Admissibility and exponential trichotomy of dynamical systems described by skew-product flows

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Admissibility and exponential trichotomy of dynamical systems described by skew-product flows Adina Lumini¸ta Sasu, Bogdan Sasu ∗ Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timi¸soara, V. Pârvan Blvd. No. 4, 300223 Timi¸soara, Romania Received 11 February 2015; revised 12 September 2015

Abstract The aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems. The investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input–output system. We prove that the uniform admissibility of the pair (Cb (R, X), L1 (R, X)) for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system. If p ∈ (1, ∞) and the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible then we obtain the uniform exponential trichotomy. Next, we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy. Supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input–output system has unique bounded solutions in certain subspaces. Finally we obtain that the uniform p-admissibility of the pair (Cb (R, X), L1 (R, X)) is a necessary and sufficient condition for uniform exponential trichotomy. © 2015 Elsevier Inc. All rights reserved.

MSC: 34D09; 34D05; 93D25 Keywords: Uniform exponential trichotomy; Skew-product flow; Admissibility; Input–output system

* Corresponding author.

E-mail addresses: [email protected], [email protected] (A.L. Sasu), [email protected], [email protected] (B. Sasu). http://dx.doi.org/10.1016/j.jde.2015.09.042 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction Exponential trichotomy represents one of the most interesting asymptotic properties of dynamical systems, being based on a complete decomposition of the state space into a direct sum of stable subspace, unstable subspace and central subspace in every point of flow’s domain. When a dynamical system admits an exponential trichotomy, the asymptotic behavior of solutions is described through exponential decay forward and backward in time on stable and unstable subspaces and respectively through the existence of uniform upper and lower bounds on central subspaces. The notion of exponential trichotomy was introduced in the remarkable work of Sacker and Sell (see [24]) for dynamical systems described by skew-product flows, marking the beginning of a new topic of wide interest in bifurcation theory. At the end of the eighties, important concepts of exponential trichotomy were identified and studied by Elaydi and Hájek (see [8,9]) both for linear and nonlinear differential equations. Over the last decades the studies devoted to trichotomy have proved to be real challenges, leading to the development of new techniques in the asymptotic theory of dynamical systems (see Alonso, Hong and Obaya [1], Elaydi and Hájek [8,9], Elaydi and Janglajew [10], López-Fenner and Pinto [11], Minh and Wu [15], Palmer [18], Pliss and Sell [20,21], Pötzsche [22], Sasu and Sasu [26,28–30,33,34], Zhu [37] and the references therein). Due to the existence of nontrivial central subspaces, the studies in the field of trichotomy were complex and the progress was done in many stages. In order to investigate the trichotomy, new methods have had to be implemented compared with the dichotomy (see e.g. Elaydi and Hájek [8], Minh and Wu [15], Palmer [18], Sasu and Sasu [26,28–30,33,34]). A first aim was to analyze the properties of various classes of dynamical systems having an exponential trichotomy. In [37] Zhu used a property of exponential trichotomy for linear systems to establish principal normal coordinates and to study the problem of heteroclinic orbits to nonhyperbolic equilibria. Various studies were devoted to the robustness of the exponential trichotomy in the presence of perturbations (see Alonso, Hong and Obaya [1], Elaydi and Hájek [8], Elaydi and Janglajew [10], López-Fenner and Pinto [11] and the references therein). A remarkable result was obtained by Palmer in [18], where the author proved that the hyperbolicity of a flow can be expressed in terms of trichotomies and thus the author used the shadowing theorem to prove Silnikov’s theorem. A notable step was done by Minh and Wu in [15], where the authors proved the existence of center-unstable and center manifolds for a nonlinear process with exponential trichotomy. The input–output techniques for exponential trichotomy were progressively developed because it was difficult to impose solvability criteria that imply the splitting of the state space into stable, unstable and central subspaces, to determine the corresponding projections and moreover to obtain the asymptotic behavior on their ranges (see Elaydi and Janglajew [10], Sasu and Sasu [26,28,29,33] and the references therein). Another class of results was represented by the Rolewicz type theorems and by the Zabczyk type criteria for exponential trichotomy which were essentially based on the convergence of some associated integrals or series of trajectories (see [30] and [34]). It is worth mentioning that the control type methods play a central role in the study of the asymptotic behavior of dynamical systems and, among them, some of the most representative techniques are based on input–output approaches (see Chow and Leiva [4], Minh [13], Minh and Huy [14], Palmer [17], Pliss and Sell [20], Sasu [31], Sasu and Sasu [25–29,32,33], Zhang [35], Zhou, Lu and Zhang [36] and the references therein). The input–output methods in the qualitative theory of differential equations or the so-called admissibility methods have the origin in the pioneering work of Perron [19]. Notable contributions were obtained in the sixties when Massera

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and Schäffer introduced and developed the admissibility techniques in the study of stability and dichotomy (see [12] and the references therein). The main idea of the admissibility techniques is to characterize an asymptotic behavior of a dynamical system in terms of the solvability of an associated input–output system (see Chow and Leiva [4], Daleckii and Krein [7], Minh [13], Minh and Huy [14], Palmer [17], Pliss and Sell [20], Sasu [31], Sasu and Sasu [26–29,32,33], Zhang [35], Zhou, Lu and Zhang [36]). In the last decades most of the studies were focused on admissibility criteria for exponential dichotomy (see Chow and Leiva [4], Minh [13], Minh and Huy [14], Sasu [31], Sasu and Sasu [27,32] and the references therein). Despite the number of substantial results concerning the connections between admissibility and exponential dichotomy, those methods couldn’t be applied to the detection of the exponential trichotomy. In order to identify the existence of a trichotomic behavior through an input–output technique, the approach is more complicated. Our first study was focused on the case of evolution families on the real line (see [26]). The case of discrete dynamical systems was studied from various perspectives in Elaydi and Janglajew [10], Sasu and Sasu [28,29,33], using admissibility techniques in terms of sequence spaces. The aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of skew-product flows. We will associate to a skew-product flow an integral input–output system and in three main stages we will impose some solvability conditions through which we deduce the trichotomic behavior. Using a new approach based on the admissibility of the pair (Cb (R, X), L1 (R, X)) we introduce and study the properties of two families of input–output operators. We propose a control type method in which the inputs are integrable functions and the outputs belong to two special spaces of continuous functions. Thus, we obtain for the first time a description of the connections between some new and general admissibility properties and the existence of the uniform exponential trichotomy of skew-product flows, as follows. In the first part, we prove that the admissibility of the pair (Cb (R, X), L1 (R, X)) for the integral input–output system associated to a skew-product flow implies the existence of three families of projections which are complemented at every point of the flow’s domain and satisfy a regularity property. We obtain unified upper bounds for the first and the third families of projections in terms of the norms of the input–output operators. At the next step the central idea is to improve the assumption on admissibility in order to determine the uniform asymptotic behavior of the skew-product flow on the ranges of the projections. Using the uniform admissibility we prove the uniform boundedness of the projections and also the reversibility on the images of the second and of the third families of projections. In this manner, we obtain a complete description of the uniform trichotomic behavior. At the third step, the main aim is to refine the hypotheses such that the admissibility can imply the uniform exponential trichotomy. The central result will establish that, if the pair (Cb (R, X), L1 (R, X)) satisfies a uniform p-admissibility condition with p ∈ (1, ∞) for the associated integral input–output system, then the initial skew-product flow admits a uniform exponential trichotomy. At this point a new question arises: whether our admissibility conditions are also necessary for the uniform exponential trichotomy of skew-product flows. In the last part of the paper we will answer this question. With this purpose we will determine the existence and the uniqueness of the solutions of the associated integral input–output system in certain subspaces, under the assumption that the initial skew-product flow admits a uniform exponential trichotomy. After that we prove that if a skew-product flow has a uniform exponential trichotomy then the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible for the associated integral input–output sys-

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tem, for every p ∈ (1, ∞). Finally we deduce new characterizations for uniform exponential trichotomy of skew-product flows. 2. Sufficient conditions for exponential trichotomy In this section we introduce three admissibility concepts and we present a new and very general method for the detection of the existence of the exponential trichotomy. Our techniques will be based on some natural solvability conditions imposed to an integral control system associated to a skew-product flow. We present a study that will be developed in several main steps: the trichotomic splitting of the state space into stable, unstable and bounded subspace in every point of the flow’s domain, next we deduce regularity properties, the uniform trichotomic behavior and finally we prove the existence of the exponential trichotomy. We start with some basic definitions and notations. Indeed, let X be a real or complex Banach space and let Id denote the identity operator on X. The norm on X and on B(X) – the space of all bounded linear operators on X, will be denoted by || · ||. Let R be the set of the real numbers, R+ = {t ∈ R : t ≥ 0} and R− = {t ∈ R : t ≤ 0}. For every A ⊂ R we denote by χA the characteristic function of A and by A∗ = A \ {0}. Let M(R, X) be the linear space of all Bochner measurable functions f : R → X, identifying the functions equal almost everywhere. For every p ∈ [1, ∞), we denote by  Lp (R, X) := {f ∈ M(R, X) : ||f (τ )||p dτ < ∞} R

 which is a Banach space with respect to the norm ||f ||p = ( R ||f (τ )||p dτ )1/p . Let C(R, X) be the linear space of all continuous functions f : R → X. Denoting by Cb (R, X) := {f ∈ C(R, X) : sup ||f (t)|| < ∞} t∈R

Cb0 (R, X) := {f ∈ Cb (R, X) : lim f (t) = 0} t→∞

C0b (R, X) := {f ∈ Cb (R, X) : lim f (t) = 0} t→−∞

we have that all these spaces are Banach spaces with respect to the norm ||f ||∞ = sup ||f (t)||. t∈R

The definitions and usual notations for flows and skew-product flows are sufficiently wellknown (see Chow and Leiva [4–6], Sacker and Sell [23,24], Pliss and Sell [20,21], Sasu and Sasu [25,28,30]), but for the sake of clarity we recall them in what follows. Let (, d) be a metric space and let E = X × . Definition 2.1. A continuous mapping σ :  × R →  is called a flow on  if σ (θ, 0) = θ and σ (θ, s + t) = σ (σ (θ, s), t), for all (θ, s, t) ∈  × R2 . Definition 2.2. A pair π = (, σ ) is called a skew-product flow on E if σ is a flow on  and the mapping  :  × R+ → B(X), called cocycle, satisfies the following properties:

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(i) (ii) (iii) (iv)

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(θ, 0) = Id , for all θ ∈ ; (θ, s + t) = (σ (θ, s), t)(θ, s), for all (θ, s, t) ∈  × R2+ (the cocycle identity); for every x ∈ X, the mapping (θ, t) → (θ, t)x is continuous; there are M ≥ 1 and ω > 0 such that ||(θ, t)|| ≤ Meωt , for all (θ, t) ∈  × R+ .

For a better understanding of the foundations of the asymptotic theory of dynamical systems described by skew-product flows and for representative examples we chronologically mention the remarkable works of Sacker and Sell [23,24], Chow and Leiva [4–6] and Pliss and Sell [20, 21] and also the references therein. Definition 2.3. An operator P ∈ B(X) is called a projection if P 2 = P . Remark 2.1. If P is a projection, then Range P and Ker P are P -invariant closed linear subspaces and X = Range P ⊕ Ker P . Moreover, if P = 0, then ||P || ≥ 1. Definition 2.4. We say that a skew-product flow π = (, σ ) has a uniform exponential trichotomy if there are three families of projections {Pk (θ )}θ∈ ⊂ B(X), k ∈ {1, 2, 3} and two constants K ≥ 1 and ν > 0 such that: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

(θ, t)Pk (θ ) = Pk (σ (θ, t))(θ, t), for all (θ, t) ∈  × R+ and all k ∈ {1, 2, 3}; Pk (θ )Pj (θ ) = 0, for all k = j and all θ ∈ ; P1 (θ ) + P2 (θ ) + P3 (θ ) = Id , for all θ ∈ ; supθ∈ ||Pk (θ )|| < ∞, for all k ∈ {1, 2, 3}; ||(θ, t)x|| ≤ Ke−νt ||x||, for all t ≥ 0, all x ∈ Range P1 (θ ) and all θ ∈ ; 1 K ||x|| ≤ ||(θ, t)x|| ≤ K ||x||, for all t ≥ 0, all x ∈ Range P2 (θ ) and all θ ∈ ; ||(θ, t)x|| ≥ K1 eνt ||x||, for all t ≥ 0, all x ∈ Range P3 (θ ) and all θ ∈ ; the restriction (θ, t)| : Range Pk (θ ) → Range Pk (σ (θ, t)) is an isomorphism, for all (θ, t) ∈  × R+ and all k ∈ {2, 3}.

Remark 2.2. The above concept of exponential trichotomy represents the trichotomy in the sense of Sacker and Sell and this was introduced for the first time in [24]. For this trichotomy concept notable achievements in case of evolution families were obtained by Palmer in [18] and by Minh and Wu in [15]. Remark 2.3. If in Definition 2.4 one considers P2 (θ ) = 0, for all θ ∈ , then one obtains the particular case of uniform exponential dichotomy (see Chow and Leiva [4–6], Pliss and Sell [20], Sacker and Sell [23] and our works [27,31]). We mention that in the dichotomy case the uniform boundedness of projections (iv) can be obtained as a consequence of the properties (i)–(iii), (v), (vii) and (viii) (see Chow and Leiva [4], Pliss and Sell [20] and our work [27]). Remark 2.4. According to Definition 2.4 (i)–(iii) it is easy to see that if π = (, σ ) has a uniform exponential trichotomy with the trichotomy projections {Pk (θ )}θ∈ , k ∈ {1, 2, 3}, then X = Range P1 (θ ) ⊕ Range P2 (θ ) ⊕ Range P3 (θ ), for all θ ∈ . As we have described in Introduction, one of the most important class of methods for the study of the asymptotic behavior of dynamical systems is represented by the admissibility methods.

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Our studies concerning the exponential trichotomy using admissibility techniques were developed in several major steps (see [26,28,29,33]). The first approach was devoted to evolution families defined on the real line (see [26]). In [26] we have defined an admissibility property with respect to an integral equation associated to an evolution family, using as inputs continuous functions with compact support, and we have deduced necessary and sufficient conditions for uniform exponential trichotomy of nonautonomous dynamical systems. The next important step was made in [28] where we have obtained the first admissibility criteria for uniform exponential trichotomy of variational difference equations using the solvability of a family of discrete input–output equations. The first study concerning the trichotomy of nonautonomous difference equations was obtained in [29], where we have used an admissibility property with respect to an associated linear difference equation such that the inputs were sequences with finite support and the outputs were well-chosen sequences from an p (Z, X)-space. The most complex study devoted to nonautonomous systems was developed in [33], where we have refined the methods by introducing new and more natural conditions of discrete admissibility for uniform exponential trichotomy of discrete dynamical systems. Moreover, using the concept of discrete admissibility introduced in [33], we have proved that the uniform exponential trichotomy of a nonautonomous system can be completely recovered from the trichotomic behavior of the associated discrete dynamical system. In the present paper, we propose a new and complex study which is completely distinct compared with those described above. Our central aim is to provide a gradual and complete description of the trichotomy properties of skew-product flows using three admissibility concepts which are introduced in three main stages. With this purpose we consider an integral control system where the inputs are integrable functions and the outputs are some bounded continuous functions. Remark 2.5. It should be mentioned that the integral techniques have a crucial role in the study of the asymptotic behavior of dynamical systems (see Chow and Lu [2], Chow, Lin and Lu [3], Chow and Leiva [6], Minh [13], Minh and Huy [14], Palmer [17], Pliss and Sell [21], Sasu and Sasu [26,30], Zhang [35]). Let π = (, σ ) be a skew-product flow on E. We associate to π the integral control system Eπ = (Eθ )θ∈ , where t f (t) = (σ (θ, s), t − s)f (s) +

(σ (θ, τ ), t − τ )v(τ ) dτ,

∀t ≥ s,

(Eθ )

s

with the input function v ∈ L1 (R, X) and the output function f ∈ C(R, X). Remark 2.6. Let θ ∈  be fixed. By considering the equation (Eθ ), it is easy to see that for every input function v (which is sufficient to be locally integrable), the corresponding solution of this equation fθ,v ∈ C(R, X). Thus the input–output system considered above is natural. Furthermore, it makes sense to introduce the following admissibility concept. Definition 2.5. The pair (Cb (R, X), L1 (R, X)) is said to be admissible for the system (Eπ ) if for every θ ∈  and for every v ∈ L1 (R, X) there exist a unique fθ,v ∈ Cb0 (R, X) and a unique gθ,v ∈ C0b (R, X) such that the pairs (fθ,v , v) and (gθ,v , v) satisfy the equation (Eθ ).

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Remark 2.7. If the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ), then for every θ ∈  it makes sense to consider the input–output operators Fθ : L1 (R, X) → Cb0 (R, X),

Fθ (v) = fθ,v

Gθ : L1 (R, X) → C0b (R, X),

Gθ (v) = gθ,v .

Lemma 2.1. If the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ), then for every θ ∈ , the input–output operators Fθ and Gθ are bounded linear operators. Proof. Let θ ∈ . We easily observe that Fθ and Gθ are linear operators, so it is sufficient to prove that these operators are closed. Let (vn ) ⊂ L1 (R, X), v ∈ L1 (R, X) and f ∈ Cb0 (R, X) be such that vn −→ v in L1 (R, X) and Fθ (vn ) −→ f in Cb0 (R, X), as n → ∞. For each n ∈ N we set fθ,vn := Fθ (vn ) and from fθ,vn −→ f in Cb0 (R, X) as n → ∞, it follows that fθ,vn (t) −→ f (t), n→∞

∀t ∈ R.

(2.1)

Let M ≥ 1 and ω > 0 be given by Definition 2.2 (iv). Then, for every t ≥ s, from t ||

t (σ (θ, τ ), t − τ )vn (τ ) dτ −

s

(σ (θ, τ ), t − τ )v(τ ) dτ || ≤ s

t ≤ Me

||vn (τ ) − v(τ )|| dτ ≤ Meω(t−s) ||vn − v||1 ,

ω(t−s)

∀n ∈ N

s

we obtain that t

t (σ (θ, τ ), t − τ )vn (τ ) dτ −→

(σ (θ, τ ), t − τ )v(τ ) dτ,

n→∞

s

∀t ≥ s.

(2.2)

s

In addition, we have that t fθ,vn (t) = (σ (θ, s), t − s)fθ,vn (s) +

(σ (θ, τ ), t − τ )vn (τ ) dτ,

∀n ∈ N, ∀t > s.

(2.3)

s

For n → ∞ in (2.3), by using relations (2.1) and (2.2) we deduce that t f (t) = (σ (θ, s), t − s)f (s) +

(σ (θ, τ ), t − τ )v(τ ) dτ,

∀t ≥ s.

s

This shows that the pair (f, v) satisfies the equation (Eθ ), so f = Fθ (v). This implies that Fθ is a closed linear operator. Using the closed graph theorem we obtain that Fθ is bounded. Using similar arguments, we deduce that Gθ is a bounded linear operator. 2

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For every θ ∈ , we denote by Q(θ ) the linear space of all functions q : R− → X with q(t) = (σ (θ, s), t − s)q(s),

∀s ≤ t ≤ 0.

Remark 2.8. If q ∈ Q(θ ), then q is continuous on R− . For every θ ∈  we consider the stable subspace S(θ ) = {x ∈ X : lim (θ, t)x = 0} t→∞

the bounded subspace B(θ ) = {x ∈ X : sup ||(θ, t)x|| < ∞ and there exists q ∈ Q(θ ) with q(0) = x t≥0

and sup ||q(t)|| < ∞} t≤0

and the unstable subspace U(θ ) = {x ∈ X : there exists q ∈ Q(θ ) with q(0) = x and lim q(t) = 0}. t→−∞

Lemma 2.2. For every (θ, t) ∈  × R+ , the following properties hold: (i) (θ, t)S(θ) ⊂ S(σ (θ, t)); (ii) (θ, t)B(θ ) = B(σ (θ, t)); (iii) (θ, t)U(θ ) = U(σ (θ, t)). Proof. See Proposition 2.1 in [30].

2

Lemma 2.3. If π = (, σ ) has a uniform exponential trichotomy with respect to the families of projections {Pk (θ )}θ∈ , k ∈ {1, 2, 3}, then Range P1 (θ ) = S(θ ),

Range P2 (θ ) = B(θ ),

Proof. See Proposition 2.2 in [30].

Range P3 (θ ) = U(θ ),

∀θ ∈ .

2

Remark 2.9. For the properties of the stable, unstable and bounded subspaces and their role in the asymptotic theory of dynamical systems we refer Chow and Leiva [4,5], Palmer [16,18], Pliss and Sell [20,21], Sacker and Sell [23,24], Sasu and Sasu [26–34], Zhou, Lu and Zhang [36], Zhu [37]. In what follows we will study the properties of the stable, unstable and bounded subspaces, assuming that the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ). Theorem 2.1. If the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ), then the following properties hold:

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(i) S(θ ) ∩ B(θ ) = S(θ ) ∩ U(θ ) = B(θ ) ∩ U(θ ) = {0}, for all θ ∈ ; (ii) S(θ ) + B(θ ) + U(θ ) = X, for all θ ∈ . Proof. (i) Let θ ∈ . Step 1. We prove that S(θ ) ∩ B(θ ) = S(θ ) ∩ U(θ ) = {0}. Let x ∈ S(θ) ∩ [B(θ ) ∪ U(θ )]. Then (θ, t)x −→ 0 as t → ∞ and there is q ∈ Q(θ ) with q(0) = x and supt≤0 ||q(t)|| < ∞. We consider the function  f : R → X,

f (t) =

(θ, t)x, q(t),

t ≥0 . t <0

We have that f ∈ Cb0 (R, X) and an easy computation shows that the pair (f, 0) satisfies the equation (Eθ ). Then we have that f = Fθ (0), so f = 0. In particular, we deduce that x = f (0) = 0. It follows that S(θ ) ∩ B(θ ) = S(θ ) ∩ U(θ ) = {0}. Step 2. We prove that B(θ ) ∩ U(θ ) = {0}. Let y ∈ B(θ ) ∩ U(θ ). From y ∈ U(θ ) we have that there is λ ∈ Q(θ ) such that λ(0) = y and λ(t) −→ 0 as t → −∞. In addition, from y ∈ B(θ ) we have that supt≥0 ||(θ, t)y|| < ∞. We consider the function  g : R → X,

g(t) =

(θ, t)x, λ(t),

t ≥0 . t <0

We have that g ∈ C0b (R, X) and an easy computation shows that the pair (g, 0) satisfies the equation (Eθ ). It follows that g = Gθ (0), so g = 0. This implies that x = g(0) = 0. So, we obtain that B(θ ) ∩ U(θ ) = {0}. 1 (ii) Let α : R → [0, 2] be a continuous function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let θ ∈  and let x ∈ X. We consider the function v : R → X,

v(t) = α(t)(θ, t)x.

We have that v is continuous with supp v ⊂ (0, 1). In particular, it follows that v ∈ L1 (R, X). Let f ∈ Cb0 (R, X) and g ∈ C0b (R, X) be such that f = Fθ (v) and g = Gθ (v). We observe that t f (t) = (θ, t)f (0) + ( α(τ ) dτ ) (θ, t)x = (θ, t)(f (0) + x),

∀t ≥ 1.

(2.4)

0

Since f ∈ Cb0 (R, X), from relation (2.4) we obtain that xs := f (0) + x ∈ S(θ ). On the other hand, from g = Gθ (v) we have that g(t) = (σ (θ, s), t − s)g(s),

∀s ≤ t ≤ 0.

This implies that g|R− ∈ Q(θ ). In addition, since g ∈ C0b it follows that

(2.5)

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xu := −g(0) ∈ U(θ ).

(2.6)

Let h : R → X, h(t) = g(t) − f (t). Then h ∈ Cb (R, X) and h(t) = (σ (θ, s), t − s)h(s),

∀t ≥ s.

(2.7)

We set xb := h(0). From (2.7) we have that h(t) = (θ, t)xb , for all t ≥ 0. This implies that sup ||(θ, t)xb || = sup ||h(t)|| ≤ ||h||∞ . t≥0

(2.8)

t≥0

Moreover, from (2.7) we have, in particular, that h|R− ∈ Q(θ ). Since h ∈ Cb (R, X) using relation (2.8) we deduce that xb ∈ B(θ ). In conclusion, using relations (2.5) and (2.6) we obtain that x = xs + xb + xu ∈ S(θ ) + B(θ ) + U(θ ) and the proof is complete. 2 Theorem 2.2. If the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ), then for every θ ∈ , the subspaces S(θ ), B(θ ) and U(θ ) are closed. 1 Proof. Let α : R → [0, 2] be a continuous function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let M, ω > 0 be given by Definition 2.2 (iv). Let θ ∈ . Step 1. We prove that S(θ ) is closed. Let (xn ) ⊂ S(θ ), xn −→ x as n → ∞. For every n ∈ N we consider the functions vn : R → X, vn (t) = α(t)(θ, t)xn ⎧ (θ, t)xn , t ≥1 ⎨ t fn : R → X, fn (t) = α(τ ) dτ (θ, t)x , t ∈ [0, 1) . n ⎩ 0 0, t <0 For every n ∈ N, we have that vn is continuous with supp vn ⊂ (0, 1). In particular, this implies that vn ∈ L1 (R, X). In addition, we observe that fn is continuous and since xn ∈ S(θ ) we have that fn (t) −→ 0 as n → ∞. In particular, we obtain that fn ∈ Cb0 (R, X). An easy computation shows that the pair (fn , vn ) satisfies the equation (Eθ ). Then, we have that fn = Fθ (vn ), for all n ∈ N. Let v : R → X, v(t) = α(t)(θ, t)x. We have that v is continuous with supp v ⊂ (0, 1), so v ∈ L1 (R, X). Let f = Fθ (v). Using Lemma 2.1 we obtain that ||fn − f ||∞ ≤ ||Fθ || ||vn − v||1 ,

∀n ∈ N.

(2.9)

Since ||vn (t) − v(t)|| ≤ α(t)Meω ||xn − x||, for all t ∈ R and all n ∈ N, we have that ||vn − v||1 ≤ Meω ||xn − x||,

∀n ∈ N.

(2.10)

From (2.9) and (2.10) it follows that ||fn − f ||∞ −→ 0 as n → ∞. In particular, this implies that f (t) = lim fn (t) = lim (θ, t)xn = (θ, t)x, n→∞

n→∞

∀t ≥ 1.

(2.11)

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Since f ∈ Cb0 (R, X) from (2.11) we obtain that (θ, t)x −→ 0 as t → ∞, so x ∈ S(θ ). This shows that S(θ ) is closed. Step 2. We prove that B(θ ) is closed. Let (yn ) ⊂ B(θ ) with yn −→ y as n → ∞. For every n ∈ N we consider the functions wn : R → X, wn (t) = α(t)(θ, t)yn ⎧ t ≥1 (θ, t)yn , ⎨ t gn : R → X, gn (t) = α(τ ) dτ (θ, t)y , t ∈ [0, 1) . n ⎩ 0 0, t <0 Using similar arguments with those considered in the proof of Step 1 we have that for every n ∈ N, wn ∈ L1 (R, X), gn ∈ C0b (R, X) and gn = Gθ (wn ). By considering the functions w : R → X,

w(t) = α(t)(θ, t)y

and g = Gθ (w) and using similar arguments with those from Step 1, we deduce that g(t) = (θ, t)y,

∀t ≥ 1.

(2.12)

Since g ∈ C0b (R, X) from (2.12) it follows, in particular, that sup ||(θ, t)y|| ≤ ||g||∞ < ∞. t≥1

Taking into account that ||(θ, t)y|| ≤ Meω ||y||,

∀t ∈ [0, 1]

we obtain that sup ||(θ, t)y|| < ∞.

(2.13)

t≥0

For every n ∈ N since yn ∈ B(θ ) we have that there is qn ∈ Q(θ ) with qn (0) = yn and supt≤0 ||qn (t)|| < ∞. For every n ∈ N we consider the function  ϕn : R → X,

ϕn (t) =

−

∞ t

α(τ ) dτ (θ, t)yn , −qn (t),

t ≥0 . t <0

Using Remark 2.8 we obtain that ϕn is continuous. Observing that ϕn (t) = 0, for all t ≥ 1 we deduce that ϕn ∈ Cb0 (R, X). An easy computation shows that the pair (ϕn , wn ) satisfies the equation (Eθ ). This implies that ϕn = Fθ (wn ), for all n ∈ N. Let ϕ = Fθ (w). Using Lemma 2.1 we have that ||ϕn − ϕ||∞ ≤ ||Fθ || ||wn − w||1 ≤ ||Fθ || Meω ||yn − y||,

∀n ∈ N.

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The above inequality implies that ϕn (t) −→ ϕ(t) as n → ∞, for each t ∈ R. In particular, it follows that qn (t) = −ϕn (t) −→ −ϕ(t), n→∞

∀t ≤ 0.

(2.14)

Let q : R− → X, q(t) = −ϕ(t). Then from (2.14) we have that q(0) = lim qn (0) = lim yn = y. n→∞

n→∞

For every n ∈ N, the function qn ∈ Q(θ ), so qn (t) = (σ (θ, s), t − s)qn (s),

∀s ≤ t ≤ 0, ∀n ∈ N.

(2.15)

Using relation (2.14), for n → ∞ in (2.15) it follows that q(t) = (σ (θ, s), t − s)q(s),

∀s ≤ t ≤ 0

which implies that q ∈ Q(θ ). In addition sup ||q(t)|| = sup ||ϕ(t)|| ≤ ||ϕ||∞ . t≤0

t≤0

So, there is q ∈ Q(θ ) with q(0) = y and supt≤0 ||q(t)|| < ∞. Using relation (2.13) we conclude that y ∈ B(θ ). It follows that B(θ ) is closed. Step 3. We prove that U(θ ) is closed. Let (zn ) ⊂ U(θ ) with zn −→ z as n → ∞. Then, for every n ∈ N there is λn ∈ Q(θ ) with λn (0) = zn and λn (t) −→ 0 as t → −∞. We consider the functions un : R → X, hn : R → X,

un (t) = −α(t)(θ, t)zn ∞ α(τ ) dτ (θ, t)zn , t ≥ 0 . hn (t) = t t <0 λn (t),

For every n ∈ N we have that un ∈ L1 (R, X), hn ∈ C0b (R, X) and by a direct computation we deduce that (hn , un ) satisfies the equation (Eθ ). So hn = Gθ (un ), for all n ∈ N. Let u : R → X, u(t) = −α(t)(θ, t)z and let h = Gθ (u). Then, using Lemma 2.1 and similar arguments with those in Step 2 we obtain that hn (t) −→ h(t) as n → ∞, for each t ∈ R. Considering the function λ : R− → X,

λ(t) = h(t)

it follows that λn (t) = hn (t) −→ λ(t), n→∞

∀t ≤ 0.

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Using similar arguments with those in the proof of Step 2 we deduce that λ(0) = z and λ ∈ Q(θ ). Since h ∈ C0b (R, X) we also have that λ(t) −→ 0 as t → −∞. This implies that z ∈ U(θ ). It follows that U(θ ) is closed and the proof is complete. 2 The first main result of this section is: Theorem 2.3 (The splitting of the state space). If the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ), then X = S(θ ) ⊕ B(θ ) ⊕ U(θ ),

∀θ ∈ .

Proof. Let θ ∈ . From Theorem 2.2 we have that the subspaces S(θ ), B(θ ) and U(θ ) are closed and from Theorem 2.1 we have that S(θ ) ∩ B(θ ) = S(θ ) ∩ U(θ ) = B(θ ) ∩ U(θ ) = {0}

(2.16)

S(θ ) + B(θ ) + U(θ ) = X.

(2.17)

and

Then, it remains to show that the decomposition obtained in (2.17) is uniquely determined. Let x ∈ X and let xθ , x˜θ ∈ S(θ ), yθ , y˜θ ∈ B(θ ) and zθ , z˜ θ ∈ U(θ ) be such that x = xθ + yθ + zθ = x˜θ + y˜θ + z˜ θ .

(2.18)

Denoting by u := xθ − x˜θ , v = y˜θ − yθ and w = z˜ θ − zθ , from (2.18) we deduce that u = v + w.

(2.19)

Since v ∈ B(θ ) there is q ∈ Q(θ ) with q(0) = v and supt≤0 ||q(t)|| < ∞. From w ∈ U(θ ) we have that there is ϕ ∈ Q(θ ) with ϕ(0) = w and ϕ(t) −→ 0 as t → −∞. Let λ : R− → X,

λ(t) = q(t) + ϕ(t).

Since q, ϕ ∈ Q(θ ) we obtain that λ ∈ Q(θ ). In addition, we have that supt≤0 ||λ(t)|| < ∞ and λ(0) = v + w. We consider the function  f : R → X,

f (t) =

(θ, t)u, λ(t),

t ≥0 . t <0

Using relation (2.19) and the property that u ∈ S(θ ), it is easy to see that f ∈ Cb0 (R, X). A simple computation shows that the pair (f, 0) satisfies the equation (Eθ ), so f = Fθ (0). This implies that f = 0. In particular, it follows that u = f (0) = 0 which implies that x˜θ = xθ . Then, from relation (2.19) we have that v = −w. Since v ∈ B(θ ) and w ∈ U(θ ), from relation (2.16) we deduce that v = w = 0. In conclusion, we obtain that x˜ θ = xθ , y˜θ = yθ and respectively z˜ θ = zθ and the proof is complete. 2

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The second main result of this section is: Theorem 2.4. If the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ), then there are three families of projections {Pk (θ )}θ∈ , k ∈ {1, 2, 3}, such that the following properties hold: Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ), for all θ ∈ ; P1 (θ ) + P2 (θ ) + P3 (θ ) = Id , for all θ ∈ ; Pk (θ )Pj (θ ) = 0, for all k = j and all θ ∈ ; (θ, t)Pk (θ ) = Pk (σ (θ, t))(θ, t), for all (θ, t) ∈  × R+ and all k ∈ {1, 2, 3}. Moreover, if M, ω > 0 are given by Definition 2.2 (iv), then the following estimations hold: (v) ||P3 (θ )|| ≤ Meω ||Gθ ||, for all θ ∈ ; (vi) ||P1 (θ )|| ≤ 1 + Meω ||Fθ ||, for all θ ∈ .

(i) (ii) (iii) (iv)

Proof. From Theorem 2.3 we have that S(θ ) ⊕ B(θ ) ⊕ U(θ ) = X,

∀θ ∈ .

(2.20)

For every θ ∈ , let P1 (θ ), P2 (θ ) and P3 (θ ) be the projections such that Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ).

(2.21)

Then, we obtain three families of projections {Pk (θ )}θ∈ , k ∈ {1, 2, 3}. According to relations (2.20) and (2.21) we have that the assertions (i)–(iii) are fulfilled. Using Lemma 2.2 we deduce that (θ, t)Pk (θ ) = Pk (σ (θ, t))(θ, t),

∀(θ, t) ∈  × R+ , ∀k ∈ {1, 2, 3}.

It remains to prove the assertions (v) and (vi). Let M, ω > 0 be given by Definition 2.2 (iv). Then ||(θ, t)|| ≤ Meωt ,

∀(θ, t) ∈  × R+ .

(2.22)

(v) Let θ ∈  and let x ∈ X. We denote by xsθ = P1 (θ )x, xbθ = P2 (θ )x and xuθ = P3 (θ )x. Then (θ, t)xsθ −→ 0 as t → ∞ and supt≥0 ||(θ, t)xbθ || < ∞. In particular, it follows that sup ||(θ, t)(xsθ + xbθ )|| < ∞.

(2.23)

t≥0

Since xuθ ∈ Range P3 (θ ) = U(θ ) there is q ∈ Q(θ ) with q(0) = xuθ and q(t) −→ 0 as t → −∞. We consider the functions v : R → X, g : R → X,

v(t) = −χ[0,1] (t)(θ, t)x ⎧ θ θ ⎪ ⎨ −(θ, t)(xs + xb ), t ≥ 1 g(t) = (θ, t)(xuθ − tx), t ∈ (0, 1) . ⎪ ⎩ q(t), t ≤0

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We have that g is continuous and using (2.23) we deduce that g ∈ C0b (R, X). On the other hand, using relation (2.22) we deduce that ||v(t)|| ≤ χ[0,1] (t)Meω ||x||,

∀t ∈ R.

(2.24)

Relation (2.24) shows that v ∈ L1 (R, X) and ||v||1 ≤ Meω ||x||.

(2.25)

Moreover we have that the pair (g, v) satisfies the equation (Eθ ), so g = Gθ (v). Then, we have that ||g(0)|| ≤ ||g||∞ ≤ ||Gθ || ||v||1 .

(2.26)

Since g(0) = xuθ = P3 (θ )x, from (2.25) and (2.26) it follows that ||P3 (θ )x|| ≤ Meω ||Gθ || ||x||.

(2.27)

Since x ∈ X and θ ∈  were arbitrary, from (2.27) we obtain that ||P3 (θ )|| ≤ Meω ||Gθ ||,

∀θ ∈ .

(vi) Let θ ∈  and let x ∈ X. We denote by xsθ = P1 (θ )x, xbθ = P2 (θ )x and xuθ = P3 (θ )x. From xbθ ∈ Range P2 (θ ) = B(θ ) we have that there is δ ∈ Q(θ ) with δ(0) = xbθ and supt≤0 ||δ(t)|| < ∞. Since xuθ ∈ Range P3 (θ ) = U(θ ) there is λ ∈ Q(θ ) with λ(0) = xuθ and λ(t) −→ 0 as t → −∞. Setting ϕ : R− → X,

ϕ(t) = δ(t) + λ(t)

we have that ϕ ∈ Q(θ ), ϕ(0) = xbθ + xuθ = x − xsθ and supt≤0 ||ϕ(t)|| < ∞. We consider the functions v : R → X, f : R → X,

⎧ ⎪ ⎨

v(t) = −χ[0,1] (t)(θ, t)x

−(θ, t)xsθ , f (t) = (θ, t)[(1 − t)x − xsθ ], ⎪ ⎩ ϕ(t),

t ≥1 t ∈ (0, 1) . t ≤0

Using similar arguments with those used at the item (v) we have that v ∈ L1 (R, X) and ||v||1 ≤ Meω ||x||. We observe that f is continuous and since xsθ ∈ Range P1 (θ ) = S(θ ) we have that f (t) −→ 0 as t → ∞. Thus, we obtain that f ∈ Cb0 (R, X). In addition, we deduce that the pair (f, v) satisfies the equation (Eθ ), so f = Fθ (v). This successively implies that ||x − xsθ || = ||ϕ(0)|| = ||f (0)|| ≤ ||f ||∞ ≤ ||Fθ || ||v||1 ≤ Meω ||Fθ || ||x||.

(2.28)

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From relation (2.28) it follows that ||P1 (θ )x|| ≤ (1 + Meω ||Fθ ||) ||x||.

(2.29)

Since x ∈ X and θ ∈  were arbitrary, from relation (2.29) we deduce that ||P1 (θ )|| ≤ 1 + Meω ||Fθ ||,

∀θ ∈ .

2

For the next step of our study, it is natural to introduce the following admissibility concept: Definition 2.6. The pair (Cb (R, X), L1 (R, X)) is said to be uniformly admissible for the system (Eπ ) if it is admissible and sup ||Fθ || < ∞ and sup ||Gθ || < ∞. θ∈

θ∈

Theorem 2.5. If the pair (Cb (R, X), L1 (R, X)) is uniformly admissible for the system (Eπ ), then there is K > 0 such that ||(θ, t)x|| ≤ K ||x||,

∀t ≥ 0, ∀x ∈ S(θ ) ∪ B(θ ), ∀θ ∈ 

(2.30)

∀t ≥ 0, ∀y ∈ B(θ ) ∪ U(θ ), ∀θ ∈ .

(2.31)

and ||(θ, t)y|| ≥

1 ||y||, K

Proof. Let L > 0 be such that sup ||Fθ || ≤ L and sup ||Gθ || ≤ L. θ∈

(2.32)

θ∈

Let M, ω > 0 be such that ||(θ, t)|| ≤ Meωt ,

∀(θ, t) ∈  × R+

(2.33)

and let K := max{Meω , LMeω }.

(2.34)

Step 1. We prove that the relation (2.30) holds. Let θ ∈  and let x ∈ S(θ ) ∪ B(θ ). Then supt≥0 ||(θ, t)x|| < ∞. We consider the function v : R → X,

v(t) = χ[0,1] (t)(θ, t)x.

Then we have that v ∈ L1 (R, X) and using (2.33) it follows that ||v||1 ≤ Meω ||x||.

(2.35)

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We consider the function g : R → X,

⎧ ⎨ (θ, t)x, g(t) = t (θ, t)x, ⎩ 0,

t ≥1 t ∈ [0, 1) . t <0

We have that g ∈ C0b (R, X) and an immediate computation shows that the pair (g, v) satisfies the equation (Eθ ). This implies that g = Gθ (v). Then, using relation (2.32) we deduce that ||g||∞ ≤ L ||v||1 .

(2.36)

From relations (2.35) and (2.36) we have that ||(θ, t)x|| = ||g(t)|| ≤ ||g||∞ ≤ LMeω ||x||,

∀t ≥ 1.

(2.37)

In addition, using relation (2.33) we have that ||(θ, t)x|| ≤ Meω ||x||,

∀t ∈ [0, 1].

(2.38)

Then, from relations (2.37), (2.38) and (2.34) we have that ||(θ, t)x|| ≤ K ||x||,

∀t ≥ 0.

Since K > 0 does not depend on θ ∈  or on x ∈ S(θ ) ∪ B(θ ), it follows that the relation (2.30) holds. Step 2. We prove that the relation (2.31) holds. Let θ ∈  and let y ∈ B(θ ) ∪ U(θ ). Then, there is q ∈ Q(θ ) with q(0) = y and supt≤0 ||q(t)|| < ∞. Let t > 0. We consider the function wt : R → X,

wt (s) = −χ[t,t+1] (s)(θ, s)y.

Using relation (2.33) we deduce that wt ∈ L1 (R, X) and ||wt ||1 ≤ Meω ||(θ, t)y||.

(2.39)

Let

ft : R → X,

⎧ ⎪ ⎪ ⎨

0, (t + 1 − s) (θ, s)y, ft (s) = (θ, s)y, ⎪ ⎪ ⎩ q(s),

s ≥t +1 s ∈ [t, t + 1) . s ∈ [0, t) s<0

We have that ft ∈ Cb0 (R, X) and we deduce that the pair (ft , wt ) satisfies the equation (Eθ ). This shows that ft = Fθ (wt ). Then, using relations (2.32), (2.39) and (2.34) we successively obtain that ||y|| = ||ft (0)|| ≤ ||ft ||∞ ≤ L ||wt ||1 ≤ L Meω ||(θ, t)y|| ≤ K ||(θ, t)y||.

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Since t > 0 was arbitrary and K ≥ 1 it follows that ||(θ, t)y|| ≥

1 ||y||, K

∀t ≥ 0.

Taking into account that θ ∈  and y ∈ B(θ ) ∪ U(θ ) were arbitrary, we conclude that the relation (2.31) holds and the proof is complete. 2 Corollary 2.1. If the pair (Cb (R, X), L1 (R, X)) is uniformly admissible for the system (Eπ ), then the following assertions hold: (i) for every (θ, t) ∈  × R+ the restriction (θ, t)| : B(θ ) → B(σ (θ, t)) is an isomorphism; (ii) for every (θ, t) ∈  × R+ the restriction (θ, t)| : U(θ ) → U(σ (θ, t)) is an isomorphism. Proof. (i) Let (θ, t) ∈  × R+ . From Lemma 2.2 (ii) we have that the restriction (θ, t)| : B(θ ) → B(σ (θ, t)) is correctly defined and surjective. From relation (2.31) it follows that (θ, t)| is also injective, so (θ, t)| is an isomorphism. (ii) This follows based on Lemma 2.2 (iii) and relation (2.31), by using similar arguments with those considered at (i). 2 The third main result of this section is the following: Theorem 2.6. If the pair (Cb (R, X), L1 (R, X)) is uniformly admissible for the system (Eπ ), then there are three families of projections {Pk (θ )}θ∈ , k ∈ {1, 2, 3} and a constant K > 0 such that the following properties hold: Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ), for all θ ∈ ; P1 (θ ) + P2 (θ ) + P3 (θ ) = Id , for all θ ∈ ; Pk (θ )Pj (θ ) = 0, for all k = j and all θ ∈ ; (θ, t)Pk (θ ) = Pk (σ (θ, t))(θ, t), for all (θ, t) ∈  × R+ and all k ∈ {1, 2, 3}; supθ∈ ||Pk (θ )|| < ∞, for all k ∈ {1, 2, 3}; ||(θ, t)x|| ≤ K ||x||, for all t ≥ 0, all x ∈ Range P1 (θ ) ∪ Range P2 (θ ) and all θ ∈ ; ||(θ, t)y|| ≥ K1 ||y||, for all t ≥ 0, all y ∈ Range P2 (θ ) ∪ Range P3 (θ ) and all θ ∈ ; for all (θ, t) ∈  × R+ and all k ∈ {2, 3} the restriction (θ, t)| : Range Pk (θ ) → Range Pk (σ (θ, t)) is an isomorphism and we denote by k (θ, t)−1 | its inverse; (ix) for every (θ, t0 ) ∈  × R, every k ∈ {2, 3} and every x ∈ Range Pk (σ (θ, t0 )), k (σ (θ, t), t0 − t)−1 | x −→ x as t  t0 ; (x) for every k ∈ {1, 2, 3} and every (x, θ ) ∈ X × , the function

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

k ϕx,θ : R → X,

k ϕx,θ (t) = Pk (σ (θ, t))x

is continuous on R. Proof. The properties (i)–(viii) follow from Theorem 2.4, Theorem 2.5 and Corollary 2.1. (ix) Let (θ, t0 ) ∈  × R, let k ∈ {2, 3} and let x ∈ Range Pk (σ (θ, t0 )). Since the operator (σ (θ, t0 − 1), 1)| : Range Pk (σ (θ, t0 − 1)) → Range Pk (σ (θ, t0 )) is an isomorphism, there is

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y ∈ Range Pk (σ (θ, t0 − 1)) such that x = (σ (θ, t0 − 1), 1)y. Then, for every t ∈ [t0 − 1, t0 ] we have that x = (σ (θ, t), t0 − t)(σ (θ, t0 − 1), t − (t0 − 1))y. This implies that k (σ (θ, t), t0 − t)−1 | x = (σ (θ, t0 − 1), t − (t0 − 1))y −→ (σ (θ, t0 − 1), 1)y = x. tt0

(x) Let γ > 0 be such that ||Pk (θ )|| ≤ γ ,

∀θ ∈ , ∀k ∈ {1, 2, 3}.

(2.40)

Let (x, θ ) ∈ X × and let k ∈ {2, 3}. Let t0 ∈ R. Using (iv), relation (2.40) and Definition 2.2 (iii) we obtain that k k ||ϕx,θ (t) − ϕx,θ (t0 )|| ≤ ||Pk (σ (θ, t))x − Pk (σ (θ, t))(σ (θ, t0 ), t − t0 )x|| +

+ ||(σ (θ, t0 ), t − t0 )Pk (σ (θ, t0 ))x − Pk (σ (θ, t0 ))x|| ≤ γ ||x − (σ (θ, t0 ), t − t0 )x|| + + ||(σ (θ, t0 ), t − t0 )Pk (σ (θ, t0 ))x − Pk (σ (θ, t0 ))x|| −→ 0. tt0

(2.41)

k is right-continuous in t . On the other hand, From relation (2.41) we have that the function ϕx,θ 0 using (iv), we deduce that k ϕx,θ (t) = k (σ (θ, t), t0 − t)−1 | (σ (θ, t), t0 − t)| Pk (σ (θ, t))x =

= k (σ (θ, t), t0 − t)−1 | Pk (σ (θ, t0 ))(σ (θ, t), t0 − t)x,

∀t < t0 .

(2.42)

Using relation (2.42), the property (vii) and respectively relation (2.40), we obtain that k k ||ϕx,θ (t) − ϕx,θ (t0 )|| ≤ ||k (σ (θ, t), t0 − t)−1 | Pk (σ (θ, t0 ))[(σ (θ, t), t0 − t)x − x]|| +

+ ||k (σ (θ, t), t0 − t)−1 | Pk (σ (θ, t0 ))x − Pk (σ (θ, t0 ))x|| ≤ Kγ ||(σ (θ, t), t0 − t)x − x|| + + ||k (σ (θ, t), t0 − t)−1 | Pk (σ (θ, t0 ))x − Pk (σ (θ, t0 ))x||,

∀t < t0 .

(2.43)

Using Definition 2.2 (iii) and the property (ix) from relation (2.43) it follows that the function k is also left-continuous in t . ϕx,θ 0 k is continuous Since t0 ∈ R was arbitrary, we deduce that for each k ∈ {2, 3} the function ϕx,θ on R. Taking into account that 1 2 3 ϕx,θ (t) = x − ϕx,θ (t) − ϕx,θ (t),

∀t ∈ R

1 is also continuous on R and the proof is complete. 2 we obtain that ϕx,θ

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Remark 2.10. According to Theorem 2.6 (i)–(viii) we deduce that the uniform admissibility of the pair (Cb (R, X), L1 (R, X)) almost provides the existence of the uniform exponential trichotomy of π . So, the natural question arises which is the minimal condition that should be added to the uniform admissibility such that this can imply the existence of the uniform exponential trichotomy. Remark 2.11. Suppose that the pair (Cb (R, X), L1 (R, X)) is uniformly admissible for the system (Eπ ) and let {Pk (θ )}θ∈ , k ∈ {1, 2, 3} be the families of projections given by Theorem 2.6. Then, from Theorem 2.6 (v) there is γ > 0 such that ||Pk (θ )|| ≤ γ ,

∀θ ∈ , ∀k ∈ {1, 2, 3}.

(2.44)

For every (k, θ ) ∈ {1, 2, 3} × , we consider the operator Pkθ : L1 (R, X) → L1 (R, X),

(Pkθ (v))(t) = Pk (σ (θ, t))v(t).

According to Theorem 2.6 (x), Pkθ is correctly defined and using relation (2.44) we have that ||(Pkθ (v))(t)|| ≤ γ ||v(t)||,

∀t ∈ R, ∀v ∈ L1 (R, X).

(2.45)

From relation (2.45) we deduce that Pkθ is a bounded linear operator and ||Pkθ || ≤ γ ,

∀θ ∈ , ∀k ∈ {1, 2, 3}.

(2.46)

For p ∈ (1, ∞) we consider the spaces Cb0 (R, X) ∩ Lp (R, X) and C0b (R, X) ∩ Lp (R, X), which are Banach spaces with respect to the norm ||v||∞,p := max{||v||∞ , ||v||p }. Definition 2.7. Let p ∈ (1, ∞). The pair (Cb (R, X), L1 (R, X)) is said to be uniformly p-admissible for the system (Eπ ) if the pair (Cb (R, X), L1 (R, X)) is uniformly admissible for it and there is λ > 0 such that the following properties hold: (i) for every (θ, v) ∈  × L1 (R, X) we have that (Fθ P1θ )(v) ∈ Cb0 (R, X) ∩ Lp (R, X) and ||(Fθ P1θ )(v)||∞,p ≤ λ ||v||1 ; (ii) for every (θ, v) ∈  × L1 (R, X) we have that (Gθ P3θ )(v) ∈ C0b (R, X) ∩ Lp (R, X) and ||(Gθ P3θ )(v)||∞,p ≤ λ ||v||1 . The central result of this section is: Theorem 2.7. Let p ∈ (1, ∞). If the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible for the system (Eπ ), then π = (, σ ) has a uniform exponential trichotomy.

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Proof. Let {Pk (θ )}θ∈ , k ∈ {1, 2, 3} be the families of projections given by Theorem 2.6 and let K > 0 be such that ||(θ, t)x|| ≤ K ||x||,

∀t ≥ 0, ∀x ∈ Range P1 (θ ) ∪ Range P2 (θ ), ∀θ ∈ 

(2.47)

1 ||y||, K

∀t ≥ 0, ∀y ∈ Range P2 (θ ) ∪ Range P3 (θ ), ∀θ ∈ .

(2.48)

and ||(θ, t)y|| ≥

Let λ > 0 be given by Definition 2.7. We consider h = (eλK 2 )p + 1

(2.49)

and we set ν = 1/ h and K˜ = Ke. Step 1. We prove that ˜ −νt ||x||, ||(θ, t)x|| ≤ Ke

∀t ≥ 0, ∀x ∈ Range P1 (θ ), ∀θ ∈ .

Let θ ∈  and let x ∈ Range P1 (θ ). If (θ, 1)x = 0, then we consider the functions v : R → X,

f : R → X,

f (t) =

v(t) = χ[0,1] (t) ⎧ ⎪ ⎨ ⎪ ⎩

(θ, t)x ||(θ, t)x||

δ (θ, t)x, t ≥1 dτ (θ, t)x, t ∈ [0, 1)

t 1 0 ||(θ,τ )x||

0,

t <0

where 1 δ := 0

1 dτ. ||(θ, τ )x||

We have that f is continuous and since x ∈ Range P1 (θ ) = S(θ ) it follows in particular that f ∈ Cb0 (R, X). Because x ∈ Range P1 (θ ) using Theorem 2.6 (iv) we deduce that v(t) ∈ Range P1 (σ (θ, t)),

∀t ∈ [0, 1].

(2.50)

Then, from relation (2.50) and Remark 2.11 we obtain that P1θ (v) = v.

(2.51)

A simple computation shows that the pair (f, v) satisfies the equation (Eθ ). Observing that v ∈ L1 (R, X) this implies that f = Fθ (v). Using the hypothesis and relation (2.51) we successively deduce that ||f ||∞,p = ||(Fθ P1θ )(v)||∞,p ≤ λ ||v||1 = λ.

(2.52)

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From relation (2.47) we have that ||(θ, h)x|| ≤ K ||(θ, t)x|| =

K ||f (t)||, δ

∀t ∈ [1, h].

(2.53)

Relation (2.53) implies that ||(θ, h)x||χ[1,h] (t) ≤

K ||f (t)||, δ

∀t ∈ R.

(2.54)

By integrating in relation (2.54) we deduce that ||(θ, h)x||(h − 1)1/p ≤

K K ||f ||p ≤ ||f ||∞,p . δ δ

(2.55)

Using relation (2.49), from relations (2.52) and (2.55) we obtain that e λ K 2 ||(θ, h)x|| ≤

λK δ

which implies that ||(θ, h)x|| ≤

1 . eKδ

(2.56)

In addition, from relation (2.47) we have that ||(θ, τ )x|| ≤ K ||x||,

∀τ ∈ [0, 1].

(2.57)

Then, using relation (2.57) we deduce that 1 ≤ δ ||x||. K

(2.58)

From relations (2.56) and (2.58) it follows that ||(θ, h)x|| ≤

1 ||x||. e

(2.59)

If (θ, 1)x = 0, then (θ, h)x = 0, so, in this case, relation (2.59) obviously holds. It follows that in both situations we have that ||(θ, h)x|| ≤

1 ||x||. e

Taking into account that h does not depend on x ∈ Range P1 (θ ) or on θ ∈  we obtain that ||(θ, h)x|| ≤

1 ||x||, e

∀x ∈ Range P1 (θ ), ∀θ ∈ .

(2.60)

Let now θ ∈  and x ∈ Range P1 (θ ). Let t ≥ 0 and let j ∈ N and r ∈ [0, h) be such that t = j h + r. Using relations (2.47) and (2.60) it follows that ˜ −νt ||x||. ||(θ, t)x|| ≤ K ||(θ, j h)x|| ≤ Ke−j ||x|| ≤ Ke

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Step 2. We prove that ||(θ, t)y|| ≥

1 νt e ||y||, K˜

∀t ≥ 0, ∀y ∈ Range P3 (θ ), ∀θ ∈ .

Let θ ∈  and let y ∈ Range P3 (θ ) \ {0}. Then, from relation (2.48) we have that (θ, t)y = 0, for all t ≥ 0. Since y ∈ Range P3 (θ ) = U(θ ), there is q ∈ Q(θ ) such that q(0) = y and q(t) −→ 0 as t → −∞. We consider the functions w : R → X,

g : R → X,

g(t) =

w(t) = −χ[h−1,h] (t) ⎧ ⎪ ⎪ ⎪ ⎨h ⎪ ⎪ ⎪ ⎩

(θ, t)y ||(θ, t)y|| t ≥h

0,

1 t ||(θ,τ )y||

dτ (θ, t)y,

t ∈ [h − 1, h) t ∈ [0, h − 1) t <0

β (θ, t)y, β q(t),

where h β := h−1

1 dτ. ||(θ, τ )y||

We have that g is continuous and obviously g ∈ C0b (R, X). Observing that ||w(t)|| = χ[h−1,h] (t),

∀t ∈ R,

we deduce that w ∈ L1 (R, X) and ||w||1 = 1. Since y ∈ Range P3 (θ ), using Theorem 2.6 (viii) it follows that w(t) ∈ Range P3 (σ (θ, t)),

∀t ≥ 0.

(2.61)

From relation (2.61) and Remark 2.11 we deduce that P3θ (w) = w. A simple computation shows that the pair (g, w) satisfies the equation (Eθ ), so g = Gθ (w). Then, we have that g = (Gθ P3θ )(w) and using the hypothesis we obtain that ||g||∞,p = ||(Gθ P3θ )(w)||∞,p ≤ λ ||w||1 = λ. From relation (2.48) we have that ||(θ, t)y|| ≥ which implies that

1 ||y||, K

∀t ∈ [0, h − 1)

(2.62)

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||y|| χ[0,h−1) (t) ≤

K ||g(t)||, β

∀t ∈ R.

(2.63)

By integrating in relation (2.63) and using relation (2.49) we successively deduce that 1

eλ K 2 ||y|| = ||y|| (h − 1) p ≤

K K ||g||p ≤ ||g||∞,p . β β

(2.64)

1 . β

(2.65)

From relations (2.62) and (2.64) we obtain that eK ||y|| ≤ From relation (2.48) we have that ||(θ, h)y|| ≥

1 ||(θ, τ )y||, K

∀τ ∈ [h − 1, h]

which implies that h β= h−1

1 1 dτ ≥ . ||(θ, τ )y|| K||(θ, h)y||

(2.66)

From relations (2.65) and (2.66) it follows that ||(θ, h)y|| ≥ e ||y||. Since h does not depend on θ ∈  or on y ∈ Range P3 (θ ) we deduce that ||(θ, h)y|| ≥ e ||y||,

∀y ∈ Range P3 (θ ), ∀θ ∈ .

(2.67)

Let now θ ∈  and let y ∈ Range P3 (θ ). Let t ≥ 0 and let k ∈ N and r ∈ [0, h) be such that t = kh + r. Using relations (2.48) and (2.67) we obtain that ||(θ, t)y|| ≥

1 1 1 ||(θ, kh)y|| ≥ ek ||y|| ≥ eνt ||y||. K K K˜

In conclusion, from Theorem 2.6, Step 1 and Step 2 it follows that π = (, σ ) has a uniform exponential trichotomy. 2 3. The equivalence between the uniform p-admissibility and the uniform exponential trichotomy The central purpose of this section is to establish whether the admissibility properties introduced in this paper are also necessary for exponential trichotomy. The main tool in our study will be to determine the existence, boundedness and uniqueness (in certain spaces) of the solutions of the integral control system (Eπ ) under the assumption that the initial skew-product flow has a uniform exponential trichotomy. Finally, we deduce new necessary and sufficient conditions for uniform exponential trichotomy of skew-product flows.

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Let X be a real or complex Banach space and let (, d) be a metric space. Let π = (, σ ) be a skew-product flow on E = X × . For every θ ∈  we consider the linear subspaces S(θ ), B(θ ) and U(θ ) associated with π (see Section 2). In what follows we suppose that π = (, σ ) has a uniform exponential trichotomy. Let {Pk (θ )}θ∈ , k ∈ {1, 2, 3} be three families of projections given by Definition 2.4. Then, according to Lemma 2.3 we have that these families of projections are uniquely determined and Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ),

∀θ ∈ .

(3.1)

Let K, ν > 0 be two constants given by Definition 2.4 and let γ > 0 be such that ||Pk (θ )|| ≤ γ ,

∀θ ∈ , ∀k ∈ {1, 2, 3}.

(3.2)

Since π has a uniform exponential trichotomy, using similar arguments with those in the proof of Theorem 2.6 (x), we obtain that for every k ∈ {1, 2, 3} and every (x, θ ) ∈ X × , the function k ϕx,θ : R → X,

k ϕx,θ (t) = Pk (σ (θ, t))x

is continuous. Using this fact, for every k ∈ {1, 2, 3} and every θ ∈  it makes sense to consider the linear operator Pkθ : L1 (R, X) → L1 (R, X),

(Pkθ (v))(t) = Pk (σ (θ, t))v(t).

Using relation (3.2) we obtain that each Pkθ is a bounded linear operator and ||Pkθ || ≤ γ ,

∀θ ∈ , ∀k ∈ {1, 2, 3}.

(3.3)

Taking into account that π has a uniform exponential trichotomy, we have that for every k ∈ {2, 3} and every (θ, t) ∈  × R+ , the restriction (θ, t)| : Range Pk (θ ) → Range Pk (σ (θ, t)) is invertible and throughout this section we denote its inverse by k (θ, t)−1 | . We start with a technical result, given by: Theorem 3.1. The following assertions hold: (i) for every (θ, v) ∈  × L1 (R, X), the function t fθ,v : R → X,

fθ,v (t) =

(σ (θ, τ ), t − τ )(P1θ (v))(τ ) dτ −

−∞

∞ −

2

 (σ (θ, t), τ

2 − t)−1 | (Pθ (v))(τ ) dτ

t

∞ −

3 3 (σ (θ, t), τ − t)−1 | (Pθ (v))(τ ) dτ

t

belongs to Cb0 (R, X) and ||fθ,v ||∞ ≤ 3Kγ ||v||1 ;

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(ii) for every (θ, v) ∈  × L1 (R, X) the unique solution of the equation (Eθ ) in Cb0 (R, X) is the function fθ,v . Proof. (i) Let (θ, v) ∈  × L1 (R, X). We consider the functions t ϕθ,v : R → X,

ϕθ,v (t) =

(σ (θ, τ ), t − τ )(P1θ (v))(τ ) dτ

−∞

∞ ψθ,v : R → X,

ψθ,v (t) = −

2 2 (σ (θ, t), τ − t)−1 | (Pθ (v))(τ ) dτ −

t

∞ −

3 3 (σ (θ, t), τ − t)−1 | (Pθ (v))(τ ) dτ.

t

From Definition 2.4 (vi), (vii) and (viii), using relation (3.2) we deduce that ∞ ||ψθ,v (t)|| ≤ 2Kγ

||v(τ )|| dτ,

∀t ∈ R.

(3.4)

t

From relation (3.4) and from v ∈ L1 (R, X) we obtain that ψθ,v ∈ Cb0 (R, X) and ||ψθ,v ||∞ ≤ 2K γ ||v||1 .

(3.5)

In addition, using Definition 2.4 (v) and relation (3.2) we have that t ||ϕθ,v (t)|| ≤ K γ

e−ν(t−τ ) ||v(τ )|| dτ.

(3.6)

−∞

In particular, from (3.6) we have that t ||ϕθ,v (t)|| ≤ Kγ

||v(τ )|| dτ ≤ Kγ ||v||1 −∞

so ϕθ,v ∈ Cb (R, X) and ||ϕθ,v ||∞ ≤ Kγ ||v||1 .

(3.7)

Next, we prove that ϕθ,v ∈ Cb0 (R, X), i.e. we show that ϕθ,v (t) −→ 0 as t → ∞. Let ε > 0. Since v ∈ L1 (R, X) there is r ∈ R such that ∞ ||v(τ )|| dτ < r

ε . 2Kγ

(3.8)

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Then, for every t > r, using relations (3.6) and (3.8) we successively obtain that

||ϕθ,v (t)|| ≤ Kγ e

−ν(t−r)

r e

−ν(r−τ )

t ||v(τ )|| dτ + Kγ

−∞

≤ Kγ ||v||1 e

−ν(t−r)

e−ν(t−τ ) ||v(τ )|| dτ ≤

r

t + Kγ

ε ||v(τ )|| dτ < Kγ ||v||1 e−ν(t−r) + , 2

∀t > r.

(3.9)

r

Let tε > r be such that e−ν(tε −r) <

ε . 2Kγ (||v||1 + 1)

(3.10)

Then, from (3.9) and (3.10) we have that ||ϕθ,v (t)|| < ε,

∀t ≥ tε .

This shows that ϕθ,v ∈ Cb0 (R, X). Then, we deduce that fθ,v = ϕθ,v + ψθ,v ∈ Cb0 (R, X) and from (3.5) and (3.7) we obtain that ||fθ,v ||∞ ≤ 3Kγ ||v||1 . (ii) Let (θ, v) ∈  × L1 (R, X) and let fθ,v be the function defined at (i). A simple computation shows that the pair (fθ,v , v) satisfies the equation (Eθ ). Let f ∈ Cb0 (R, X) be such that the pair (f, v) satisfies the equation (Eθ ). Denoting by h : R → X,

h(t) = f (t) − fθ,v (t)

we deduce that h ∈ Cb0 (R, X) and h(t) = (σ (θ, s), t − s)h(s),

∀t ≥ s.

(3.11)

For every k ∈ {1, 2, 3} we consider the function hk : R → X,

hk (t) = (Pkθ (h))(t).

We have that h = h1 + h2 + h3 and using Definition 2.4 (i), from relation (3.11) we have that hk (t) = (σ (θ, s), t − s)hk (s),

∀t ≥ s, ∀k ∈ {1, 2, 3}.

(3.12)

Let t0 ∈ R and let k ∈ {2, 3}. Using relations (3.12) and (3.2) and Definition 2.4 (vi) and (vii) we obtain that 1 ||hk (t0 )|| ≤ ||hk (t)|| ≤ γ ||h(t)||, K

∀t ≥ t0 .

(3.13)

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Since h ∈ Cb0 (R, X), for t → ∞ in (3.13) we deduce that hk (t0 ) = 0. It follows that h2 (t0 ) = h3 (t0 ) = 0. In addition, using Definition 2.4 (v) we have that ||h1 (t0 )|| ≤ K e−ν(t0 −s) ||h1 (s)|| ≤ Kγ e−ν(t0 −s) ||h(s)|| ≤ ≤ Kγ ||h||∞ e−ν(t0 −s) ,

∀s ≤ t0 .

(3.14)

For s → −∞ in (3.14) we also obtain that h1 (t0 ) = 0. In conclusion, we have that h(t0 ) = h1 (t0 ) + h2 (t0 ) + h3 (t0 ) = 0. Since t0 ∈ R was arbitrary it follows that h = 0, so f = fθ,v and the proof is complete. 2 Theorem 3.2. The following assertions hold: (i) for every (θ, v) ∈  × L1 (R, X) the function t gθ,v : R → X,

gθ,v (t) =

(σ (θ, τ ), t − τ )(P1θ (v))(τ ) dτ +

−∞

t +

∞ (σ (θ, τ ), t

− τ )(P2θ (v))(τ ) dτ

−∞

−

3 3 (σ (θ, t), τ − t)−1 | (Pθ (v))(τ ) dτ

t

belongs to C0b (R, X) and ||gθ,v ||∞ ≤ 3Kγ ||v||1 ; (ii) for every (θ, v) ∈  × L1 (R, X) the unique solution of the equation (Eθ ) in C0b (R, X) is the function gθ,v . Proof. (i) Let (θ, v) ∈  × L1 (R, X). We consider the functions t δθ,v : R → X,

δθ,v (t) =

(σ (θ, τ ), t − τ )(P1θ (v))(τ ) dτ +

−∞

t (σ (θ, τ ), t − τ )(P2θ (v))(τ ) dτ

+ −∞

∞ λθ,v : R → X,

λθ,v (t) = −

3 3 (σ (θ, t), τ − t)−1 | (Pθ (v))(τ ) dτ.

t

From Definition 2.4 (v) and (vi) and using relation (3.2) we have that t ||δθ,v (t)|| ≤ 2Kγ

||v(τ )|| dτ, −∞

∀t ∈ R.

(3.15)

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Since v ∈ L1 (R, X), from relation (3.15) we have that δθ,v ∈ C0b (R, X) and ||δθ,v ||∞ ≤ 2Kγ ||v||1 .

(3.16)

From Definition 2.4 (vii) and relation (3.2) we obtain that ∞ ||λθ,v (t)|| ≤ Kγ

e−ν(τ −t) ||v(τ )|| dτ.

(3.17)

t

Using relation (3.17) and similar arguments with those in the proof of Theorem 3.1 (i) we deduce that λθ,v ∈ C0b (R, X). Then, we have that gθ,v = δθ,v + λθ,v ∈ C0b (R, X). Moreover, from relations (3.16) and (3.17) it follows that ||gθ,v ||∞ ≤ 3Kγ ||v||1 . (ii) This follows using similar arguments with those in the proof of Theorem 3.1 (ii).

2

Before presenting the first main result of this section, we need an auxiliary lemma: Lemma 3.1. Let p ∈ (1, ∞) and let ν > 0. If v ∈ L1 (R, R), then the associated functions t ϕv : R → R,

e−ν(t−τ ) v(τ ) dτ

ϕv (t) = −∞

and ∞ ψv : R → R,

ψv (t) =

e−ν(τ −t) v(τ ) dτ

t

belong to Lp (R, R). Moreover, denoting by αp,ν := (2/νp)1/p we have that ||ϕv ||p ≤ αp,ν ||v||1 ,

∀v ∈ L1 (R, R)

||ψv ||p ≤ αp,ν ||v||1 ,

∀v ∈ L1 (R, R).

and

Proof. The idea of the proof is standard, but for the sake of clarity we expose all the details. Let v ∈ L1 (R, R). Using Hölder’s inequality we successively deduce that t |ϕv (t)| ≤

1

ν

ν

(e− 2 (t−τ ) |v(τ )| p ) (e− 2 (t−τ ) |v(τ )|

p−1 p

) dτ ≤

−∞

⎛ ≤⎝

⎞ p1 ⎛

t

−∞

e

− νp 2

(t−τ )

|v(τ )| dτ ⎠ ⎝

⎞ p−1 p

t

−∞

e

νp − 2(p−1)

(t−τ )

|v(τ )| dτ ⎠

≤

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⎛ p−1 p

≤ ||v||1

⎝

t

⎞ p1 νp

e− 2 (t−τ ) |v(τ )| dτ ⎠ ,

∀t ∈ R.

(3.18)

−∞

From relation (3.18), using Fubini’s theorem we obtain that 

p−1



|ϕv (t)|p dt ≤ ||v||1 R

⎛ ⎝

p−1

⎞ νp

e− 2 (t−τ ) |v(τ )| dτ ⎠ dt =

−∞

R

= ||v||1

t

⎞ ⎛  ∞ νp ⎝ e− 2 (t−τ ) |v(τ )| dt ⎠ dτ = 2 ||v||p . 1 νp

R

(3.19)

τ

The relation (3.19) shows that ϕv ∈ Lp (R, R) and ||ϕv ||p ≤ αp,ν ||v||1 . Using similar arguments we deduce that ψv ∈ Lp (R, R) and ||ψv ||p ≤ αp,ν ||v||1 .

2

The first main result of this section is: Theorem 3.3. Let p ∈ (1, ∞). If the skew-product flow π = (, σ ) has a uniform exponential trichotomy, then the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible for the system (Eπ ). Proof. Let {Pk (θ )}θ∈ , k ∈ {1, 2, 3} be the families of projections and let K, ν > 0 be two constants given by Definition 2.4. Let γ > 0 be such that ||Pk (θ )|| ≤ γ ,

∀θ ∈ , ∀k ∈ {1, 2, 3}.

For every k ∈ {1, 2, 3} and every θ ∈  we consider the bounded linear operator Pkθ : L1 (R, X) → L1 (R, X),

(Pkθ (v))(t) = Pk (σ (θ, t))v(t).

From Theorem 3.1 and Theorem 3.2 it follows that for every (θ, v) ∈  × L1 (R, X) there exists a unique fθ,v ∈ Cb0 (R, X) and a unique gθ,v ∈ C0b (R, X) such that the pairs (fθ,v , v) and (gθ,v , v) satisfy the equation (Eθ ). This shows that the pair (Cb (R, X), L1 (R, X)) is admissible for the system (Eπ ). For every θ ∈  we consider the linear operators Fθ : L1 (R, X) → Cb0 (R, X),

Fθ (v) = fθ,v

Gθ : L (R, X) → C0b (R, X),

Gθ (v) = gθ,v .

1

Then, from Theorem 3.1 we have that ||Fθ (v)||∞ = ||fθ,v ||∞ ≤ 3Kγ ||v||1 ,

∀v ∈ L1 (R, X)

(3.20)

∀v ∈ L1 (R, X).

(3.21)

and from Theorem 3.2 we have that ||Gθ (v)||∞ = ||gθ,v ||∞ ≤ 3Kγ ||v||1 ,

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From relations (3.20) and (3.21) we obtain that ||Fθ || ≤ 3Kγ and ||Gθ || ≤ 3Kγ ,

∀θ ∈ .

(3.22)

From relation (3.22) it follows that the pair (Cb (R, X), L1 (R, X)) is uniformly admissible for the system (Eπ ). Let now (θ, v) ∈  × L1 (R, X). Then, from Theorem 3.1 and from Definition 2.4 (ii) we have that t ((Fθ P1θ )(v))(t) = fθ,P1 (v) (t) = θ

(σ (θ, τ ), t − τ )(P1θ (v))(τ ) dτ,

∀t ∈ R.

(3.23)

−∞

Then, from Definition 2.4 (v) and relation (3.23) we obtain that t ||((Fθ P1θ )(v))(t)|| ≤ Kγ

e−ν(t−τ ) ||v(τ )|| dτ,

∀t ∈ R.

(3.24)

−∞

Let αp,ν := (2/νp)1/p . Then, from relation (3.24) and Lemma 3.1 we deduce that (Fθ P1θ )(v) ∈ Lp (R, X) and ||(Fθ P1θ )(v)||p ≤ Kγ αp,ν ||v||1 .

(3.25)

On the other hand, from relation (3.24) we have that t ||((Fθ P1θ )(v))(t)|| ≤ Kγ

e−ν(t−τ ) ||v(τ )|| dτ ≤ Kγ ||v||1 ,

∀t ∈ R

−∞

which implies that ||(Fθ P1θ )(v)||∞ ≤ Kγ ||v||1 .

(3.26)

Setting λ := Kγ max{αp,ν , 1} and using relations (3.25) and (3.26) we obtain that (Fθ P1θ )(v) ∈ Cb0 (R, X) ∩ Lp (R, X) and ||(Fθ P1θ )(v)||∞,p = max{||(Fθ P1θ )(v)||∞ , ||(Fθ P1θ )(v)||p } ≤ λ ||v||1 .

(3.27)

On the other hand, from Theorem 3.2 and Definition 2.4 (ii) we deduce that ∞ ((Gθ P3θ )(v))(t) = gθ,P3 (v) (t) = − θ

t

3 3 (σ (θ, t), τ − t)−1 | (Pθ (v))(τ ) dτ,

∀t ∈ R.

(3.28)

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Using Definition 2.4 (vii), (viii) and relation (3.28) we have that ∞ ||((Gθ P3θ )(v))(t)|| ≤ Kγ

e−ν(τ −t) ||v(τ )|| dτ,

∀t ∈ R.

(3.29)

t

Using relation (3.29), Lemma 3.1 and similar arguments with those presented above we obtain that (Gθ P3θ )(v) ∈ C0b (R, X) ∩ Lp (R, X) and ||(Gθ P3θ )(v)||∞,p ≤ λ ||v||1 .

(3.30)

Taking into account that λ does not depend on θ ∈  or on v ∈ L1 (R, X), from relations (3.27) and (3.30) it follows that the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible for the system (Eπ ). 2 Remark 3.1. Relation (3.1) is valid even for skew-product flows which do not necessarily satisfy condition (iv) from Definition 2.2 (see the proof of Proposition 2.2 in [30]). Remark 3.2. (i) From the proofs of Theorems 3.1–3.3 it follows that the results obtained in these theorems hold for skew-product flows which satisfy only conditions (i)–(iii) from Definition 2.2, without requiring property (iv). (ii) In order to obtain the central result of Section 2 (Theorem 2.7) all the properties given by Definition 2.2 were necessary. The main result of this paper is: Theorem 3.4. Let π = (, σ ) be a skew-product flow on X × . The following assertions are equivalent: (i) π has a uniform exponential trichotomy; (ii) for every p ∈ (1, ∞) the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible for the system (Eπ ); (iii) there exists p ∈ (1, ∞) such that the pair (Cb (R, X), L1 (R, X)) is uniformly p-admissible for the system (Eπ ). Proof. (i) ⇒ (ii) This follows from the proof of Theorem 3.3. (ii) ⇒ (iii) This is obvious. (iii) ⇒ (i) This follows from Theorem 2.7. 2 Acknowledgments The authors would like to thank the referee for carefully reading the paper and for his suggestions and comments, which led to the improvement of the paper. The first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2011-3-0103.

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