω0. We define the family of projections associated with the decomposition E(ω)⊕F(ω) =X as{P(ω)}ω∈Ω0, whereP(ω) is the linear
projection ofX ontoF(ω) alongE(ω), for eachω∈Ω0.
The family of projections associated with the decompositionE(ω)⊕F(ω) =X is called strongly measurable if for eachu∈X the mapping [ Ω03ω 7→P(ω)u∈X] is (F,B(X))-measurable.
We say that the decomposition E(ω)⊕F(ω) = X, with {E(ω)}ω∈Ω0
A strongly measurable family of projections associated with the invariant de-compositionE(ω)⊕F(ω) =X is referred to astempered if
lim
t→±∞
lnkP(θtω)k
t = 0 P-a.e. on Ω0. (2.5)
2.2. Ordered Banach spaces. Let X be a Banach space with norm k · k. We say thatX is an ordered Banach space if there is a closed convex cone, that is, a nonempty closed subsetX+⊂X satisfying
(O1) X++X+⊂X+. (O2) R+X+⊂X+. (O3) X+∩(−X+) ={0}.
Then a partial ordering inX is defined by
u≤v ⇐⇒ v−u∈X+;
u < v ⇐⇒ v−u∈X+ and u6=v .
The cone X+ is said to be reproducing if X+−X+ =X. The cone X+ is said to be normalif the norm of the Banach space X issemimonotone, i.e., there is a positive constant k >0 such that 0 ≤u≤v implieskuk ≤kkvk. In such a case, the Banach space can be renormed so that for anyu, v ∈ X, 0 ≤u≤ v implies kuk ≤ kvk(see Schaefer [35, V.3.1, p. 216]). Such a norm is calledmonotone.
For an ordered Banach spaceX denote by (X∗)+ the set of allu∗ ∈ X∗ such thathu, u∗i ≥0 for allu∈X+. The set (X∗)+ has the properties of a cone, except that (X∗)+∩(−(X∗)+) ={0}need not be satisfied (such sets are called wedges).
If (X∗)+ is a cone we call it the dual cone. This happens, for instance, when
X+ is total (that is,X+−X+ is dense inX, which in particular holds when X+ is reproducing and this will be one of our hypothesis).
Nonzero elements ofX+ (resp. of (X∗)+) are called positive. We say that two positive vectorsu,v∈X+\ {0}arecomparable, writtenu∼v, if there are positive numbers, α, α, such thatα v ≤u≤α v. For a nonzero vectoru∈X+ we call the
component ofu
Cu={v∈X+\ {0} |v∼u}, (2.6)
i.e., the equivalence class ofu.
We now recall the concept of the Hilbert projective metric. Definition 2.1. Letu,v∈X.
(1) If{α∈R|α v≤u}is nonempty, define
m(u/v) := sup{α∈R|α v≤u}.
If{α∈R|u≤α v}is nonempty, define
M(u/v) := inf{α∈R|u≤α v}.
(2) If bothm(u/v) andM(u/v) exist, define theoscillationofuoverv, and if
u∼v theprojective distance betweenuandvas
osc(u/v) :=M(u/v)−m(u/v) and d(u, v) := lnM(u/v)
Lemma 2.2. Assume that X+ is normal. Then for anyu, v ∈X+,u∼v, with
kuk=kvk= 1, there holds
ku−vk ≤3 ed(u,v)−1.
2.3. Assumptions. Throughout the paper we will assume that X is an ordered separable Banach space with dimX ≥2 such that its dualX∗ is separable, with positive coneX+normal and reproducing. It follows then that the dual cone (X∗)+ is normal and reproducing, too (see [35, V.3]). We always assume that the norms onX and onX∗ are monotone.
Let Φ = ((Uω(t)),(θt)) be a measurable linear skew-product semidynamical
sys-tem onX covering an ergodic metric dynamical system (θt) on (Ω,F,P), and its dual Φ∗= ((Uω∗(t))ω∈Ω,t∈R+,(θ−t)t∈R) onX
∗ covering (θ −t)t∈R.
We now list assumptions we will make at various points in the sequel. (A1) (Integrability) The functions
Ω3ω7→ sup 0≤s≤1
ln+kUω(s)k ∈[0,∞)∈L1(Ω,F,P) and
Ω3ω7→ sup 0≤s≤1
ln+kUθsω(1−s)k ∈[0,∞)
∈L1(Ω,F,P). (A1)* (Integrability) The functions
Ω3ω7→ sup 0≤s≤1
ln+kUω∗(s)k ∈[0,∞)
∈L1(Ω,F,P) and
Ω3ω7→ sup 0≤s≤1
ln+kUθ∗sω(1−s)k ∈[0,∞)
∈L1(Ω,F,P). (A2) (Positivity) For anyω∈Ω,t≥0 andu1, u2∈X withu1≤u2
Uω(t)u1≤Uω(t)u2.
(A2)* (Positivity) For any ω∈Ω,t≥0 andu∗1, u2∗∈X∗ withu∗1 ≤u∗2 Uω∗(t)u∗1≤Uω∗(t)u∗2.
Notice that in our case, as explained in [20], (A1)* follows from (A1) and (A2)* follows from (A2).
For a measurable linear skew-product semidynamical system Φ satisfying (A2) we say that an entire orbitvω of Uω is positive ifvω(t)∈X+\ {0} for all t ∈R. Similarly, when (A2)* is satisfied, an entire orbit v∗
ω of Uω∗ is positive if vω∗(t) ∈
(X∗)+\ {0}for allt∈ R.
Next we introduce focusing conditions (A3) and (A3)* in the following way. (A3) (Focusing) (A2) is satisfied and there aree∈X+withkek= 1 andU
ω(1)e6=
0 for all ω ∈ Ω, and an (F,B(R))-measurable function κ: Ω → [1,∞) with ln+ln
κ∈L1((Ω,F,P)) such that for anyω∈Ω and any nonzerou∈X+ • Uω(1)u= 0, or
• Uω(1)u6= 0 and there isβ(ω, u)>0 with the property that
β(ω, u)e≤Uω(1)u≤κ(ω)β(ω, u)e. (2.8) Remark 2.3. Under (A3), by the cocycle property (3.4), foru∈X+the following dichotomy holds:
• Uω(t)u >0 for allt≥0.
(A3)* (Focusing forX∗) (A2)* is satisfied and there aree∗∈(X∗)+withhe,e∗i= 1 and ke∗k = 1 and an (F,B(
R))-measurable function κ∗: Ω → [1,∞) with ln+ln
κ∗∈L1((Ω,F,P)) such that for anyω∈Ω there holdsUω∗(1)e∗6= 0, and for
any ω ∈Ω and any nonzerou∗ ∈(X∗)+ there is β∗(ω, u∗)>0 with the property that
β∗(ω, u∗)Uω∗(1)e∗≤Uω∗(1)u∗≤κ∗(ω)β∗(ω, u∗)U∗
ω(1)e∗. (2.9)
It should be remarked that our condition (A3) is weaker than condition (A3) in [20]. We write the latter as:
(A3-O) (A2) is satisfied and there are e∈X+ with kek = 1 andU
ω(1)e6= 0 for
all ω ∈ Ω, and an (F,B(R))-measurable function κ: Ω → [1,∞) with ln+lnκ ∈
L1((Ω,F,P)) such that for any ω ∈ Ω and any nonzero u ∈ X+ there holds
Uω(1)u6= 0 and there isβ(ω, u)>0 with the property that
β(ω, u)e≤Uω(1)u≤κ(ω)β(ω, u)e. (2.10) Condition (A3)* follows from (A3), as we prove now.
Proposition 2.4. (A3)⇒(A3)*. Proof. From (2.8) we have
β(θ−1ω, u)e≤Uθ−1ω(1)u≤κ(θ−1ω)β(θ−1ω, u)e
provided that Uθ−1ω(1)u6= 0. Fix anye
∗ ∈(X∗)+ withhe,e∗i= 1 and ke∗k= 1 (the existence of such ane∗follows from [35, Theorem 5.4]). Hence,
β(θ−1ω, u)he,e∗i ≤ hUθ−1ω(1)u,e
∗i=hu, U∗
ω(1)e
∗i
(in particular,Uω∗(1)e∗6= 0) and
hUθ−1ω(1)u, u
∗i ≤
κ(θ−1ω)β(θ−1ω, u)he, u∗i he,e∗i. Combining these two inequalities and denoting
β∗(ω, u∗) = he, u ∗i
κ(θ−1ω)
and κ∗(ω) =κ2(θ−1ω), we obtain
hUθ−1ω(1)u, u
∗i ≤
κ∗(ω)β∗(ω, u∗)hu, Uω∗(1)e∗i,
which also holds ifUθ−1ω(1)u= 0, and then
Uω∗(1)u∗≤κ∗(ω)β∗(ω, u∗)U∗
ω(1)e∗. (2.11)
Analogously, from (2.8) andhe,e∗i= 1 we deduce that hUθ−1ω(1)u, u
∗i ≥β(θ
−1ω, u)he, u∗i=
β(θ−1ω, u) κ(θ−1ω)
he, u∗iκ(θ−1ω)
≥ he, u ∗i
κ(θ−1ω)
hUθ−1ω(1)u,e
∗i=β∗(ω, u∗)hU
θ−1ω(1)u,e
∗i,
that is,
β∗(ω, u∗)Uω∗(1)e∗≤Uω∗(1)u∗,
which together with (2.11) finishes the proof.
Lemma 2.5. Assume (A3). Then for any ω ∈ Ω, any t ≥ 0 and any nonzero u∗∈(X∗)+ there holdsUω∗(t)u∗6= 0.
Proof. It follows from Proposition 2.4, by induction, that U∗
ω(n)u∗ 6= 0 for each n ∈ N. Suppose to the contrary that Uω∗(t)u∗ = 0 for some ω ∈ Ω, t > 0 and u∗ ∈(X∗)+. Then, by (2.4), U∗
ω(btc+ 1)u∗ =Uθ−∗ tω(btc+ 1−t)U ∗
ω(t)u∗ = 0, a
contradiction.
Remark 2.6. We can replace time 1 with some nonzero T ∈ R+ in (A1), (A3), and (A1)*, (A3)*.
3. Generalized principal Floquet subspaces
The main result of the paper is the proof of the existence, under the new focusing assumption (A3), of a new version of the concept of generalized exponential separa-tion for a measurable linear skew-product semidynamical system. We will preserve the structure and follow the arguments in [20], just introducing the modifications which are required in this new situation. This section will be devoted to the proof of the existence of a family of generalized principal Floquet subspaces.
As stated before, X is an ordered separable Banach space such that X∗ is separable, with positive cone X+ normal and reproducing, and recall that, from Lemma2.4, once (A3) is assumed, assumption (A3)* holds.
Definition 3.1. A family of one-dimensional subspaces{Ee(ω)}ω∈
e
ΩofX is called a family of generalized principal Floquet subspacesof Φ = ((Uω(t)),(θt)) if Ωe ⊂Ω
is invariant,P(Ω) = 1, ande
(i) Ee(ω) = span{w(ω)}withw:Ωe →X+\ {0}being (F,B(X))-measurable,
(ii) Uω(t)Ee(ω) =Ee(θtω), for anyω∈Ω and anye t >0,
(iii) there iseλ∈[−∞,∞) such that
e
λ= lim
t→∞
lnkUω(t)w(ω)k
t for eachω∈Ωe,
(iv) for eachω∈Ω withe Uω(1)u6= 0
lim sup
t→∞
lnkUω(t)uk t ≤eλ .
e
λis called thegeneralized principal Lyapunov exponentof Φ associated to the gen-eralized principal Floquet subspaces{Ee(ω)}ω∈
e
Ω.
We recall the definitions of oscillation ratio, Birkhoff contraction ratio and pro-jective diameter, needed in the proofs of our main theorems. See Definition2.1and Subsection2.2for previous definitions and notations.
Definition 3.2. Assume (A2) and let ω∈Ω. (1) The oscillation ratioofUω(1) is defined as
p(ω) := sup
u, v∈X+
u∼v , u6=α v
(2) The Birkhoff contraction ratioofUω(1) is defined as q(ω) := sup
u, v∈X+
u∼v , u6=α v
d(Uω(1)u, Uω(1)v) d(u, v) .
(3) The projective diameterofUω(1) is defined as τ(ω) := sup
u, v∈X+
Uω(1)u∼Uω(1)v
d(Uω(1)u, Uω(1)v).
The functionsp∗,q∗ andτ∗ for the dual Φ∗ are defined in an analogous way. The following lemmas, proved in [20] as Lemma 4.10 and 4.11 for different fo-cusing conditions, hold for the new fofo-cusing conditions (A3) and (A3)* with small changes, and hence their proofs are omitted. Recall thate∈X+ ande∗ ∈(X∗)+ are the positive vectors of condition (A3) and (A3)*, respectively, and notice that it is easy to check that
τ(ω) = sup
u, v∈X+
Uω(1)u∼Uω(1)v Uω(1)u∼e
d(Uω(1)u, Uω(1)v). (3.1)
Lemma 3.3. Under assumption(A3), for eachω∈Ωand eachu∈X+, such that
Uω(1)u∼eandu∗∈(X∗)+\ {0} there holdsUω∗(1)u∗∼Uω∗(1)e∗ and d(Uω(1)u,e)≤lnκ(ω), d(Uω∗(1)u
∗, U∗
ω(1)e
∗)≤ln κ∗(ω).
Consequently,τ(ω)≤2 lnκ(ω)and τ∗(ω)≤2 lnκ∗(ω) for anyω∈Ω. Lemma 3.4. Under assumption(A3), for eachω∈Ω
τ(ω)<∞, τ∗(ω)<∞, and
p(ω) =q(ω) = tanhτ(ω)
4 (<1), p
∗(ω) =q∗(ω) = tanhτ∗(ω)
4 (<1). As in Lemma 4.13 of [20], taking into account (3.1) and changing the dense countable subsets of the proof by (vj+e/n)j,n, the following result is proved.
Lemma 3.5. Under assumption(A3), the functions
Ω3ω7→τ(ω)∈R
,
Ω3ω7→τ∗(ω)∈R
Ω3ω7→p(ω)∈R, Ω3ω7→p∗(ω)∈R
Ω3ω7→q(ω)∈R, Ω3ω7→q∗(ω)∈R
are(F,B(R))-measurable.
Next we consider the set of positive vectors
Σ :={u∈Ce| kuk= 1}=Ce∩S1(X+),
whereCe denotes the component ofedefined in (2.6).
Proof. As stated in Eveson [8], since X+ is normal (hence almost Archimedean) and the norm on X is monotone, any two vectors u, v of Σ ⊂ Ce are regularly comparable, i.e., they are comparable andm(u/v)≤1≤M(u/v) (see Definition2.1 for notation). Consequently, from [8, Theorem 1.2.1] the projective distance d
defined in (2.7) is a metric on Σ and the metric space (Σ, d) is complete. Lemma 3.7. Under assumption(A3),
Uω(t)u∈Ce for each t≥0, ω∈Ω and u∈Ce.
Proof. First we check thatUω(t)e∈Ce for eacht ≥0. From (A3) we know that Uω(1)e 6= 0, and then (2.8) implies that Uω(1)e∈Ce for each ω ∈ Ω. From the
cocycle property (2.2) we deduce thatUω(2)e= Uθ1ω(1)Uω(1)e, which together
with (A2) yieldsUω(2)e∈Ce. In a recursive way we obtain that
Uω(n)e∈Ce for eachω∈Ω andn∈N. (3.2) Now taket≥1. SinceUω(t)e=Uθt−1ω(1)Uω(t−1)e,from the focusing condition
(A3) we have two options Uω(t)e= 0 or Uω(t)e ∈Ce. Assume that Uω(t)e= 0
and taken1∈Nsuch thatn1≤t≤n1+ 1. Again (2.2) provides
Uω(n1+ 1)e=Uθtω(n1+ 1−t)Uω(t)e= 0, which contradicts (3.2) and proves thatUω(t)e∈Ce fort≥1.
Next, ift≥0, we have proved thatUθ−1ω(1)eandUθ−1ω(t+ 1)e∈Ce, that is, e
αe≤Uθ−1ω(1)e≤βee and αe≤Uθ−1ω(t+ 1)e≤βe,
and consequently, from Uθ−1ω(t+ 1)e = Uω(t)Uθ−1ω(1)e and the monotonicity
property (A2) we deduce that
αe≤β Ue ω(t)e and βe≥α Ue ω(t)e
i.e.,Uω(t)e∈Ce, as claimed.
Finally, if u ∈ Ce, it is immediate to check fromUω(t)e ∈ Ce and (A2) that
Uω(t)u∈Ce, which finishes the proof.
As a consequence of this lemma, under assumption (A3), the map Uω(t) : Σ −→ Σ
u 7→ Uω(t)u kUω(t)uk
(3.3)
is well defined for each t ≥ 0 and ω ∈ Ω. Moreover, it is continuous for the projective distance becaused(Uω(t)u,Uω(t)v)≤d(u, v), as can be easily deduced
from Lemmas 4.5 and 4.9 of [20]. Furthermore, as a consequence of (2.2) Uθsω(t)◦ Uω(s) =Uω(t+s), for each ω∈Ω ands, t∈R
+. (3.4) We omit the proof of the next result which is completely analogous to the first part of the proof of Proposition 5.3 of [20]. It follows from (3.4), the properties of the distance, the definition ofq(see Definition3.2) and Lemma3.3. As before, we denote bybtcthe integer part of the real numbert.
Proposition 3.8. Under assumption(A3),
(i) for eachω∈Ω,t≥2andu,eu∈Σ
d Uθ−tω(t)u,Uθ−tω(t)eu
(ii) for eachω∈Ω,t≥2andu,eu∈Σ
d Uω(t)u,Uω(t)ue
≤2 lnκ(ω)q θ1ω
· · ·q θbtc−1ω
.
As a consequence, the following contraction property follows, whose proof is also omitted because it follows the arguments of Proposition 5.4 of [20].
Proposition 3.9. Under assumption (A3), let I:=R
Ωlnq dP<0. Then, there is
an invariant setΩ¯1⊂ΩwithP( ¯Ω1) = 1such that
(1) for eachI < J <0 andω∈Ω1¯ , there is aC1(J, ω)>0 such that d Uθ−tω(t)u,Uθ−tω(t)eu
≤C1(J, ω)eJ t whenevert≥3 andu,eu∈Σ,
(2) for eachI < J <0 andω∈Ω1¯ , there is aC2(J, ω)>0 such that d Uω(t)u,Uω(t)ue
≤C2(J, ω)eJ t
whenevert≥2 andu,eu∈Σ.
Proposition3.9(1) ensures that for any ω∈Ω1¯ the following exists
w(ω) := lim
s→∞Uθ−sω(s)e, (3.5) where the limit is taken in d. Since, by Lemma 3.6, (Σ, d) is a complete metric space,w(ω) belongs to Σ. Further, it follows from Lemma2.2that the above limit can be taken in the X-norm. Moreover, since the functions [ω 7→ Uθ−nω(n)e] are (F,B(X))-measurable, the functionw: ¯Ω1→X is measurable.
The next theorem shows the existence of generalized Floquet subspaces and principal Lyapunov exponent, and the uniqueness of entire positive orbits, which is the equivalent result to Theorem 3.6 of [20] for the new focusing.
Theorem 3.10. Under assumptions (A1) and (A3), if Ω¯1 is the invariant set of Proposition3.9, there is an(F,B(X))-measurable function
w: ¯Ω1→Σ =Ce∩S1(X+), ω7→w(ω)
defined by (3.5)such that
(1) for eachω∈Ω¯1 andt≥0,
w(θtω) =
Uω(t)w(ω)
kUω(t)w(ω)k
; (3.6)
(2) for eachω∈Ω¯1, the map wω:R→X+ defined by
wω(t) =
w(θtω)
kUθtω(−t)w(θtω)k
fort≤0, Uω(t)w(ω) fort≥0 ;
(3.7)
is a positive entire orbit of Uω, unique up to multiplication by a positive scalar;
(3) there are an invariant set Ωe1 ⊂Ω¯1 with P(Ωe1) = 1 and a λ¯1 ∈ [−∞,∞)
such that
¯
λ1= lim
t→±∞ 1
t lnkwω(t)k=
Z
Ω
(4) for eachω∈Ω1e andu∈X+ with Uω(1)u6= 0
lim
t→∞ 1
t lnkUω(t)uk= ¯λ1; (3.8)
(5) for eachω∈Ω1e andu∈X
lim sup
t→∞ 1
t lnkUω(t)uk ≤
¯
λ1; (3.9)
that is,{E1(ω)}ω∈Ωe1, withE1(ω) = span{w(ω)}, is a family of generalized
principal Floquet subspaces, and ¯λ1 is the generalized principal Lyapunov
exponent.
Proof. (1) From relation (3.4) and the definition ofw(ω) we deduce that Uω(t)w(ω) =Uω(t)
lim
s→∞Uθ−sω(s)e
= lim
s→∞ Uω(t)◦ Uθ−sω(s)
e = lim
s→∞Uθ−sω(s+t)e= lims→∞ Uθt−sω(s)◦ Uθ−sω(t)
e,
for eachω∈Ω1¯ andt≥0. Moreover, from Proposition3.9(1)
d Uθt−sω(s)e,Uθt−sω(s) Uθ−sω(t)e
≤C1(J, θtω)eJ s,
from which it follows thatUω(t)w(ω) = lims→∞Uθt−sω(s)e=w(θtω), as stated. (2) First notice that ift≤0 andω∈Ω1¯
Uθtω(−t)wω(t) =
Uθtω(−t)w(θtω) kUθtω(−t)w(θtω)k
=w(ω). (3.10) Now we check that wω is an entire orbit, i.e.,wω(s+t) =Uθsω(t)wω(s) for each
s∈Randt≥0. Ift≥0 ands≥0 it is immediate. Ift≥0,s≤0 andt+s≥0, from (2.2) and (3.10)
Uθsω(t)wω(s) =Uω(t+s)(Uθ−sω(−s)wω(s)) =Uω(t+s)w(ω) =wω(t+s). Next, sinceθt+sω=θt(θsω), ift≥0,s≤0 andt+s≤0, from
wω(t+s) =
w(θt+sω)
kUθt+sω(−t−s)w(θt+sω)k
, w(θt+sω) =
Uθsω(t)w(θs) kUθsω(t)w(θs)k
,
andUθt+sω(−t−s)◦Uθsω(t) =Uθsω(−s), we deduce that
wω(t+s) =
Uθsω(t)w(θsω) kUθsω(−s)w(θsω)k
=Uθsω(t)ωω(s), andwωis an entire positive orbit, as claimed.
Finally we check the uniqueness. Letvω be another entire positive orbit ofUω.
From vω(s+t) = Uθsω(t)vω(s) we deduce that vω(s) = Uθs−1ω(1)vω(s−1), and
sincevω(s)∈X+\{0}for eachs∈R, the focusing condition (A3) yieldsvω(s)∈Ce
for eachs∈R. Therefore,
u= vω(−s+t) kvω(−s+t)k
∈Σ,
and from Proposition3.9(1) and the definition ofw(ω) it follows that 0 = lim
s→∞d Uθ−s(θtω)(s)u, w(θtω)
= lim
s→∞d
v
ω(t)
kvω(t)k
, w(θtω)
for each t ∈ R and ω ∈ Ω1. Since, by Lemma¯ 3.6, d is a metric on Σ, vω(t) =
kvω(t)kw(θtω) for each t ∈ R. From this we deduce that vω(0) = kvω(0)kw(ω),
and hence
vω(t) =Uω(t)vω(0) =kvω(0)kUω(t)w(ω) =kvω(0)kwω(t),
i.e., they coincide up to multiplication by a positive scalar, as stated.
(3) Since lnkwω(1)k = lnkUω(1)w(ω)k, from assumption (A1) we deduce that
the map [ Ω3ω7→ln+kw
ω(1)k]∈L1(Ω,F,P). Moreover,
lnkwω(t+s)k= lnkwθsω(t)k+ lnkwω(s)k for eacht, s∈R.
Therefore, the application of Birkhoff ergodic theorem to ((Ω,F,P),(θn)n∈Z) and lnkwω(1)k provides a subset Ω01 ⊂Ω1¯ with θ1(Ω01) = Ω01 and P(Ω01) = 1, and an invariant measurable mapg withg+ ∈L1(Ω,F,
P) such that lim
n→±∞ 1
n n
X
k=1
lnkwθkω(1)k=n→±∞lim 1
nlnkwω(n)k=g(ω)
for eachω∈Ω01 and
Z
Ω
lnkwω0(1)kdP(ω0) =
Z
Ω
g(ω0)dP(ω0). (3.11) Next we take the invariant setΩe1=∪s∈[0,1]θsΩ01. From assumption (A1), as in Lemma 3.4 of Lian and Lu [16], we check that ifω ∈Ω1e withω =θsωωe for some
sω∈[0,1) and eω∈Ω
0 1
lim sup
t→±∞ 1
t lnkwω(t)k ≤n→±∞lim 1
nlnkwωe(n)k=g(ωe),
lim inf
t→±∞ 1
t lnkwω(t)k ≥n→±∞lim 1
nlnkwωe(n)k=g(ωe),
that is, there exists the limit and coincide withg(ωe), lim
t→±∞ 1
t lnkwω(t)k=g(eω).
Finally, since the function on the left is invariant and hence constant a.e., from (3.11) we conclude that
lim
t→±∞ 1
t lnkwω(t)k=
Z
Ω
lnkwω0(1)kdP(ω0)
for eachω∈Ω1, as stated.e
(4) The focusing condition (A3) yieldsβ(ω, u)e≤Uω(1)u≤κ(ω)β(ω, u)eand, together withβ(ω, w(ω))e≤Uω(1)w(ω)≤κ(ω)β(ω, w(ω))e, we deduce that
0< Uω(1)u≤ κ
(ω)β(ω, u)
β(ω, w(ω)) Uω(1)w(ω)≤κ 2(ω)U
ω(1)u .
The monotonicity assumption (A2) andUω(t) =Uθ1ω(t−1)◦Uω(1) implies that
0< Uω(t)u≤ κ
(ω)β(ω, u)
β(ω, w(ω)) Uω(t)w(ω)≤κ 2(ω)U
ω(t)u fort≥1,
and the normal character of the positive coneX+ provides kUω(t)uk ≤ κ
(ω)β(ω, u)
β(ω, w(ω)) kUω(t)w(ω)k ≤κ 2(ω)
from which we conclude that lim
t→∞ 1
tlnkUω(t)uk= limt→∞ 1
tlnkUω(t)w(ω)k= ¯λ1,
as claimed.
(5) Since we are assuming that the coneX+ is reproducing, i.e.,X =X+−X+, we can decompose u ∈ X as u = u+ −v+ for some u+ and v+ ∈ X+. Thus, denoting|u|=u++v+∈X+, we have−|u| ≤u≤ |u|, and again the monotonicity assumption (A2) yields
−Uω(t)|u| ≤Uω(t)u≤Uω(t)|u|, t≥0.
Therefore, we deduce that 0≤Uω(t)|u|+Uω(t)u≤2Uω(t)|u|, hence, the normal
character of the cone provideskUω(t)uk ≤3kUω(t)|u|k, and inequality (3.9) follows
from relation (3.8).
Remark 3.11. When one replaces (A3) with (A3-O), part (4) of Theorem 3.10 takes the form
(4) for eachω∈Ω1e andu∈X+\ {0}
lim
t→∞ 1
t lnkUω(t)uk= ¯λ1. (3.12)
Consequently, Theorem3.10is an extension of Theorem 2.2 in [22].
Remark 3.12. Note that as in Proposition 2.2 of Mierczy´nski and Shen [22], from the existence of the family of generalized Floquet subspaces and the generalized principal Lyapunov exponent ¯λ1 obtained in the previous Theorem3.10, it follows that
lim
t→∞
lnkUω(t)k t = ¯λ1
for eachω∈Ωe1.
The following theorem provides a counterpart of Theorem 3.10 for the dual system. As in [20], we define, fort≥0 andω∈Ω
U∗
ω(t) : S1((X∗)+) −→ S1((X∗)+)
u∗ 7→ U
∗
ω(t)u∗
kU∗
ω(t)u∗k
. (3.13)
By Lemma2.5, the above mappings are well defined. Further, it follows from (2.4) that
Uθ−∗ tω(s)◦ U ∗
ω(t) =Uω∗(t+s), for each ω∈Ω ands, t∈R
+. (3.14) Lemma 3.13. Assume (A3)*. For any s≥1 and any nonzero u∗ ∈(X∗)+ there
holdsUθ∗
sω(s)u ∗∈C∗
U∗
ω(1)e∗, where
CU∗∗
ω(1)e∗:={v
∗∈(X∗)+
\ {0} |v∗∼Uω∗(1)e∗} (3.15)
denotes the component of Uω∗(1)e∗ in (X∗)+.
Proof. By the cocycle property (2.4),Uθ∗
sω(s)u ∗=U∗
ω(1) Uθ∗s−1ω(s−1)u
∗
. Lemma 2.5gives thatU∗
θs−1ω(s−1)u
∗∈(X∗)+\ {0}. Now we need to apply (2.9).
(i) for eachω∈Ω,t≥2andu∗,eu∗∈S1((X∗)+)
d Uθ∗tω(t)u ∗,U∗
θtω(t)ue
∗
≤2 lnκ∗ θbtcω
q∗ θbtc−1ω
· · ·q∗ θ1ω
,
(ii) for eachω∈Ω,t≥2andu∗,eu∗∈S1((X∗)+)
d Uω∗(t)u∗,Uω∗(t)ue∗
≤2 lnκ∗(ω)q∗ θ−[t]+1ω
· · ·q∗ θ−1ω.
Proposition 3.15. Under assumption (A3)*, letI :=R
Ωlnq dP<0. Then, there
is an invariant setΩ¯∗1⊂Ωwith P( ¯Ω∗1) = 1 such that
(1) for eachI < J <0 andω∈Ω¯∗1, there is a C3(J, ω)>0 such that
d Uθ∗tω(t)u,Uθ∗tω(t)ue∗
≤C3(J, ω)eJ t whenevert≥3 andu∗,
e
u∗∈S
1((X∗)+), (2) for eachI < J <0 andω∈Ω¯∗
1, there is a C4(J, ω)>0 such that
d Uω∗(t)u∗, Uω∗(t)ue∗
≤C4(J, ω)eJ t whenevert≥2 andu∗,eu∗∈S1((X∗)+).
Proposition3.15(1) ensures that for anyω∈Ω¯∗1 the following exists
w∗(ω) := lim
s→∞U ∗
θsω(s)e
∗, (3.16)
where the limit is taken in d. Since, by Lemma 3.13, U∗
θsω(s)e
∗ belongs, for any s ≥ 1, to CU∗∗
ω(1)e∗∩S1((X
∗)+), and since, by a counterpart of Lemma 3.6, (CU∗∗
ω(1)e∗∩S1((X
∗)+), d) is a complete metric space,w∗(ω) belongs to (C∗
U∗
ω(1)e∗∩
S1((X∗)+). Further, it follows from a counterpart of Lemma 2.2 that the above limit can be taken in theX∗-norm. Moreover, since the functions [ω7→ U∗
θnω(n)e ∗] are (F,B(X∗))-measurable, the functionw∗: ¯Ω
1→X∗ is measurable. We want to remark that noww∗(ω) does not necessarily belong toC
e∗, because of the different
focusing condition (2.9).
Moreover, we claim that, if ¯Ω1 is the invariant set of Proposition 3.9 and w is defined by (3.5), for each ω∈Ω1¯ ∩Ω¯∗1
0<hw(ω), w∗(ω)i ≤1. (3.17) The right inequality is immediate because they are unitary vectors. Concerning the left one, sincew(ω)∈Ce there is anα >0 such thatαe≤w(ω) and, consequently
hw(ω), w∗(ω)i ≥ αhe, w∗(ω)i. Next we check that he, w∗(ω)i > 0. Otherwise, from he, w∗(ω)i = 0 we would deduce that he, Uω∗(1)w∗(ω)i = 0, and from rela-tion (2.9) that he, Uω∗(1)e∗i = 0, that is, hUθ−1ω(1)e,e
∗i = 0. This contradicts thatUθ−1ω(1)e∈Ce (see (2.8)) andhe,e
∗i= 1, and proves the assertion.
Theorem 3.16. Under assumptions (A1)* and (A3)*, if Ω¯∗1 is the invariant set of (3.16), there is an (F,B(X∗))-measurable function
w∗: ¯Ω∗1→S1((X∗)+), ω7→w∗(ω)
defined by (3.16)such that
(1) for eachω∈Ω¯∗1 andt≥0, w∗(θ−tω) =
Uω∗(t)w∗(ω) kU∗
ω(t)w∗(ω)k
(2) for eachω∈Ω¯∗1, the map w∗ω:R→X+ defined by
is a positive entire orbit of Uω∗, unique up to multiplication by a positive scalar;
(4) The generalized principal Lyapunov exponentλ¯1 obtained inTheorem3.10
coincides with ¯λ∗1.
Proof. (1) From relation (3.14) and the definition ofw∗(ω) we deduce that Uω∗(t)w(ω) =Uω∗(t)
from which it follows that U∗
ω(t)w∗(ω) = lims→∞Uθ∗s−tω(s)e
∗ = w∗(θ
−tω), as
stated.
(2) The fact thatwω∗ is a positive entire orbit follows along the lines of the proof of Theorem3.10(2).
We check the uniqueness. Let vω∗ be another entire positive orbit of Uω∗. It follows from Proposition3.15(1) and the definition of w∗ that
0 = lim
(4) Letω∈Ω1e ∩Ωe∗1. First note that, with the help of (3.7) and (3.19), Moreover, from relations (2.3), (3.6) and (3.18) we obtain
kUw(t)w(ω)k hw(θtω), w∗(θtω)i=hUω(t)w(ω), w∗(θtω)i Remark 3.17. In view of Proposition 2.4, (A1)* and (A3)* can be replaced in Theorem3.16by (A1) and (A3). Therefore, Theorem3.16is new even in the case of assumption (A3-O).
4. Generalized exponential separation
As stated before,X is an ordered separable Banach space such thatX∗ is sep-arable, with positive coneX+ normal and reproducing, and recall that, once (A3) is assumed, assumptions (A2), (A2)* and (A3)* hold.
In this section we will prove the existence of a generalized exponential sepa-ration of type II, now introduced and important for cases in which the previous concept of generalized exponential separation does not apply, as measurable linear skew-product semidynamical systems induced by delay differential equations. Definition 4.1. The measurable linear skew-product semidynamical system Φ = ((Uω(t)),(θt)) is said to admit ageneralized exponential separation of typeII if there
are a family of generalized principal Floquet subspaces {Ee(ω)}ω∈
e
Ω, and a family of one-codimensional closed vector subspaces{Fe(ω)}ω∈
e
ΩofX, satisfying (i) Fe(ω)∩X+={u∈X+|Uω(1)u= 0},
(ii) X =Ee(ω)⊕Fe(ω) for any ω ∈ Ω, where the decomposition is invariant,e
(iii) there existsσe∈(0,∞] such that
lim
t→∞ 1
t ln
kUω(t)|Fe(ω)k
kUω(t)w(ω)k
=−eσ
for eachω∈Ω.e
We say that{Ee(·),Fe(·),eσ}generates a generalized exponential separation of typeII.
Note that the only difference with the definition of generalized exponential sep-aration given in [20] is that in this caseFe(ω) contains those positive vectorsu >0
for whichUω(1)u= 0 becauseUω(1) is not assumed to be injective.
Next we considerΩe1 and Ωe∗1, the invariant sets of Theorem 3.10and 3.16, and
we define for eachω∈Ω1e ∩Ωe∗1
F1(ω) ={u∈X | hu, w∗(ω)i= 0}.
From (3.17), eachu∈X can be decomposed asu=α w(ω) +u−α w(ω) with
α= hu, w ∗(ω)i hw(ω), w∗(ω)i, and hence,
X =E1(ω)⊕F1(ω),
where, as denoted in Theorem3.10, E1(ω) = span{w(ω)}. As a consequence, the following result holds.
Lemma 4.2. The family{P(ω)}ω∈
e
Ω1∩Ωe∗1 of projections associated with the decom-positionE1(ω)⊕F1(ω) =X is given by the formula
P(ω)u=u− hu, w ∗(ω)i
hw(ω), w∗(ω)iw(ω), ω∈Ω1e ∩Ωe∗1. (4.1) We omit the proof of the next result, based in Lemma 5.10 of [20] with the cor-responding modifications due to the different definition of the maps (3.3) and (3.5). See Definition3.2for the oscillation of two vectors.
Proposition 4.3. Under assumptions (A1) and (A3), there exists an invariant subset Ω2e ⊂ Ω1e ∩Ωe∗1 of full measure P(Ω2) = 1e , with the property that for each
0> J >RΩlnp dPand each ω∈Ω2e , there is a C5(ω, J)>0such that
osc
U
ω(t)u
kwω(t)k
/w(θtω)
≤C5(J, ω)eJ t
wheneveru∈Σ =Ce∩S1(X+)andt≥1.
Lemma 4.4. Under assumptions(A1)and(A3), let¯λ1be the generalized principal
Lyapunov exponent ofTheorem3.10. Ifλ¯1>−∞then lim
t→±∞ 1
t lnhw(θtω), w
∗(θ
Proof. We prove the result for t → ∞, the other limit is completely analogous. Since ¯λ1= ¯λ∗1, from inequality (3.25) we deduce that
lim sup
t→∞ 1
tlnhw(θtω), w
∗(θ
tω)i= 0.
Assume now on the contrary to the assertion of the lemma that there is a sequence
tn ↑ ∞such that
lim
n→∞ 1
tn
lnhw(θtnω), w ∗(θ
tnω)i=a <0.
Again from (3.25) we would obtain ¯λ∗1= ¯λ1+a, a contradiction. We include part of the proof of the next result, although similar to the proof of Proposition 5.11 of [20], to remark the differences derived from the new assumption (A3) and the fact that we do not have a Banach lattice but an ordered separable Banach space with positive coneX+ normal and reproducing.
Proposition 4.5. Under assumptions(A1)and(A3), letΩ2e be the invariant subset
of Proposition4.3. Ifλ¯1>−∞, there exists aσ¯2>0 such that lim sup
t→∞ 1
t lnu∈X,supkuk=1
Uω(t)u
kwω(t)k
− hu, w ∗(ω)i
hw(ω), w∗(ω)iw(θtω)
≤ −¯σ2
for eachω∈Ω2e .
Proof. Denote Ueω(t)u := Uω(t)u/kwω(t)k. As in Proposition 5.11 of [20], from
Proposition 4.3 we prove that m(Ueω(t)u/w(θtω)) and M(Ueω(t)u/w(θtω)) both
converge to a common limit denoted byµ(u, ω) and
µ(u, ω)−m(Ueω(t)u/w(θtω))≤C5(J, ω)eJ t, M(Ueω(t)u/w(θtω))−µ(u, ω)≤C5(J, ω)eJ t for eacht≥1 andu∈Σ =Ce∩S1(X+). Moreover, since
m(Ueω(t)u/w(θtω))−µ(u, ω)
w(θtω)≤Ueω(t)u−µ(t, ω)w(θtω)
≤ M(Ueω(t)u/w(θtω))−µ(u, ω)
w(θtω),
andX+ is normal we deduce that
Uω(t)u
kwω(t)k
−µ(u, ω)w(θtω)
≤3C5(J, ω)eJ t (4.2)
for eacht≥1 andu∈Σ =Ce∩S1(X+). Next we fixu∈Σ andω∈
e
Ω2. It is not hard to check, as in Proposition 5.11 of [20], that
U
ω(t)u
kwω(t)k
−µ(u, ω)w(θtω), w∗(θtω)
=
hu, w∗(ω)i
hw(ω), ω∗(ω)i−µ(u, ω)
hw(θtω), w∗(θtω)i,
which together with the exponential decay (4.2) and Lemma4.4 provides
Therefore, we have proved that
from (3.18) and (2.3) we deduce that
hu, w∗(ω)i=
Next, a straightforward computation shows that, ifeu=Uω(1)uwe have Uω(t)u
which finishes the proof.
The next theorem shows the existence of a generalized exponential separation of type II. We maintain the notation of the previous results.
Theorem 4.6. Under assumptions(A1) and (A3), letλ¯1 be the generalized
prin-cipal Lyapunov exponent ofTheorem3.10 and assume thatλ¯1>−∞. Then there
is an invariant setΩ0e of full measure P(Ω0) = 1e such that
(1) The family {P(ω)}ω∈
e
Ω0 of projections associated with invariant
and
lim
t→∞ 1
t lnkUω(t)|F1(ω)k= ¯λ2
for eachω ∈Ω0e , that is, Φadmits a generalized exponential separation of
typeII.
Proof. (1) The strong measurability follows from (4.1) and the measurability ofw
and w∗. Next we show that it is a tempered family, i.e. (2.5) holds. Let Ω1e and the definition ofw∗ω we deduce that
hu, w∗(ω)i= Moreover, from the focusing condition (A3) we deduce that ifu∈X+\ {0}we have two optionsUω(1)u= 0 and henceu∈F1(ω)∩X+, orUω(1)u∈Ce. In this case, From the focusing condition (A3)* forX∗, i.e., inequality (2.9),
Uθ∗2ω(1)w∗(θ2ω)≥β∗(θ2ω, w∗(θ2ω))Uθ∗2ω(1)e
∗,
and hence, together with (4.7), (3.18), and (3.19) yields hu, w∗(ω)i ≥β(ω, u)β
and, therefore hu, w∗(ω)i > 0, as claimed, which finishes the proof of (2) when
ω∈Ωe1∩Ωe∗1.
(3) From Remark3.12we know that lim
t→∞ 1
tlnkUω(t)k= ¯λ1 for eachω∈Ωe1.
Now, let Ω2e ⊂Ω1e ∩Ωe∗1 be the invariant subset of Proposition4.3, fixω∈Ω2e and
letu∈X\F1(ω) withUω(1)u6= 0. We can decomposeu=u1+u2 with
u2=
hu, w∗(ω)i
hw(ω), w∗(ω)iw(ω),
and sincehu, w∗(ω)i 6= 0, thenku2k>0. From (3.6) we deduce that for eacht≥0
Uω(t)u2=
hu, w∗(ω)i
hw(ω), w∗(ω)iUω(t)w(ω) =
hu, w∗(ω)i
hw(ω), w∗(ω)ikUω(t)w(ω)kw(θtω), and hence, kUω(t)u2k =kUω(t)w(ω)k ku2k>0. Moreover, from Proposition 4.5,
i.e. relation (4.5), and (3.6) kUω(t)u1k=
Uω(t)u−
hu, w∗(ω)i
hw(ω), ω∗(ω)iw(θtω)
≤b(ω, J)kUω(t)w(ω)k kukeJ t
for eacht≥2 and, consequently,
kUω(t)uk ≥ kUω(t)u2k − kUω(t)u1k ≥ kUω(t)w(ω)k(ku2k −b(ω, J)kukeJ t). Therefore,
lim inf
t→∞ 1
t lnkUω(t)uk ≥lim inft→∞ 1
t lnkUω(t)w(ω)k ≥
¯
λ1,
which together with Theorem 3.10(5), i.e. relation (3.9), finishes the proof of (3) whenω∈Ω2e ⊂Ω1e ∩Ωe∗1.
(4) We omit the proof of this part of the theorem because it follows step by step the proof of Theorem 3.8(4) in [20]. An invariant subsetΩ0e of full measureP(Ω0) =e
1, and contained in Ω2e ⊂Ω1e ∩Ωe∗1 where (4) holds is obtained. Consequently, all
the previous results (1-3) apply forω∈Ωe0.
Remark 4.7. The definition ofgeneralized exponential separation of typeI resem-bles Definition4.1, the only difference being that (i) is replaced by
e
F(ω)∩X+ ={0}.
In particular, generalized exponential separation of type I implies generalized ex-ponential separation of type II.
Under (A1) and (A3-O), and assuming additionally that ¯λ1>−∞, Theorem4.6 gives the existence of generalized exponential separation of type I:
Theorem 4.8. Under assumptions (A1) and (A3-O), let λ¯1 be the generalized
principal Lyapunov exponent of Theorem 3.10 and assume that λ¯1 > −∞. Then
there is an invariant set Ωe0 of full measureP(Ωe0) = 1such that (1) The family {P(ω)}ω∈
e
Ω0 of projections associated with invariant
decompo-sition E1(ω)⊕F1(ω) =X is strongly measurable and tempered.
(3) For any ω∈Ω0e andu∈X\F1(ω)there holds
lim
t→∞ 1
t lnkUω(t)k= limt→∞ 1
t lnkUω(t)uk= ¯λ1.
(4) There existseσ∈(0,∞] and¯λ2= ¯λ1−eσsuch that
lim
t→∞ 1
tln
kUω(t)|F1(ω)k
kUω(t)w(ω)k
=−eσ and
lim
t→∞ 1
t lnkUω(t)|F1(ω)k= ¯λ2
for eachω ∈Ωe0, that is, Φadmits a generalized exponential separation of
typeI.
Remark 4.9. Theorem 4.8 is new even in the case of generalized exponential separation of type I. Indeed, under an additional assumption that ¯λ1 >−∞it is stronger than [22, Thm. 2.4]: the latter requires that lnκ,lnκ∗ ∈L1(Ω,F,P) and he,e∗i>0.
5. Scalar linear random delay differential equations
This section is devoted to show the applications of the previous theory to random dynamical systems generated by scalar linear random delay differential equations of the form
z0(t) =a(θtω)z(t) +b(θtω)z(t−1), ω∈Ω. (5.1)
Let 1< p <∞. We consider the separable Banach spaceX =R×Lp([−1,0],R) with the norm
kukX=|u1|+ku2kp=|u1|+
Z 0
−1
|u2(s)|pds
1/p
for anyu= (u1, u2) withu1∈Randu2∈Lp([−1,0],R). The positive cone
X+={u= (u1, u2)∈X |u1≥0 andu2(s)≥0 for Lebesgue-a.e. s∈[0,1]} is normal and reproducing, and the dualX∗=R×Lq([−1,0],R) with 1/q+1/p= 1 is also separable.
Now we introduce the assumptions on the coefficients of the family (5.1): (S1) the (F,B(R))-measurable functionsaandb have the properties:
Ω3ω7→a(ω)∈R
∈L1(Ω,F,P), and
h
Ω3ω7→ln+
Z 1
0
|b(θrω)|qdr∈R
i
∈L1(Ω,F,P). (S2) b(ω)≥0 for eachω∈Ω.
Remark 5.1. The following is sufficient for the fulfillment of the second condition in (S1):
Ω3ω7→b(ω)∈R∈Lq(Ω,F,P). Indeed, since|b|q ∈L
1(Ω,F,P) and the measurePis invariant, for anyt∈R
Z
Ω
|b(θtω0)|qdP(ω0) =
Z
Ω
|b(ω0)|qd
and an application of Fubini’s theorem gives that the map
belongs toL1(Ω,F,P), from which the required statement follows immediately. In order to define the measurable linear skew-product semidynamical system we are going to deal with, for eachu= (u1, u2)∈X andω∈Ω we consider the initial and an application of Fubini’s theorem gives that the map
R3t7→a(θtω)∈R∈L1,loc(R) (5.3) forω ∈Ω0⊂Ω, invariant set of full measure. Then we can put the value ofa(ω) forω ∈ Ω\Ω0 to be equal to zero to obtain (5.3) for allω ∈Ω. Analogously, by changing the value ofbto zero in a set of null measure, the map
R3t7→b(θtω)∈R∈Lq,loc(R)⊂L1,loc(R), (5.4) for all ω ∈ Ω. Therefore, for a fixed ω ∈ Ω and 0 ≤ t ≤ 1 the system (5.2) of Carath´eodory type has a unique solution, as shown by Coddington and Levinson [6, Theorem 2.1], which can be written as
Lemma 5.3. Under assumptions (S1) and (S2), for each ω ∈Ω, 0 ≤t ≤ 1 and u∈X there holds
|z(t, ω, u)| ≤c(ω) (1 +d(ω))kukX.
Proof. From (5.5), (5.4), b ≥ 0, u2 ∈ Lp([−1,0],R) and H¨older inequality, we deduce that if 0≤t≤1 andu= (u1, u2)∈R×Lp([−1,0],R)
|z(t, ω, u)| ≤c(ω)
|u1|+
Z t
0
b(θs)|u2(s−1)|ds
=c(ω)
|u1|+
Z t−1
−1
b(θs+1)|u2(s)|ds
≤c(ω)
|u1|+
Z 0
−1
b(θs+1)|u2(s)|ds
≤c(ω)
|u1|+
Z 0
−1
bq(θs+1ω)ds
1/q
ku2kp
≤c(ω) (1 +d(ω))kukX,
as stated.
Moreover, it can be checked that for eachtand r≥0
z(t+r, ω, u) =z(t, θrω,(z(r, ω, u), zr(ω, u))), (5.8)
wherezt(ω, u) : [−1,0]→R,s7→z(t+s, ω, u), which together with Lemma5.3show thatzt(ω, u)∈Lp([−1,0],R) for each t≥0 and we can define the linear operator
Uω(t) : X −→ X
u 7→ (z(t, ω, u), zt(ω, u))
(5.9)
Proposition 5.4. Under assumptions (S1) and (S2), Uω(t) satisfies (2.1), (2.2) andUω(t)∈ L(X)for eacht≥0 andω∈Ω.
Proof. Relation (2.1) is immediate and (2.2) follows from (5.8). Once that this cocycle property is shown, to prove that Uω(t) ∈ L(X) for t ≥ 0, it is enough
to check that Uω(t) is a bounded operator for t ∈ [0,1] and ω ∈ Ω, which is a
consequence of Lemma5.3because kUω(t)ukX=|z(t, ω, u)|+
Z 0
−1
|z(t+s, ω, u)|pds
1/p
≤c(ω) (1 +d(ω))kukX
+
Z −t
−1
|u2(t+s)|pds
1/p
+
Z 0
−t
|z(t+s, ω, u)|pds
1/p
≤3c(ω) (1 +d(ω))kukX,
that is,kUω(t)k ≤3c(ω) (1 +d(ω)) fort∈[0,1], which finishes the proof.
In order to show that Φ = ((Uω(t))ω∈Ω,t∈R+,(θt)t∈R) is a measurable linear
skew-product semidynamical system we start with the following auxiliary lemma. Lemma 5.5. Under (S1) and(S2), for eachu∈X andt >0 the mapping
Ω3ω7→Uω(t)u∈X
Proof. It follows from (2.2) that it suffices to prove the result for t ∈ (0,1] only. Since X is separable, from Pettis’ Theorem (see Hille and Phillips [10, Theorem 3.5.3 and Corollary 2 on pp. 72–73]) the weak and strong measurability notions are equivalent and therefore, it is enough to check that for eachu∗∈X∗ the mapping
Ω3ω7→u∗(Uω(t)u)∈Ris (F,B(R))-measurable. (5.10) Fixing u∗ = (u∗1, u∗2) ∈ R×Lq([−1,0],R), u= (u1, u2) ∈ R×Lp([−1,0],R) and
t∈(0,1], we have
u∗(Uω(t)u) =z(t, ω, u)u∗1+
Z 0
−1
z(t+s, ω, u)u∗2(s)ds . (5.11) The measurability of the map
Ω×[0, t] → Ω → R (ω, r) 7→ θrω 7→ a(θrω)
and an application of Fubini’s theorem show that
Ω3ω7→Rt
0a(θrω)dr∈R
is (F,B(R))-measurable. Therefore,
Ω3ω7→exp R0ta(θrω)dr
u1∈Ris (F,B(R))-measurable. (5.12) Since Rsta(θrω)dr =
Rt
0χ[0,t](t−s)a(θr+sω)dr, again the measurability of the mapping
Ω×[0, t]×[0, t] → R
(ω, r, s) 7→ χ[0,t](t−s)a(θr+sω)
and Fubini’s theorem prove that the mapsΩ×[0, t]3(ω, s)7→Rt
sa(θrω)dr∈R
and
Ω×[0, t]3(ω, s)7→exp Rt
sa(θrω)dr
b(θsω)u2(s−1)∈R(taking a Borel representation of the function u2 ∈ Lp([−1,0],R)) are (F⊗B([0, t]),B(R ))-mea-surable. From this, as before, we deduce that
Ω3ω7→Rt
0exp
Rt
sa(θrω)dr
b(θsω)u2(s−1)ds∈R is (F,B(R))-measurable, which together with (5.12) and formula (5.5) prove that
Ω3ω7→z(t, ω, u)∈R is (F,B(R))-measurable. Finally from this fact, the formula
Z 0
−1
z(t+s, ω, u)u∗2(s)ds=
Z −t
−1
u2(t+s)u∗2(s)ds+
Z 0
−t
z(t+s, ω, u)u∗2(s)ds
=
Z −t
−1
u2(t+s)u∗2(s)ds+
Z t
0
z(s, ω, u)u∗2(s−1)ds ,
a similar argument, and (5.11) we conclude that (5.10) holds, as claimed. In view of the above, we call Φ = ((Uω(t))ω∈Ω,t∈R+,(θt)t∈R) as defined by (5.9)
the measurable linear skew-product semidynamical systemgenerated by (5.1). We finish this section showing that the skew-product semidynamical system, generated by the family of scalar linear random delay differential equations of the form (5.1), satisfies all the requirements for the existence of a generalized expo-nential separation. Notice that according to Remark 2.6we can take time T = 2 instead of 1 to check conditions (A1) and (A3).
Lemma 5.6. Let x1, . . . , xn>0. Then
Proof. Applying the Jensen inequality to the convex function f(x) = xlnx we obtain
As the right-hand side of the above inequality is nonnegative, we obtain the desired
result.
Proposition 5.7. Under (S1) and(S2),Φ = ((Uω(t))ω∈Ω,t∈R+,(θt)t∈R)is a
mea-surable linear skew-product semidynamical system satisfying properties (A1), (A2)
and (A3) for time T = 2. Moreover, the generalized Lyapunov exponent satisfies
¯
λ1≥RΩa dP.
Proof. Since forω ∈Ω andu∈X fixed the mappingR+ 3t7→Uω(t)u∈X
is easily seen to be continuous, the fact that the mapping
R+×Ω×X3(t, ω, u)7→Uω(t)u∈X
is (B(R+)⊗F⊗B(X),B(X))-measurable follows from Proposition5.4, Lemma5.5 and Aliprantis and Border [2, Lemma 4.51 on pp. 153]. The rest of the prop-erties have been already checked, so that Φ is a measurable linear skew-product semidynamical system, as claimed.
Concerning the first part of (A1) withT = 2 notice that sup
As shown in Proposition5.4,kUω(t)k ≤3c(ω) (1 +d(ω)) for eachω∈Ω andt∈ of (S1), Fubini’s theorem and the invariance of P. Analogously, ln+d(ω) belongs toL1(Ω,F,P) because of (S1), and therefore, with the help of Lemma5.6,
and hence, an analogous argument using (S1), Fubini’s theorem and the invariance of the measurePproves that
and the first assertion of (A1) holds. We omit the second part of (A1) because it is analogous. It is immediate to check that (A2) follows from (S2).
We will finish by verifying that (A3) holds for timeT = 2. We consider the vector e= (1/2, u0)∈X+ withu0(s) = 1/2 for eachs∈[−1,0]. We havekekX = 1 and
A straightforward computation shows that
Consequently, again (5.6) yields In order to finish the proof we have to check that ln+ln
κ∈L1((Ω,F,P)). However,
from which, as before, together with (S1), Fubini’s theorem and the invariance of P, we conclude that lnκ∈L1((Ω,F,P)) and (A3) holds, as stated. The inequality ¯
λ1≥
R
Ωa dPfollows from Remark5.2and Birkhoff ergodic theorem. To sum up, we have proved the following.
Theorem 5.8. Assume (S1) and (S2). Then the measurable linear skew-product semidynamical system Φ = ((Uω(t))ω∈Ω,t∈R+,(θt)t∈R) generated by (5.1)admits a
generalized exponential separation of type II, with ˜λ1>−∞.
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(Janusz Mierczy´nski)Faculty of Pure and Applied Mathematics, Wroc law University of Science and Technology, Wybrze ˙ze Wyspia´nskiego 27, PL-50-370 Wroc law, Poland.
E-mail address, Janusz Mierczy´nski:[email protected]
(Sylvia Novo, Rafael Obaya)Departamento de Matem´atica Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain.
E-mail address, Sylvia Novo:[email protected]
