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Nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces
Nonlinear Analysis 74 (2011) 5097–5110
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Nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces Bogdan Sasu ∗ , Adina Luminiţa Sasu Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara, Romania
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Article history: Received 5 February 2011 Accepted 3 May 2011 Communicated by S. Carl MSC: 34D09 40C10 45G15 Keywords: Exponential trichotomy Invariant subspace Skew-product flow
abstract In this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew-product flows in infinite dimensional spaces. We treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families. We show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary. Our results generalize the previous studies on this topic and provide as particular cases many interesting situations, among which we mention the detection of the exponential trichotomy of general non-autonomous systems. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Exponential trichotomy is the most complex asymptotic property of dynamical systems arising from the central manifold theory (see [1–15]). Starting from the idea that the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold, it was clear that the concept of exponential dichotomy describes a rather idealistic situation when the solution is either exponentially stable on the stable subspaces or exponentially unstable on the unstable subspaces (see [16,17,7,8,18]). Thus, with motivation from the properties arising in bifurcation theory (see [10], Chapter 6), it was the moment to introduce a new asymptotic concept called exponential trichotomy which reflects a deeper analysis of the behavior of solutions of dynamical systems. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable and central subspaces at each point from the flow’s domain. The concept of exponential trichotomy was introduced in the pioneering works of Elaydi and Hájek (see [6,7]), where the authors analyzed this property for the first time, both in the case of differential systems and in the case of nonlinear differential systems, pointing out significant properties of systems with exponential trichotomy. In the past few years, notable progress was made in the study of the exponential trichotomy of dynamical systems (see [1–5,8,9,11–14]) using diverse methods situated on the boundary between bifurcation theory and control, for various classes of trichotomy concepts. An important step was made by Zhu and Xu in [14], where they proved the Fredholm Alternative Lemma for a differential equation with exponential trichotomies, and, as a consequence, the main results could then be applied in order to obtain the persistence condition for heteroclinic orbits connecting non-hyperbolic equilibria. In [9] López-Fenner and Pinto proposed a nice study of discrete non-autonomous nonlinear systems possessing (h, k)-trichotomies. In [8]
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Corresponding author. E-mail addresses: [email protected] (B. Sasu), [email protected] (A.L. Sasu).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.05.004
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Elaydi and Janglajew obtained a valuable contribution to the study of the exponential trichotomy, providing the first input–output characterization for exponential trichotomy (see Theorem 4, pp. 423). An interesting analysis of the robustness of exponential trichotomy of difference equations was presented by Alonso et al. in [1]. In recent years, the exponential trichotomy of discrete-time equations became a front-line topic and the techniques associated with this asymptotic concept were widely diversified (see [2,4,5,8,12] and the references therein). In [4] Cuevas and Vidal pointed out new mechanisms arising in the subtle connections between weighted exponential trichotomy and the (h, k) trichotomy on Z+ and Z− . The techniques developed in [4] were extended in [5] where Vidal, Cuevas and Del Campo presented a systematic study of the asymptotic properties of the solutions of nonlinear difference equations proceeding from the perturbation of systems with weighted compensated exponential trichotomy. An inedit approach was proposed by Barreira and Valls in [2] where the authors give a complete characterization of nonuniform exponential trichotomies in terms of strict Lyapunov sequences for a linear cocycle with discrete time. The connections between the existence of exponential trichotomy of variational difference equations and the solvability of the associated variational control system was established for the first time in [12]. The case of differential equations and evolution families was considered in [3,11] from distinct perspectives. The exponential trichotomy of evolution families on the real line was studied in [11], where we proved premiere input–output characterizations in terms of the admissibility with respect to an integral equation between certain spaces of continuous functions. An interesting analysis of the robustness of nonuniform exponential trichotomies of differential equations under sufficiently small linear perturbations was recently presented in [3], establishing the continuous dependence on the perturbation of the constants in the notion of trichotomy. In contrast with the techniques developed in the above mentioned references, the aim of the present paper is to propose a new and distinct perspective in the study of the existence of exponential trichotomy. The starting points of our investigation are several famous theorems which played a crucial role in the development of the stability theory of differential equations (see [19–26]). One of the most notable criteria in the stability theory of evolution equations is the Datko–Pazy theorem, which states that an evolution family U = {U (t , s)}t ≥s≥0 on a Banach space X is exponentially stable (i.e. there exist K , ν > 0 such that ‖U (t , s)‖ ≤ K e−ν(t −s) for all t ≥ s ≥ 0) if and only if there is p ∈ [1, ∞) such that ∞
∫
‖U (t , s)x‖p dt < ∞,
sup s≥0
∀x ∈ X .
s
For the case p = 2 this result was proved for the first time by Datko in [19] for semigroups in Hilbert spaces and later in [20] for evolution families. Using distinct techniques, Pazy extended the framework and proved that the result holds also for any p ∈ [1, ∞) (see [23,24]). An interesting study in the nonlinear case was presented by Ichikawa in [21], where the author proved a Datko type theorem for a family of nonlinear operators T = {T (t , s)}t ≥s≥0 on a Banach space X such that T (t , s) : Ys → Yt , T (t , t ) = IYt is the identity on Yt , T (t , u)T (u, s) = T (t , s) on Ys and T (·, s)y is continuous on [s, ∞) for each y ∈ Ys . One of the main results in [21] states that if there is a positive continuous function g : [0, ∞) → [0, ∞) such that ‖T (t , s)y‖ ≤ g (t − s)‖y‖ for all y ∈ Ys and all t ≥ s ≥ 0, then T is uniformly exponentially stable (i.e. there are K , ν > 0 such that ‖T (t , s)y‖ ≤ K e−ν(t −s) ‖y‖ for all t ≥ s ≥ 0 and all y ∈ Ys ) if and only if there are p ∈ (0, ∞) and M > 0 such that ∞
∫
‖T (t , s)y‖p dt ≤ M ‖y‖p ,
∀y ∈ Ys , ∀s ≥ 0.
s
A decisive step in this framework was made when Rolewicz obtained a new nonlinear condition for stability (see [25,26]). The notable theorem of Rolewicz may be stated as: Theorem 1.1. Let N : R∗+ × R+ → R+ be a function such that for every t > 0, s → N (t , s) is continuous and nondecreasing with N (t , 0) = 0, N (t , s) > 0 for all s > 0, and for every s ≥ 0, t → N (t , s) is nondecreasing. If U = {U (t , s)}t ≥s≥0 is an evolution family on the Banach space X such that for every x ∈ X , there is α(x) > 0 with ∞
∫
N (α(x), ‖U (t , s)x‖) dt < ∞
sup s≥0
s
then U is exponentially stable. In the original proof Rolewicz proposed an approach based on category arguments. Using other methods, based on the theory of Banach function spaces, the theorem of Rolewicz was recently deduced as a consequence of the main result in [27]. In the last few years, due to the impressive progress in asymptotic theory of nonlinear equations (see [28,29]) the stability and dichotomy properties of new classes of dynamical systems were studied using Rolewicz type conditions (see e.g. [30,31,27,13] and the references therein). It worth mentioning that the main tools used in the proofs of the Rolewicz type results relied either on category arguments or on the behavior of some associated functionals on certain function spaces. In this context the natural question arises whether the Rolewicz type conditions may be extended to the most complex asymptotic behavior like the phenomenon described by exponential trichotomy. The aim of this paper is to answer this question. The main idea is not to extend a theory from stability or dichotomy to trichotomy, but to find proper criteria that will completely model the trichotomic behavior of a dynamical system. In this paper we are interested in finding nonlinear conditions for the existence of exponential trichotomy of skewproduct flows in infinite dimensional spaces, providing a direct analysis of the subtle mechanisms which govern the splitting
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of the main space and generate the asymptotic behavior on each invariant closed subspace. We consider the most general class of evolutionary systems described by skew-product flows on X × Θ , where X is a Banach space and Θ is a metric space (see Definition 2.3) and we determine a set of nonlinear integral conditions which ensure, for every θ ∈ Θ , the decomposition of the space X into a direct sum of three invariant subspaces such that the behavior of trajectories starting in the first and the third subspaces is described by exponential decay backward and forward in time and the trajectories starting in the second subspace have uniform upper and lower bounds (see Definition 2.5). We obtain a new method in the study of the exponential trichotomy applicable to any class of skew-product flows, working in the most flexible context, without assuming a priori the existence of the projection families, without any additional requirements concerning the invertibility of the cocycle and without any other invariance property. We show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties. Moreover, we prove that the imposed conditions are also necessary. As particular cases we deduce many interesting situations, among we mention the remarkable stability theorem due to Rolewicz. Moreover, the main results are applied to the study of the exponential trichotomy of non-autonomous systems in infinite dimensional spaces. 2. Basic definitions and preliminaries In [32] the authors proved that classical equations like the Navier–Stokes, Taylor–Couette, and Bubnov–Galerkin ones can be considered and modelled in the unified setting of skew-product flows and also it was pointed out that the skew-product flows often proceed from the linearization of nonlinear equations. Classical examples of skew-product flows arise as operator solutions for variational equations and therefore, in the past few years, the asymptotic behavior of skew-product flows became a topic of great interest with various applications in nonlinear problems (see [16,17,32,18,33] and the references therein). In this section, for the sake of clarity, we present some basic definitions as well as some auxiliary results. Indeed, we denote by R the set of the real numbers, by R+ the set of all t ∈ R, t ≥ 0 and by R− the set of all t ∈ R, t ≤ 0. For every A ⊂ R we denote by χA the characteristic function of the set A. Let X be a real or a complex Banach space. The norm on X and on L(X )—the Banach algebra of all bounded linear operators on X —will be denoted by ‖ · ‖. The identity operator on X will be denoted by I. Let (Θ , d) be a metric space. We denote by E = X × Θ . Definition 2.1. A continuous mapping σ : Θ × R → Θ is called a flow on Θ if σ (θ , 0) = θ and σ (θ , s + t ) = σ (σ (θ , s), t ) for all (θ , s, t ) ∈ Θ × R2 . Definition 2.2. A mapping Φ : Θ × R+ → L(X ) is called a cocycle over the flow σ if it satisfies the following conditions: (i) (ii) (iii) (iv)
Φ (θ , 0) = I for all θ ∈ Θ ; Φ (θ , s + t ) = Φ (σ (θ , s), t )Φ (θ , s) for all (θ , t , s) ∈ Θ × R2+ (the cocycle identity); there are M ≥ 1 and ω > 0 such that ‖Φ (θ , t )‖ ≤ Meωt for all (θ , t ) ∈ Θ × R+ ; for every x ∈ X the mapping (θ , t ) → Φ (θ , t )x is continuous.
Definition 2.3. If σ is a flow on Θ and Φ is a cocycle over the flow σ , then the dynamical system π = (Φ , σ ) is called a skew-product flow on E . Remark 2.1. We note that generally the operators in the cocycle are not assumed to be invertible and for this reason the cocycle is parameterized by t ≥ 0 but not by t ∈ R. Definition 2.4. A family U = {U (t , s)}t ≥s ⊂ L(X ) is called an evolution family if the following properties hold: (i) U (t0 , t0 ) = I and U (t , s)U (s, t0 ) = U (t , t0 ) for all t ≥ s ≥ t0 ; (ii) for every x ∈ X the mapping (t , s) → U (t , s)x is continuous; (iii) there are M ≥ 1 and ω > 0 such that ‖U (t , t0 )‖ ≤ Meω(t −t0 ) for all t ≥ t0 . Remark 2.2. The class of skew-product flows generalizes the setting of non-autonomous differential equations (see [24]). Indeed, if U = {U (t , s)}t ≥s is an evolution family on the Banach space X , then by considering the translation flow σ : R × R → R, σ (θ , t ) = θ + t, and Φ : R × R+ → L(X ), ΦU (θ , t ) = U (θ + t , θ ), it is easily seen that ΦU is a cocycle over the flow σ . Then πU = (ΦU , σ ) is called the skew-product flow associated with U. Example 2.1 (The Variational Equation). Let Θ be a locally compact metric space, let σ be a flow on Θ and let {A(θ )}θ∈Θ be a family of densely defined closed operators on a Banach space X . We consider the variational equation
(A)
x˙ (t ) = A(σ (θ , t ))x(t ),
(θ , t ) ∈ Θ × R+ .
A cocycle Φ : Θ × R+ → L(X ) over the flow σ is said to be a solution of the equation (A) if for every θ ∈ Θ , there is a dense subset Dθ ⊂ D(A(θ )) such that for every xθ ∈ Dθ the mapping t → x(t ) := Φ (θ , t )xθ is differentiable on R+ , for every t ∈ R+ , x(t ) ∈ D(A(σ (θ , t ))) and the mapping t → x(t ) satisfies the equation (A).
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Example 2.2. If π = (Φ , σ ) is a skew-product flow on E and D : Θ → L(X ) is a strongly continuous mapping with supθ∈Θ ‖D(θ )‖ < ∞ then there exists a unique cocycle ΦD over the flow σ such that
ΦD (θ , t )x = Φ (θ , t )x +
t
∫
Φ (σ (θ , s), t − s) D(σ (θ , s)) ΦD (θ , s)x ds,
∀t ≥ 0, ∀θ ∈ Θ , ∀x ∈ X
0
(see [34] Theorem 2.1, [17] Section 4). Example 2.3. Let a : R → (0, 1] be a continuous function and for every s ∈ R, denote by as (t ) = a(t + s) the translation of the function a by s. We define Θ = {as : s ∈ R} and on Θ we consider the metric d(θ , θ˜ ) = sup |θ (s) − θ˜ (s)|. s∈R
Let X be a Banach space and let {T (t )}t ≥0 be a C0 -semigroup on X with the infinitesimal generator A : D(A) ⊂ X → X . For every θ ∈ Θ suppose that A(θ ) := θ (0)A. It easily checked that σ : Θ × R → Θ , σ (θ , t )(s) := θ (t + s), is a flow on Θ . We consider the variational equation x˙ (t ) = A(σ (θ , t ))x(t ), x(0) = x0 .
[A]
t ≥0
θ (s) ds x, which is a cocycle over the flow σ . Then π = (Φ , σ ) is a skew-product flow on E = X × Θ . Moreover, for every x0 ∈ D(A), the mapping defined by x(t ) := Φ (θ , t )x0 for all t ≥ 0 is We define Φ : Θ × R+ → L(X ), Φ (θ , t )x = T
t 0
the strong solution of the equation [A]. Remark 2.3. We recall that an operator P ∈ L(X ) is called a projection if P 2 = P. If P is a projection, then Range P and Ker P are closed linear subspaces and X = Range P ⊕ Ker P. It obvious that if P is a projection then I − P is also a projection and X = Range P ⊕ Range (I − P ). It is also well known that if X0 is a closed linear subspace of X there need not exist a complementary closed subspace Y0 such that X = X0 ⊕ Y0 (although in Hilbert spaces Y0 exists, being the orthogonal complement). The concept of exponential trichotomy is firmly rooted in the central manifold theory for dynamical systems. This extends the classical dichotomic properties and describes at each point θ ∈ Θ the asymptotic behavior on three complemented closed subspaces: Definition 2.5. A skew-product flow π = (Φ , σ ) is said to be exponentially trichotomic if there are three families of projections {Pk (θ )}θ∈Θ ⊂ L(X ), k ∈ {1, 2, 3} and two constants K ≥ 1 and ν > 0 such that: (i) (ii) (iii) (iv) (v) (vi)
Pk (θ )Pj (θ ) = 0 for all k ̸= j and all θ ∈ Θ ; P1 (θ ) + P2 (θ ) + P3 (θ ) = I for all θ ∈ Θ ; supθ∈Θ ‖Pk (θ )‖ < ∞ for all k ∈ {1, 2, 3}; Φ (θ , t )Pk (θ ) = Pk (σ (θ , t ))Φ (θ , t ) for all (θ , t ) ∈ Θ × R+ and all k ∈ {1, 2, 3}; ‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖ for all t ≥ 0, x ∈ Range P1 (θ ) and all θ ∈ Θ ; 1 ‖x‖ ≤ ‖Φ (θ , t )x‖ ≤ K ‖x‖ for all t ≥ 0, x ∈ Range P2 (θ ) and all θ ∈ Θ ; K
(vii) ‖Φ (θ , t )x‖ ≥ K1 eν t ‖x‖ for all t ≥ 0, x ∈ Range P3 (θ ) and all θ ∈ Θ ; (viii) the restriction Φ (θ , t )| : Range Pk (θ ) → Range Pk (σ (θ , t )) is an isomorphism for all (θ , t ) ∈ Θ × R+ and all k ∈ {2, 3}. Remark 2.4. We note that according to Definition 2.5 all the fundamental invariant subspaces (stable, unstable or central) given by Range Pk (θ ), with k ∈ {1, 2, 3} and θ ∈ Θ , are allowed to have infinite dimension. Example 2.4. Suppose that a : R → [−1, 1], a(t ) = sin t, and for every s ∈ R suppose that as (t ) = a(t + s) for all t ∈ R. We define Θ = {as : s ∈ R} and we consider on Θ the metric d(θ , θ˜ ) = sups∈R |θ (s) − θ˜ (s)|. Let (V , ‖ · ‖) be an infinite dimensional Banach space and suppose that X = V 3 with respect to the norm ‖(v1 , v2 , v3 )‖ = ‖v1 ‖ + ‖v2 ‖ + ‖v3 ‖. For every (θ , t ) ∈ Θ × R+ we consider the operator Φ (θ , t ) : X → X defined by
t t t Φ (θ , t )(v1 , v2 , v3 ) = e−t + 0 θ(τ ) dτ v1 , e 0 θ(τ ) dτ v2 , et + 0 θ (τ ) dτ v3 . Then π = (Φ , σ ) is a skew-product flow on X × Θ . We consider the projections P1 : X → X , P1 (v1 , v2 , v3 ) = (v1 , 0, 0), P2 : X → X , P2 (v1 , v2 , v3 ) = (0, v2 , 0) and P3 : X → X , P3 (v1 , v2 , v3 ) = (0, 0, v3 ). Then, an easy computation shows that π is exponentially trichotomic with respect to the families of projections Pk (θ ) = Pk ,
∀θ ∈ Θ , ∀k ∈ {1, 2, 3}
and with the constants K = e2 and ν = 1.
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Remark 2.5. If a skew-product flow is exponentially trichotomic with respect to the families of projections {Pk (θ )}θ∈Θ , k ∈ {1, 2, 3}, then: (i) Φ (θ , t )Range P1 (θ ) ⊂ Range P1 (σ (θ , t )) for all (θ , t ) ∈ Θ × R+ ; (ii) Φ (θ , t )Range Pk (θ ) = Range Pk (σ (θ , t )) for all (θ , t ) ∈ Θ × R+ and all k ∈ {2, 3}. Notation. Let π = (Φ , σ ) be a skew-product flow on E . At every point θ ∈ Θ we associate with π three fundamental subspaces, which have an essential role in the study of exponential trichotomy. The main idea starts from the previous remark and from the necessity of finding suitable conditions for the existence of exponential trichotomy. Indeed, for every θ ∈ Θ we denote by N(θ ) the linear space of all functions ϕ : R− → X with ϕ(s) = Φ (σ (θ , τ ), s − τ )ϕ(τ ) for all τ ≤ s ≤ 0. Remark 2.6. If θ ∈ Θ and ϕ ∈ N(θ ), then ϕ is a continuous function on R− . For every θ ∈ Θ we consider the sets
S(θ ) = x ∈ X : lim Φ (θ , t )x = 0
t →∞
B(θ ) =
x ∈ X : sup ‖Φ (θ , t )x‖ < ∞ and there is ϕ ∈ N(θ ) with ϕ(0) = x and sup ‖ϕ(t )‖ < ∞ t ≥0
U(θ ) =
t ≤0
x ∈ X : there is ϕ ∈ N(θ ) with ϕ(0) = x and
lim ϕ(t ) = 0 .
t →−∞
Remark 2.7. It is easy to verify that S(θ ), B(θ ) and U(θ) are linear subspaces of X for each θ ∈ Θ . Usually, for every θ ∈ Θ , S(θ ) is called the stable subspace, B(θ ) is called the bounded subspace and U(θ ) is called the unstable subspace. In what follows we discuss several properties of these subspaces, which explain the appellations given in the previous remark for each subspace. Proposition 2.1. The following assertions hold: (i) Φ (θ , t )S(θ ) ⊆ S(σ (θ , t )) for all (θ , t ) ∈ Θ × R+ ; (ii) Φ (θ , t )B(θ ) = B(σ (θ , t )) for all (θ , t ) ∈ Θ × R+ ; (iii) Φ (θ , t )U(θ ) = U(σ (θ , t )) for all (θ , t ) ∈ Θ × R+ . Proof. The assertion (i) is immediate. To prove (ii) let (θ, t ) ∈ Θ × (0, ∞) and let x ∈ B(θ ). Setting yx = Φ (θ , t )x and taking into account that x ∈ B(θ ) we have that sup ‖Φ (σ (θ , t ), s)yx ‖ = sup ‖Φ (θ , s + t )x‖ < ∞. s ≥0
(2.1)
s ≥0
Since x ∈ B(θ ) there is ϕ ∈ N(θ ) with ϕ(0) = x and sups≤0 ‖ϕ(s)‖ < ∞. We consider the function
ψ : (−∞, t ] → X ,
ψ(s) =
Φ (θ , s)x, ϕ(s),
s ∈ [0, t ] s < 0.
An easy computation shows that ψ(s) = Φ (σ (θ , τ ), s − τ )ψ(τ ) for all τ ≤ s ≤ t, which implies that
ψ(s + t ) = Φ (σ (θ , τ + t ), s − τ )ψ(τ + t ),
∀τ ≤ s ≤ 0.
(2.2)
Taking ζ : R− → X , ζ (s) = ψ(s + t ), from (2.2) it follows that ζ ∈ N(σ (θ , t )). Moreover, ζ (0) = ψ(t ) = yx , and hence using also relation (2.1) we deduce that yx ∈ B(σ (θ , t )). This shows that Φ (θ , t )B(θ ) ⊂ B(σ (θ , t )). Conversely, let y ∈ B(σ (θ , t )). Then there is λ ∈ N(σ (θ , t )) with λ(0) = y and sups≤0 ‖λ(s)‖ < ∞. We denote by xy = λ(−t ). From
λ(s) = Φ (σ (θ , t + τ ), s − τ )λ(τ ),
∀τ ≤ s ≤ 0
(2.3)
we obtain that y = λ(0) = Φ (θ , t )xy . We consider the function δ : R− → X , δ(s) = λ(s − t ), and from (2.3) we deduce that δ ∈ N(θ ). In addition, we observe that δ(0) = xy and sups≤0 ‖δ(s)‖ = sups≤−t ‖λ(s)‖ < ∞. Since y ∈ B(σ (θ , t )) we have that sup ‖Φ (θ , s)xy ‖ = sup ‖Φ (σ (θ , t ), s − t )Φ (θ , t )xy ‖ = sup ‖Φ (σ (θ , t ), s − t )y‖ s≥t
s≥t
= sup ‖Φ (σ (θ , t ), τ )y‖ < ∞. τ ≥0
s≥t
(2.4)
Let M , ω > 0 be given by Definition 2.2(iii). Then ‖Φ (θ , s)xy ‖ ≤ Meωt ‖xy ‖ for all s ∈ [0, t ]. Hence using relation (2.4) it follows that sups≥0 ‖Φ (θ , s)xy ‖ < ∞. In conclusion, we obtain that xy ∈ B(θ ) and so y = Φ (θ , t )xy ∈ Φ (θ , t )B(θ ). This implies that B(σ (θ , t )) ⊂ Φ (θ , t )B(θ ) and the proof is complete. (iii) This follows using arguments similar to those used in the proof of (ii).
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In the next proposition we briefly prove the structure of the trichotomy projections in order to clarify the intrinsic mechanisms of exponential trichotomy. Proposition 2.2. If the skew-product flow π = (Φ , σ ) is exponentially trichotomic with respect to three families of projections {Pk (θ)}θ ∈Θ ⊂ L(X ), k ∈ {1, 2, 3}, then: (i) Range P1 (θ ) = S(θ ) for all θ ∈ Θ ; (ii) Range P2 (θ ) = B(θ ) for all θ ∈ Θ ; (iii) Range P3 (θ ) = U(θ ) for all θ ∈ Θ . Proof. Let K , ν > 0 be given by Definition 2.5 and let Mk = supθ∈Θ ‖Pk (θ )‖, k ∈ {1, 2, 3}. Let θ ∈ Θ . We observe that if x ∈ S(θ ) ∪ B(θ ), then setting αx := supt ≥0 ‖Φ (θ , t )x‖ < ∞ and using properties (vii) and (ii) from Definition 2.5, we successively deduce that
‖P3 (θ )x‖ ≤ K e−ν t ‖Φ (θ , t )(x − P1 (θ )x − P2 (θ )x)‖ ≤ K e−ν t (1 + M1 + M2 )αx −→ 0, t →∞
which yields P3 (θ )x = 0. (i) It is easy to see that Range P1 (θ ) ⊂ S(θ ). Let x ∈ S(θ ). According to Definition 2.5(iv) and (vi) we have that 1 K
‖P2 (θ )x‖ ≤ ‖Φ (θ , t )P2 (θ )x‖ = ‖P2 (σ (θ , t ))Φ (θ , t )x‖ ≤ M2 ‖Φ (θ , t )x‖ −→ 0 t →∞
which shows that P2 (θ )x = 0. Since also P3 (θ )x = 0 it follows that x = P1 (θ )x ∈ Range P1 (θ ). (ii) Let x ∈ B(θ ) and let ϕ ∈ N(θ ) be such that ϕ(0) = x and βx := sups≤0 ‖ϕ(s)‖ < ∞. From Definition 2.5(iv) and (v) we obtain that
‖P1 (θ )x‖ = ‖P1 (θ )ϕ(0)‖ = ‖Φ (σ (θ , s), −s)P1 (σ (θ , s))ϕ(s)‖ ≤ K eν s ‖P1 (σ (θ , s))‖ ‖ϕ(s)‖ ≤ M1 K βx eν s −→ 0 s→−∞
so P1 (θ )x = 0. Since also P3 (θ )x = 0 it follows that x = P2 (θ )x ∈ Range P2 (θ ), so B(θ ) ⊂ Range P2 (θ ). Conversely, let y ∈ Range P2 (θ ). Then, from Definition 2.5(vi) we have that supt ≥0 ‖Φ (θ , t )y‖ < ∞. We consider the function
ϕy : R− → X ,
1 ϕy (t ) = Φ2 (σ (θ , t ), −t )− | y
1 where for every t ≤ 0, Φ2 (σ (θ , t ), −t )− denotes the inverse of the operator Φ (σ (θ , t ), −t )| : Range P2 (σ (θ , t )) → | Range P2 (θ ). Then ϕy ∈ N(θ ) and taking into account that ‖ϕy (t )‖ ≤ K ‖y‖ for all t ≤ 0, we deduce that y = ϕy (0) ∈ B(θ ). This shows that B(θ ) = Range P2 (θ ). (iii) Let x ∈ U(θ ). Then there is δ ∈ N(θ ) with δ(0) = x and lims→−∞ δ(s) = 0. Let k ∈ {1, 2}. Since δ ∈ N(θ ) we deduce that
‖Pk (θ )x‖ = ‖Pk (θ )δ(0)‖ = ‖Φ (σ (θ , s), −s)Pk (σ (θ , s))δ(s)‖ ≤ KMk ‖δ(s)‖ −→ 0 s→−∞
which shows that Pk (θ )x = 0 for k ∈ {1, 2}, so x = P3 (θ )x. It follows that U(θ ) ⊂ Range P3 (θ ). Conversely, let y ∈ Range P3 (θ ). We define
ψy : R− → X ,
1 ψy (t ) = Φ3 (σ (θ , t ), −t )− | y
1 where for every t ≤ 0, Φ3 (σ (θ , t ), −t )− denotes the inverse of the operator Φ (σ (θ , t ), −t )| : Range P3 (σ (θ , t )) → | Range P3 (θ ). It is easy to see that ψy ∈ N(θ ) and ‖ψy (t )‖ ≤ K eν t ‖y‖ for all t ≤ 0. This shows that y = ψy (0) ∈ U(θ ), so Range P3 (θ ) ⊂ U(θ ) and the proof is complete.
Remark 2.8. From Proposition 2.2 it follows that if a skew-product flow is exponentially trichotomic with respect to three families of projections, then these projections are uniquely determined. 3. The main results In what follows we are interested in obtaining necessary and sufficient nonlinear conditions for the existence of the exponential trichotomy of skew-product flows. We emphasize that we are working in the most general case, without any additional assumption on the flow or on the cocycle. It is also very important to point out that we do not assume a priori the existence of the projection families or any invertibility properties on the bounded subspace or on the unstable subspace. We show that an inedit assembly of integral nonlinear conditions are necessary and sufficient conditions for the existence of the exponential trichotomy. Let X be a Banach space. Let (Θ , d) be a metric space. We denote by E = X × Θ . We denote by V the set of all continuous nondecreasing functions V : R+ → R+ with V (0) = 0, V (t ) > 0 for all t > 0 and limt →∞ V (t ) = ∞.
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Theorem 3.1. Let π = (Φ , σ ) be a skew-product flow on E . If there exist V ∈ V and λ > 0 with the following properties:
t +1
(a) λ1 V (‖x‖) ≤ t V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖) for all t ≥ 0, all x ∈ B(θ ) and all θ ∈ Θ ; t +1 1 (b) t V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all t ≥ 0, all x ∈ B(θ ) \ {0} and all θ ∈ Θ ; then the following assertions hold: (i) for every (θ , t ) ∈ Θ × R+ , the operator Φ (θ , t )| : B(θ ) → B(σ (θ , t )) is an isomorphism; (ii) there is K > 0 such that 1
‖x‖ ≤ ‖Φ (θ , t )x‖ ≤ K ‖x‖, ∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ ; K (iii) B(θ ) is a closed linear subspace for all θ ∈ Θ . Proof. Let M , ω > 0 be given by Definition 2.2(iii). (i) Let (θ , t ) ∈ Θ × R+ . From Proposition 2.1(ii) we have that the operator Φ (θ , t )| : B(θ ) → B(σ (θ , t )) is surjective. Let x ∈ B(θ ) with Φ (θ , t )x = 0. Then Φ (θ , τ )x = 0 for all τ ∈ [t , t + 1]. Then from (a) it follows that V (‖x‖) = 0 and since V ∈ V we deduce that x = 0, so Φ (θ , t )| is also injective. (ii) From limt →∞ V (t ) = ∞ it follows that there is h ≥ 1 such that V (h) > λ V (1). We denote by K = hMeω . Step 1. We prove that
‖Φ (θ , t )x‖ ≤ K ‖x‖,
∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.1)
Since
‖Φ (θ , t )x‖ ≤ Meω ‖x‖ ≤ K ‖x‖,
∀t ∈ [0, 1], ∀x ∈ X , ∀θ ∈ Θ
(3.2)
it remains to prove that (3.1) holds for t > 1. Suppose to the contrary that there are t > 1, θ ∈ Θ and x ∈ B(θ ) such that
‖Φ (θ , t )x‖ > K ‖x‖.
(3.3)
From (3.3) we deduce that x ̸= 0 and setting yx = x/‖x‖ we have that yx ∈ B(θ ). On the other hand
‖Φ (θ , t )yx ‖ ≤ Meω ‖Φ (θ , τ )yx ‖,
∀τ ∈ [t − 1, t ].
(3.4)
From relations (3.3) and (3.4) we have that h < ‖Φ (θ , τ )yx ‖,
∀τ ∈ [t − 1, t ].
(3.5)
Since V is nondecreasing, from hypothesis (a) and relation (3.5) we deduce that V (h) <
t
∫
V (‖Φ (θ , τ )yx ‖) dτ ≤ λ V (1) t −1
which contradicts the way in which h was chosen. It follows that the assumption (3.3) is false. This implies that
‖Φ (θ , t )x‖ ≤ K ‖x‖,
∀t > 1, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.6)
From (3.2) and (3.6) we deduce that relation (3.1) holds. Step 2. We prove that
‖Φ (θ , t )x‖ ≥
1
‖x‖,
K
∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.7)
Suppose to the contrary that there are θ ∈ Θ , x ∈ B(θ ) and t ≥ 0 such that
‖Φ (θ , t )x‖ <
1 K
‖x‖.
(3.8)
From (3.8) we deduce that x ̸= 0. Setting yx = (x/‖x‖) we have that yx ∈ B(θ ). From
‖Φ (θ , τ )yx ‖ ≤ Meω ‖Φ (θ , t )yx ‖,
∀τ ∈ [t , t + 1]
using (3.8) we obtain that
‖Φ (θ , τ )yx ‖ <
1 h
,
∀τ ∈ [t , t + 1].
Since V is nondecreasing, from hypothesis (b) and relation (3.9) we deduce that V (h) <
t +1
∫
V
t
1
‖Φ (θ , τ )yx ‖
dτ ≤ λ V (1)
which contradicts the choice of h. It follows that assertion (3.8) is false, so (3.7) holds.
(3.9)
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(iii) Let K > 0 be given by (ii). Let θ ∈ Θ and let {xn } ⊂ B(θ ) with xn −→ x. n→∞
Step 1. According to hypothesis (a) we have that t +1
∫
V (‖Φ (θ , τ )xn ‖) dτ ≤ λ V (‖xn ‖),
∀t ≥ 0 , ∀n ∈ N .
(3.10)
t
Since V is continuous, for n → ∞ in (3.10) we obtain that t +1
∫
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖),
∀t ≥ 0.
(3.11)
t
Let q ≥ 1 be such that V (q) > λ V (‖x‖). We prove that
‖Φ (θ , t )x‖ ≤ qMeω ,
∀t ≥ 1.
(3.12)
Suppose to the contrary that there is t ≥ 1 such that
‖Φ (θ , t )x‖ > qMeω .
(3.13)
ω
From ‖Φ (θ , t )x‖ ≤ Me ‖Φ (θ , τ )x‖ for all τ ∈ [t − 1, t ], using (3.13) we deduce that q < ‖Φ (θ , τ )x‖,
∀τ ∈ [t − 1, t ].
(3.14)
Since V is nondecreasing, from (3.14) and (3.11) it follows that V (q) <
∫
t
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖).
(3.15)
t −1
The relation (3.15) contradicts the choice of q. This shows that assertion (3.13) is false, so relation (3.12) holds. Since
‖Φ (θ , t )x‖ ≤ Meω ‖x‖,
∀t ∈ [0, 1)
(3.16)
from (3.12) and (3.16) we deduce that supt ≥0 ‖Φ (θ , t )x‖ < ∞. Step 2. For every n ∈ N, xn ∈ B(θ ) so there is ϕn ∈ N(θ ) with ϕn (0) = xn and supt ≤0 ‖ϕn (t )‖ < ∞. We observe that ϕn (t ) ∈ B(σ (θ , t )) and then using (ii) we have that
‖xn − xm ‖ = ‖ϕn (0) − ϕm (0)‖ = ‖Φ (σ (θ , t ), −t )(ϕn (t ) − ϕm (t ))‖ ≥
1 K
‖ϕn (t ) − ϕm (t )‖,
∀t ≤ 0 , ∀m, n ∈ N .
(3.17)
Relation (3.17) shows that for every t ≤ 0 the sequence {ϕn (t )} is convergent. We define ϕ : R− → X , ϕ(t ) = limn→∞ ϕn (t ). Since {ϕn } ⊂ N(θ ) it follows that ϕ ∈ N(θ ). For m → ∞ in relation (3.17) we also obtain that
‖ϕ(t )‖ ≤ ‖ϕn (t ) − ϕ(t )‖ + ‖ϕn (t )‖ ≤ K ‖xn − x‖ + ‖ϕn (t )‖,
∀t ≤ 0 , ∀n ∈ N .
Since supt ≤0 ‖ϕn (t )‖ < ∞ for every n ∈ N, from the above inequality we deduce that supt ≤0 ‖ϕ(t )‖ < ∞. Moreover ϕ(0) = limn→∞ ϕn (0) = limn→∞ xn = x. In conclusion, from Steps 1 and 2 it follows that x ∈ B(θ ), so B(θ ) is closed. Theorem 3.2. Let π = (Φ , σ ) be a skew-product flow on E . If there exist V ∈ V and λ > 0 such that ∞
∫
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖),
∀x ∈ S(θ ), ∀θ ∈ Θ
0
then the following assertions hold: (i) there are K , ν > 0 such that ‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖ for all t ≥ 0, all x ∈ S(θ ) and all θ ∈ Θ ; (ii) S(θ ) is a closed linear subspace for all θ ∈ Θ ; (iii) S(θ ) ∩ B(θ ) = {0} for all θ ∈ Θ . Proof. (i) According to our hypothesis, we observe that t +1
∫
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖),
∀t ≥ 0, ∀x ∈ S(θ ), ∀θ ∈ Θ .
(3.18)
t
Using arguments similar to those used in the proof of Theorem 3.1(ii) Step 1, we deduce that there is δ > 0 such that
‖Φ (θ , t )x‖ ≤ δ ‖x‖,
∀t ≥ 0, ∀x ∈ S(θ ), ∀θ ∈ Θ .
(3.19)
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We prove that there is r > 0 such that
‖Φ (θ , r )x‖ ≤
1
∀x ∈ S(θ ), ∀θ ∈ Θ .
‖x‖,
e
(3.20)
Indeed, suppose that r > (λV (1))/V (1/δ e). Let θ ∈ Θ and let x ∈ S(θ ) with ‖x‖ = 1. From Proposition 2.1(i) and relation (3.19) we have that
‖Φ (θ , r )x‖ ≤ δ ‖Φ (θ , τ )x‖,
∀τ ∈ [0, r ].
Since V is nondecreasing we obtain that
x rV Φ (θ , r ) ≤
r
∫
δ
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (1).
(3.21)
0
Taking into account the way in which r was chosen, from relation (3.21) we deduce that
1 x . V Φ (θ , r ) < V δ δe
(3.22)
Since V is nondecreasing, from (3.22) it follows that
‖Φ (θ , r )x‖ ≤
1 e
.
Taking into account that r does not depend on θ or x ∈ S(θ ) with ‖x‖ = 1 we obtain that
‖Φ (θ , r )x‖ ≤
1 e
∀x ∈ S(θ ), ∀θ ∈ Θ .
‖x‖,
(3.23)
We denote by K = δ e and by ν = 1/r. Let θ ∈ Θ , x ∈ S(θ ) and t ≥ 0. Then there are n ∈ N and τ ∈ [0, r ) such that t = nr + τ . Using Proposition 2.1(i) and relations (3.19) and (3.23) we successively deduce that
‖Φ (θ , t )x‖ ≤ δ ‖Φ (θ , nr )x‖ ≤ δ e−n ‖x‖ ≤ K e−ν t ‖x‖. (ii) Let θ ∈ Θ and let {xn } ⊂ S(θ ) with xn −→ x. According to (i) we have that n→∞
‖Φ (θ , t )xn ‖ ≤ K e−ν t ‖xn ‖,
∀t ≥ 0, ∀n ∈ N.
(3.24)
For n → ∞ in (3.24) we obtain that
‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖,
∀t ≥ 0.
(3.25)
In particular, from (3.25) it follows that limt →∞ Φ (θ , t )x = 0, so x ∈ S(θ ). This shows that S(θ ) is closed. (iii) Let θ ∈ Θ and let x ∈ S(θ ) ∩ B(θ ). From (i) we have that
‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖,
t ≥ 0.
(3.26)
Since x ∈ B(θ ), from Theorem 3.1 it follows that there is K1 > 0 such that 1 K1
‖x‖ ≤ ‖Φ (θ , t )x‖,
∀t ≥ 0.
From relations (3.26) and (3.27) we obtain that x = 0, so S(θ ) ∩ B(θ ) = {0}.
(3.27)
Theorem 3.3. Let π = (Φ , σ ) be a skew-product flow on E . If there exist V ∈ V and λ > 0 such that:
t +1
(a) λ1 V (‖x‖) ≤ t V (‖Φ (θ , τ )x‖) dτ for all t ≥ 0, all x ∈ U(θ ) and all θ ∈ Θ ; ∞ 1 (b) 0 V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all x ∈ U(θ ) \ {0} and all θ ∈ Θ ; then the following assertions hold: (i) for every (θ , t ) ∈ Θ × R+ the operator Φ (θ , t )| : U(θ ) → U(σ (θ , t )) is invertible; (ii) there are K , ν > 0 such that
‖Φ (θ , t )x‖ ≥
1 K
eν t ‖x‖,
∀t ≥ 0, ∀x ∈ U(θ ), ∀θ ∈ Θ ;
(iii) U(θ ) is a closed linear subspace for all θ ∈ Θ ; (iv) U(θ ) ∩ B(θ ) = U(θ) ∩ S(θ ) = {0}.
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Proof. (i) This follows using arguments similar to those used in the proof of Theorem 3.1(i). (ii) According to hypothesis (b) we have that t +1
∫
V
t
1
dτ ≤ λ V
‖Φ (θ , τ )x‖
1
‖x‖
,
∀t ≥ 0, ∀x ∈ U(θ ) \ {0}, ∀θ ∈ Θ .
Using arguments analogous to those used to prove Theorem 3.1(ii) Step 2, we deduce that there is δ > 0 such that
‖Φ (θ , t )x‖ ≥
1
δ
‖x‖,
∀t ≥ 0, ∀x ∈ U(θ ), ∀θ ∈ Θ .
(3.28)
We prove that there is h > 0 such that
‖Φ (θ , h)x‖ ≥ e ‖x‖,
∀x ∈ U(θ ), ∀θ ∈ Θ .
(3.29)
Indeed, suppose that h > λV (1)/V (1/eδ). Let θ ∈ Θ and x ∈ U(θ ) with ‖x‖ = 1. According to Proposition 2.1(iii) and to relation (3.28) we obtain that
‖Φ (θ , h)x‖ ≥
1
δ
‖Φ (θ , τ )x‖,
∀τ ∈ [0, h].
(3.30)
Since V is nondecreasing, from (3.30) we deduce that
hV
1
δ ‖Φ (θ , h)x‖
h
∫ ≤
V
0
1
‖Φ (θ , τ )x‖
dτ ≤ λ V (1).
(3.31)
From (3.31) it follows that ‖Φ (θ , h)x‖ ≥ e. Taking into account that h does not depend on θ or x we have that (3.29) holds. We denote by K = δ e and ν = 1/h. Let θ ∈ Θ , x ∈ U(θ ) and t ≥ 0. Let n ∈ N and s ∈ [0, h) be such that t = nh + s. Then using Proposition 2.1(iii) and relations (3.28) and (3.29) we deduce that
‖Φ (θ , t )x‖ ≥
1
δ
‖Φ (θ , nh)x‖ ≥
1
δ
en ‖ x ‖ ≥
1 νt e ‖x‖.
K
(iii) Let θ ∈ Θ and let {xn } ⊂ U(θ ) with xn −→ x. For every n ∈ N suppose that ϕn ∈ N(θ ) with ϕn (0) = xn and n→∞
limt →−∞ ϕn (t ) = 0. Using arguments similar to those from the proof of Theorem 3.1(iii) Step 2, we obtain that for every t ≤ 0, the sequence {ϕn (t )} is convergent. We define ϕ : R− → X , ϕ(t ) = limn→∞ ϕn (t ), and we have that ϕ ∈ N(θ ) and ϕ(0) = limn→∞ xn = x. Moreover, in the same manner as in the proof of Theorem 3.1(iii), we deduce that
‖ϕ(t )‖ ≤ K eν t ‖xn − x‖ + ‖ϕn (t )‖,
∀t ≤ 0 , ∀n ∈ N .
(3.32)
In particular, from (3.32) we have that limt →−∞ ϕ(t ) = 0, so x ∈ U(θ ). This shows that U(θ ) is closed. (iv) Let θ ∈ Θ and let x ∈ U(θ ) ∩ (B(θ ) ∪ S(θ )). Then, from (ii) we have that 1 K
eν t ‖x‖ ≤ ‖Φ (θ , t )x‖,
∀t ≥ 0.
(3.33)
From Theorem 3.1(ii) and Theorem 3.2(i) we deduce that there is γ > 0 such that
‖Φ (θ , t )x‖ ≤ γ ‖x‖,
∀t ≥ 0.
From (3.33) and (3.34) it follows that x = 0, so U(θ ) ∩ B(θ ) = U(θ ) ∩ S(θ ) = {0}.
(3.34)
The main result of this paper is given as follows: Theorem 3.4. A skew-product flow π = (Φ , σ ) is exponentially trichotomic if and only if there are V ∈ V and λ > 0 such that: (i)
∞ 0 1
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖) for all x ∈ S(θ ) and all θ ∈ Θ ;
t +1 V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖) for all t ≥ 0, all x ∈ B(θ ) and all θ ∈ Θ ; t t +1 1 (iii) t V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all t ≥ 0, all x ∈ B(θ ) \ {0} and all θ ∈ Θ ; t +1 (iv) λ1 V (‖x‖) ≤ t V (‖Φ (θ , τ )x‖) dτ for all t ≥ 0, all x ∈ U(θ ) and all θ ∈ Θ ; ∞ 1 (v) 0 V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all x ∈ U(θ ) \ {0} and all θ ∈ Θ ; (ii) λ V (‖x‖) ≤
(vi) for every (x, θ ) ∈ X × Θ there are xθs ∈ S(θ ), xθb ∈ B(θ ) and xθu ∈ U(θ ) such that x = xθs + xθb + xθu and V (‖xθs ‖) + V (‖xθb ‖) + V (‖xθu ‖) ≤ λ V (‖x‖).
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Proof. Necessity. If π is exponentially trichotomic let {Pk (θ )}θ∈Θ , k ∈ {1, 2, 3} be the families of projections and let K , ν > 0 be given by Definition 2.5. Since supθ∈Θ ‖Pk (θ )‖ < ∞ for all k ∈ {1, 2, 3}, let M > 0 be such that
‖Pk (θ )‖ ≤ M ,
∀θ ∈ Θ , ∀k ∈ {1, 2, 3}.
We define V (t ) = t for all t ≥ 0 and λ = max{K , K /ν, 3M }. From Proposition 2.2 we have that Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ) for all θ ∈ Θ . It is easy to see that all the conditions (i)–(v) are fulfilled. In addition, for every (x, θ ) ∈ X × Θ , taking xθs = P1 (θ )x, xθb = P2 (θ )x and xθu = P3 (θ )x we have that the condition (vi) also holds. Sufficiency. According to Theorems 3.1–3.3 we have that the hypotheses (i)–(v) imply the existence of two constants K , ν > 0 such that
‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖, 1 K
∀t ≥ 0, ∀x ∈ S(θ ), ∀θ ∈ Θ .
‖x‖ ≤ ‖Φ (θ , t )x‖ ≤ K ‖x‖,
‖Φ (θ , t )x‖ ≥
1 K
eν t ‖x‖,
(3.35)
∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.36)
∀t ≥ 0, ∀x ∈ U(θ ), ∀θ ∈ Θ .
(3.37)
Let θ ∈ Θ . From hypothesis (vi) we have that S(θ ) + B(θ) + U(θ ) = X . From Theorems 3.1–3.3 we obtain that S(θ ), B(θ ) and U(θ ) are closed linear subspaces and S(θ ) ∩ B(θ ) = S(θ ) ∩ U(θ ) = B(θ ) ∩ U(θ ) = {0}. We prove that S(θ ) ⊕ B(θ ) ⊕ U(θ ) = X .
Indeed, let x ∈ X and let x1s , x2s ∈ S(θ ), x1b , x2b ∈ B(θ ), x1u , x2u ∈ U(θ ) be such that x = x1s + x1b + x1u = x2s + x2b + x2u .
(3.38)
From relations (3.35)–(3.38) we successively deduce that 1 K
eν t ‖x1u − x2u ‖ ≤ ‖Φ (θ , t )(x1u − x2u )‖ = ‖Φ (θ , t )[(x2s − x1s ) + (x2b − x1b )]‖
≤ K [‖x2s − x1s ‖ + ‖x2b − x1b ‖],
∀t ≥ 0.
(3.39)
Relation (3.39) implies that = − Since B(θ ) ∩ S(θ ) = {0} we obtain that = From (3.38) it follows that − x1b = x2b and x1s = x2s . So we have proved that for every x ∈ X the representation x = xθs + xθb + xθu is uniquely determined. Thus, we conclude that x1u
x1b
x2u .
S(θ ) ⊕ B(θ ) ⊕ U(θ ) = X ,
x2b
x2s
x1s .
∀θ ∈ Θ .
For every θ ∈ Θ , let Pk (θ ), k ∈ {1, 2, 3} be the projections such that Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ). Then, we have that Pk (θ )Pj (θ ) = 0,
∀k ̸= j, ∀θ ∈ Θ
(3.40)
and P1 (θ ) + P2 (θ ) + P3 (θ ) = I ,
∀θ ∈ Θ .
(3.41)
Let h > 0 be such that V (h) > λ V (1). Let θ ∈ Θ , k ∈ {1, 2, 3} and let x ∈ X with ‖x‖ = 1. According to hypothesis (vi) we deduce that V (‖Pk (θ )x‖) ≤ λ V (1) < V (h).
(3.42)
Since V is nondecreasing, from (3.42) it follows that
‖Pk (θ )x‖ ≤ h,
∀x ∈ X , ‖x‖ = 1.
(3.43)
Taking into account that h does not depend on x, θ or k, from (3.43) we obtain that
‖Pk (θ )x‖ ≤ h ‖x‖,
∀x ∈ X , ∀θ ∈ Θ , ∀k ∈ {1, 2, 3}
which shows that sup ‖Pk (θ )‖ ≤ h, θ∈Θ
∀k ∈ {1, 2, 3}.
Finally, from relations (3.35)–(3.37) and from Theorem 3.1(i) and Theorem 3.3(i) we conclude that π is exponentially trichotomic.
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Remark 3.1. The above result may be regarded as a generalization in two directions of a famous nonlinear integral condition for stability due to Rolewicz. Theorem 3.4 extends the Rolewicz type results in two directions: the framework is extended from the non-autonomous case of evolution families to skew-product flows and the main subject is expanded from stability to the most general asymptotic property described by trichotomy. The methods used in the present paper are new and rely on direct and constructive techniques, emphasizing the fundamental role of the subspaces associated with a skew-product flow. As we mentioned at the beginning of this section, the study is done in the most general case: without assuming the existence of the projection families, and without any requirement concerning the invertibility of the cocycle or the flow or any other invariance properties. Thus, in order to prove the existence of the exponential trichotomy we provided a set of nonlinear integral conditions which imply the direct splitting of the main space X at every θ ∈ Θ , the existence of the projections and the qualitative behavior on the range of each projection. 4. Applications to non-autonomous systems Starting from the ideas pointed out in Remark 3.1, in what follows we apply our main results to the study of the existence of exponential trichotomy in the non-autonomous case. We show that the techniques from the previous section do break fresh ground, bringing into focus new situations and adding some interesting perspectives to the subject concerning the asymptotic behavior of evolution families in infinite dimensional spaces. Let X be a real or complex Banach space. Definition 4.1. An evolution family U = {U (t , s)}t ≥s is said to be exponentially trichotomic if there exist three families of projections {Pk (t )}t ∈R ⊂ L(X ), k ∈ {1, 2, 3} and two constants K ≥ 1 and ν > 0 such that: (i) (ii) (iii) (iv) (v) (vi)
Pk (t )Pj (t ) = 0 for all t ∈ R and k ̸= j; P1 (t ) + P2 (t ) + P3 (t ) = I for all t ∈ R; supt ∈R ‖Pk (t )‖ < ∞ for each k ∈ {1, 2, 3}; U (t , t0 )Pk (t0 ) = Pk (t )U (t , t0 ) for all t ≥ t0 and k ∈ {1, 2, 3}; ‖U (t , t0 )x‖ ≤ K e−ν(t −t0 ) ‖x‖ for all x ∈ Range P1 (t0 ) and all t ≥ t0 ; 1 ‖x‖ ≤ ‖U (t , t0 )x‖ ≤ K ‖x‖ for all x ∈ Range P2 (t0 ) and all t ≥ t0 ; K
(vii) ‖U (t , t0 )x‖ ≥ K1 eν(t −t0 ) ‖x‖ for all x ∈ Range P3 (t0 ) and all t ≥ t0 ; (viii) the restriction Uk (t , t0 )| : Range Pk (t0 ) → Range Pk (t ) is an isomorphism for all t ≥ t0 and all k ∈ {2, 3}. Remark 4.1. If U = {U (t , s)}t ≥s is an evolution family on X and πU = (ΦU , σ ) is the skew-product flow associated with U (see Remark 2.2), then U is exponentially trichotomic if and only if πU is exponentially trichotomic. Let U = {U (t , s)}t ≥s be an evolution family on X . For every t0 ∈ R we denote by N(t0 ) the linear space of all functions ϕ : R− → X with the property that
ϕ(s) = U (t0 + s, t0 + τ )ϕ(τ ),
∀τ ≤ s ≤ 0.
For every t0 ∈ R we consider the following subspaces: the stable subspace
Xs (t0 ) = x ∈ X :
lim U (t0 + t , t0 )x = 0 ,
t →∞
the bounded subspace Xb (t0 ) =
x ∈ X : sup ‖U (t0 + t , t0 )x‖ < ∞ and there is ϕ ∈ N(t0 ) with ϕ(0) = x and sup ‖ϕ(t )‖ < ∞ t ≥0
t ≤0
and the unstable subspace Xu (t0 ) =
x ∈ X : there is ϕ ∈ N(t0 ) with ϕ(0) = x and
lim ϕ(t ) = 0 .
t →−∞
Let V denote the set of all continuous nondecreasing functions V : R+ → R+ with V (0) = 0, V (t ) > 0 for all t > 0 and limt →∞ V (t ) = ∞. Theorem 4.1. Let U = {U (t , s)}t ≥s be an evolution family on X . U is exponentially trichotomic if and only if there are V ∈ V and λ > 0 such that: (i)
∞ t0 1
V (‖U (τ , t0 )x‖) dτ ≤ λ V (‖x‖) for all x ∈ Xs (t0 ) and all t0 ∈ R;
t +1
(ii) λ V (‖x‖) ≤ t V (‖U (τ , t0 )x‖) dτ ≤ λ V (‖x‖) for all t ≥ t0 , all x ∈ Xb (t0 ) and all t0 ∈ R; t +1 1 (iii) t V ‖U (τ ,t )x‖ dτ ≤ λ V ‖1x‖ for all t ≥ t0 , all x ∈ Xb (t0 ) \ {0} and all t0 ∈ R; (iv) λ1 V (‖x‖) ≤
0t +1 t
V (‖U (τ , t0 )x‖) dτ for all t ≥ t0 , all x ∈ Xu (t0 ) and all t0 ∈ R;
B. Sasu, A.L. Sasu / Nonlinear Analysis 74 (2011) 5097–5110
(v)
∞ t0
V
1
‖U (τ ,t0 )x‖
dτ ≤ λ V
1
‖x‖
5109
for all x ∈ Xu (t0 ) \ {0} and all t0 ∈ R; t
t
t
t
t
t
(vi) for every (x, t0 ) ∈ X × R there are xs0 ∈ Xs (t0 ), xb0 ∈ Xb (t0 ) and xu0 ∈ Xu (t0 ) such that x = xs0 + xb0 + xu0 and V (‖
t xs0
‖) + V (‖
t xb0
‖) + V (‖
t xu0
‖) ≤ λ V (‖x‖).
Proof. This follows from Theorem 3.4 and Remark 4.1.
Corollary 4.1. Let U = {U (t , s)}t ≥s be an evolution family on X . U is exponentially trichotomic if and only if there are p ∈ [1, ∞) and λ > 0 such that: (i)
∞
(ii) λ
(iii)
(iv) (v)
‖U (τ , t0 )x‖p dτ ≤ λ ‖x‖p for all x ∈ Xs (t0 ) and all t0 ∈ R; t +1 ‖x‖p ≤ t ‖U (τ , t0 )x‖p dτ ≤ λ ‖x‖p for all t ≥ t0 , all x ∈ Xb (t0 ) and all t0 ∈ R;
t0 1
t +1 1 d t ‖U (τ ,t0 )x‖p t +1 1 p x U t λ ∞ 1 d t0 ‖U (τ ,t0 )x‖p
τ ≤λ
‖ ‖ ≤
for all t ≥ t0 , all x ∈ Xb (t0 ) \ {0} and all t0 ∈ R;
1
‖x‖p
‖ (τ , t0 )x‖ dτ for all t ≥ t0 , all x ∈ Xu (t0 ) and all t0 ∈ R; τ ≤ λ ‖x1‖p for all x ∈ Xu (t0 ) \ {0} and all t0 ∈ R;
p
t
t
t
t
t
t
(vi) for every (x, t0 ) ∈ X × R there are xs0 ∈ Xs (t0 ), xb0 ∈ Xb (t0 ) and xu0 ∈ Xu (t0 ) such that x = xs0 + xb0 + xu0 and
‖
t xs0 p
‖ +‖
t xb0 p
‖ +‖
t xu0 p
‖ ≤ λ ‖x‖p .
Remark 4.2. The characterization for exponential trichotomy given by Corollary 4.1 may be regarded as a generalization of the Datko–Pazy theorem to the case of exponential trichotomy of evolution families on the real line. Remark 4.3. It still remains an open problem whether the integral nonlinear conditions may be also used in the study of the existence of the exponential trichotomy of evolution families on the half-line. Thus far, it is clear that this new case requires a direct approach, because this situation cannot be treated as an application proceeding from the variational case. Furthermore, the resolution of this new problem will generate an individual and distinct analysis starting with the structure of the expected trichotomy subspaces and ending with the fulfillment of the appropriate integral estimations on each associated subspace. Acknowledgments The authors wish to thank the anonymous referee for carefully reading the manuscript and for his/her observations. The work was supported by CNCS-UEFISCDI, Research Project PN II ID 1081 code 550/2009 and by the application PN-II-RU-TE2011 code 0148. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
A.I. Alonso, J. Hong, R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl. 38 (1999) 41–49. L. Barreira, C. Valls, Lyapunov sequences for exponential trichotomies, Nonlinear Anal. 72 (2010) 192–203. L. Barreira, C. Valls, Robustness of nonuniform exponential trichotomies in Banach spaces, J. Math. Anal. Appl. 351 (2009) 373–381. C. Cuevas, C. Vidal, Weighted exponential trichotomy of difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15 (2008) 353–379. C. Vidal, C. Cuevas, Claudio, L. del Campo, Weighted exponential trichotomy of difference equations and asymptotic behavior of nonlinear systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010) 377–400. S. Elaydi, O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl. 129 (1988) 362–374. S. Elaydi, O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ. 3 (1990) 1201–1224. S. Elaydi, K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equ. Appl. 3 (1998) 417–448. J. López-Fenner, M. Pinto, (h, k)-trichotomies and asymptotics of nonautonomous difference equations, Comput. Math. Appl. 33 (1997) 105–124. Dingjun Luo, Xian Wang, Deming Zhu, Maoan Han, Biffurcation Theory and Methods of Dynamical Systems, in: Advanced Series in Dynamical Systems, vol. 15, World Scientific, Publishing Co. Inc., River Edge, NJ, 1997. B. Sasu, A.L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z. 253 (2006) 515–536. A.L. Sasu, B. Sasu, Exponential trichotomy for variational difference equations, J. Differ. Equ. Appl. 15 (2009) 693–718. Bin-Guo Wang, Wan-Tong Li, Limit set trichotomy and dichotomy for skew-product semiflows with applications, Nonlinear Anal. 10 (2009) 2062–2071. Deming Zhu, Min Xu, Exponential trichotomy, orthogonality and their applications, Chin. Ann. of Math. 18 (1997) 55–64. M. Megan, C. Stoica, On uniform exponential trichotomy of evolution operators in Banach spaces, Integral Equ. Oper. Theory 60 (2008) 499–506. S.N. Chow, H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations 120 (1995) 429–477. S.N. Chow, H. Leiva, Unbounded perturbation of the exponential dichotomy for evolution equations, J. Differential Equations 129 (1996) 509–531. B. Sasu, A.L. Sasu, Input–output conditions for the asymptotic behavior of linear skew-product flows and applications, Commun. Pure Appl. Anal. 5 (2006) 551–569. R. Datko, Extending a theorem of Liapunov to Hilbert spaces, J. Math. Anal. Appl. 32 (1970) 610–616. R. Datko, Uniform asymptotic stability of evolutionary processes in Banach spaces, SIAM J. Math. Anal. 3 (1972) 428–445. A. Ichikawa, Equivalence of Lp and uniform exponential stability for a class of nonlinear semigroups, Nonlinear Anal. TMA 8 (1984) 805–810. A. Ichikawa, H. Katayama, Linear Time Varying Systems and Sampled-Data Systems, in: Lecture Notes in Control and Information Sciences, Springer, 2001.
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B. Sasu, A.L. Sasu / Nonlinear Analysis 74 (2011) 5097–5110 A. Pazy, On the applicability of Liapunov′ s theorem in Hilbert spaces, SIAM J. Math. Anal. Appl. 3 (1972) 291–294. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1983. S. Rolewicz, On uniform N-equistability, J. Math. Anal. Appl. 115 (1986) 434–441. S. Rolewicz, Functional Analysis and Control Theory. Linear systems, Kluwer Academic Press, 1987. B. Sasu, Generalizations of a theorem of Rolewicz, Appl. Anal. 84 (2005) 1165–1172. D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, 1996. V. Lakshmikantham, X. Liu, Stability Analysis in Terms of Two Measures, World Scientific, Singapore, 1993. M. Megan, A.L. Sasu, B. Sasu, On uniform exponential stability of linear skew-product semiflows in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin 9 (2002) 143–154. A.L. Sasu, Integral characterizations for stability of linear skew-product flows, Math. Inequal. Appl. 7 (2004) 535–541. C. Foiaş, G. Sell, R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988) 309–353. B. Sasu, New criteria for exponential expansiveness of variational difference equations, J. Math. Anal. Appl. 327 (2007) 287–297. M. Megan, A.L. Sasu, B. Sasu, Stabilizability and controllability of systems associated to linear skew-product semiflows, Rev. Mat. Complut. 15 (2002) 599–618.
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Nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces Bogdan Sasu ∗ , Adina Luminiţa Sasu Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara, Romania
article
info
Article history: Received 5 February 2011 Accepted 3 May 2011 Communicated by S. Carl MSC: 34D09 40C10 45G15 Keywords: Exponential trichotomy Invariant subspace Skew-product flow
abstract In this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew-product flows in infinite dimensional spaces. We treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families. We show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary. Our results generalize the previous studies on this topic and provide as particular cases many interesting situations, among which we mention the detection of the exponential trichotomy of general non-autonomous systems. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Exponential trichotomy is the most complex asymptotic property of dynamical systems arising from the central manifold theory (see [1–15]). Starting from the idea that the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold, it was clear that the concept of exponential dichotomy describes a rather idealistic situation when the solution is either exponentially stable on the stable subspaces or exponentially unstable on the unstable subspaces (see [16,17,7,8,18]). Thus, with motivation from the properties arising in bifurcation theory (see [10], Chapter 6), it was the moment to introduce a new asymptotic concept called exponential trichotomy which reflects a deeper analysis of the behavior of solutions of dynamical systems. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable and central subspaces at each point from the flow’s domain. The concept of exponential trichotomy was introduced in the pioneering works of Elaydi and Hájek (see [6,7]), where the authors analyzed this property for the first time, both in the case of differential systems and in the case of nonlinear differential systems, pointing out significant properties of systems with exponential trichotomy. In the past few years, notable progress was made in the study of the exponential trichotomy of dynamical systems (see [1–5,8,9,11–14]) using diverse methods situated on the boundary between bifurcation theory and control, for various classes of trichotomy concepts. An important step was made by Zhu and Xu in [14], where they proved the Fredholm Alternative Lemma for a differential equation with exponential trichotomies, and, as a consequence, the main results could then be applied in order to obtain the persistence condition for heteroclinic orbits connecting non-hyperbolic equilibria. In [9] López-Fenner and Pinto proposed a nice study of discrete non-autonomous nonlinear systems possessing (h, k)-trichotomies. In [8]
∗
Corresponding author. E-mail addresses: [email protected] (B. Sasu), [email protected] (A.L. Sasu).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.05.004
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Elaydi and Janglajew obtained a valuable contribution to the study of the exponential trichotomy, providing the first input–output characterization for exponential trichotomy (see Theorem 4, pp. 423). An interesting analysis of the robustness of exponential trichotomy of difference equations was presented by Alonso et al. in [1]. In recent years, the exponential trichotomy of discrete-time equations became a front-line topic and the techniques associated with this asymptotic concept were widely diversified (see [2,4,5,8,12] and the references therein). In [4] Cuevas and Vidal pointed out new mechanisms arising in the subtle connections between weighted exponential trichotomy and the (h, k) trichotomy on Z+ and Z− . The techniques developed in [4] were extended in [5] where Vidal, Cuevas and Del Campo presented a systematic study of the asymptotic properties of the solutions of nonlinear difference equations proceeding from the perturbation of systems with weighted compensated exponential trichotomy. An inedit approach was proposed by Barreira and Valls in [2] where the authors give a complete characterization of nonuniform exponential trichotomies in terms of strict Lyapunov sequences for a linear cocycle with discrete time. The connections between the existence of exponential trichotomy of variational difference equations and the solvability of the associated variational control system was established for the first time in [12]. The case of differential equations and evolution families was considered in [3,11] from distinct perspectives. The exponential trichotomy of evolution families on the real line was studied in [11], where we proved premiere input–output characterizations in terms of the admissibility with respect to an integral equation between certain spaces of continuous functions. An interesting analysis of the robustness of nonuniform exponential trichotomies of differential equations under sufficiently small linear perturbations was recently presented in [3], establishing the continuous dependence on the perturbation of the constants in the notion of trichotomy. In contrast with the techniques developed in the above mentioned references, the aim of the present paper is to propose a new and distinct perspective in the study of the existence of exponential trichotomy. The starting points of our investigation are several famous theorems which played a crucial role in the development of the stability theory of differential equations (see [19–26]). One of the most notable criteria in the stability theory of evolution equations is the Datko–Pazy theorem, which states that an evolution family U = {U (t , s)}t ≥s≥0 on a Banach space X is exponentially stable (i.e. there exist K , ν > 0 such that ‖U (t , s)‖ ≤ K e−ν(t −s) for all t ≥ s ≥ 0) if and only if there is p ∈ [1, ∞) such that ∞
∫
‖U (t , s)x‖p dt < ∞,
sup s≥0
∀x ∈ X .
s
For the case p = 2 this result was proved for the first time by Datko in [19] for semigroups in Hilbert spaces and later in [20] for evolution families. Using distinct techniques, Pazy extended the framework and proved that the result holds also for any p ∈ [1, ∞) (see [23,24]). An interesting study in the nonlinear case was presented by Ichikawa in [21], where the author proved a Datko type theorem for a family of nonlinear operators T = {T (t , s)}t ≥s≥0 on a Banach space X such that T (t , s) : Ys → Yt , T (t , t ) = IYt is the identity on Yt , T (t , u)T (u, s) = T (t , s) on Ys and T (·, s)y is continuous on [s, ∞) for each y ∈ Ys . One of the main results in [21] states that if there is a positive continuous function g : [0, ∞) → [0, ∞) such that ‖T (t , s)y‖ ≤ g (t − s)‖y‖ for all y ∈ Ys and all t ≥ s ≥ 0, then T is uniformly exponentially stable (i.e. there are K , ν > 0 such that ‖T (t , s)y‖ ≤ K e−ν(t −s) ‖y‖ for all t ≥ s ≥ 0 and all y ∈ Ys ) if and only if there are p ∈ (0, ∞) and M > 0 such that ∞
∫
‖T (t , s)y‖p dt ≤ M ‖y‖p ,
∀y ∈ Ys , ∀s ≥ 0.
s
A decisive step in this framework was made when Rolewicz obtained a new nonlinear condition for stability (see [25,26]). The notable theorem of Rolewicz may be stated as: Theorem 1.1. Let N : R∗+ × R+ → R+ be a function such that for every t > 0, s → N (t , s) is continuous and nondecreasing with N (t , 0) = 0, N (t , s) > 0 for all s > 0, and for every s ≥ 0, t → N (t , s) is nondecreasing. If U = {U (t , s)}t ≥s≥0 is an evolution family on the Banach space X such that for every x ∈ X , there is α(x) > 0 with ∞
∫
N (α(x), ‖U (t , s)x‖) dt < ∞
sup s≥0
s
then U is exponentially stable. In the original proof Rolewicz proposed an approach based on category arguments. Using other methods, based on the theory of Banach function spaces, the theorem of Rolewicz was recently deduced as a consequence of the main result in [27]. In the last few years, due to the impressive progress in asymptotic theory of nonlinear equations (see [28,29]) the stability and dichotomy properties of new classes of dynamical systems were studied using Rolewicz type conditions (see e.g. [30,31,27,13] and the references therein). It worth mentioning that the main tools used in the proofs of the Rolewicz type results relied either on category arguments or on the behavior of some associated functionals on certain function spaces. In this context the natural question arises whether the Rolewicz type conditions may be extended to the most complex asymptotic behavior like the phenomenon described by exponential trichotomy. The aim of this paper is to answer this question. The main idea is not to extend a theory from stability or dichotomy to trichotomy, but to find proper criteria that will completely model the trichotomic behavior of a dynamical system. In this paper we are interested in finding nonlinear conditions for the existence of exponential trichotomy of skewproduct flows in infinite dimensional spaces, providing a direct analysis of the subtle mechanisms which govern the splitting
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of the main space and generate the asymptotic behavior on each invariant closed subspace. We consider the most general class of evolutionary systems described by skew-product flows on X × Θ , where X is a Banach space and Θ is a metric space (see Definition 2.3) and we determine a set of nonlinear integral conditions which ensure, for every θ ∈ Θ , the decomposition of the space X into a direct sum of three invariant subspaces such that the behavior of trajectories starting in the first and the third subspaces is described by exponential decay backward and forward in time and the trajectories starting in the second subspace have uniform upper and lower bounds (see Definition 2.5). We obtain a new method in the study of the exponential trichotomy applicable to any class of skew-product flows, working in the most flexible context, without assuming a priori the existence of the projection families, without any additional requirements concerning the invertibility of the cocycle and without any other invariance property. We show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties. Moreover, we prove that the imposed conditions are also necessary. As particular cases we deduce many interesting situations, among we mention the remarkable stability theorem due to Rolewicz. Moreover, the main results are applied to the study of the exponential trichotomy of non-autonomous systems in infinite dimensional spaces. 2. Basic definitions and preliminaries In [32] the authors proved that classical equations like the Navier–Stokes, Taylor–Couette, and Bubnov–Galerkin ones can be considered and modelled in the unified setting of skew-product flows and also it was pointed out that the skew-product flows often proceed from the linearization of nonlinear equations. Classical examples of skew-product flows arise as operator solutions for variational equations and therefore, in the past few years, the asymptotic behavior of skew-product flows became a topic of great interest with various applications in nonlinear problems (see [16,17,32,18,33] and the references therein). In this section, for the sake of clarity, we present some basic definitions as well as some auxiliary results. Indeed, we denote by R the set of the real numbers, by R+ the set of all t ∈ R, t ≥ 0 and by R− the set of all t ∈ R, t ≤ 0. For every A ⊂ R we denote by χA the characteristic function of the set A. Let X be a real or a complex Banach space. The norm on X and on L(X )—the Banach algebra of all bounded linear operators on X —will be denoted by ‖ · ‖. The identity operator on X will be denoted by I. Let (Θ , d) be a metric space. We denote by E = X × Θ . Definition 2.1. A continuous mapping σ : Θ × R → Θ is called a flow on Θ if σ (θ , 0) = θ and σ (θ , s + t ) = σ (σ (θ , s), t ) for all (θ , s, t ) ∈ Θ × R2 . Definition 2.2. A mapping Φ : Θ × R+ → L(X ) is called a cocycle over the flow σ if it satisfies the following conditions: (i) (ii) (iii) (iv)
Φ (θ , 0) = I for all θ ∈ Θ ; Φ (θ , s + t ) = Φ (σ (θ , s), t )Φ (θ , s) for all (θ , t , s) ∈ Θ × R2+ (the cocycle identity); there are M ≥ 1 and ω > 0 such that ‖Φ (θ , t )‖ ≤ Meωt for all (θ , t ) ∈ Θ × R+ ; for every x ∈ X the mapping (θ , t ) → Φ (θ , t )x is continuous.
Definition 2.3. If σ is a flow on Θ and Φ is a cocycle over the flow σ , then the dynamical system π = (Φ , σ ) is called a skew-product flow on E . Remark 2.1. We note that generally the operators in the cocycle are not assumed to be invertible and for this reason the cocycle is parameterized by t ≥ 0 but not by t ∈ R. Definition 2.4. A family U = {U (t , s)}t ≥s ⊂ L(X ) is called an evolution family if the following properties hold: (i) U (t0 , t0 ) = I and U (t , s)U (s, t0 ) = U (t , t0 ) for all t ≥ s ≥ t0 ; (ii) for every x ∈ X the mapping (t , s) → U (t , s)x is continuous; (iii) there are M ≥ 1 and ω > 0 such that ‖U (t , t0 )‖ ≤ Meω(t −t0 ) for all t ≥ t0 . Remark 2.2. The class of skew-product flows generalizes the setting of non-autonomous differential equations (see [24]). Indeed, if U = {U (t , s)}t ≥s is an evolution family on the Banach space X , then by considering the translation flow σ : R × R → R, σ (θ , t ) = θ + t, and Φ : R × R+ → L(X ), ΦU (θ , t ) = U (θ + t , θ ), it is easily seen that ΦU is a cocycle over the flow σ . Then πU = (ΦU , σ ) is called the skew-product flow associated with U. Example 2.1 (The Variational Equation). Let Θ be a locally compact metric space, let σ be a flow on Θ and let {A(θ )}θ∈Θ be a family of densely defined closed operators on a Banach space X . We consider the variational equation
(A)
x˙ (t ) = A(σ (θ , t ))x(t ),
(θ , t ) ∈ Θ × R+ .
A cocycle Φ : Θ × R+ → L(X ) over the flow σ is said to be a solution of the equation (A) if for every θ ∈ Θ , there is a dense subset Dθ ⊂ D(A(θ )) such that for every xθ ∈ Dθ the mapping t → x(t ) := Φ (θ , t )xθ is differentiable on R+ , for every t ∈ R+ , x(t ) ∈ D(A(σ (θ , t ))) and the mapping t → x(t ) satisfies the equation (A).
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Example 2.2. If π = (Φ , σ ) is a skew-product flow on E and D : Θ → L(X ) is a strongly continuous mapping with supθ∈Θ ‖D(θ )‖ < ∞ then there exists a unique cocycle ΦD over the flow σ such that
ΦD (θ , t )x = Φ (θ , t )x +
t
∫
Φ (σ (θ , s), t − s) D(σ (θ , s)) ΦD (θ , s)x ds,
∀t ≥ 0, ∀θ ∈ Θ , ∀x ∈ X
0
(see [34] Theorem 2.1, [17] Section 4). Example 2.3. Let a : R → (0, 1] be a continuous function and for every s ∈ R, denote by as (t ) = a(t + s) the translation of the function a by s. We define Θ = {as : s ∈ R} and on Θ we consider the metric d(θ , θ˜ ) = sup |θ (s) − θ˜ (s)|. s∈R
Let X be a Banach space and let {T (t )}t ≥0 be a C0 -semigroup on X with the infinitesimal generator A : D(A) ⊂ X → X . For every θ ∈ Θ suppose that A(θ ) := θ (0)A. It easily checked that σ : Θ × R → Θ , σ (θ , t )(s) := θ (t + s), is a flow on Θ . We consider the variational equation x˙ (t ) = A(σ (θ , t ))x(t ), x(0) = x0 .
[A]
t ≥0
θ (s) ds x, which is a cocycle over the flow σ . Then π = (Φ , σ ) is a skew-product flow on E = X × Θ . Moreover, for every x0 ∈ D(A), the mapping defined by x(t ) := Φ (θ , t )x0 for all t ≥ 0 is We define Φ : Θ × R+ → L(X ), Φ (θ , t )x = T
t 0
the strong solution of the equation [A]. Remark 2.3. We recall that an operator P ∈ L(X ) is called a projection if P 2 = P. If P is a projection, then Range P and Ker P are closed linear subspaces and X = Range P ⊕ Ker P. It obvious that if P is a projection then I − P is also a projection and X = Range P ⊕ Range (I − P ). It is also well known that if X0 is a closed linear subspace of X there need not exist a complementary closed subspace Y0 such that X = X0 ⊕ Y0 (although in Hilbert spaces Y0 exists, being the orthogonal complement). The concept of exponential trichotomy is firmly rooted in the central manifold theory for dynamical systems. This extends the classical dichotomic properties and describes at each point θ ∈ Θ the asymptotic behavior on three complemented closed subspaces: Definition 2.5. A skew-product flow π = (Φ , σ ) is said to be exponentially trichotomic if there are three families of projections {Pk (θ )}θ∈Θ ⊂ L(X ), k ∈ {1, 2, 3} and two constants K ≥ 1 and ν > 0 such that: (i) (ii) (iii) (iv) (v) (vi)
Pk (θ )Pj (θ ) = 0 for all k ̸= j and all θ ∈ Θ ; P1 (θ ) + P2 (θ ) + P3 (θ ) = I for all θ ∈ Θ ; supθ∈Θ ‖Pk (θ )‖ < ∞ for all k ∈ {1, 2, 3}; Φ (θ , t )Pk (θ ) = Pk (σ (θ , t ))Φ (θ , t ) for all (θ , t ) ∈ Θ × R+ and all k ∈ {1, 2, 3}; ‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖ for all t ≥ 0, x ∈ Range P1 (θ ) and all θ ∈ Θ ; 1 ‖x‖ ≤ ‖Φ (θ , t )x‖ ≤ K ‖x‖ for all t ≥ 0, x ∈ Range P2 (θ ) and all θ ∈ Θ ; K
(vii) ‖Φ (θ , t )x‖ ≥ K1 eν t ‖x‖ for all t ≥ 0, x ∈ Range P3 (θ ) and all θ ∈ Θ ; (viii) the restriction Φ (θ , t )| : Range Pk (θ ) → Range Pk (σ (θ , t )) is an isomorphism for all (θ , t ) ∈ Θ × R+ and all k ∈ {2, 3}. Remark 2.4. We note that according to Definition 2.5 all the fundamental invariant subspaces (stable, unstable or central) given by Range Pk (θ ), with k ∈ {1, 2, 3} and θ ∈ Θ , are allowed to have infinite dimension. Example 2.4. Suppose that a : R → [−1, 1], a(t ) = sin t, and for every s ∈ R suppose that as (t ) = a(t + s) for all t ∈ R. We define Θ = {as : s ∈ R} and we consider on Θ the metric d(θ , θ˜ ) = sups∈R |θ (s) − θ˜ (s)|. Let (V , ‖ · ‖) be an infinite dimensional Banach space and suppose that X = V 3 with respect to the norm ‖(v1 , v2 , v3 )‖ = ‖v1 ‖ + ‖v2 ‖ + ‖v3 ‖. For every (θ , t ) ∈ Θ × R+ we consider the operator Φ (θ , t ) : X → X defined by
t t t Φ (θ , t )(v1 , v2 , v3 ) = e−t + 0 θ(τ ) dτ v1 , e 0 θ(τ ) dτ v2 , et + 0 θ (τ ) dτ v3 . Then π = (Φ , σ ) is a skew-product flow on X × Θ . We consider the projections P1 : X → X , P1 (v1 , v2 , v3 ) = (v1 , 0, 0), P2 : X → X , P2 (v1 , v2 , v3 ) = (0, v2 , 0) and P3 : X → X , P3 (v1 , v2 , v3 ) = (0, 0, v3 ). Then, an easy computation shows that π is exponentially trichotomic with respect to the families of projections Pk (θ ) = Pk ,
∀θ ∈ Θ , ∀k ∈ {1, 2, 3}
and with the constants K = e2 and ν = 1.
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Remark 2.5. If a skew-product flow is exponentially trichotomic with respect to the families of projections {Pk (θ )}θ∈Θ , k ∈ {1, 2, 3}, then: (i) Φ (θ , t )Range P1 (θ ) ⊂ Range P1 (σ (θ , t )) for all (θ , t ) ∈ Θ × R+ ; (ii) Φ (θ , t )Range Pk (θ ) = Range Pk (σ (θ , t )) for all (θ , t ) ∈ Θ × R+ and all k ∈ {2, 3}. Notation. Let π = (Φ , σ ) be a skew-product flow on E . At every point θ ∈ Θ we associate with π three fundamental subspaces, which have an essential role in the study of exponential trichotomy. The main idea starts from the previous remark and from the necessity of finding suitable conditions for the existence of exponential trichotomy. Indeed, for every θ ∈ Θ we denote by N(θ ) the linear space of all functions ϕ : R− → X with ϕ(s) = Φ (σ (θ , τ ), s − τ )ϕ(τ ) for all τ ≤ s ≤ 0. Remark 2.6. If θ ∈ Θ and ϕ ∈ N(θ ), then ϕ is a continuous function on R− . For every θ ∈ Θ we consider the sets
S(θ ) = x ∈ X : lim Φ (θ , t )x = 0
t →∞
B(θ ) =
x ∈ X : sup ‖Φ (θ , t )x‖ < ∞ and there is ϕ ∈ N(θ ) with ϕ(0) = x and sup ‖ϕ(t )‖ < ∞ t ≥0
U(θ ) =
t ≤0
x ∈ X : there is ϕ ∈ N(θ ) with ϕ(0) = x and
lim ϕ(t ) = 0 .
t →−∞
Remark 2.7. It is easy to verify that S(θ ), B(θ ) and U(θ) are linear subspaces of X for each θ ∈ Θ . Usually, for every θ ∈ Θ , S(θ ) is called the stable subspace, B(θ ) is called the bounded subspace and U(θ ) is called the unstable subspace. In what follows we discuss several properties of these subspaces, which explain the appellations given in the previous remark for each subspace. Proposition 2.1. The following assertions hold: (i) Φ (θ , t )S(θ ) ⊆ S(σ (θ , t )) for all (θ , t ) ∈ Θ × R+ ; (ii) Φ (θ , t )B(θ ) = B(σ (θ , t )) for all (θ , t ) ∈ Θ × R+ ; (iii) Φ (θ , t )U(θ ) = U(σ (θ , t )) for all (θ , t ) ∈ Θ × R+ . Proof. The assertion (i) is immediate. To prove (ii) let (θ, t ) ∈ Θ × (0, ∞) and let x ∈ B(θ ). Setting yx = Φ (θ , t )x and taking into account that x ∈ B(θ ) we have that sup ‖Φ (σ (θ , t ), s)yx ‖ = sup ‖Φ (θ , s + t )x‖ < ∞. s ≥0
(2.1)
s ≥0
Since x ∈ B(θ ) there is ϕ ∈ N(θ ) with ϕ(0) = x and sups≤0 ‖ϕ(s)‖ < ∞. We consider the function
ψ : (−∞, t ] → X ,
ψ(s) =
Φ (θ , s)x, ϕ(s),
s ∈ [0, t ] s < 0.
An easy computation shows that ψ(s) = Φ (σ (θ , τ ), s − τ )ψ(τ ) for all τ ≤ s ≤ t, which implies that
ψ(s + t ) = Φ (σ (θ , τ + t ), s − τ )ψ(τ + t ),
∀τ ≤ s ≤ 0.
(2.2)
Taking ζ : R− → X , ζ (s) = ψ(s + t ), from (2.2) it follows that ζ ∈ N(σ (θ , t )). Moreover, ζ (0) = ψ(t ) = yx , and hence using also relation (2.1) we deduce that yx ∈ B(σ (θ , t )). This shows that Φ (θ , t )B(θ ) ⊂ B(σ (θ , t )). Conversely, let y ∈ B(σ (θ , t )). Then there is λ ∈ N(σ (θ , t )) with λ(0) = y and sups≤0 ‖λ(s)‖ < ∞. We denote by xy = λ(−t ). From
λ(s) = Φ (σ (θ , t + τ ), s − τ )λ(τ ),
∀τ ≤ s ≤ 0
(2.3)
we obtain that y = λ(0) = Φ (θ , t )xy . We consider the function δ : R− → X , δ(s) = λ(s − t ), and from (2.3) we deduce that δ ∈ N(θ ). In addition, we observe that δ(0) = xy and sups≤0 ‖δ(s)‖ = sups≤−t ‖λ(s)‖ < ∞. Since y ∈ B(σ (θ , t )) we have that sup ‖Φ (θ , s)xy ‖ = sup ‖Φ (σ (θ , t ), s − t )Φ (θ , t )xy ‖ = sup ‖Φ (σ (θ , t ), s − t )y‖ s≥t
s≥t
= sup ‖Φ (σ (θ , t ), τ )y‖ < ∞. τ ≥0
s≥t
(2.4)
Let M , ω > 0 be given by Definition 2.2(iii). Then ‖Φ (θ , s)xy ‖ ≤ Meωt ‖xy ‖ for all s ∈ [0, t ]. Hence using relation (2.4) it follows that sups≥0 ‖Φ (θ , s)xy ‖ < ∞. In conclusion, we obtain that xy ∈ B(θ ) and so y = Φ (θ , t )xy ∈ Φ (θ , t )B(θ ). This implies that B(σ (θ , t )) ⊂ Φ (θ , t )B(θ ) and the proof is complete. (iii) This follows using arguments similar to those used in the proof of (ii).
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In the next proposition we briefly prove the structure of the trichotomy projections in order to clarify the intrinsic mechanisms of exponential trichotomy. Proposition 2.2. If the skew-product flow π = (Φ , σ ) is exponentially trichotomic with respect to three families of projections {Pk (θ)}θ ∈Θ ⊂ L(X ), k ∈ {1, 2, 3}, then: (i) Range P1 (θ ) = S(θ ) for all θ ∈ Θ ; (ii) Range P2 (θ ) = B(θ ) for all θ ∈ Θ ; (iii) Range P3 (θ ) = U(θ ) for all θ ∈ Θ . Proof. Let K , ν > 0 be given by Definition 2.5 and let Mk = supθ∈Θ ‖Pk (θ )‖, k ∈ {1, 2, 3}. Let θ ∈ Θ . We observe that if x ∈ S(θ ) ∪ B(θ ), then setting αx := supt ≥0 ‖Φ (θ , t )x‖ < ∞ and using properties (vii) and (ii) from Definition 2.5, we successively deduce that
‖P3 (θ )x‖ ≤ K e−ν t ‖Φ (θ , t )(x − P1 (θ )x − P2 (θ )x)‖ ≤ K e−ν t (1 + M1 + M2 )αx −→ 0, t →∞
which yields P3 (θ )x = 0. (i) It is easy to see that Range P1 (θ ) ⊂ S(θ ). Let x ∈ S(θ ). According to Definition 2.5(iv) and (vi) we have that 1 K
‖P2 (θ )x‖ ≤ ‖Φ (θ , t )P2 (θ )x‖ = ‖P2 (σ (θ , t ))Φ (θ , t )x‖ ≤ M2 ‖Φ (θ , t )x‖ −→ 0 t →∞
which shows that P2 (θ )x = 0. Since also P3 (θ )x = 0 it follows that x = P1 (θ )x ∈ Range P1 (θ ). (ii) Let x ∈ B(θ ) and let ϕ ∈ N(θ ) be such that ϕ(0) = x and βx := sups≤0 ‖ϕ(s)‖ < ∞. From Definition 2.5(iv) and (v) we obtain that
‖P1 (θ )x‖ = ‖P1 (θ )ϕ(0)‖ = ‖Φ (σ (θ , s), −s)P1 (σ (θ , s))ϕ(s)‖ ≤ K eν s ‖P1 (σ (θ , s))‖ ‖ϕ(s)‖ ≤ M1 K βx eν s −→ 0 s→−∞
so P1 (θ )x = 0. Since also P3 (θ )x = 0 it follows that x = P2 (θ )x ∈ Range P2 (θ ), so B(θ ) ⊂ Range P2 (θ ). Conversely, let y ∈ Range P2 (θ ). Then, from Definition 2.5(vi) we have that supt ≥0 ‖Φ (θ , t )y‖ < ∞. We consider the function
ϕy : R− → X ,
1 ϕy (t ) = Φ2 (σ (θ , t ), −t )− | y
1 where for every t ≤ 0, Φ2 (σ (θ , t ), −t )− denotes the inverse of the operator Φ (σ (θ , t ), −t )| : Range P2 (σ (θ , t )) → | Range P2 (θ ). Then ϕy ∈ N(θ ) and taking into account that ‖ϕy (t )‖ ≤ K ‖y‖ for all t ≤ 0, we deduce that y = ϕy (0) ∈ B(θ ). This shows that B(θ ) = Range P2 (θ ). (iii) Let x ∈ U(θ ). Then there is δ ∈ N(θ ) with δ(0) = x and lims→−∞ δ(s) = 0. Let k ∈ {1, 2}. Since δ ∈ N(θ ) we deduce that
‖Pk (θ )x‖ = ‖Pk (θ )δ(0)‖ = ‖Φ (σ (θ , s), −s)Pk (σ (θ , s))δ(s)‖ ≤ KMk ‖δ(s)‖ −→ 0 s→−∞
which shows that Pk (θ )x = 0 for k ∈ {1, 2}, so x = P3 (θ )x. It follows that U(θ ) ⊂ Range P3 (θ ). Conversely, let y ∈ Range P3 (θ ). We define
ψy : R− → X ,
1 ψy (t ) = Φ3 (σ (θ , t ), −t )− | y
1 where for every t ≤ 0, Φ3 (σ (θ , t ), −t )− denotes the inverse of the operator Φ (σ (θ , t ), −t )| : Range P3 (σ (θ , t )) → | Range P3 (θ ). It is easy to see that ψy ∈ N(θ ) and ‖ψy (t )‖ ≤ K eν t ‖y‖ for all t ≤ 0. This shows that y = ψy (0) ∈ U(θ ), so Range P3 (θ ) ⊂ U(θ ) and the proof is complete.
Remark 2.8. From Proposition 2.2 it follows that if a skew-product flow is exponentially trichotomic with respect to three families of projections, then these projections are uniquely determined. 3. The main results In what follows we are interested in obtaining necessary and sufficient nonlinear conditions for the existence of the exponential trichotomy of skew-product flows. We emphasize that we are working in the most general case, without any additional assumption on the flow or on the cocycle. It is also very important to point out that we do not assume a priori the existence of the projection families or any invertibility properties on the bounded subspace or on the unstable subspace. We show that an inedit assembly of integral nonlinear conditions are necessary and sufficient conditions for the existence of the exponential trichotomy. Let X be a Banach space. Let (Θ , d) be a metric space. We denote by E = X × Θ . We denote by V the set of all continuous nondecreasing functions V : R+ → R+ with V (0) = 0, V (t ) > 0 for all t > 0 and limt →∞ V (t ) = ∞.
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Theorem 3.1. Let π = (Φ , σ ) be a skew-product flow on E . If there exist V ∈ V and λ > 0 with the following properties:
t +1
(a) λ1 V (‖x‖) ≤ t V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖) for all t ≥ 0, all x ∈ B(θ ) and all θ ∈ Θ ; t +1 1 (b) t V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all t ≥ 0, all x ∈ B(θ ) \ {0} and all θ ∈ Θ ; then the following assertions hold: (i) for every (θ , t ) ∈ Θ × R+ , the operator Φ (θ , t )| : B(θ ) → B(σ (θ , t )) is an isomorphism; (ii) there is K > 0 such that 1
‖x‖ ≤ ‖Φ (θ , t )x‖ ≤ K ‖x‖, ∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ ; K (iii) B(θ ) is a closed linear subspace for all θ ∈ Θ . Proof. Let M , ω > 0 be given by Definition 2.2(iii). (i) Let (θ , t ) ∈ Θ × R+ . From Proposition 2.1(ii) we have that the operator Φ (θ , t )| : B(θ ) → B(σ (θ , t )) is surjective. Let x ∈ B(θ ) with Φ (θ , t )x = 0. Then Φ (θ , τ )x = 0 for all τ ∈ [t , t + 1]. Then from (a) it follows that V (‖x‖) = 0 and since V ∈ V we deduce that x = 0, so Φ (θ , t )| is also injective. (ii) From limt →∞ V (t ) = ∞ it follows that there is h ≥ 1 such that V (h) > λ V (1). We denote by K = hMeω . Step 1. We prove that
‖Φ (θ , t )x‖ ≤ K ‖x‖,
∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.1)
Since
‖Φ (θ , t )x‖ ≤ Meω ‖x‖ ≤ K ‖x‖,
∀t ∈ [0, 1], ∀x ∈ X , ∀θ ∈ Θ
(3.2)
it remains to prove that (3.1) holds for t > 1. Suppose to the contrary that there are t > 1, θ ∈ Θ and x ∈ B(θ ) such that
‖Φ (θ , t )x‖ > K ‖x‖.
(3.3)
From (3.3) we deduce that x ̸= 0 and setting yx = x/‖x‖ we have that yx ∈ B(θ ). On the other hand
‖Φ (θ , t )yx ‖ ≤ Meω ‖Φ (θ , τ )yx ‖,
∀τ ∈ [t − 1, t ].
(3.4)
From relations (3.3) and (3.4) we have that h < ‖Φ (θ , τ )yx ‖,
∀τ ∈ [t − 1, t ].
(3.5)
Since V is nondecreasing, from hypothesis (a) and relation (3.5) we deduce that V (h) <
t
∫
V (‖Φ (θ , τ )yx ‖) dτ ≤ λ V (1) t −1
which contradicts the way in which h was chosen. It follows that the assumption (3.3) is false. This implies that
‖Φ (θ , t )x‖ ≤ K ‖x‖,
∀t > 1, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.6)
From (3.2) and (3.6) we deduce that relation (3.1) holds. Step 2. We prove that
‖Φ (θ , t )x‖ ≥
1
‖x‖,
K
∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.7)
Suppose to the contrary that there are θ ∈ Θ , x ∈ B(θ ) and t ≥ 0 such that
‖Φ (θ , t )x‖ <
1 K
‖x‖.
(3.8)
From (3.8) we deduce that x ̸= 0. Setting yx = (x/‖x‖) we have that yx ∈ B(θ ). From
‖Φ (θ , τ )yx ‖ ≤ Meω ‖Φ (θ , t )yx ‖,
∀τ ∈ [t , t + 1]
using (3.8) we obtain that
‖Φ (θ , τ )yx ‖ <
1 h
,
∀τ ∈ [t , t + 1].
Since V is nondecreasing, from hypothesis (b) and relation (3.9) we deduce that V (h) <
t +1
∫
V
t
1
‖Φ (θ , τ )yx ‖
dτ ≤ λ V (1)
which contradicts the choice of h. It follows that assertion (3.8) is false, so (3.7) holds.
(3.9)
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(iii) Let K > 0 be given by (ii). Let θ ∈ Θ and let {xn } ⊂ B(θ ) with xn −→ x. n→∞
Step 1. According to hypothesis (a) we have that t +1
∫
V (‖Φ (θ , τ )xn ‖) dτ ≤ λ V (‖xn ‖),
∀t ≥ 0 , ∀n ∈ N .
(3.10)
t
Since V is continuous, for n → ∞ in (3.10) we obtain that t +1
∫
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖),
∀t ≥ 0.
(3.11)
t
Let q ≥ 1 be such that V (q) > λ V (‖x‖). We prove that
‖Φ (θ , t )x‖ ≤ qMeω ,
∀t ≥ 1.
(3.12)
Suppose to the contrary that there is t ≥ 1 such that
‖Φ (θ , t )x‖ > qMeω .
(3.13)
ω
From ‖Φ (θ , t )x‖ ≤ Me ‖Φ (θ , τ )x‖ for all τ ∈ [t − 1, t ], using (3.13) we deduce that q < ‖Φ (θ , τ )x‖,
∀τ ∈ [t − 1, t ].
(3.14)
Since V is nondecreasing, from (3.14) and (3.11) it follows that V (q) <
∫
t
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖).
(3.15)
t −1
The relation (3.15) contradicts the choice of q. This shows that assertion (3.13) is false, so relation (3.12) holds. Since
‖Φ (θ , t )x‖ ≤ Meω ‖x‖,
∀t ∈ [0, 1)
(3.16)
from (3.12) and (3.16) we deduce that supt ≥0 ‖Φ (θ , t )x‖ < ∞. Step 2. For every n ∈ N, xn ∈ B(θ ) so there is ϕn ∈ N(θ ) with ϕn (0) = xn and supt ≤0 ‖ϕn (t )‖ < ∞. We observe that ϕn (t ) ∈ B(σ (θ , t )) and then using (ii) we have that
‖xn − xm ‖ = ‖ϕn (0) − ϕm (0)‖ = ‖Φ (σ (θ , t ), −t )(ϕn (t ) − ϕm (t ))‖ ≥
1 K
‖ϕn (t ) − ϕm (t )‖,
∀t ≤ 0 , ∀m, n ∈ N .
(3.17)
Relation (3.17) shows that for every t ≤ 0 the sequence {ϕn (t )} is convergent. We define ϕ : R− → X , ϕ(t ) = limn→∞ ϕn (t ). Since {ϕn } ⊂ N(θ ) it follows that ϕ ∈ N(θ ). For m → ∞ in relation (3.17) we also obtain that
‖ϕ(t )‖ ≤ ‖ϕn (t ) − ϕ(t )‖ + ‖ϕn (t )‖ ≤ K ‖xn − x‖ + ‖ϕn (t )‖,
∀t ≤ 0 , ∀n ∈ N .
Since supt ≤0 ‖ϕn (t )‖ < ∞ for every n ∈ N, from the above inequality we deduce that supt ≤0 ‖ϕ(t )‖ < ∞. Moreover ϕ(0) = limn→∞ ϕn (0) = limn→∞ xn = x. In conclusion, from Steps 1 and 2 it follows that x ∈ B(θ ), so B(θ ) is closed. Theorem 3.2. Let π = (Φ , σ ) be a skew-product flow on E . If there exist V ∈ V and λ > 0 such that ∞
∫
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖),
∀x ∈ S(θ ), ∀θ ∈ Θ
0
then the following assertions hold: (i) there are K , ν > 0 such that ‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖ for all t ≥ 0, all x ∈ S(θ ) and all θ ∈ Θ ; (ii) S(θ ) is a closed linear subspace for all θ ∈ Θ ; (iii) S(θ ) ∩ B(θ ) = {0} for all θ ∈ Θ . Proof. (i) According to our hypothesis, we observe that t +1
∫
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖),
∀t ≥ 0, ∀x ∈ S(θ ), ∀θ ∈ Θ .
(3.18)
t
Using arguments similar to those used in the proof of Theorem 3.1(ii) Step 1, we deduce that there is δ > 0 such that
‖Φ (θ , t )x‖ ≤ δ ‖x‖,
∀t ≥ 0, ∀x ∈ S(θ ), ∀θ ∈ Θ .
(3.19)
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We prove that there is r > 0 such that
‖Φ (θ , r )x‖ ≤
1
∀x ∈ S(θ ), ∀θ ∈ Θ .
‖x‖,
e
(3.20)
Indeed, suppose that r > (λV (1))/V (1/δ e). Let θ ∈ Θ and let x ∈ S(θ ) with ‖x‖ = 1. From Proposition 2.1(i) and relation (3.19) we have that
‖Φ (θ , r )x‖ ≤ δ ‖Φ (θ , τ )x‖,
∀τ ∈ [0, r ].
Since V is nondecreasing we obtain that
x rV Φ (θ , r ) ≤
r
∫
δ
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (1).
(3.21)
0
Taking into account the way in which r was chosen, from relation (3.21) we deduce that
1 x . V Φ (θ , r ) < V δ δe
(3.22)
Since V is nondecreasing, from (3.22) it follows that
‖Φ (θ , r )x‖ ≤
1 e
.
Taking into account that r does not depend on θ or x ∈ S(θ ) with ‖x‖ = 1 we obtain that
‖Φ (θ , r )x‖ ≤
1 e
∀x ∈ S(θ ), ∀θ ∈ Θ .
‖x‖,
(3.23)
We denote by K = δ e and by ν = 1/r. Let θ ∈ Θ , x ∈ S(θ ) and t ≥ 0. Then there are n ∈ N and τ ∈ [0, r ) such that t = nr + τ . Using Proposition 2.1(i) and relations (3.19) and (3.23) we successively deduce that
‖Φ (θ , t )x‖ ≤ δ ‖Φ (θ , nr )x‖ ≤ δ e−n ‖x‖ ≤ K e−ν t ‖x‖. (ii) Let θ ∈ Θ and let {xn } ⊂ S(θ ) with xn −→ x. According to (i) we have that n→∞
‖Φ (θ , t )xn ‖ ≤ K e−ν t ‖xn ‖,
∀t ≥ 0, ∀n ∈ N.
(3.24)
For n → ∞ in (3.24) we obtain that
‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖,
∀t ≥ 0.
(3.25)
In particular, from (3.25) it follows that limt →∞ Φ (θ , t )x = 0, so x ∈ S(θ ). This shows that S(θ ) is closed. (iii) Let θ ∈ Θ and let x ∈ S(θ ) ∩ B(θ ). From (i) we have that
‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖,
t ≥ 0.
(3.26)
Since x ∈ B(θ ), from Theorem 3.1 it follows that there is K1 > 0 such that 1 K1
‖x‖ ≤ ‖Φ (θ , t )x‖,
∀t ≥ 0.
From relations (3.26) and (3.27) we obtain that x = 0, so S(θ ) ∩ B(θ ) = {0}.
(3.27)
Theorem 3.3. Let π = (Φ , σ ) be a skew-product flow on E . If there exist V ∈ V and λ > 0 such that:
t +1
(a) λ1 V (‖x‖) ≤ t V (‖Φ (θ , τ )x‖) dτ for all t ≥ 0, all x ∈ U(θ ) and all θ ∈ Θ ; ∞ 1 (b) 0 V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all x ∈ U(θ ) \ {0} and all θ ∈ Θ ; then the following assertions hold: (i) for every (θ , t ) ∈ Θ × R+ the operator Φ (θ , t )| : U(θ ) → U(σ (θ , t )) is invertible; (ii) there are K , ν > 0 such that
‖Φ (θ , t )x‖ ≥
1 K
eν t ‖x‖,
∀t ≥ 0, ∀x ∈ U(θ ), ∀θ ∈ Θ ;
(iii) U(θ ) is a closed linear subspace for all θ ∈ Θ ; (iv) U(θ ) ∩ B(θ ) = U(θ) ∩ S(θ ) = {0}.
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B. Sasu, A.L. Sasu / Nonlinear Analysis 74 (2011) 5097–5110
Proof. (i) This follows using arguments similar to those used in the proof of Theorem 3.1(i). (ii) According to hypothesis (b) we have that t +1
∫
V
t
1
dτ ≤ λ V
‖Φ (θ , τ )x‖
1
‖x‖
,
∀t ≥ 0, ∀x ∈ U(θ ) \ {0}, ∀θ ∈ Θ .
Using arguments analogous to those used to prove Theorem 3.1(ii) Step 2, we deduce that there is δ > 0 such that
‖Φ (θ , t )x‖ ≥
1
δ
‖x‖,
∀t ≥ 0, ∀x ∈ U(θ ), ∀θ ∈ Θ .
(3.28)
We prove that there is h > 0 such that
‖Φ (θ , h)x‖ ≥ e ‖x‖,
∀x ∈ U(θ ), ∀θ ∈ Θ .
(3.29)
Indeed, suppose that h > λV (1)/V (1/eδ). Let θ ∈ Θ and x ∈ U(θ ) with ‖x‖ = 1. According to Proposition 2.1(iii) and to relation (3.28) we obtain that
‖Φ (θ , h)x‖ ≥
1
δ
‖Φ (θ , τ )x‖,
∀τ ∈ [0, h].
(3.30)
Since V is nondecreasing, from (3.30) we deduce that
hV
1
δ ‖Φ (θ , h)x‖
h
∫ ≤
V
0
1
‖Φ (θ , τ )x‖
dτ ≤ λ V (1).
(3.31)
From (3.31) it follows that ‖Φ (θ , h)x‖ ≥ e. Taking into account that h does not depend on θ or x we have that (3.29) holds. We denote by K = δ e and ν = 1/h. Let θ ∈ Θ , x ∈ U(θ ) and t ≥ 0. Let n ∈ N and s ∈ [0, h) be such that t = nh + s. Then using Proposition 2.1(iii) and relations (3.28) and (3.29) we deduce that
‖Φ (θ , t )x‖ ≥
1
δ
‖Φ (θ , nh)x‖ ≥
1
δ
en ‖ x ‖ ≥
1 νt e ‖x‖.
K
(iii) Let θ ∈ Θ and let {xn } ⊂ U(θ ) with xn −→ x. For every n ∈ N suppose that ϕn ∈ N(θ ) with ϕn (0) = xn and n→∞
limt →−∞ ϕn (t ) = 0. Using arguments similar to those from the proof of Theorem 3.1(iii) Step 2, we obtain that for every t ≤ 0, the sequence {ϕn (t )} is convergent. We define ϕ : R− → X , ϕ(t ) = limn→∞ ϕn (t ), and we have that ϕ ∈ N(θ ) and ϕ(0) = limn→∞ xn = x. Moreover, in the same manner as in the proof of Theorem 3.1(iii), we deduce that
‖ϕ(t )‖ ≤ K eν t ‖xn − x‖ + ‖ϕn (t )‖,
∀t ≤ 0 , ∀n ∈ N .
(3.32)
In particular, from (3.32) we have that limt →−∞ ϕ(t ) = 0, so x ∈ U(θ ). This shows that U(θ ) is closed. (iv) Let θ ∈ Θ and let x ∈ U(θ ) ∩ (B(θ ) ∪ S(θ )). Then, from (ii) we have that 1 K
eν t ‖x‖ ≤ ‖Φ (θ , t )x‖,
∀t ≥ 0.
(3.33)
From Theorem 3.1(ii) and Theorem 3.2(i) we deduce that there is γ > 0 such that
‖Φ (θ , t )x‖ ≤ γ ‖x‖,
∀t ≥ 0.
From (3.33) and (3.34) it follows that x = 0, so U(θ ) ∩ B(θ ) = U(θ ) ∩ S(θ ) = {0}.
(3.34)
The main result of this paper is given as follows: Theorem 3.4. A skew-product flow π = (Φ , σ ) is exponentially trichotomic if and only if there are V ∈ V and λ > 0 such that: (i)
∞ 0 1
V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖) for all x ∈ S(θ ) and all θ ∈ Θ ;
t +1 V (‖Φ (θ , τ )x‖) dτ ≤ λ V (‖x‖) for all t ≥ 0, all x ∈ B(θ ) and all θ ∈ Θ ; t t +1 1 (iii) t V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all t ≥ 0, all x ∈ B(θ ) \ {0} and all θ ∈ Θ ; t +1 (iv) λ1 V (‖x‖) ≤ t V (‖Φ (θ , τ )x‖) dτ for all t ≥ 0, all x ∈ U(θ ) and all θ ∈ Θ ; ∞ 1 (v) 0 V ‖Φ (θ,τ )x‖ dτ ≤ λ V ‖1x‖ for all x ∈ U(θ ) \ {0} and all θ ∈ Θ ; (ii) λ V (‖x‖) ≤
(vi) for every (x, θ ) ∈ X × Θ there are xθs ∈ S(θ ), xθb ∈ B(θ ) and xθu ∈ U(θ ) such that x = xθs + xθb + xθu and V (‖xθs ‖) + V (‖xθb ‖) + V (‖xθu ‖) ≤ λ V (‖x‖).
B. Sasu, A.L. Sasu / Nonlinear Analysis 74 (2011) 5097–5110
5107
Proof. Necessity. If π is exponentially trichotomic let {Pk (θ )}θ∈Θ , k ∈ {1, 2, 3} be the families of projections and let K , ν > 0 be given by Definition 2.5. Since supθ∈Θ ‖Pk (θ )‖ < ∞ for all k ∈ {1, 2, 3}, let M > 0 be such that
‖Pk (θ )‖ ≤ M ,
∀θ ∈ Θ , ∀k ∈ {1, 2, 3}.
We define V (t ) = t for all t ≥ 0 and λ = max{K , K /ν, 3M }. From Proposition 2.2 we have that Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ) for all θ ∈ Θ . It is easy to see that all the conditions (i)–(v) are fulfilled. In addition, for every (x, θ ) ∈ X × Θ , taking xθs = P1 (θ )x, xθb = P2 (θ )x and xθu = P3 (θ )x we have that the condition (vi) also holds. Sufficiency. According to Theorems 3.1–3.3 we have that the hypotheses (i)–(v) imply the existence of two constants K , ν > 0 such that
‖Φ (θ , t )x‖ ≤ K e−ν t ‖x‖, 1 K
∀t ≥ 0, ∀x ∈ S(θ ), ∀θ ∈ Θ .
‖x‖ ≤ ‖Φ (θ , t )x‖ ≤ K ‖x‖,
‖Φ (θ , t )x‖ ≥
1 K
eν t ‖x‖,
(3.35)
∀t ≥ 0, ∀x ∈ B(θ ), ∀θ ∈ Θ .
(3.36)
∀t ≥ 0, ∀x ∈ U(θ ), ∀θ ∈ Θ .
(3.37)
Let θ ∈ Θ . From hypothesis (vi) we have that S(θ ) + B(θ) + U(θ ) = X . From Theorems 3.1–3.3 we obtain that S(θ ), B(θ ) and U(θ ) are closed linear subspaces and S(θ ) ∩ B(θ ) = S(θ ) ∩ U(θ ) = B(θ ) ∩ U(θ ) = {0}. We prove that S(θ ) ⊕ B(θ ) ⊕ U(θ ) = X .
Indeed, let x ∈ X and let x1s , x2s ∈ S(θ ), x1b , x2b ∈ B(θ ), x1u , x2u ∈ U(θ ) be such that x = x1s + x1b + x1u = x2s + x2b + x2u .
(3.38)
From relations (3.35)–(3.38) we successively deduce that 1 K
eν t ‖x1u − x2u ‖ ≤ ‖Φ (θ , t )(x1u − x2u )‖ = ‖Φ (θ , t )[(x2s − x1s ) + (x2b − x1b )]‖
≤ K [‖x2s − x1s ‖ + ‖x2b − x1b ‖],
∀t ≥ 0.
(3.39)
Relation (3.39) implies that = − Since B(θ ) ∩ S(θ ) = {0} we obtain that = From (3.38) it follows that − x1b = x2b and x1s = x2s . So we have proved that for every x ∈ X the representation x = xθs + xθb + xθu is uniquely determined. Thus, we conclude that x1u
x1b
x2u .
S(θ ) ⊕ B(θ ) ⊕ U(θ ) = X ,
x2b
x2s
x1s .
∀θ ∈ Θ .
For every θ ∈ Θ , let Pk (θ ), k ∈ {1, 2, 3} be the projections such that Range P1 (θ ) = S(θ ), Range P2 (θ ) = B(θ ) and Range P3 (θ ) = U(θ ). Then, we have that Pk (θ )Pj (θ ) = 0,
∀k ̸= j, ∀θ ∈ Θ
(3.40)
and P1 (θ ) + P2 (θ ) + P3 (θ ) = I ,
∀θ ∈ Θ .
(3.41)
Let h > 0 be such that V (h) > λ V (1). Let θ ∈ Θ , k ∈ {1, 2, 3} and let x ∈ X with ‖x‖ = 1. According to hypothesis (vi) we deduce that V (‖Pk (θ )x‖) ≤ λ V (1) < V (h).
(3.42)
Since V is nondecreasing, from (3.42) it follows that
‖Pk (θ )x‖ ≤ h,
∀x ∈ X , ‖x‖ = 1.
(3.43)
Taking into account that h does not depend on x, θ or k, from (3.43) we obtain that
‖Pk (θ )x‖ ≤ h ‖x‖,
∀x ∈ X , ∀θ ∈ Θ , ∀k ∈ {1, 2, 3}
which shows that sup ‖Pk (θ )‖ ≤ h, θ∈Θ
∀k ∈ {1, 2, 3}.
Finally, from relations (3.35)–(3.37) and from Theorem 3.1(i) and Theorem 3.3(i) we conclude that π is exponentially trichotomic.
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B. Sasu, A.L. Sasu / Nonlinear Analysis 74 (2011) 5097–5110
Remark 3.1. The above result may be regarded as a generalization in two directions of a famous nonlinear integral condition for stability due to Rolewicz. Theorem 3.4 extends the Rolewicz type results in two directions: the framework is extended from the non-autonomous case of evolution families to skew-product flows and the main subject is expanded from stability to the most general asymptotic property described by trichotomy. The methods used in the present paper are new and rely on direct and constructive techniques, emphasizing the fundamental role of the subspaces associated with a skew-product flow. As we mentioned at the beginning of this section, the study is done in the most general case: without assuming the existence of the projection families, and without any requirement concerning the invertibility of the cocycle or the flow or any other invariance properties. Thus, in order to prove the existence of the exponential trichotomy we provided a set of nonlinear integral conditions which imply the direct splitting of the main space X at every θ ∈ Θ , the existence of the projections and the qualitative behavior on the range of each projection. 4. Applications to non-autonomous systems Starting from the ideas pointed out in Remark 3.1, in what follows we apply our main results to the study of the existence of exponential trichotomy in the non-autonomous case. We show that the techniques from the previous section do break fresh ground, bringing into focus new situations and adding some interesting perspectives to the subject concerning the asymptotic behavior of evolution families in infinite dimensional spaces. Let X be a real or complex Banach space. Definition 4.1. An evolution family U = {U (t , s)}t ≥s is said to be exponentially trichotomic if there exist three families of projections {Pk (t )}t ∈R ⊂ L(X ), k ∈ {1, 2, 3} and two constants K ≥ 1 and ν > 0 such that: (i) (ii) (iii) (iv) (v) (vi)
Pk (t )Pj (t ) = 0 for all t ∈ R and k ̸= j; P1 (t ) + P2 (t ) + P3 (t ) = I for all t ∈ R; supt ∈R ‖Pk (t )‖ < ∞ for each k ∈ {1, 2, 3}; U (t , t0 )Pk (t0 ) = Pk (t )U (t , t0 ) for all t ≥ t0 and k ∈ {1, 2, 3}; ‖U (t , t0 )x‖ ≤ K e−ν(t −t0 ) ‖x‖ for all x ∈ Range P1 (t0 ) and all t ≥ t0 ; 1 ‖x‖ ≤ ‖U (t , t0 )x‖ ≤ K ‖x‖ for all x ∈ Range P2 (t0 ) and all t ≥ t0 ; K
(vii) ‖U (t , t0 )x‖ ≥ K1 eν(t −t0 ) ‖x‖ for all x ∈ Range P3 (t0 ) and all t ≥ t0 ; (viii) the restriction Uk (t , t0 )| : Range Pk (t0 ) → Range Pk (t ) is an isomorphism for all t ≥ t0 and all k ∈ {2, 3}. Remark 4.1. If U = {U (t , s)}t ≥s is an evolution family on X and πU = (ΦU , σ ) is the skew-product flow associated with U (see Remark 2.2), then U is exponentially trichotomic if and only if πU is exponentially trichotomic. Let U = {U (t , s)}t ≥s be an evolution family on X . For every t0 ∈ R we denote by N(t0 ) the linear space of all functions ϕ : R− → X with the property that
ϕ(s) = U (t0 + s, t0 + τ )ϕ(τ ),
∀τ ≤ s ≤ 0.
For every t0 ∈ R we consider the following subspaces: the stable subspace
Xs (t0 ) = x ∈ X :
lim U (t0 + t , t0 )x = 0 ,
t →∞
the bounded subspace Xb (t0 ) =
x ∈ X : sup ‖U (t0 + t , t0 )x‖ < ∞ and there is ϕ ∈ N(t0 ) with ϕ(0) = x and sup ‖ϕ(t )‖ < ∞ t ≥0
t ≤0
and the unstable subspace Xu (t0 ) =
x ∈ X : there is ϕ ∈ N(t0 ) with ϕ(0) = x and
lim ϕ(t ) = 0 .
t →−∞
Let V denote the set of all continuous nondecreasing functions V : R+ → R+ with V (0) = 0, V (t ) > 0 for all t > 0 and limt →∞ V (t ) = ∞. Theorem 4.1. Let U = {U (t , s)}t ≥s be an evolution family on X . U is exponentially trichotomic if and only if there are V ∈ V and λ > 0 such that: (i)
∞ t0 1
V (‖U (τ , t0 )x‖) dτ ≤ λ V (‖x‖) for all x ∈ Xs (t0 ) and all t0 ∈ R;
t +1
(ii) λ V (‖x‖) ≤ t V (‖U (τ , t0 )x‖) dτ ≤ λ V (‖x‖) for all t ≥ t0 , all x ∈ Xb (t0 ) and all t0 ∈ R; t +1 1 (iii) t V ‖U (τ ,t )x‖ dτ ≤ λ V ‖1x‖ for all t ≥ t0 , all x ∈ Xb (t0 ) \ {0} and all t0 ∈ R; (iv) λ1 V (‖x‖) ≤
0t +1 t
V (‖U (τ , t0 )x‖) dτ for all t ≥ t0 , all x ∈ Xu (t0 ) and all t0 ∈ R;
B. Sasu, A.L. Sasu / Nonlinear Analysis 74 (2011) 5097–5110
(v)
∞ t0
V
1
‖U (τ ,t0 )x‖
dτ ≤ λ V
1
‖x‖
5109
for all x ∈ Xu (t0 ) \ {0} and all t0 ∈ R; t
t
t
t
t
t
(vi) for every (x, t0 ) ∈ X × R there are xs0 ∈ Xs (t0 ), xb0 ∈ Xb (t0 ) and xu0 ∈ Xu (t0 ) such that x = xs0 + xb0 + xu0 and V (‖
t xs0
‖) + V (‖
t xb0
‖) + V (‖
t xu0
‖) ≤ λ V (‖x‖).
Proof. This follows from Theorem 3.4 and Remark 4.1.
Corollary 4.1. Let U = {U (t , s)}t ≥s be an evolution family on X . U is exponentially trichotomic if and only if there are p ∈ [1, ∞) and λ > 0 such that: (i)
∞
(ii) λ
(iii)
(iv) (v)
‖U (τ , t0 )x‖p dτ ≤ λ ‖x‖p for all x ∈ Xs (t0 ) and all t0 ∈ R; t +1 ‖x‖p ≤ t ‖U (τ , t0 )x‖p dτ ≤ λ ‖x‖p for all t ≥ t0 , all x ∈ Xb (t0 ) and all t0 ∈ R;
t0 1
t +1 1 d t ‖U (τ ,t0 )x‖p t +1 1 p x U t λ ∞ 1 d t0 ‖U (τ ,t0 )x‖p
τ ≤λ
‖ ‖ ≤
for all t ≥ t0 , all x ∈ Xb (t0 ) \ {0} and all t0 ∈ R;
1
‖x‖p
‖ (τ , t0 )x‖ dτ for all t ≥ t0 , all x ∈ Xu (t0 ) and all t0 ∈ R; τ ≤ λ ‖x1‖p for all x ∈ Xu (t0 ) \ {0} and all t0 ∈ R;
p
t
t
t
t
t
t
(vi) for every (x, t0 ) ∈ X × R there are xs0 ∈ Xs (t0 ), xb0 ∈ Xb (t0 ) and xu0 ∈ Xu (t0 ) such that x = xs0 + xb0 + xu0 and
‖
t xs0 p
‖ +‖
t xb0 p
‖ +‖
t xu0 p
‖ ≤ λ ‖x‖p .
Remark 4.2. The characterization for exponential trichotomy given by Corollary 4.1 may be regarded as a generalization of the Datko–Pazy theorem to the case of exponential trichotomy of evolution families on the real line. Remark 4.3. It still remains an open problem whether the integral nonlinear conditions may be also used in the study of the existence of the exponential trichotomy of evolution families on the half-line. Thus far, it is clear that this new case requires a direct approach, because this situation cannot be treated as an application proceeding from the variational case. Furthermore, the resolution of this new problem will generate an individual and distinct analysis starting with the structure of the expected trichotomy subspaces and ending with the fulfillment of the appropriate integral estimations on each associated subspace. Acknowledgments The authors wish to thank the anonymous referee for carefully reading the manuscript and for his/her observations. The work was supported by CNCS-UEFISCDI, Research Project PN II ID 1081 code 550/2009 and by the application PN-II-RU-TE2011 code 0148. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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